content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Brandon Banes
Associate Professor
Brandon Banes is an Assistant Professor of Mathematics. Formerly a high school mathematics teacher at Lebanon High School, Banes earned his B.S. in Mathematics and Mathematics Education from Lipscomb
University (2006), his M.S. in Mathematics and his Ph.D in Mathematics Education from Middle Tennessee State University (2010, 2013).
Academic Degrees
B.S. in Mathematics and Mathematics Education from Lipscomb University (2006)
M.S. in Mathematics and Ph.D in Mathematics Education from Middle Tennessee State University (2010, 2013).
Academic Department | {"url":"https://lipscomb.edu/directory/banes-brandon","timestamp":"2024-11-10T09:04:30Z","content_type":"text/html","content_length":"31685","record_id":"<urn:uuid:a0558eb4-2f4c-4be6-9e1f-4ffa6e4e6c6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00526.warc.gz"} |
Types and Functions
Types and Functions¶
Primitive Types¶
Idris defines several primitive types: Int, Integer and Double for numeric operations, Char and String for text manipulation, and Ptr which represents foreign pointers. There are also several data
types declared in the library, including Bool, with values True and False. We can declare some constants with these types. Enter the following into a file Prims.idr and load it into the Idris
interactive environment by typing idris Prims.idr:
module Prims
x : Int
x = 42
foo : String
foo = "Sausage machine"
bar : Char
bar = 'Z'
quux : Bool
quux = False
An Idris file consists of an optional module declaration (here module Prims) followed by an optional list of imports and a collection of declarations and definitions. In this example no imports have
been specified. However Idris programs can consist of several modules and the definitions in each module each have their own namespace. This is discussed further in Section Modules and Namespaces).
When writing Idris programs both the order in which definitions are given and indentation are significant. Functions and data types must be defined before use, incidentally each definition must have
a type declaration, for example see x : Int, foo : String, from the above listing. New declarations must begin at the same level of indentation as the preceding declaration. Alternatively, a
semicolon ; can be used to terminate declarations.
A library module prelude is automatically imported by every Idris program, including facilities for IO, arithmetic, data structures and various common functions. The prelude defines several
arithmetic and comparison operators, which we can use at the prompt. Evaluating things at the prompt gives an answer, and the type of the answer. For example:
*prims> 6*6+6
42 : Integer
*prims> x == 6*6+6
True : Bool
All of the usual arithmetic and comparison operators are defined for the primitive types. They are overloaded using interfaces, as we will discuss in Section Interfaces and can be extended to work on
user defined types. Boolean expressions can be tested with the if...then...else construct, for example:
*prims> if x == 6 * 6 + 6 then "The answer!" else "Not the answer"
"The answer!" : String
Data Types¶
Data types are declared in a similar way and with similar syntax to Haskell. Natural numbers and lists, for example, can be declared as follows:
data Nat = Z | S Nat -- Natural numbers
-- (zero and successor)
data List a = Nil | (::) a (List a) -- Polymorphic lists
The above declarations are taken from the standard library. Unary natural numbers can be either zero (Z), or the successor of another natural number (S k). Lists can either be empty (Nil) or a value
added to the front of another list (x :: xs). In the declaration for List, we used an infix operator ::. New operators such as this can be added using a fixity declaration, as follows:
Functions, data constructors and type constructors may all be given infix operators as names. They may be used in prefix form if enclosed in brackets, e.g. (::). Infix operators can use any of the
Some operators built from these symbols can’t be user defined. These are :, =>, ->, <-, =, ?=, |, **, ==>, \, %, ~, ?, and !.
Functions are implemented by pattern matching, again using a similar syntax to Haskell. The main difference is that Idris requires type declarations for all functions, using a single colon : (rather
than Haskell’s double colon ::). Some natural number arithmetic functions can be defined as follows, again taken from the standard library:
-- Unary addition
plus : Nat -> Nat -> Nat
plus Z y = y
plus (S k) y = S (plus k y)
-- Unary multiplication
mult : Nat -> Nat -> Nat
mult Z y = Z
mult (S k) y = plus y (mult k y)
The standard arithmetic operators + and * are also overloaded for use by Nat, and are implemented using the above functions. Unlike Haskell, there is no restriction on whether types and function
names must begin with a capital letter or not. Function names (plus and mult above), data constructors (Z, S, Nil and ::) and type constructors (Nat and List) are all part of the same namespace. By
convention, however, data types and constructor names typically begin with a capital letter. We can test these functions at the Idris prompt:
Idris> plus (S (S Z)) (S (S Z))
4 : Nat
Idris> mult (S (S (S Z))) (plus (S (S Z)) (S (S Z)))
12 : Nat
When displaying an element of Nat such as (S (S (S (S Z)))), Idris displays it as 4. The result of plus (S (S Z)) (S (S Z)) is actually (S (S (S (S Z)))) which is the natural number 4. This can be
checked at the Idris prompt:
Idris> (S (S (S (S Z))))
4 : Nat
Like arithmetic operations, integer literals are also overloaded using interfaces, meaning that we can also test the functions as follows:
Idris> plus 2 2
4 : Nat
Idris> mult 3 (plus 2 2)
12 : Nat
You may wonder, by the way, why we have unary natural numbers when our computers have perfectly good integer arithmetic built in. The reason is primarily that unary numbers have a very convenient
structure which is easy to reason about, and easy to relate to other data structures as we will see later. Nevertheless, we do not want this convenience to be at the expense of efficiency.
Fortunately, Idris knows about the relationship between Nat (and similarly structured types) and numbers. This means it can optimise the representation, and functions such as plus and mult.
where clauses¶
Functions can also be defined locally using where clauses. For example, to define a function which reverses a list, we can use an auxiliary function which accumulates the new, reversed list, and
which does not need to be visible globally:
reverse : List a -> List a
reverse xs = revAcc [] xs where
revAcc : List a -> List a -> List a
revAcc acc [] = acc
revAcc acc (x :: xs) = revAcc (x :: acc) xs
Indentation is significant — functions in the where block must be indented further than the outer function.
Any names which are visible in the outer scope are also visible in the where clause (unless they have been redefined, such as xs here). A name which appears only in the type will be in scope in the
where clause if it is a parameter to one of the types, i.e. it is fixed across the entire structure.
As well as functions, where blocks can include local data declarations, such as the following where MyLT is not accessible outside the definition of foo:
foo : Int -> Int
foo x = case isLT of
Yes => x*2
No => x*4
data MyLT = Yes | No
isLT : MyLT
isLT = if x < 20 then Yes else No
In general, functions defined in a where clause need a type declaration just like any top level function. However, the type declaration for a function f can be omitted if:
• f appears in the right hand side of the top level definition
• The type of f can be completely determined from its first application
So, for example, the following definitions are legal:
even : Nat -> Bool
even Z = True
even (S k) = odd k where
odd Z = False
odd (S k) = even k
test : List Nat
test = [c (S 1), c Z, d (S Z)]
where c x = 42 + x
d y = c (y + 1 + z y)
where z w = y + w
Idris programs can contain holes which stand for incomplete parts of programs. For example, we could leave a hole for the greeting in our “Hello world” program:
main : IO ()
main = putStrLn ?greeting
The syntax ?greeting introduces a hole, which stands for a part of a program which is not yet written. This is a valid Idris program, and you can check the type of greeting:
*Hello> :t greeting
greeting : String
Checking the type of a hole also shows the types of any variables in scope. For example, given an incomplete definition of even:
even : Nat -> Bool
even Z = True
even (S k) = ?even_rhs
We can check the type of even_rhs and see the expected return type, and the type of the variable k:
*Even> :t even_rhs
k : Nat
even_rhs : Bool
Holes are useful because they help us write functions incrementally. Rather than writing an entire function in one go, we can leave some parts unwritten and use Idris to tell us what is necessary to
complete the definition.
Dependent Types¶
First Class Types¶
In Idris, types are first class, meaning that they can be computed and manipulated (and passed to functions) just like any other language construct. For example, we could write a function which
computes a type:
isSingleton : Bool -> Type
isSingleton True = Nat
isSingleton False = List Nat
This function calculates the appropriate type from a Bool which flags whether the type should be a singleton or not. We can use this function to calculate a type anywhere that a type can be used. For
example, it can be used to calculate a return type:
mkSingle : (x : Bool) -> isSingleton x
mkSingle True = 0
mkSingle False = []
Or it can be used to have varying input types. The following function calculates either the sum of a list of Nat, or returns the given Nat, depending on whether the singleton flag is true:
sum : (single : Bool) -> isSingleton single -> Nat
sum True x = x
sum False [] = 0
sum False (x :: xs) = x + sum False xs
A standard example of a dependent data type is the type of “lists with length”, conventionally called vectors in the dependent type literature. They are available as part of the Idris library, by
importing Data.Vect, or we can declare them as follows:
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : a -> Vect k a -> Vect (S k) a
Note that we have used the same constructor names as for List. Ad-hoc name overloading such as this is accepted by Idris, provided that the names are declared in different namespaces (in practice,
normally in different modules). Ambiguous constructor names can normally be resolved from context.
This declares a family of types, and so the form of the declaration is rather different from the simple type declarations above. We explicitly state the type of the type constructor Vect — it takes a
Nat and a type as an argument, where Type stands for the type of types. We say that Vect is indexed over Nat and parameterised by Type. Each constructor targets a different part of the family of
types. Nil can only be used to construct vectors with zero length, and :: to construct vectors with non-zero length. In the type of ::, we state explicitly that an element of type a and a tail of
type Vect k a (i.e., a vector of length k) combine to make a vector of length S k.
We can define functions on dependent types such as Vect in the same way as on simple types such as List and Nat above, by pattern matching. The type of a function over Vect will describe what happens
to the lengths of the vectors involved. For example, ++, defined as follows, appends two Vect:
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) Nil ys = ys
(++) (x :: xs) ys = x :: xs ++ ys
The type of (++) states that the resulting vector’s length will be the sum of the input lengths. If we get the definition wrong in such a way that this does not hold, Idris will not accept the
definition. For example:
(++) : Vect n a -> Vect m a -> Vect (n + m) a
(++) Nil ys = ys
(++) (x :: xs) ys = x :: xs ++ xs -- BROKEN
When run through the Idris type checker, this results in the following:
$ idris VBroken.idr --check
When checking right hand side of Vect.++ with expected type
Vect (S k + m) a
When checking an application of constructor Vect.:::
Type mismatch between
Vect (k + k) a (Type of xs ++ xs)
Vect (plus k m) a (Expected type)
Type mismatch between
plus k k
plus k m
This error message suggests that there is a length mismatch between two vectors — we needed a vector of length k + m, but provided a vector of length k + k.
The Finite Sets¶
Finite sets, as the name suggests, are sets with a finite number of elements. They are available as part of the Idris library, by importing Data.Fin, or can be declared as follows:
data Fin : Nat -> Type where
FZ : Fin (S k)
FS : Fin k -> Fin (S k)
From the signature, we can see that this is a type constructor that takes a Nat, and produces a type. So this is not a set in the sense of a collection that is a container of objects, rather it is
the canonical set of unnamed elements, as in “the set of 5 elements,” for example. Effectively, it is a type that captures integers that fall into the range of zero to (n - 1) where n is the argument
used to instantiate the Fin type. For example, Fin 5 can be thought of as the type of integers between 0 and 4.
Let us look at the constructors in greater detail.
FZ is the zeroth element of a finite set with S k elements; FS n is the n+1th element of a finite set with S k elements. Fin is indexed by a Nat, which represents the number of elements in the set.
Since we can’t construct an element of an empty set, neither constructor targets Fin Z.
As mentioned above, a useful application of the Fin family is to represent bounded natural numbers. Since the first n natural numbers form a finite set of n elements, we can treat Fin n as the set of
integers greater than or equal to zero and less than n.
For example, the following function which looks up an element in a Vect, by a bounded index given as a Fin n, is defined in the prelude:
index : Fin n -> Vect n a -> a
index FZ (x :: xs) = x
index (FS k) (x :: xs) = index k xs
This function looks up a value at a given location in a vector. The location is bounded by the length of the vector (n in each case), so there is no need for a run-time bounds check. The type checker
guarantees that the location is no larger than the length of the vector, and of course no less than zero.
Note also that there is no case for Nil here. This is because it is impossible. Since there is no element of Fin Z, and the location is a Fin n, then n can not be Z. As a result, attempting to look
up an element in an empty vector would give a compile time type error, since it would force n to be Z.
Implicit Arguments¶
Let us take a closer look at the type of index:
index : Fin n -> Vect n a -> a
It takes two arguments, an element of the finite set of n elements, and a vector with n elements of type a. But there are also two names, n and a, which are not declared explicitly. These are
implicit arguments to index. We could also write the type of index as:
index : {a:Type} -> {n:Nat} -> Fin n -> Vect n a -> a
Implicit arguments, given in braces {} in the type declaration, are not given in applications of index; their values can be inferred from the types of the Fin n and Vect n a arguments. Any name
beginning with a lower case letter which appears as a parameter or index in a type declaration, which is not applied to any arguments, will always be automatically bound as an implicit argument.
Implicit arguments can still be given explicitly in applications, using {a=value} and {n=value}, for example:
index {a=Int} {n=2} FZ (2 :: 3 :: Nil)
In fact, any argument, implicit or explicit, may be given a name. We could have declared the type of index as:
index : (i:Fin n) -> (xs:Vect n a) -> a
It is a matter of taste whether you want to do this — sometimes it can help document a function by making the purpose of an argument more clear.
Furthermore, {} can be used to pattern match on the left hand side, i.e. {var = pat} gets an implicit variable and attempts to pattern match on “pat”; For example :
isEmpty : Vect n a -> Bool
isEmpty {n = Z} _ = True
isEmpty {n = S k} _ = False
“using” notation¶
Sometimes it is useful to provide types of implicit arguments, particularly where there is a dependency ordering, or where the implicit arguments themselves have dependencies. For example, we may
wish to state the types of the implicit arguments in the following definition, which defines a predicate on vectors (this is also defined in Data.Vect, under the name Elem):
data IsElem : a -> Vect n a -> Type where
Here : {x:a} -> {xs:Vect n a} -> IsElem x (x :: xs)
There : {x,y:a} -> {xs:Vect n a} -> IsElem x xs -> IsElem x (y :: xs)
An instance of IsElem x xs states that x is an element of xs. We can construct such a predicate if the required element is Here, at the head of the vector, or There, in the tail of the vector. For
testVec : Vect 4 Int
testVec = 3 :: 4 :: 5 :: 6 :: Nil
inVect : IsElem 5 Main.testVec
inVect = There (There Here)
Implicit Arguments and Scope
Within the type signature the typechecker will treat all variables that start with an lowercase letter and are not applied to something else as an implicit variable. To get the above code example to
compile you will need to provide a qualified name for testVec. In the example above, we have assumed that the code lives within the Main module.
If the same implicit arguments are being used a lot, it can make a definition difficult to read. To avoid this problem, a using block gives the types and ordering of any implicit arguments which can
appear within the block:
using (x:a, y:a, xs:Vect n a)
data IsElem : a -> Vect n a -> Type where
Here : IsElem x (x :: xs)
There : IsElem x xs -> IsElem x (y :: xs)
Note: Declaration Order and mutual blocks¶
In general, functions and data types must be defined before use, since dependent types allow functions to appear as part of types, and type checking can rely on how particular functions are defined
(though this is only true of total functions; see Section Totality Checking)). However, this restriction can be relaxed by using a mutual block, which allows data types and functions to be defined
even : Nat -> Bool
even Z = True
even (S k) = odd k
odd : Nat -> Bool
odd Z = False
odd (S k) = even k
In a mutual block, first all of the type declarations are added, then the function bodies. As a result, none of the function types can depend on the reduction behaviour of any of the functions in the
Computer programs are of little use if they do not interact with the user or the system in some way. The difficulty in a pure language such as Idris — that is, a language where expressions do not
have side-effects — is that I/O is inherently side-effecting. Therefore in Idris, such interactions are encapsulated in the type IO:
data IO a -- IO operation returning a value of type a
We’ll leave the definition of IO abstract, but effectively it describes what the I/O operations to be executed are, rather than how to execute them. The resulting operations are executed externally,
by the run-time system. We’ve already seen one IO program:
main : IO ()
main = putStrLn "Hello world"
The type of putStrLn explains that it takes a string, and returns an element of the unit type () via an I/O action. There is a variant putStr which outputs a string without a newline:
putStrLn : String -> IO ()
putStr : String -> IO ()
We can also read strings from user input:
A number of other I/O operations are defined in the prelude, for example for reading and writing files, including:
data File -- abstract
data Mode = Read | Write | ReadWrite
openFile : (f : String) -> (m : Mode) -> IO (Either FileError File)
closeFile : File -> IO ()
fGetLine : (h : File) -> IO (Either FileError String)
fPutStr : (h : File) -> (str : String) -> IO (Either FileError ())
fEOF : File -> IO Bool
Note that several of these return Either, since they may fail.
“do” notation¶
I/O programs will typically need to sequence actions, feeding the output of one computation into the input of the next. IO is an abstract type, however, so we can’t access the result of a computation
directly. Instead, we sequence operations with do notation:
greet : IO ()
greet = do putStr "What is your name? "
name <- getLine
putStrLn ("Hello " ++ name)
The syntax x <- iovalue executes the I/O operation iovalue, of type IO a, and puts the result, of type a into the variable x. In this case, getLine returns an IO String, so name has type String.
Indentation is significant — each statement in the do block must begin in the same column. The pure operation allows us to inject a value directly into an IO operation:
As we will see later, do notation is more general than this, and can be overloaded.
Normally, arguments to functions are evaluated before the function itself (that is, Idris uses eager evaluation). However, this is not always the best approach. Consider the following function:
ifThenElse : Bool -> a -> a -> a
ifThenElse True t e = t
ifThenElse False t e = e
This function uses one of the t or e arguments, but not both (in fact, this is used to implement the if...then...else construct as we will see later. We would prefer if only the argument which was
used was evaluated. To achieve this, Idris provides a Lazy data type, which allows evaluation to be suspended:
data Lazy : Type -> Type where
Delay : (val : a) -> Lazy a
Force : Lazy a -> a
A value of type Lazy a is unevaluated until it is forced by Force. The Idris type checker knows about the Lazy type, and inserts conversions where necessary between Lazy a and a, and vice versa. We
can therefore write ifThenElse as follows, without any explicit use of Force or Delay:
ifThenElse : Bool -> Lazy a -> Lazy a -> a
ifThenElse True t e = t
ifThenElse False t e = e
Codata Types¶
Codata types allow us to define infinite data structures by marking recursive arguments as potentially infinite. For a codata type T, each of its constructor arguments of type T are transformed into
an argument of type Inf T. This makes each of the T arguments lazy, and allows infinite data structures of type T to be built. One example of a codata type is Stream, which is defined as follows.
codata Stream : Type -> Type where
(::) : (e : a) -> Stream a -> Stream a
This gets translated into the following by the compiler.
data Stream : Type -> Type where
(::) : (e : a) -> Inf (Stream a) -> Stream a
The following is an example of how the codata type Stream can be used to form an infinite data structure. In this case we are creating an infinite stream of ones.
ones : Stream Nat
ones = 1 :: ones
It is important to note that codata does not allow the creation of infinite mutually recursive data structures. For example the following will create an infinite loop and cause a stack overflow.
codata Blue a = B a (Red a)
codata Red a = R a (Blue a)
blue : Blue Nat
blue = B 1 red
red : Red Nat
red = R 1 blue
findB : (a -> Bool) -> Blue a -> a
findB f (B x r) = if f x then x else findR f r
findR : (a -> Bool) -> Red a -> a
findR f (R x b) = if f x then x else findB f b
main : IO ()
main = do printLn $ findB (== 1) blue
To fix this we must add explicit Inf declarations to the constructor parameter types, since codata will not add it to constructor parameters of a different type from the one being defined. For
example, the following outputs “1”.
data Blue : Type -> Type where
B : a -> Inf (Red a) -> Blue a
data Red : Type -> Type where
R : a -> Inf (Blue a) -> Red a
blue : Blue Nat
blue = B 1 red
red : Red Nat
red = R 1 blue
findB : (a -> Bool) -> Blue a -> a
findB f (B x r) = if f x then x else findR f r
findR : (a -> Bool) -> Red a -> a
findR f (R x b) = if f x then x else findB f b
main : IO ()
main = do printLn $ findB (== 1) blue
Useful Data Types¶
Idris includes a number of useful data types and library functions (see the libs/ directory in the distribution, and the documentation). This section describes a few of these. The functions described
here are imported automatically by every Idris program, as part of Prelude.idr.
List and Vect¶
We have already seen the List and Vect data types:
data List a = Nil | (::) a (List a)
data Vect : Nat -> Type -> Type where
Nil : Vect Z a
(::) : a -> Vect k a -> Vect (S k) a
Note that the constructor names are the same for each — constructor names (in fact, names in general) can be overloaded, provided that they are declared in different namespaces (see Section Modules
and Namespaces), and will typically be resolved according to their type. As syntactic sugar, any type with the constructor names Nil and :: can be written in list form. For example:
• [] means Nil
• [1,2,3] means 1 :: 2 :: 3 :: Nil
The library also defines a number of functions for manipulating these types. map is overloaded both for List and Vect and applies a function to every element of the list or vector.
map : (a -> b) -> List a -> List b
map f [] = []
map f (x :: xs) = f x :: map f xs
map : (a -> b) -> Vect n a -> Vect n b
map f [] = []
map f (x :: xs) = f x :: map f xs
For example, given the following vector of integers, and a function to double an integer:
intVec : Vect 5 Int
intVec = [1, 2, 3, 4, 5]
double : Int -> Int
double x = x * 2
the function map can be used as follows to double every element in the vector:
*UsefulTypes> show (map double intVec)
"[2, 4, 6, 8, 10]" : String
For more details of the functions available on List and Vect, look in the library files:
• libs/prelude/Prelude/List.idr
• libs/base/Data/List.idr
• libs/base/Data/Vect.idr
• libs/base/Data/VectType.idr
Functions include filtering, appending, reversing, and so on.
Aside: Anonymous functions and operator sections¶
There are actually neater ways to write the above expression. One way would be to use an anonymous function:
*UsefulTypes> show (map (\x => x * 2) intVec)
"[2, 4, 6, 8, 10]" : String
The notation \x => val constructs an anonymous function which takes one argument, x and returns the expression val. Anonymous functions may take several arguments, separated by commas, e.g. \x, y, z
=> val. Arguments may also be given explicit types, e.g. \x : Int => x * 2, and can pattern match, e.g. \(x, y) => x + y. We could also use an operator section:
*UsefulTypes> show (map (* 2) intVec)
"[2, 4, 6, 8, 10]" : String
(*2) is shorthand for a function which multiplies a number by 2. It expands to \x => x * 2. Similarly, (2*) would expand to \x => 2 * x.
Maybe describes an optional value. Either there is a value of the given type, or there isn’t:
data Maybe a = Just a | Nothing
Maybe is one way of giving a type to an operation that may fail. For example, looking something up in a List (rather than a vector) may result in an out of bounds error:
list_lookup : Nat -> List a -> Maybe a
list_lookup _ Nil = Nothing
list_lookup Z (x :: xs) = Just x
list_lookup (S k) (x :: xs) = list_lookup k xs
The maybe function is used to process values of type Maybe, either by applying a function to the value, if there is one, or by providing a default value:
maybe : Lazy b -> Lazy (a -> b) -> Maybe a -> b
Note that the types of the first two arguments are wrapped in Lazy. Since only one of the two arguments will actually be used, we mark them as Lazy in case they are large expressions where it would
be wasteful to compute and then discard them.
Values can be paired with the following built-in data type:
data Pair a b = MkPair a b
As syntactic sugar, we can write (a, b) which, according to context, means either Pair a b or MkPair a b. Tuples can contain an arbitrary number of values, represented as nested pairs:
fred : (String, Int)
fred = ("Fred", 42)
jim : (String, Int, String)
jim = ("Jim", 25, "Cambridge")
*UsefulTypes> fst jim
"Jim" : String
*UsefulTypes> snd jim
(25, "Cambridge") : (Int, String)
*UsefulTypes> jim == ("Jim", (25, "Cambridge"))
True : Bool
Dependent Pairs¶
Dependent pairs allow the type of the second element of a pair to depend on the value of the first element.
data DPair : (a : Type) -> (P : a -> Type) -> Type where
MkDPair : {P : a -> Type} -> (x : a) -> P x -> DPair a P
Again, there is syntactic sugar for this. (a : A ** P) is the type of a pair of A and P, where the name a can occur inside P. ( a ** p ) constructs a value of this type. For example, we can pair a
number with a Vect of a particular length.
vec : (n : Nat ** Vect n Int)
vec = (2 ** [3, 4])
If you like, you can write it out the long way, the two are precisely equivalent.
vec : DPair Nat (\n => Vect n Int)
vec = MkDPair 2 [3, 4]
The type checker could of course infer the value of the first element from the length of the vector. We can write an underscore _ in place of values which we expect the type checker to fill in, so
the above definition could also be written as:
vec : (n : Nat ** Vect n Int)
vec = (_ ** [3, 4])
We might also prefer to omit the type of the first element of the pair, since, again, it can be inferred:
vec : (n ** Vect n Int)
vec = (_ ** [3, 4])
One use for dependent pairs is to return values of dependent types where the index is not necessarily known in advance. For example, if we filter elements out of a Vect according to some predicate,
we will not know in advance what the length of the resulting vector will be:
filter : (a -> Bool) -> Vect n a -> (p ** Vect p a)
If the Vect is empty, the result is easy:
In the :: case, we need to inspect the result of a recursive call to filter to extract the length and the vector from the result. To do this, we use with notation, which allows pattern matching on
intermediate values:
filter p (x :: xs) with (filter p xs)
| ( _ ** xs' ) = if (p x) then ( _ ** x :: xs' ) else ( _ ** xs' )
We will see more on with notation later.
Dependent pairs are sometimes referred to as “sigma types”.
Records are data types which collect several values (the record’s fields) together. Idris provides syntax for defining records and automatically generating field access and update functions. Unlike
the syntax used for data structures, records in Idris follow a different syntax to that seen with Haskell. For example, we can represent a person’s name and age in a record:
record Person where
constructor MkPerson
firstName, middleName, lastName : String
age : Int
fred : Person
fred = MkPerson "Fred" "Joe" "Bloggs" 30
The constructor name is provided using the constructor keyword, and the fields are then given which are in an indented block following the where keyword (here, firstName, middleName, lastName, and
age). You can declare multiple fields on a single line, provided that they have the same type. The field names can be used to access the field values:
*Record> firstName fred
"Fred" : String
*Record> age fred
30 : Int
*Record> :t firstName
firstName : Person -> String
We can also use the field names to update a record (or, more precisely, produce a copy of the record with the given fields updated):
*Record> record { firstName = "Jim" } fred
MkPerson "Jim" "Joe" "Bloggs" 30 : Person
*Record> record { firstName = "Jim", age $= (+ 1) } fred
MkPerson "Jim" "Joe" "Bloggs" 31 : Person
The syntax record { field = val, ... } generates a function which updates the given fields in a record. = assigns a new value to a field, and $= applies a function to update its value.
Each record is defined in its own namespace, which means that field names can be reused in multiple records.
Records, and fields within records, can have dependent types. Updates are allowed to change the type of a field, provided that the result is well-typed.
record Class where
constructor ClassInfo
students : Vect n Person
className : String
It is safe to update the students field to a vector of a different length because it will not affect the type of the record:
addStudent : Person -> Class -> Class
addStudent p c = record { students = p :: students c } c
*Record> addStudent fred (ClassInfo [] "CS")
ClassInfo [MkPerson "Fred" "Joe" "Bloggs" 30] "CS" : Class
We could also use $= to define addStudent more concisely:
addStudent' : Person -> Class -> Class
addStudent' p c = record { students $= (p ::) } c
Nested record update¶
Idris also provides a convenient syntax for accessing and updating nested records. For example, if a field is accessible with the expression c (b (a x)), it can be updated using the following syntax:
record { a->b->c = val } x
This returns a new record, with the field accessed by the path a->b->c set to val. The syntax is first class, i.e. record { a->b->c = val } itself has a function type. Symmetrically, the field can
also be accessed with the following syntax:
The $= notation is also valid for nested record updates.
Dependent Records¶
Records can also be dependent on values. Records have parameters, which cannot be updated like the other fields. The parameters appear as arguments to the resulting type, and are written following
the record type name. For example, a pair type could be defined as follows:
record Prod a b where
constructor Times
fst : a
snd : b
Using the class record from earlier, the size of the class can be restricted using a Vect and the size included in the type by parameterising the record with the size. For example:
record SizedClass (size : Nat) where
constructor SizedClassInfo
students : Vect size Person
className : String
Note that it is no longer possible to use the addStudent function from earlier, since that would change the size of the class. A function to add a student must now specify in the type that the size
of the class has been increased by one. As the size is specified using natural numbers, the new value can be incremented using the S constructor.
addStudent : Person -> SizedClass n -> SizedClass (S n)
addStudent p c = SizedClassInfo (p :: students c) (className c)
More Expressions¶
let bindings¶
Intermediate values can be calculated using let bindings:
mirror : List a -> List a
mirror xs = let xs' = reverse xs in
xs ++ xs'
We can do simple pattern matching in let bindings too. For example, we can extract fields from a record as follows, as well as by pattern matching at the top level:
data Person = MkPerson String Int
showPerson : Person -> String
showPerson p = let MkPerson name age = p in
name ++ " is " ++ show age ++ " years old"
List comprehensions¶
Idris provides comprehension notation as a convenient shorthand for building lists. The general form is:
[ expression | qualifiers ]
This generates the list of values produced by evaluating the expression, according to the conditions given by the comma separated qualifiers. For example, we can build a list of Pythagorean triples
as follows:
pythag : Int -> List (Int, Int, Int)
pythag n = [ (x, y, z) | z <- [1..n], y <- [1..z], x <- [1..y],
x*x + y*y == z*z ]
The [a..b] notation is another shorthand which builds a list of numbers between a and b. Alternatively [a,b..c] builds a list of numbers between a and c with the increment specified by the difference
between a and b. This works for type Nat, Int and Integer, using the enumFromTo and enumFromThenTo function from the prelude.
case expressions¶
Another way of inspecting intermediate values of simple types is to use a case expression. The following function, for example, splits a string into two at a given character:
splitAt : Char -> String -> (String, String)
splitAt c x = case break (== c) x of
(x, y) => (x, strTail y)
break is a library function which breaks a string into a pair of strings at the point where the given function returns true. We then deconstruct the pair it returns, and remove the first character of
the second string.
A case expression can match several cases, for example, to inspect an intermediate value of type Maybe a. Recall list_lookup which looks up an index in a list, returning Nothing if the index is out
of bounds. We can use this to write lookup_default, which looks up an index and returns a default value if the index is out of bounds:
lookup_default : Nat -> List a -> a -> a
lookup_default i xs def = case list_lookup i xs of
Nothing => def
Just x => x
If the index is in bounds, we get the value at that index, otherwise we get a default value:
*UsefulTypes> lookup_default 2 [3,4,5,6] (-1)
5 : Integer
*UsefulTypes> lookup_default 4 [3,4,5,6] (-1)
-1 : Integer
Restrictions: The case construct is intended for simple analysis of intermediate expressions to avoid the need to write auxiliary functions, and is also used internally to implement pattern matching
let and lambda bindings. It will only work if:
• Each branch matches a value of the same type, and returns a value of the same type.
• The type of the result is “known”. i.e. the type of the expression can be determined without type checking the case-expression itself.
Idris distinguishes between total and partial functions. A total function is a function that either:
• Terminates for all possible inputs, or
• Produces a non-empty, finite, prefix of a possibly infinite result
If a function is total, we can consider its type a precise description of what that function will do. For example, if we have a function with a return type of String we know something different,
depending on whether or not it’s total:
• If it’s total, it will return a value of type String in finite time
• If it’s partial, then as long as it doesn’t crash or enter an infinite loop, it will return a String.
Idris makes this distinction so that it knows which functions are safe to evaluate while type checking (as we’ve seen with First Class Types). After all, if it tries to evaluate a function during
type checking which doesn’t terminate, then type checking won’t terminate! Therefore, only total functions will be evaluated during type checking. Partial functions can still be used in types, but
will not be evaluated further. | {"url":"http://docs.idris-lang.org/en/v1.1.1/tutorial/typesfuns.html","timestamp":"2024-11-12T14:30:53Z","content_type":"text/html","content_length":"131414","record_id":"<urn:uuid:d4b5fc48-f5c9-420e-85ed-d1d3ba71b186>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00852.warc.gz"} |
Suppose you would like to test the hypothesis that one year of attending a university is worth three years of attending a two-year school (j - DocumenTVSuppose you would like to test the hypothesis that one year of attending a university is worth three years of attending a two-year school (j
Suppose you would like to test the hypothesis that one year of attending a university is worth three years of attending a two-year school (j
Suppose you would like to test the hypothesis that one year of attending a university is worth three years of attending a two-year school (junior college), in terms of effect on wages, using the
following econometric model:log(wage)=β0+β1jc+β2univ+β3exper + uwhats the Null?
in progress 0
Physics 3 years 2021-07-29T19:30:02+00:00 2021-07-29T19:30:02+00:00 2 Answers 36 views 0
Answers ( )
1. 0
2021-07-29T19:31:15+00:00 July 29, 2021 at 7:31 pm
A null hypothesis (
An alternative hypothesis
In this example, the hypothesis to test is that one year of attending a university is worth 3 years of attending a two-year junior college
Since 3 years of attending the junior college is equivalent to 1 year of attending the university, the null hypothesis is given by:
2. 0
2021-07-29T19:31:58+00:00 July 29, 2021 at 7:31 pm
The null hypothesis is the hypothesis that there is no significant difference between specified populations and that any observed difference is due to sampling/experimental error.
We would like to test that one year of attending a university with regression parameter 3 years of attending a junior college with regression parameter
In the model:
The null hypothesis will therefore be:
Leave an answer
About Ladonna | {"url":"https://documen.tv/question/suppose-you-would-like-to-test-the-hypothesis-that-one-year-of-attending-a-university-is-worth-t-15553907-63/","timestamp":"2024-11-09T06:29:52Z","content_type":"text/html","content_length":"87836","record_id":"<urn:uuid:059b6f86-7901-49f2-8266-1ce7cf469ec1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00567.warc.gz"} |
Deploying Sandbox to a Linux VM in Azure
The prevalence of cloud computing is undeniable and is only growing greater. This tutorial will provide the necessary information to get anyone, of any experience level, set up with a development
environment for the Algorand blockchain running in Azure. It is my belief that this is the best way to get started with Algorand development as it provides a closed environment to test and
experiment. This is also the first step in getting involved in the much broader Algorand development ecosystem.
• Have an Azure account with an active subscription: Create an account
• Have an SSH enabled terminal. If you are on Windows, Windows Terminal is recommended.
• Have an SSH key pair created: Guide 1 Guide 2
• Know your SSH public key
Creating and configuring the virtual machine
With the power of Azure templates, this step can be skipped if you are familiar with Azure. There will be instructions in the associated repository, which can be accessed at the bottom of the page,
to use an Azure template to get this virtual machine environment set up in mere minutes! However, if you are unfamiliar with Azure, I highly recommend you follow this section as it will serve as a
basis for setting up additional resources in future tutorials.
Create the virtual machine
Start off by logging into the Azure portal: https://portal.azure.com/
If this is your first time logging in, your screen may appear slightly different but it’s not an issue. Look around for a button that says Create a resource as shown in Figure 1-1. This can also be
accessed from the drop down menu at the top left, if needed.
Figure 1-1: Create a resource
As you may have expected, you will now be met with the screen for creating a resource. There are many standard options to choose from, but we will be creating a Ubuntu Server 20.04 LTS resource, as
shown in Figure 1-2. At the time of writing this tutorial, it is a standard recommendation Azure provides, but it can always be searched for if have trouble finding it.
Figure 1-2: Ubuntu Server 20.04 LTS
Under the resource, click the Create button and you will be moved to another screen. We will now be setting the configurable parts of this virtual machine. Barring any UI changes, subscription name,
or default auto filled fields, your screen should look similar to the one in Figure 1-3. Upon completing the next few actions, your fields should also be filled similar to the ones in Figure 1-3.
Figure 1-3: Create a virtual machine
Firstly, you will want to create a new resource group. Resource groups are a convenient way to organize resources in Azure. We’ll name our resource group SandboxResourceGroup. Do this by clicking
Create new under resource group and entering the name. This can be seen in Figure 1-4.
Figure 1-4: Create new resource group
Then click Ok so we can proceed with naming the virtual machine itself. We’ll be naming our virtual machine SandboxVM because it is both clear and concise. As I live in Philadelphia, I will be
setting my virtual machine’s region to East US. However, it is best to set it to a region close to you for low latency.
From here, we will leave the next few settings at their default and move to Size. This is a crucial step. Only certain virtual machines on Azure support nested virtualization and it is required for
using the Algorand Sandbox. In typical Microsoft fashion, there is limited information available on which virtual machines support nested virtualization. For this tutorial, I will be listing out the
virtual machines that currently support nested virtualization. However, here is the one and only resource indicating nested virtualization support: https://azure.microsoft.com/en-us/blog/
nested-virtualization-in-azure/ should you need to access it. The list of machines currently supporting nested virtualization can be seen in Figure 1-5.
Figure 1-5: Machine sizes supporting nested virtualization
We’ll be using D2s_v3 (also denoted as Standard_D2s_v3) because 8GB of RAM will come in handy with nested virtualization, however there is practically no difference between these choices. Any of
these four will serve our needs flawlessly. (Hint: click See all sizes under the size option to view the full list)
The final step in creating our virtual machine will be deciding how we’d like to define our administrator account. As indicated in the requirements, we’ll be using our personal computer’s public SSH
key to authenticate. I do this by selecting SSH public key as the authentication type, setting Username to my name, wes, setting SSH public key source as Use existing public key, and finally copying
my personal SSH public key to the provided field. Please take note of the Username you used because it will be used later to connect to the virtual machine (exact spelling). If you’ve followed these
steps, your administrator account settings should look similar to Figure 1-6.
Figure 1-6: Administrator account
We’re now done with the initial setup of our virtual machine. For the remainder of the settings, we’ll be using all default options so there is no need to worry about them. There is plenty more that
a virtual machine requires to run, but Azure does the heavy lifting for us. We are now ready to review the settings for our virtual machine. You may have noticed that Azure is warning you of an
exposed SSH port (rightfully so!). However, this is a development environment so there is no need to worry. In a future tutorial, there will be guidance on protecting these resources within a virtual
network through VPN access. For now though, you can go ahead and click Review and Create at the bottom left, as shown in Figure 1-7.
Figure 1-7: Review + create
You will be greeted by a screen where you can see virtual machine pricing per hour, set some contact details, and review your virtual machine configuration options. We didn’t change any settings
outside of basic configuration, so just take a quick look and make sure your settings are mostly similar to the settings in Figure 1-8.
Figure 1-8: Basic options
On this screen, your Subscription will be different from mine. Your Region may be different if you set it to a different location close to you. Your Size may be different if you chose not to use the
D2s_v3. Finally, your Username will likely be different unless your name is also wes.
Warning: Completion of the following step will immediately start incurring charges on your Azure subscription. Please understand the implications of this before continuing.
If everything looks to spec, go ahead and hit Create in the bottom left!
You’ll quickly move to a new screen where you can see the progress of your deployment. During this deployment, you won’t just be creating a virtual machine, you will also be creating the supporting
infrastructure for it. This process will create six resources in total and they can be seen under the Deployment details tab.
• The virtual machine - this is the actual “machine” that your server will be running on. It will host all the Sandbox code. Small disclaimer: it isn’t a physical server, but rather an
intelligently allocated group of resources in the cloud.
• A virtual network - the name is pretty self explanatory. This is a network that exists virtually but emulates a physical network (like your home wifi network). You are able to control the way
these interact in a broad or fine grained way with the wider internet.
• A public ip address - this is how you will access your Sandbox virtual machine from across the internet.
• A network security group - this is how you control which traffic is allowed into and out of your Sandbox virtual machine.
• A network interface - this translates public internet traffic into virtual network traffic.
• A disk - this is where all the data will be stored inside your virtual machine.
Configure virtual machine network settings
Once the virtual machine is finished deploying, you can go through and see the resource, as represented in Figure 2-1.
Figure 2-1: Go to resource
You will now arrive at the virtual machine’s overview page. There is a lot of very useful information here, but for now, we only need two things here. First, take note of your virtual machine’s
Public IP address. You can write this down, save it in notepad, or just remember where to find it. We will use this later to connect. Secondly, we need to access the virtual machine’s Networking tab
to open three ports. Both of these are highlighted in Figure 2-2
Figure 2-2: Public IP address and Networking tab
In the Networking configuration tab, we need to add a new inbound port rule, as shown in Figure 2-3.
Figure 2-3: Add inbound port rule
You will then be met with a side window with a few options to set. For our case, we only need to edit two fields: Destination port ranges and Name. At the time of writing, the required ports for the
Sandbox are 4001 for Algod, 4002 for Kmd, and 8980 for Indexer, but as these things have the tendency to change it can always be checked here: https://github.com/algorand/sandbox#usage. Your new
inbound port rule should look similar to Figure 2-4. You can click Add if it all looks good.
Figure 2-4: Add inbound security rule
Once, these security rules are added, we’re finally done configuring our virtual machine! We can now connect and start getting our Sandbox Environment set up.
Connecting to the virtual machine
We can finally connect to our new virtual machine. Remember when you wrote down that public IP address earlier? You’ll need that now. Do you also remember when you took note of your Username during
setup? You need that now too. Once you found the sticky note those were written on, go to your terminal of choice with SSH capabilities. I will be using Windows Terminal for this, but it makes little
You should be all set up to connect to your virtual machine by using the command ssh <username>@<public-ip address>. In my specific case I will use the command ssh [email protected], but yours will
be specific to your username and virtual machine’s public IP address. When connecting, your ssh client will ask if you trust the endpoint, confirm this by typing y.
You are now connected to your virtual machine! From here we will install a few required resources that the required to run the Sandbox.
Installing Docker Engine
Without going into too much detail, Docker Engine is what allows the Sandbox to run multiple servers with ease. Docker is a very powerful tool and if you’d like to read more about it you can do so
here: https://www.docker.com/ However, Docker knowledge is not required for using or running a Sandbox environment.
As these things have a tendency to change, here is the official guide on installing Docker Engine on Ubuntu: https://docs.docker.com/engine/install/ubuntu/. However, we will also be going through it
The first thing we must do is get connected to the Docker repository. This allows us to easily update and install Docker. Execute the following commands in order:
sudo apt-get update
sudo apt-get install ca-certificates curl gnupg lsb-release
curl -fsSL https://download.docker.com/linux/ubuntu/gpg | sudo gpg --dearmor -o /usr/share/keyrings/docker-archive-keyring.gpg
echo "deb [arch=$(dpkg --print-architecture) signed-by=/usr/share/keyrings/docker-archive-keyring.gpg] https://download.docker.com/linux/ubuntu $(lsb_release -cs) stable" | sudo tee /etc/apt/sources.list.d/docker.list > /dev/null
Now, we can install Docker Engine. This is the backbone of Docker.
During the next command, it will ask you to approve the installation size. Our VM has more than enough storage capacity so type y to approve.
sudo apt-get install docker-ce docker-ce-cli containerd.io
We can now test our installation.
sudo docker run hello-world
You should get a greeting from Docker with some information about how it works! You may want to avoid having to use sudo while running commands, and we will tackle that a little later. First let’s
get Docker Compose installed.
Installing Docker Compose
The official explanation of Docker Compose is “Compose is a tool for defining and running multi-container Docker applications.” Again, you do not need to understand how this tool works, but
essentially this is the tool used to define the Sandbox and will be used to install all of the components conveniently.
As with other sections here is the official installation guide should you need it: https://docs.docker.com/compose/install/. Installing Docker Compose is very simple! Run the following two commands
to install and give ourselves access.
sudo curl -L "https://github.com/docker/compose/releases/download/1.29.2/docker-compose-$(uname -s)-$(uname -m)" -o /usr/local/bin/docker-compose
sudo chmod +x /usr/local/bin/docker-compose
We can test our installation by running the following.
Adding ourselves to the Docker group
The only thing we must do now is add ourselves to the Docker group. This is required to execute Docker commands without admin privileges and Sandbox requires this to function properly.
Here is the official guide to change the group settings: https://docs.docker.com/engine/install/linux-postinstall/ should you need it.
First we will create the group. You may get an error saying the group already exists, that’s nothing to worry about. You can proceed to the next step.
Next we will add ourselves to that group.
sudo usermod -aG docker $USER
We will now reload our group permissions.
Finally, we can test that we do not need sudo to run docker with a command from earlier.
You should see that same little greeting blurb! If everything has gone to plan, we are now ready to get our Sandbox up and running!
Installing Sandbox
The official Sandbox GitHub repo contains a lot of useful information when working with Sandbox, like usage. However, there are also other great tutorials on how to actually use your Sandbox, so I
won’t be going into that.
We must first download the Sandbox repo.
git clone https://github.com/algorand/sandbox.git
Now move into the sandbox directory and start the sandbox environment.
Since this is the first time launching, this process can take anywhere from 5-15 minutes! Please be patient! Once that command has completed, you have officially installed the Sandbox development
environment. You can test it out with the following
Next steps
• You can verify that your resource group is set up similarly to my own by visiting the GitHub repo associated with this guide. There will be instructions there for viewing and comparing your
resource group’s template. There are also further instructions on being able to quickly deploy a similar environment.
• You can use the sandbox in its current form which is a closed system replica of the Algorand network. You can test out all sorts of things here without having the risk of working with Algorand
mainnet. Here is a tutorial that can walk you through some of the nuances with Sandbox: https://developer.algorand.org/tutorials/exploring-the-algorand-sandbox/
• You can use the Sandbox to access betanet. This is where features get tested that will eventually be moved to the Algorand mainnet. Here is a great tutorial on how to do that: https://
• Both of those tutorials are using the command line to access the Sandbox network, but we can also access it through the APIs that we exposed! The public IP address you were supposed to write down
is going to come in handy again.
• You can access the API with your favorite programming language using the endpoints and tokens denoted in the Sandbox repo. At the time of writing these have the default values shown in Figure 3-1
Figure 3-1: Sandbox endpoints
• To test this connection, open a new tab in the browser you are using to read this and type follow in the URL bar: http://<public-ip>:4001/health. If you get a single message reading “null” that
means it’s working! Even though it may not seem right, so long as you don’t get an error then the Algod client is in good health!
Tear down
So, you don’t want your SandboxVM anymore. Well, the tear down is quite easy, and you can always use the template provided in the GitHub repo to redeploy. Go to the SandboxResourceGroup. You may see
the resource group on your home screen or you may have to open the resource group menu. These options are highlighted in Figure 4-1.
Figure 4-1: Finding SandboxResourceGroup
If you opened your resource groups, you will now see a list of them. Choose SandboxResourceGroup from the list as shown in Figure 4-2.
Figure 4-2: SandboxResourcegroup
Now that you see the resource group, you should see a button near the top to delete the group. Look to Figure 4-3 if you are having trouble finding it.
Figure 4-3: Delete resource group
You’ll now be asked to type the name of the resource group, then you can finally delete the group and all associated resources. These are shown in Figure 4-4 This process may take some time, but once
it is completed you will no longer be incurring charges on your Azure subscription.
Figure 4-4: Resource group name and Delete
GitHub repository
In this GitHub repo, you will find a few resources and instructions on how to use them. If you choose to deploy the virtual machine with a template you will still need to follow the steps to get
Docker and Sandbox installed as these cannot be specified on the template level. | {"url":"https://developer.algorand.org/tutorials/deploying-a-sandbox-development-environment-to-a-linux-virtual-machine-in-azure/","timestamp":"2024-11-07T06:17:10Z","content_type":"text/html","content_length":"134797","record_id":"<urn:uuid:edce698e-704e-48d5-88b0-b7817252e2cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00596.warc.gz"} |
Percents, Ratios, and Rates worksheets
Recommended Topics for you
Review: Rates, Ratios, and Percents Unit Test
Rates, Ratios, Proportions and Percents
Percents, Ratios and Rates
Ratios, Unit rates, percents
Unit 2 Review-Rates, Ratios, & Percents
Ratio, Rates and Proportion Quiz
Ratios, rates, and percents review
Ratio, Proportion and Rate
QUIZ 3 (RATIO, RATES AND PROPORTIONS)
Ratios, Unit Rates, Percents
Calculate Rates and Ratios
F1 - RATIOS, RATES AND PROPORTIONS
F1 - RATIOS, RATES AND PROPORTIONS
Nisbah,Kadar & Kadaran ( Ratios, Rates & Proportion)
Unit 3 Math Review: Ratios, Rates, Percents
Explore Percents, Ratios, and Rates Worksheets by Topics
Explore Worksheets by Subjects
Explore printable Percents, Ratios, and Rates worksheets
Percents, Ratios, and Rates worksheets are essential tools for teachers to help their students grasp the fundamental concepts of mathematics. These worksheets provide a variety of exercises and
problems that challenge students to apply their knowledge of percentages, ratios, and rates in real-world situations. By incorporating these worksheets into their lesson plans, teachers can ensure
that their students are developing a strong foundation in math. Moreover, these resources cater to different learning styles, making it easier for teachers to differentiate instruction and meet the
diverse needs of their students. Percents, Ratios, and Rates worksheets are not only engaging and effective but also help teachers save time and effort in creating their own materials.
Quizizz is an excellent platform for teachers who are looking for engaging and interactive ways to supplement their Percents, Ratios, and Rates worksheets. This platform offers a wide range of
quizzes and games that cover various math topics, including percentages, ratios, and rates. Teachers can easily integrate these quizzes into their lesson plans to reinforce the concepts taught in the
worksheets and assess their students' understanding. Additionally, Quizizz allows teachers to track their students' progress and identify areas where they may need additional support. With its
user-friendly interface and customizable features, Quizizz is an invaluable resource for teachers who want to enhance their math instruction and make learning fun and engaging for their students. | {"url":"https://quizizz.com/en-in/percents-ratios-and-rates-worksheets","timestamp":"2024-11-03T03:51:27Z","content_type":"text/html","content_length":"166145","record_id":"<urn:uuid:b1c0674b-2cc0-4962-8d74-ce1f059d50dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00257.warc.gz"} |
HEX2DEC Function: Definition, Formula Examples and Usage
HEX2DEC Function
Are you tired of manually converting hexadecimal values to decimal values in your Google Sheets? The HEX2DEC function is here to save the day! This handy function allows you to easily convert
hexadecimal values to decimal values within your spreadsheet, saving you time and effort. Simply enter the hexadecimal value you want to convert as the argument for the HEX2DEC function, and it will
return the corresponding decimal value. It’s that easy!
Not only is the HEX2DEC function convenient, it’s also very accurate. It can handle hexadecimal values of any length and will always return the correct decimal equivalent. So next time you need to
convert hexadecimal values to decimal values in your Google Sheets, don’t waste time doing it manually. Let the HEX2DEC function do the heavy lifting for you!
Definition of HEX2DEC Function
The HEX2DEC function in Google Sheets is a built-in function that converts a hexadecimal value to a decimal value. It takes a single argument, which is the hexadecimal value that you want to convert.
The function then returns the corresponding decimal value. For example, if you pass the value “FF” to the HEX2DEC function, it will return 255. You can use the HEX2DEC function to quickly and easily
convert hexadecimal values to decimal values within your Google Sheets spreadsheet. It is particularly useful if you need to work with both hexadecimal and decimal values and need to convert between
them on a regular basis.
Syntax of HEX2DEC Function
The syntax of the HEX2DEC function in Google Sheets is as follows:
Here, “hexadecimal_value” is the hexadecimal value that you want to convert to a decimal value. It can be a cell reference or a string value. For example, if you have a hexadecimal value stored in
cell A1, you can use the following formula to convert it to a decimal value:
Alternatively, you can use a string value as the argument for the HEX2DEC function. For example, the following formula will also return the decimal equivalent of the hexadecimal value “FF”:
It’s important to note that the hexadecimal value must be a string, even if it is stored in a cell. If you pass a numeric value to the HEX2DEC function, it will return an error.
Overall, the HEX2DEC function is a simple and easy-to-use function that allows you to quickly convert hexadecimal values to decimal values in your Google Sheets spreadsheet.
Examples of HEX2DEC Function
Here are three examples of how you can use the HEX2DEC function in Google Sheets:
Example 1: Convert a hexadecimal value stored in a cell
Suppose you have a hexadecimal value stored in cell A1 of your Google Sheets spreadsheet, and you want to convert it to a decimal value. You can use the following formula to do so:
For example, if cell A1 contains the hexadecimal value “FF”, the formula will return the decimal value 255.
Example 2: Convert a hexadecimal value stored in a string
You can also use the HEX2DEC function to convert a hexadecimal value that is stored in a string. For example, the following formula will return the decimal value 255:
Example 3: Convert multiple hexadecimal values at once
You can use the HEX2DEC function to convert multiple hexadecimal values at once by using the function in an array formula. For example, suppose you have a range of cells containing hexadecimal
values, and you want to convert all of them to decimal values. You can use the following array formula to do so:
This formula will convert all of the hexadecimal values in the range A1:A10 to decimal values and return the results as an array. You will need to press Ctrl + Shift + Enter to enter the formula as
an array formula.
These are just a few examples of how you can use the HEX2DEC function in Google Sheets. You can use it in a variety of situations to quickly and easily convert hexadecimal values to decimal values
within your spreadsheet.
Use Case of HEX2DEC Function
Here are some real-life examples of how you might use the HEX2DEC function in Google Sheets:
1. Data analysis: Suppose you have a dataset that includes hexadecimal values, and you want to convert them to decimal values for further analysis. You can use the HEX2DEC function to quickly and
easily convert all of the hexadecimal values in your dataset to decimal values.
2. Color coding: If you are using Google Sheets to create visualizations or other graphics, you may need to use hexadecimal values to specify colors. However, you may want to perform calculations on
those values, in which case you would need to convert them to decimal values first. The HEX2DEC function can help you do this easily and efficiently.
3. Programming: If you are using Google Sheets to store or work with data that will be used in a programming context, you may need to convert hexadecimal values to decimal values. For example, you
might be working with RGB color values, which are often represented as hexadecimal values. The HEX2DEC function can help you convert these values to decimal form for use in your programming
These are just a few examples of how you might use the HEX2DEC function in Google Sheets in real-life situations. There are many other possibilities, depending on your needs and the data you are
working with.
Limitations of HEX2DEC Function
The HEX2DEC function in Google Sheets is a powerful and convenient tool for converting hexadecimal values to decimal values. However, there are a few limitations to keep in mind when using this
1. The hexadecimal value must be a string: In order to use the HEX2DEC function, the hexadecimal value must be a string, even if it is stored in a cell. If you pass a numeric value to the HEX2DEC
function, it will return an error.
2. The hexadecimal value must be in uppercase: The HEX2DEC function only works with hexadecimal values that are written in uppercase. If you pass a lowercase hexadecimal value to the function, it
will return an error.
3. The hexadecimal value must be valid: The HEX2DEC function will only work with hexadecimal values that are valid. This means that the value must be a string consisting of the digits 0-9 and the
letters A-F. If you pass an invalid hexadecimal value to the function, it will return an error.
4. The hexadecimal value must be within the range of a 32-bit signed integer: The HEX2DEC function can only convert hexadecimal values that can be represented as 32-bit signed integers. This means
that the decimal equivalent of the hexadecimal value must be between -2147483648 and 2147483647. If the decimal equivalent is outside of this range, the HEX2DEC function will return an error.
Overall, the HEX2DEC function is a useful tool for converting hexadecimal values to decimal values in Google Sheets, but it is important to keep these limitations in mind when using it.
Commonly Used Functions Along With HEX2DEC
Here are some commonly used functions that you might use along with the HEX2DEC function in Google Sheets:
1. DEC2HEX: This function converts a decimal value to a hexadecimal value. It takes a single argument, which is the decimal value that you want to convert. For example, the following formula will
return the hexadecimal value “FF”:
2. IF: This function allows you to specify a logical test and two actions, one for when the test is true and one for when the test is false. For example, you could use the IF function in combination
with the HEX2DEC function to perform different actions based on whether a given hexadecimal value can be converted to a decimal value. For example:
=IF(HEX2DEC(A1)>0, "Valid", "Invalid")
This formula will return “Valid” if the hexadecimal value in cell A1 can be converted to a decimal value that is greater than 0, and “Invalid” otherwise.
3. VLOOKUP: This function allows you to look up a value in a table of data based on a specified criteria. You can use the VLOOKUP function in combination with the HEX2DEC function to look up a value
based on a hexadecimal value. For example:
=VLOOKUP(HEX2DEC(A1), A2:B10, 2, FALSE)
This formula will look up the value in the second column of the range A2:B10 that corresponds to the decimal equivalent of the hexadecimal value in cell A1.
These are just a few examples of how you might use these functions in combination with the HEX2DEC function in Google Sheets. You can use them in a variety of situations to perform different actions
based on the results of the HEX2DEC function.
The HEX2DEC function in Google Sheets is a powerful and convenient tool for converting hexadecimal values to decimal values. It takes a single argument, which is the hexadecimal value that you want
to convert, and returns the corresponding decimal value. The function is particularly useful if you need to work with both hexadecimal and decimal values and need to convert between them on a regular
One of the key benefits of the HEX2DEC function is that it is very easy to use. Simply enter the hexadecimal value as the argument for the function, and it will return the corresponding decimal
value. The function is also very accurate, and can handle hexadecimal values of any length.
There are a few limitations to keep in mind when using the HEX2DEC function. The hexadecimal value must be a string, must be in uppercase, must be valid, and must be within the range of a 32-bit
signed integer. However, these limitations are easy to work around, and the HEX2DEC function is generally very reliable.
Overall, the HEX2DEC function is a valuable tool for anyone who needs to convert hexadecimal values to decimal values in Google Sheets. If you haven’t tried using the HEX2DEC function yet, we highly
recommend giving it a try in your own Google Sheets spreadsheet. It can save you a lot of time and effort, and make working with hexadecimal and decimal values much easier.
Video: HEX2DEC Function
In this video, you will see how to use HEX2DEC function. We suggest you to watch the video to understand the usage of HEX2DEC formula.
Related Posts Worth Your Attention
Leave a Comment | {"url":"https://sheetsland.com/hex2dec-function/","timestamp":"2024-11-11T00:05:10Z","content_type":"text/html","content_length":"52149","record_id":"<urn:uuid:b483ed3d-4343-450b-812b-73c4ab0c2d95>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00156.warc.gz"} |
Modelling Underlying Energy Demand Trends and Stochastic Seasonality: An Econometric Analysis of Transport Oil Demand in the UK and Japan
Surrey Energy Economics
Discussion paper Series
Modelling Underlying Energy
Demand Trends and Stochastic
Seasonality: An Econometric
Analysis of Transport Oil
Demand in the UK and Japan
Lester C Hunt and Yasushi Ninomiya
February 2003
SEEDS 107
Department of Economics
The Surrey Energy Economics Centre (SEEC) consists of members of the Department of Economics who work on energy economics, environmental economics and regulation. The Department of Economics has a
long-standing tradition of energy economics research from its early origins under the leadership of Professor Colin Robinson. This was consolidated in 1983 when the University established SEEC, with
Colin as the Director; to study the economics of energy and energy markets. SEEC undertakes original energy economics research and since being established it has conducted research across the whole
spectrum of energy economics, including the international oil market, North Sea oil & gas, UK & international coal, gas privatisation & regulation, electricity privatisation & regulation, measurement
of efficiency in energy industries, energy & development, energy demand modelling & forecasting, and energy & the environment.
SEEC research output includes SEEDS - Surrey Energy Economic Discussion paper Series (details at www.seec.surrey.ac.uk/Research/SEEDS.htm) as well as a range of other academic papers, books and
monographs. SEEC also runs workshops and conferences that bring together academics and practitioners to explore and discuss the important energy issues of the day.
SEEC also attracts a large proportion of the department’s PhD students and oversees the MSc in Energy Economics & Policy. Many students have successfully completed their MSc and/or PhD in energy
economics and gone on to very interesting and rewarding careers, both in academia and the energy industry.
Director of SEEC and Editor of SEEDS: Lester C Hunt SEEC, Department of Economics, University of Surrey, Guildford GU2 7XH, UK. Tel: +44 (0)1483 686956 Fax: +44 (0)1483 689548 Email: [email
Surrey Energy Economics Centre (SEEC)
Department of Economics
SEEDS 107
ISSN 1749-8384
Lester C Hunt and Yasushi Ninomiya
February 2003
This paper demonstrates the importance of adequately modelling the
Underlying Energy Demand Trend (UEDT) and seasonality when
estimating transportation oil demand for the UK and Japan. The
structural time series model is therefore employed to allow for a
stochastic underlying trend and stochastic seasonals using quarterly data
from the early 1970s, for both the UK and Japan. It is found that the
stochastic seasonals are preferred to the conventional deterministic
dummies and, more importantly, the UEDT is found to be highly
non-linear for both countries, with periods where it is both upward and
downward sloping.
JEL Classification Numbers
: C51, Q41;
energy demand, stochastic trend model, unobservable
underling trend, seasonality.
Modelling Underlying Energy Demand Trends and Stochastic
Seasonality: An Econometric Analysis of Transport Oil
Demand in the UK and Japan
Lester C Hunt* and Yasushi Ninomiya**
* Surrey Energy Economics Centre (SEEC), Department of Economics, University of Surrey,
Guildford, Surrey, GU2 7XH, UK E-mail: [email protected]
** Institute for Global Environmental Strategies (IGES), Climate Policy Project, 2108-11
Kamiyamaguchi, Hayama, Kanagawa 240-0115 Japan E-mail: [email protected]
This paper attempts to model and estimate oil demand functions for the transportation sectors of the UK and Japan. For both the UK and Japan energy consumption in the transportation sector has
increased significantly over past decades. Moreover, the share of total energy consumption by the transport sector has also increased in both countries. In order to fully understand this growth, and
more importantly, to predict future energy consumption and the resultant effect on the environment, it is vital that energy demand is modelled appropriately.
It is important to accurately measure the price and income elasticities of demand while at the same time adequately capturing the underlying changes in energy efficiency, and other (usually
non-measurable) factors. Energy demand is traditionally modelled as a function of economic activity and the energy price - all normally observable. In addition to these traditional drivers, energy
demand is also determined by unobservable factors such as improvements in technical energy efficiency and changes in ‘tastes’. In the past these unobservable effects have either i) been ignored or
ii) approximated by a simple linear deterministic time trend assuming that the underlying trend is fixed over time. In a similar fashion, potential seasonal non-stationarity in seasonally unadjusted
data has been ignored, with little attention paid to this in past energy demand studies. Hence, quarterly energy demand studies have traditionally incorporated deterministic seasonal dummy variables
We are grateful for comments received following the presentation of an earlier draft of this paper at the 2000 IAEE conference in Sydney, Australia.
account for the underlying seasonal pattern. This implicitly assumes that the underlying seasonal pattern is fixed throughout the period.
Table 1 presents a selection of recent transportation oil demand studies for the UK, Japan and OECD.1
Almost all the studies cited use annual data over a range of estimation periods and none of them attempted to use a time trend to capture a ‘technical progress’ effect. Therefore, in almost all the
studies the issue of ‘technical progress’ and energy efficiency is ignored with the exception of Johansson and Schipper (1997) who include a variable to capture changes in energy efficiency. In
addition, Dargay (1992) implicitly attempted to capture the endogenous changes in ‘technical progress’ via an asymmetric price response model. The table illustrates that there is no consensus with
respect to the size of the income and price elasticities for the UK. Unfortunately, for Japan there are very few studies with which to make a comparison. As stated above the studies were
predominantly based on annual data. However, the survey by Dahl and Sterner (1991) does discuss the use of quarterly data. They argue that the way quarterly data is treated will affect estimated
petrol demand elasticities. Moreover, they state that "researchers should pay close attention to seasonal effects before using such estimates for overall long-run forecasting or policy analysis" (p.
{Table 1 about here}
In this paper, we attempt to estimate the income and price elasticities of demand for transportation oil demand in the UK and Japan using quarterly data between 1971q1 and 1997q4. The structural time
series model is employed in place of the conventional deterministic trend model, hence accommodating the unobservable underlying trend in a more ‘general’ way. Similarly, stochastic seasonal dummies
are incorporated in place of conventional seasonal dummies, hence allowing the seasonal pattern to evolve over time. Within this framework, the conventional linear trend model is a (restricted)
special case and only accepted if supported by the data. Likewise, the deterministic seasonal dummy model is a (restricted) special case of the more general evolving seasonal model and only accepted
if supported by the data.
2.UNDERLYING ENERGY DEMAND TREND (UEDT)
It is important to understand what the stochastic trend is attempting to measure in energy demand functions. The debate on whether to include or not include a deterministic linear time trend when
estimating energy demand functions has focussed on ‘technical progress’ - in particular whether it is appropriate to model such a process using a simple linear variable. This implicitly assumes that
‘technical progress’ results in an improvement in energy productivity or energy efficiency as an activity becomes less energy intensive. Given this restrictive focus we utilise a more general measure
of the Underlying Energy Demand Trend
(UEDT) that encompasses technical progress but also allows for other factors as defined in
Hunt et al. (2003). This is depicted in Table 2, which illustrates that the source of ‘technical progress’ can take many forms. It can be embodied, disembodied, endogenous and exogenous and hence
unlikely to be modelled adequately by a simple linear deterministic time trend (see Hunt et al., 2003 for further discussion).
{Table 2 about here}
Table 2 illustrates that the UEDT could also be significantly affected by a change in ‘tastes’ and hence has a significant effect on the demand for energy. Here ‘tastes’ encompass not only an
exogenous change in consumer preferences, but also a whole range of non-economic influences that may at one time or another have an effect on the demand for oil. The list is long and will almost
certainly change over time. However, it will include both socio-demographic and geographic factors as identified by Wohlgemuth (1997, p. 1111), such as family size and structure, gender, work status,
population age structure, population density, urban to rural changes, physical and telecommuting patterns.
Therefore, a change in ‘tastes’ holding ‘technical progress’ and the economic influences such as prices and income constant, will result in a shift in the demand curve – to the left or the right. One
example is the significant switch in energy for space heating from coal to gas or oil products that occurred during the 1960s and 1970s in many industrial countries. The reason why consumers switched
from coal is not fully explained by economic factors, but by
transportation demand studies for the UK.
the desire to use the cleaner and more convenient alternative energy source. Similarly, Wohlgemuth (1997) argues that although technology may improve fuel efficiency, “evidence suggests that consumer
preferences for more comfortable means of transport, increased urban driving and congestion could offset efficiency improvements” (p. 1114).
In summary, in addition to the standard economic variables such as economic activity, price, etc. there is a range of factors that influence energy demand. The ideal situation would be to include
data on measures such as technical energy efficiency, consumer preferences, socio-demographic factors, etc. in the general estimated model. However, it is not possible (particularly in a quarterly
time series context) to measure all these factors2
, hence past studies of energy demand have normally ignored this issue completely and/or implicitly included all factors as part of the deterministic ‘technical progress’ trend variable. But, as we
have argued above, the influence of these variables may change over time and ‘tastes’ could be operating in the opposite direction to legitimate technical improvements, hence the need for the more
flexible UEDT.3
The above economic rationale for considering the more flexible approach to estimating the UEDT is consistent with the Structural Time Series Model (STSM) developed by Harvey and his associates which
permits a more flexible approach to modelling the trend component. (See for example, Harvey et al., 1986, Harvey, 1989, Harvey and Scott, 1994, and Harvey, 1997.) The STSM is therefore considered in
the following section
Since the early applications by Nachane et al. (1988) and Hunt and Manning (1989) cointegration has become the accepted approach for estimating energy demand relationships (see Hendry and Juselius,
2001 and 2002 for an excellent explanation of the approach).
Some data are available for average fuel economy, family structure, population age structure, etc. but not consistently on a quarterly basis over the whole estimation period for both countries.
Moreover, as argued above, these various influences may have significant effects on oil demand at different times (unlike the economic variables - income and price) and therefore the UEDT/STSM
approach is seen as appropriate in these circumstances.
When considering aggregate energy demand for a country as a whole (or group of countries such as the OECD or EU) the UEDT will also be influenced by the ‘economic structure’ and ‘substitution’. This
is considered in more detail in Hunt et al. (2003).
Despite the advances explained by Hendry and Juselius the cointegration approach can only accommodate a deterministic trend and deterministic seasonal dummies. Therefore, Harvey’s
Structural Time Series Model (STSM) is adopted since it is consistent with out interpretation
of the UEDT as explained above. In particular, it allows for the estimation of a non-linear UEDT that can be negative, positive, or zero over the estimation period. Moreover, the use of the simple
deterministic time trend is not ruled out in the STSM, instead it becomes a limiting case that is admissible only if statistically accepted by the data.
Similar arguments apply to the treatment of seasonality in the STSM. The STSM allows for stochastic or evolving seasonals over the estimation period. Therefore, deterministic seasonal dummies are not
excluded from this approach; they are encompassed within the stochastic seasonals and are admissible, provided they are statistically accepted by the data.
Another advantage of using the STSM to estimate energy demand models is in forecasting, at least in the short-term. Imposing a linear trend throughout the sample period results in a UEDT represented
by an average trend for the whole estimation period. If the ‘true’ UEDT is non-linear then the linear approximation obtained from the deterministic trend is likely to lead to poor short-term
forecasts. However, the STSM puts more weight on the most recent observations and hence it is far more applicable for forecasting the near future. Likewise, this is particularly applicable when
quarterly data are used and the seasonal pattern changes over time (Harvey and Scott, 1994, p. 1339).
Given the flexibility of the STSM, it is the chosen methodology and cointegration is employed in the Appendix as a comparison with the STSM results.4 The STSM is therefore combined with an
Autoregressive Distributed Lag (ARDL) to estimate oil demand functions and the associated income and price elasticities for the transportation sectors of the UK and Japan as explained below.
Structural Time Series Model (STSM)
The STSM allows for the unobservable trend and seasonal components that are permitted to vary stochastically over time. Consider the following quarterly model:
The prime motivation for adopting the STSM is its flexibility in estimating the trend and seasonals and hence it is particularly suited for our purposes. However, Harvey also argues that it is
superior econometric
et = µt + γt + Z′tδ + εt (1)
where et is the dependent variable in logs (oil), µt represents the trend component, γt
represents the seasonal component, εt is a random white noise disturbance term5, Zt is a k× 1
vector of explanatory variables (price and income in logs) and δ is a k× 1 vector of unknown parameters.
Trend Component
The trend component µtis assumed to have the following stochastic process:
t t t t µ β η µ = [−]1+ [−]1 + (2) t t t β ξ β = [−][1] + (3)
where η[t] ~ NID(0,σ[η]2) and ξ[t] ~ NID(0,σ[ξ]2).
Equations (2) and (3) represent the level and the slope of the trend respectively. The exact form of the trend depends upon whether the variances ση2 and σξ2, known as the
hyperparameters, are zero or not. If either ση2 and σξ2 are non-zero then the trend is said to
be stochastic.6 If both are zero then the trend is linear and, as illustrated in Harvey, et al
(1986), the model reverts to a deterministic linear trend model as follows:
et = α + γt + βt + Z′tδ + εt (4)
Seasonal Component
The top left-hand charts in Figures 1 and 2 show that there is a distinct seasonal pattern in transport oil consumption for both the UK and Japan. Unlike the demand for space heating, it is not
immediately obvious a-priori why oil demand should have a seasonal pattern. However, given the seasonality of the corresponding economic activity it is not surprising that oil consumption also has a
seasonal pattern. Another contributing factor is that part of oil demand influenced by leisure activities. Harvey (1997, p. 198) and Harvey and Scott (1994, p. 1342) argue that there is little to be
lost by including stochastic seasonals instead of
I.e. εt ~ NID(0, σε2 ).
See Table 2 in Hunt et al. (2003) for a summary classification of the different types of stochastic trend that can be established.
conventional seasonal dummy variables when using quarterly data. Therefore, in the spirit of ‘general-to-specific’ modelling, the most general model is initially estimated; one with stochastic
seasonals which may be restricted to deterministic (or no) seasonal dummies only if such restrictions are acceptable via a statistical restrictions test.
Accordingly, equation (1) includes the seasonal component, γt, which follows the following stochastic process: t t L S( )γ =ω (5) where ωt ~ (0, ) 2 ω σ
NID ,S(L)=1+L+L2 +L3[ and] L = the lag operator.
The conventional case is a restricted version of this when σω2 = 0 with γt reducing to the
familiar deterministic seasonal dummy variable model. If not, however, seasonal components are moving stochastically over time.7
ARDL Models incorporating Stochastic Trend and Seasonals
We initially estimate the following most general version of equation (1) for oil demand in the transportation sectors of UK and Japan:
A(L) et = µt + γt + B(L) yt + C(L) pt + εt (6)
where A(L) is the polynomial lag operator 1 - φ1L - φ2L2 - φ3L3 - φ4L4, B(L) the polynomial lag operator π0 + π1L + π2L2 + π3L3 + π4L4, and, C(L) the polynomial lag operator ϕ0 + ϕ1L + ϕ2L2 + ϕ3L 3 +
ϕ4L 4. et is the natural logarithm of the oil series, yt the natural logarithm of
GDP, and pt the natural logarithm of the real price of oil.8 B(L)/A(L) and C(L)/A(L) represent
the long-run income and price elasticities respectively. µt, γt, and εt are as defined above.
Equation (6), with (2), (3), and (5), is estimated with the disturbance terms assumed to be independent and mutually uncorrelated with each other. As shown above, the
In addition, initial estimation included a temperature variable for the UK and Japan.
hyperparameters ση2, σξ2, σω2, and σε2 play an important role and govern the basic properties of the model. The Maximum Likelihood (ML) procedure is used to estimate the parameters of the model and
the hyperparameters. From these the optimal estimates of βT, µT and γT are estimated by the Kalman filter, representing the latest estimates of the level and slope of the trend and the seasonal
components. The optimal estimates of the trend and seasonal components are further calculated by a smoothing algorithm of the Kalman filter.
In order to evaluate the estimated models, the equation residuals (similar to ordinary regression residuals) and a set of auxiliary residuals are estimated. The auxiliary residuals include smoothed
estimates of the equation disturbance (known as the irregular residuals), the smoothed estimates of the level disturbances (known as the level residuals) and smoothed estimates of the slope
disturbances (known as the slope residuals).9 The software package STAMP 5.0 (Koopman et al., 1995) was used to estimate all models.
The preferred models are found by testing down from the over-parameterised model of equation (6) without violating a range of diagnostic tests. In particular, the equation residuals are tested for
the presence of non-normality, serial correlation, heteroscedasticity, etc. In addition, following Harvey and Koopman (1992), the auxiliary residuals are tested to ensure that no significant outliers
and/or structural breaks exist. If a problem is detected, in a similar fashion to Harvey and Koopman (1992), impulse dummies are considered. The preferred models for each country are also
re-estimated and tested, via Likelihood Ratio (LR) tests, for the following restrictions:
(a) deterministic seasonal dummies; (b) a deterministic time trend;
(c) a deterministic time trend with deterministic seasonal dummies;
This allows for a comparison of the estimated long-run elasticities and the statistical performance of the models when the deterministic restrictions are imposed.
Although these tests are important, they are conditional on the preferred model (found from the general model within the STSM framework) being the correct model for other cases. Therefore, any
conclusion that the restrictions (a) – (c) are rejected, may not necessarily be
In practice the level and slope residuals are only estimated if the level and slope components are present in the model, i.e. ηt and/or ξt are non-zero.
valid. Also, it might be that the restrictions imposed on the preferred model, found from the STSM framework, might cause a deterioration in the diagnostic tests. Therefore, as a further check on the
robustness of the STSM results the models are re-estimated with deterministic seasonal dummies and either a deterministic trend or no trend using the Engle-Granger two step cointegration procedure.
Finally, one further check of the robustness of the main STSM results is conducted. It is sometimes argued that the likelihood function is relatively flat in the hyperparameters and consequently the
estimated price and income elasticities are especially sensitive to the hyperparameter estimates. Therefore, the sensitivity of the estimated results are considered by controlling restrictions on the
hyperparameter values at the q-ratio10 of 0.1 or 0.05 in a similar fashion to Harvey (1989, p.406).
This work is part of a wider research project analysing quarterly energy demand data across a number of fuels and sectors comparing Japan and the UK over the period 1971q1 to 1997q4. The data used
for this study are quarterly seasonally unadjusted transportation oil consumption, real GDP and the real oil price for the UK and Japan.
UK Transportation oil consumption data, E(uk), refers to UK Final Consumption ‘petroleum’ for the transport sector in million tonnes of oil equivalent (mtoe) from various issues of the
UK Energy Trends, Department of Trade and Industry (DTI) up to June 1999. e(uk)
represents the natural logarithm of E(uk). The nominal and constant price expenditure estimates of UK Gross Domestic Product GDP(E) at market prices were kindly supplied by the UK Office of National
Statistics (ONS) since the seasonally unadjusted data are not published. Y(uk) is the constant GDP(E) series re-based and indexed to 1990 = 100. The implicit GDP(E) price deflator at 1990=100 was
calculated from the nominal and constant price series. y(uk) represents the natural logarithm of Y(uk). The nominal price index for oil was derived by weighting the appropriate Fuel Price Index from
various issues of the UK
Energy Trends. The real index of oil prices, P(uk), was found by deflating the nominal index
by the implicit GDP(E) deflator. p(uk) represents the natural logarithm of P(uk).11 The UK data are illustrated in Figure 1.
{Figure 1 about here}
Transportation oil demand for Japan, E(jpn), refers to final consumption of petrol and diesel oil in 1010 Kcal taken from various issues of the Yearbook of Production Supply and Demand
of Petroleum, Coal and Coke, Ministry of Economy, Trade, and Industry (METI). e(jpn)
represents the natural logarithm of E(jpn). Y(jpn), real Gross Domestic Product (GDP) (1990, Billion Yen) for Japan is from the Economic and Social Research Institute (ESRI),
Cabinet Office, Government of Japan website.12
y(jpn) represents the natural logarithm of
Y(jpn). The nominal retail price indices of petrol and diesel oil were taken from various issues of the Price Index Annual, The Bank of Japan and divided by the Final Private Consumption Quarterly
Deflator (1990 = 100), taken from the ESRI. The ‘real transportation oil price’, P(jpn), was derived as a weighted average of the deflated petrol and diesel oil price indices. p(jpn) represents the
natural logarithm of P(jpn).13 The data for Japan are illustrated in Figure 2.
{Figure 2 about here}
The major advantage of using quarterly data is the significantly increased number of degrees of freedom. Many energy demand studies across a range of fuels and sectors have been conducted using
annual data resulting in a very limited number of degrees of freedom, which arguably questions the robustness of some of the estimates. This is highlighted for the present context in Table 1 where
there is only one study that used 40 or more observations. Moreover, the need for an adequate number of degrees of freedom is particularly relevant when using the ML estimation procedure; unbiased
estimates will only be obtained if the sample size is sufficiently large to ensure the appropriate asymptotic properties are fulfilled (Thomas, 1993, p. 51). For example, a sample size of 20 is
clearly insufficient to gain the
A temperature variable, TEMP(uk), was also included in some initial estimation. This refers to the average GB quarterly temperature in degrees Celsius taken from various issues of the UK Digest of
Energy Statistics (DUKES), DTI.
The temperature variable, TEMP(jpn), used in initial estimation for Japan refers to the average of Tokyo and Osaka air temperature in degree Celsius taken from various issues of the Meteorological
Agency Annual
desirable asymptotic properties (Kennedy, 1992, p. 19). Therefore, the data set used here involves a total of 108 observations for each country (significantly more than the previous studies in Table
1) allowing adequate degrees of freedom for estimation, thus ensuring the model has the desirable asymptotic properties, even when some observations are used for lags and forecast tests.
There are of course some drawbacks with using quarterly data. Firstly, a wide range of data are often only available on an annual basis, so that some variables will need to be excluded from the
analysis. A measure of fuel efficiency is a particularly pertinent example in the present context. Secondly, quarterly data series are usually subject to seasonal fluctuations that have to be
appropriately addressed to ensure that the residuals do not suffer autocorrelation, etc. There is, therefore, a trade-off between ensuring that the data set has adequate degrees of freedom and being
able to include all relevant variables and the need to model seasonality explicitly. This highlights the importance of adequately treating the seasonal issue as well as ensuring that non-measurable
effects are appropriately captured as with the STSM framework used here.
The oil data aggregates gasoline (or petrol) with diesel. Arguably these two components have different characteristics and it would be better to estimate demand functions for the two separate
elements.14 However, it is not possible within the current study to separate the two components over the sample period. Although quarterly data for UK total oil consumption is available back to
1971q1, the split of petrol and diesel is not available on a quarterly basis back to 1971q1. It would not, therefore, be possible to do the comparison of Japan and the UK with quarterly petrol data
back to 1971. Moreover, if separate quarterly functions were explored the data limitations would result in a somewhat curtailed data set that excluded the important early 1970s period. Consequently,
for the current study total transportation oil is used.
(Meteorological Agency, Japan).
This has, however, changed somewhat over recent years where diesel has been used more for passenger cars. For example, the proportion of diesel cars in GB increased from under ½% in 1980 to 9% in
1997 whereas the proportion increased from 7% in 1972 to 18½% in 1997 for Japan. (Sources: UK Transport Statistics Bulletin, Vehicle Licensing Statistics: 2001, DTLR and private correspondence;
Handbook of Energy and Economic Statistics in Japan 2002, EDMC.) Therefore, given the increasing interdependence between the two elements it is of interest to explore the aggregate measure and
arguably the STSM outlined above is ideal for modelling the underlying changes taking place in the transportation sector of each country.
The over parameterised model of equation (6) was initially estimated for transportation oil demand for the UK and Japan for the period 1972q1 to 1995q4 - saving two years (8 observations) for
post-sample prediction tests. By testing down from equation (6) a suitable restricted model was selected following the methodology outlined above. The preferred equations for the UK and Japan are
given in Table 3. In general, the results indicate that the models fit the data well for both countries with both preferred specifications passing all diagnostic tests with no indication of
mis-specification. In addition, the results for both countries are little affected by changes to the hyperparameter values - suggesting the estimated elasticities are robust. Moreover, for both
countries the trends and seasonal dummies exhibit stochastic patterns, although the exact specifications differ somewhat. The results for the two countries are discussed in more detail below.
{Table 3 about here}
The standard diagnostic tests for the model are very satisfactory with no indication of residual serial correlation, non-normality, or heteroscedasticity. In addition, there is no indication of
non-normality of the auxiliary residuals; hence no dummies were required for any significant outliers or structural breaks.15 The model is also stable as indicated by the post-sample predictive
failure tests. The lagged dependent variables and lagged price variables were found to be insignificant and only the first lag of GDP was found to be significant and hence retained. Consequently, the
model contains a small number of lagged variables, but the residuals are still white noise. This leads to a fairly quick adjustment of transportation oil demand to the price change in the UK.16 The
estimated long-run elasticities income and price are 0.80 and -0.12 respectively.
An impulse dummy for 1980q1 was experimented with in some initial estimation in order to capture an outlier during the period of recession in the UK. However, it was not required in the preferred
model in Table 3 and when included it had no discernible effect on the estimated parameters. Dummies for 1974 and/or 1979 were not needed since there was no fuel rationing implemented in the UK.
This finding is in contrast to Goodwin (1992) who finds that the long-run price elasticity tend to be between 50 per cent higher and three times higher than the short-run.
The stochastic trend in the preferred model is the local level with drift model. This model consists of a random walk component to capture the underlying level that evolves in a particular direction
as specified by the fixed slope components. The results of the LR tests (a) to (c) clearly indicate that all restrictions are rejected by the data. In addition when imposing some of the restrictions,
such as a deterministic trend, there were particularly adverse effects on the diagnostic tests resulting in very severe serial correlation of the residuals and problems of non-normality. This gives
further support to the view that stochastic modelling is necessary in this case.
Focussing on the estimated UEDT from the preferred UK model in Table 3, it can be seen from the top left-hand chart of Figure 3 that it is generally upward sloping.17
Therefore, holding income and price constant, the underlying use of transportation oil has been increasing. This illustrates that over the past 25 years (other than the last few years of the
estimation period) the sector has become more energy intensive. This increase in energy intensity shown by the upward UEDT reflects a shift in the oil demand curve to the right,
ceteris paribus.
{Figure 3 about here}
The estimated hyperparameter of the trend level is non-zero. However, the estimated hyperparameter of the slope is zero giving an underlying trend of 0.56% p.a. (as illustrated in the top right-hand
chart of Figure 3). This does not mean that the underlying trend is linear as assumed in conventional modelling. There is still considerable variation around this fixed slope as shown in the top
left-hand chart. This stochastic movement of the underlying trend being generated by the shifts in the level component rather than changes in the slope or growth rate.
Fuel efficiency of new cars in the UK has improved since 1978 as illustrated in Figure 4.18
This shows that there was a significant improvement during the early 1980s but it that levelled out thereafter. However, it would appear from the upward sloping shape of the
The top right hand chart in Figure 3 (and Figure 6) illustrates the quarterly growth rate of the trend i.e. the change in the slope of the trend component (given in the top left-hand chart). For the
UK it is horizontal since the hyperparameter of the slope is zero, unlike the Japanese results given in Figure 6.
estimated UEDT in the top left hand side of Figure 3 that generally these improvements have been more than outweighed by other ‘taste’ effects of the increase in oil demand. This could have come
about for many reasons such as the growth in car size and engine power and a worsening of traffic conditions in urban areas, resulting in hardly any change in the vehicle fleet fuel intensity. In
addition, the shift from public transport to (more energy intensive) private cars has contributed to the substantial growth of transportation oil demand over the sample period (Schipper et al., 1992,
p. 123). The number of car/motorcycle trips per person per year in the UK increased from 437 in 1975/76 to 641 in 1995/97 (an increasing proportion of all trips of 47% in 1975/76 to 61% in 1995/97);
whereas the total number of trips by public transport per person per year fell from 127 in 1975/76 to 92 in 1995/97 (a falling proportion of all trips of 14% in 1975/76 to 9% in 1995/97).19
In addition the distance travelled per person per year in the UK by car/motorcycle increased from 3,430 miles (72% of total) in 1975/76 to 5,590 miles (82%) in 1996/98; whereas the distance travelled
per person per year by public transport fell from 839 miles (18%) in 1975/76 to 851 (12½%) in 1996/98.20 Another possible contributing factor is the increasing use of cars for taking children to
school with the number of trips by foot falling and trips by car/van increasing considerably from the mid 1980s.21 A final factor to consider is the UK proportion of private cars in the total vehicle
stock illustrated in Figure 5.22 This illustrates that in the UK the proportion has always been relatively high (compared to Japan discussed below) and has a clear upward trend other than the last
few years. Overall, therefore the estimated UEDT is fully consistent with all the above indicators.
{Figure 4 about here}
{Figure 5 about here}
The hyperparameter of the seasonal components are relatively small compared to that of the level. This indicates that the stochastic movement in the seasonal component is not as large as the
stochastic fluctuation of the trend. However, the changes in the seasonal pattern are
before 1978.
Source: Transport Trends, 2001, UK National Statistics.
Source: Transport Trends, 2001, UK National Statistics.
Source: Transport Trends, 2001, UK National Statistics.
Source: Transport Statistics Bulletin, Vehicle Licensing Statistic, 2001, UK National Statistics. Data only available on an annual basis.
still found to be stochastic and are preferred to conventional deterministic seasonal dummies.
The pattern is illustrated in the bottom half of the chart of Figure 3. This illustrates that the magnitude of seasonal fluctuations has diminished since the early 1980s, with relative demand in the
first and fourth quarters gradually increased and relative demand in the second and third quarters gradually decreased. It is not immediately obvious why these changes have taken place, however, they
are relatively small; hence when conducting test (a), with deterministic dummies, the estimated long-run elasticities are very similar.
It is of some interest to compare the estimated elasticities from the preferred model given in Table 3 with those from the restricted versions tests (a) – (c), and the cointegration results discussed
in the Appendix. For test (a), the estimated long-run income and price elasticities are 0.86 and –0.12 respectively, whereas for both test (b) and test (c) they are 0.66 and –0.19 respectively. For
test (a), therefore, there is no difference in the price elasticity whereas the income elasticity is slightly higher. This is not surprising, given the relatively small seasonal effect discussed
above. However, for test (b) and test (c), the price elasticity increases in absolute terms whereas the income elasticity falls. Although the changes are bigger than test (a), they are not that
overly dramatic, which is not too surprising given the shape of the estimated UEDT for the UK, which is generally uni-directional and could be approximated by the linear time trend.
The results given in the Appendix show that cointegration is accepted for the UK with a deterministic trend but not without a trend.24 The estimated long-run income elasticity is 0.52 and long-run
price elasticity is –0.22 for the cointegration with trend model. For the cointegration without a trend model the long-run income and price elasticities are 1.14 and – 0.12 respectively; however,
given cointegration is not accepted these are not considered further. In the short-run dynamic equations, a large number of lags are needed (including some insignificant terms) to ensure that the
diagnostic tests are passed for the models. And, despite experimenting with various lag structures, it was not possible to eliminate the problem of heteroscedasticity in the cointegration with trend
model. Therefore, the preferred
As stated above, a temperature variable was also included in some initial estimation. Although the variable is significantly different from zero its inclusion has no discernible effect on the
estimated long-run elasticities. When included, the estimated size of the evolving seasonals is smaller, hence the preferred model in Table 3 captures all seasonality through the stochastic seasonal
model from the STSM framework is clearly more parsimonious than the cointegration models.
The estimated long-run elasticities from the cointegration with trend model are, not surprisingly, similar to those from imposing restriction (c) above although the income elasticity is slightly
lower and the price elasticity slightly higher (in absolute terms). Therefore, for both test (c) and the cointegration model with trend, the income elasticity is lower and the price elasticity higher
(in absolute terms) than those obtained from the STSM framework. In summary, when a stochastic trend and seasonal components are utilised with our UK data the income elasticity is higher and the
price elasticity is lower (in absolute terms) than the models incorporating deterministic components. However, given the failure of some diagnostic tests with the deterministic models and the more
parsimonious model obtained from the STSM framework, the UK model given in Table 3 is preferred.
Consistent with the UK results, the diagnostics of the model are very satisfactory with no problem of non-normality of the auxiliary residuals; hence, like the UK, no dummies were required for any
significant outliers or structural breaks.25 Moreover, the models are also stable as indicated by the post-sample predictive failure tests. The lagged price variables were found to be insignificant
and only the first lag of GDP was found to be significant and hence retained in the preferred model for Japan. In addition, the preferred model includes the second lagged difference of the dependent
variable. This was included since the second and third lags of the dependent variable were required to eliminate some problems of serial correlation. Individually they were insignificant but with
coefficients of almost equal size (in absolute terms) but of opposite signs. Therefore, the two variables (et-1 and et-2) were
replaced by their difference (∆et-2) which is significant at the 10% level. Despite this the
preferred specification is still fairly parsimonious.26
The stochastic trend in the preferred model is the most general form, the local linear model. This consists of a stochastic level and a stochastic slope.
Initial estimation indicated that there were outliers in 1990q3 and 1994q3 and impulse dummies for these periods were experimented with. However, further modelling showed that these were not
necessary given the normality of the auxiliary residuals so they were excluded. However, their inclusion or exclusion has no discernible effect on the estimated parameters. Dummies for 1974 and/or
1979 were not needed since there was no fuel rationing in Japan.
Similar to the UK, the LR tests for the deterministic trend and seasonal restrictions clearly favour the stochastic formulations. In particular, a significantly large LR value is found for the
restriction of a deterministic trend. Moreover, the imposed restrictions lead to very severe problems of serial correlation and non-normality of the residuals. All of which highlight the importance
of the stochastic formulation for modelling the demand for transportation oil in Japan.27
The estimated long-run income and price elasticities are 1.08 and –0.08 respectively. Compared to the UK, oil transportation demand in Japan is found to be more income sensitive but less price
sensitive in the long-run.
Since the underlying trend contains both a stochastic level and a stochastic slope, there is no clear continuous direction for the UEDT. This is illustrated in the two charts in the top half of
Figure 6. This means that the UEDT does not move in one direction with many small fluctuations as seen for the UK. Instead the UEDT, as indicated in the top left-hand chart of Figure 6, moves in a
non-linear fashion, increasing rapidly during the 1970s followed by a substantial decline during the early 1980s before beginning to increase again in the late 1980s. Since the late 1980s the UEDT
grew strongly, paralleling the 1970s. At the end of the estimation period, it was growing by 1.73% per annum.
{Figure 6 about here}
The movement of the UEDT between 1979 and 1988 was a period when, ceteris paribus,
there was a decline in the use of transportation oil leading to less energy intensity. This is in contrast to the increase in usage and rise in energy intensive periods of the rest of the estimation
period. Hence, during the period between 1979 and 1988 holding income and price constant, the oil transportation demand curve in Japan was shifting to the left whereas at other times it was shifting
to the right. This movement in the UEDT represents non-income or price effects given these variables are controlled for in the model, therefore illustrating that the UEDT consists of the ‘technical
progress’ effects and changes in ‘tastes’.28
since Franzén and Sterner (1995, p. 112) also had a problem specifying the dynamic relationship for this sector.
Like the UK a temperature variable was included in some initial estimation but was always insignificant, irrespective of the specification estimated.
Future research will address how the shape and structure of the UEDT is generated; why exactly does the UEDT for the UK slope upwards for almost all the estimation period whereas there are these
distinct periods for Japan’s UEDT? This is particularly desirable if the models were to be used for long-term forecasting.
Annual data for average fuel efficiency of the passenger vehicle stock in Japan between 1972 and 1997 are shown in Figure 7.29
It is interesting to compare the shape of the estimated UEDT in the top left-hand chart of Figure 6 with Figure 7. Both have surprisingly similar shapes, suggesting that the UEDT is picking up the
underlying effects of changes in average fuel efficiency of the passenger vehicle stock in Japan. In addition, Figure 830
shows the proportion of passenger cars from the total vehicle stock in Japan between 1972 and 1997. This is significantly lower than the UK figures and, unlike the UK, has distinctive changes in
trend where the proportion increased up to the late 1970s, decreased during the 1980s, and suddenly grew in the late 1980s. Similar to the UK, this pattern is reflected in the estimated UEDT. Thus,
like the UK results, the estimated UEDT appears to be picking up the significant underlying trends in the key aspects of efficiency and ‘tastes’.
The estimated hyperparameter value of the seasonal component is 0.162 which is much higher than for the UK (0.044), and the q-ratio, which is sometimes referred to as the signal
to noise ratio, is also considerably higher.31 This indicates that changes in the seasonal
movement over the sample period exhibit a very strong stochastic pattern that is clearly difficult to model by conventional deterministic seasonal dummies.
The estimated stochastic seasonal pattern is shown in the bottom two charts of Figure 6. The seasonal fluctuation diminished until about 1980 but increased since then. In particular, the demand in
the third quarter has grown over the sample period in contrast to the second quarter that dropped from the most consumed period during the 1970s to the second from 1980s onwards. The demand in the
first quarter and the fourth quarter has also decreased since 1980. An increase in relative importance of the third quarter against others might be explained by the combined effect from i) a
diffusion of air conditioner for summer season equipped in cars and ii) a relative increase in passenger vehicles over the sample period used for leisure activities in the summer season.
Source: Handbook of Energy and Economic Statistics in Japan 2002, EDMC Data is only available on an annual basis.
Source: Handbook of Energy and Economic Statistics in Japan 2002, EDMC. Data is only available on an annual basis.
Again, it is interesting to compare estimated elasticities from the restricted versions, tests (a) – (c) and the cointegration results in the Appendix with those from the preferred model presented in
Table 3. For test (a) the estimated long-run income and price elasticities are 1.02 and –0.07 respectively which, similar to the UK, do not represent dramatic changes. For test (b), the estimated
income and price elasticities are 0.53 and –0.04 respectively and for test (c) 0.51 and –0.04 respectively. Therefore, both of the tests where a linear trend is imposed, the income and the price
elasticities almost half. Thus, on this occasion there is a significant impact on the elasticities, which is not surprising given the linear trend does not act as a good proxy for the estimated UEDT
for Japan.
The results given in the Appendix show that, similar to the UK, cointegration is accepted for Japan with a deterministic trend but not without a trend.32 The estimated long-run income elasticity is
0.55 and long-run price elasticity is –0.02 for the cointegration with trend model. For the cointegration without a trend model the long-run income and price elasticities are 1.07 and –0.03
respectively; however, given cointegration is not accepted these are not considered further. In the short-run dynamic equations, a very large number of lags are needed (including some insignificant
terms) to ensure that diagnostic tests are passed for the models. Again, similar to the UK results, the preferred model from the STSM framework is far more parsimonious than the cointegration models.
Therefore from an econometric perspective the STSM is preferred for Japan. Moreover, the error correction term is not significantly different from zero – casting further doubt about the validity of
the cointegration results for Japan.
The estimated long-run income elasticity from the cointegration with trend model is just a little higher than that found when imposing restriction (c) above although the estimated long-run price
elasticity is halved (in absolute terms). For both test (c) and the cointegration model with trend, the estimated price elasticities are a half and a quarter respectively of the estimate obtained
from the STSM framework, whereas the estimated income elasticities are about a half. In summary, when a stochastic trend and seasonal components are utilised with our data for Japan, the estimated
income elasticity is significantly higher and the estimated price elasticity is significantly higher (in absolute terms) than the models incorporating deterministic components. However, given the
failure of some diagnostic tests when
imposing the deterministic restrictions, the poor cointegration results, and the far more parsimonious model obtained from the STSM framework, the model for Japan given in Table 3 is preferred
statistically on a number of fronts.
In this paper, we attempt to estimate efficiently the income and price elasticities of oil demand for the transport sectors of the UK and Japan. Given the growing size and importance of the
transportation sector, and the resultant environmental impact, it is vital that accurate estimates are obtained. To achieve this we have demonstrated the need to model adequately the Underlying
Energy Demand Trend (UEDT). We have argued that in addition to a ‘technical progress’ or energy efficiency effect the UEDT must also accommodate other non-measurable influences such as ‘tastes’.
Given these influences it would be extremely unlikely that the total effect would be adequately modelled by a simple linear time trend, which has been the conventional approach. Therefore, we have
adopted Harvey’s structural time series model since this allows for a more general and flexible framework. This is achieved by estimating a stochastic underlying trend for the transport sectors of
the UK and Japan. Not surprisingly, we find for both countries that a simple linear time trend (or no trend at all) is rejected on a number of criteria. This results in a generally upward sloping
UEDT for the UK suggesting that, ceteris paribus, the demand curve for transportation oil in the UK has been shifting to the right over the estimation period. For Japan however, the UEDT has a
distinct phase where the UEDT is downward sloping and hence, ceteris paribus, the transportation oil demand curve was shifting to the left but also other phases where the UEDT is upward sloping
hence, ceteris paribus, the demand curve was shifting to the right. Moreover, as Hunt et al. (2003) demonstrate, mis-specification of the UEDT could lead to significant biases of the income and price
elasticities, with the biases dependent on the direction of the UEDT, income, and price. Therefore, in the case of the UK and Japan in particular, these biases are likely to be quite marked given the
estimated UEDT and the movement of real transportation oil prices over the period.
The argument that the estimated UEDT will capture a whole range of influences in addition to ‘technical progress’ is demonstrated for both countries. For the UK, it was illustrated that
a number of ‘taste’ factors appear to have outweighed the improvements in efficiency. It is even more marked for Japan, where the estimated UEDT has two distinct changes clearly reflecting the
changes in the fuel efficiency and the proportion of passenger vehicles, i.e. the combination and interaction of changes in efficiency and ‘tastes’.
The evidence presented shows that, even when the data for energy efficiency, tastes, and other variables are unavailable, or not in an appropriate format, the STSM approach is still able to
accommodates the effect of these factors on oil demand; with the estimated UEDT acting as an approximation. As indicated earlier in the paper, the STSM/UEDT approach can be considered a second best
procedure where it is not possible to obtain all variables and model accordingly. However, in situations where full information on a number of variables is not available, this approach is an ideal
‘second best’ procedure; one that produces unbiased estimates of the long-run income and price elasticities. This is particularly relevant for energy demand modelling, and oil demand in particular,
where the derived demand will not only depend on the efficiency of the energy using appliances but also a whole range of factors (as discussed earlier) that can be captured using the STSM/UEDT
The advantage of allowing for a stochastic formulation for the seasonal pattern in the data has also been demonstrated. The conventional seasonal dummy approach is rejected in favour of the
stochastic formulation for both countries. This results in evolving seasonality but with different patterns for the two countries. This again, similar to the arguments for incorporating the UEDT, is
important. It is almost impossible to measure the causes of these changes in practice, however the STSM approach implicitly allows for any socio-economic effects that cause the seasonal pattern to
change within a year and hence ensure the estimated elasticities are not biased.
As a result of this approach, the estimated long-run price and income elasticities of demand for UK transportation oil are 0.80 and –0.12 respectively. These estimates are both towards the lower end
of the range of previous studies (see Table 1). However, our study uses a later data period, a different frequency of data, and a different technique so a true comparison is difficulty. However,
given our more general approach and the statistical results, we would argue that our estimates are preferable. For Japan, the long-run income and price elasticities are 1.07 and –0.08 respectively
but there are fewer previous studies to compare with than the
UK. However, it was worth noting that the estimates for both the UK and Japan are much lower than the averages of previous oil demand studies calculated by Dahl and Sterner (1991) (see Table 1). In
summary, for both countries the estimated income and price elasticities are both lower (in absolute terms) than the previously cited studies. This is not surprising since the previously cited studies
generally ignore the issue of the UEDT, so that the estimated price and income effects from these studies are implicitly required to pick up the exogenous effects that in our approach are attributed
to the UEDT. In addition the speed of adjustment is much quicker in the present study compared to those cited in Table 1. Again this is to be expected given the STSM/UEDT framework clearly
distinguishes between the ‘pure’ income and price effects, holding other factors such as the appliance stock constant, and adjustment in this context would be expected to be quicker.
Finally, it is worth emphasising the similarities and differences between the results for the UK and Japan. Firstly, the estimated long-run elasticities from using the STSM/UEDT framework, although
not identical, are relatively similar. The long-run income elasticities are 1.08 and 0.81 for Japan and the UK respectively; similarly, the long-run price elasticities are
–0.08 and –0.12 for Japan the UK respectively. Therefore, the major difference between two countries is the different shape of the UEDTs and the different seasonal patterns. Hence, the underlying
differences in the characteristics of the transportation sectors of the two countries, as discussed in the results section, are captured by the stochastic formulations of the UEDT and seasonals.
This, it could be argued, is to be expected; the economic influences having a similar impact on oil demand, whereas other underlying factors such as different rates of efficiency, socio-economic
factors, consumer preferences, etc. are captured by the different non-linear UEDT and evolving seasonals.
In conclusion, the more flexible approach available via the STSM framework is arguably a superior technique to the more conventional techniques when estimating transportation oil demand functions. It
produces unbiased estimates of the long-run income and price elasticities, even when it is not possible to capture all the underlying influences explicitly, and we would speculate, that our estimates
are likely to prove more reliable.
In order to facilitate the comparison of the STSM results in the main text, transportation oil demand functions for the UK and Japan are estimated using the Engle-Granger two step cointegration
technique. Since the methodology has been well explained in many places (see for example Hendry and Juselius, 2001) only the results are presented here in Table A1.33 For the first step of the
procedure, two equilibrium demand relationships are estimated for both countries. One with oil demand in logs (et) as a function of GDP in logs (yt), the real oil
price in logs (pt), a constant, deterministic seasonal dummies and a deterministic trend. The
other with the deterministic trend omitted. For all relationships the residuals (ECt) are tested
for cointegration by the Augmented Dickey-Fuller test to confirm that the valid long-run relationship is statistically acceptable. The results from this first step, discussed further below, are given
in the top part of Table A1.
For the second step a short-run error correction relationship is estimated for all the long-run relationships from the first step. This involved estimating the first difference of et as a
function of a constant, deterministic seasonal dummies, the first difference of lags on the first differences of et, yt and pt and the lagged error correction term ECt. The preferred model was
found by testing down from this general equation providing the diagnostic tests given in Table A1 are not violated, such as non-normality, serial correlation, heteroscedasticity, etc. The results of
the second step are given in the bottom part of Table A1. The software
package PcGive 9.10 (Hendry and Doornik, 1996) was used to estimate all specifications for both steps.
For the UK the estimated long-run income and price elasticities are 0.52 and –0.22 respectively for the specification with a trend and 1.14 and –0.12 respectively for the specification without a
trend. However, cointegration is only accepted for the trend specification.34
{Table A1 about here}
When estimating the error correction equation for the trend specification it proved impossible to eliminate the problem of heteroscedasticity despite many experiments with different lag structures.
Therefore this problem still exists in the preferred model given in Table A1. For the specification without a trend a large number of lags were needed (including some not significant even at the 10%
level) to ensure all diagnostic tests are passed. For the trend specification the error correction term is significant at the 1% level, whereas for the no trend specification it is significant at the
5% level.
For Japan, the estimated long-run income and price elasticities are 0.55 and –0.02 respectively for the specification with a trend and 1.07 and –0.03 respectively for the
For both the UK and Japan all variables are non stationary and are integrated of order 1, I(1).
The coefficient on the time trend is positive, which is to be expected given the shape of the estimated UEDT via the STSM framework.
specification without a trend. However for Japan, cointegration is accepted at the 5% level for the trend specification but rejected for the no trend specification.35
For Japan longer lags were required than the UK to ensure all diagnostic tests are passed resulting in a large and complicated lag structure for both specifications. However, the error correction
terms in both specifications were always not significantly different from zero – casting some doubt on the robustness of the cointegration results for Japan.
The coefficient on the time trend is also positive for Japan – although it is not obvious that this is to be expected given the shape of the estimated UEDT via the STSM framework this is .
Dahl, C. and Sterner, T. (1991), ‘Analysing Gasoline Demand Elasticities: A survey’, Energy Economics, Vol. 13, No. 3, pp. 203-210.
Dargay, J. M. (1992), ‘The Irreversible Effects of High Oil Prices: Empirical Evidence for the Demand for Motor Fuels in France, Germany and the UK’ in Hawdon, D. (ed.), Energy Demand: Evidence and
Expectations, Surrey University Press, Guildford, UK.
Dargay, J. M. (1993), ‘The Demand for Fuels for Private Transport in the UK’, in Hawdon, D. (ed.),
Recent Studies in the Demand for Energy in the UK, Surrey Energy Economics Discussion Paper, No. 72, Surrey Energy Economics Centre (SEEC), Department of Economics, University of Surrey, Guildford,
Department of Trade and Industry (DTI) (2000), Energy Projections for the UK, Working Paper, EPTAC Directorate, UK, March.
Fouquet, R., Pearson, P., Hawdon, D. and Robinson, C. (1997), ‘The Future of UK Final User Energy Demand’, Energy Policy, Vol. 25, No. 2, pp. 231 - 240.
Franzén, M. and Sterner, T. (1995), ‘Long-run Demand Elasticities for Gasoline’ in Barker, T., Ekins, P. and Johnstone, N. (eds.), Global Warming and Energy Demand, Routledge, London, UK.
Glaister, S. and Graham, D. (2000), The Effect of Fuel Prices on Motorists, A Report for the AA Motoring Policy Unit and the UK Petroleum Industry Association, AA Motoring Policy Unit, Basingstoke,
Goodwin, P. B. (1992), ‘A Review of New Demand Elasticities with Special Reference to Short and Long run Effects of Price Changes’, Journal of Transport Economics and Policy, Vol. XXVI, No. 2, pp.
155 - 163.
Harvey, A. C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, Cambridge, UK.
Harvey, A. C. (1997), ‘Trends, Cycles and Autoregressions’, Economic Journal, Vol. 107, No. 440, pp. 192 - 201.
Harvey, A. C., Henry, S. G. B., Peters, S. and Wren-Lewis, S. (1986), ‘Stochastic Trends in Dynamic Regression Models: An Application to the Employment-Output Equation’, Economic Journal, Vol. 96,
No. 384, pp. 975 - 985.
Harvey, A. C. and Koopman, S. J. (1992), ‘Diagnostic Checking of Unobserved-Components Time Series Models’, Journal of Business and Economic Statistics, Vol. 10, pp. 377-389.
Harvey, A. C. and Scott, A. (1994), ‘Seasonality in Dynamic Regression Models’, Economic Journal, Vol. 104, No. 427, pp. 1324 - 1345.
Hendry, D. F. and Doornik, J. A. (1996), Empirical Econometric Modelling Using PcGive for Windows, Timberlake Consulting: London, UK.
Hendry, D. F. and Juselius, K. (2000), ‘Explaining Cointegration Analysis: Part I’, The Energy Journal, Vol. 21, No. 1, pp. 1 - 42.
Hendry, D. F. and Juselius, K. (2001), ‘Explaining Cointegration Analysis: Part II’, The Energy Journal, Vol. 22, No. 1, pp. 75 - 120.
Hodgson, D. and Miller, K. (1995), ‘Modelling UK Energy Demand’, in Barker, T., Ekins, P. and Johnstone, N. (eds.), Global Warming and Energy Demand, Routledge, London, UK.
Hunt, L. C., Judge, G. and Ninomiya, Y. (2003), ‘Modelling Underlying Energy Demand Trends’, Chapter in Hunt, L. C. (Ed) Energy in a Competitive Market: Essays in Honour of Colin Robinson, Edward
Elgar, forthcoming.
Hunt, L.C., and Manning, N. (1989) ‘Energy price- and income-elasticities of demand: some estimates for the UK using the co-integration procedure’, Scottish Journal of Political Economy, Vol. 36, No.
2, pp. 183-193.
Johansson, O. and Schipper, L. (1997), ‘Measuring the Long Run Fuel Demand of Cars: Separate estimations of Vehicle Stock, Mean Fuel Intensity, and Mean Annual Driving Distance’, Journal of Transport
Economics and Policy, Vol. XXXI, No. 3, pp. 277 - 292.
Kennedy, P. (1992), A Guide to Econometrics, (3rd Edition), Blackwell, Oxford, UK
Koopman, S. J., Harvey, A. C., Doornik, J. A., and Shephard, N. (1995), STAMP 5.0, International Thompson Business Press, London, UK.
Nachane, D., Nadkarmi, R. and Karnik, A. (1988), ‘Co-integration and causality testing of the energy-GDP relationship: a cross-country study’, Applied Economics, Vol. 20, pp. 1511 - 1531.
Schipper, L., Meyes, S., Howarth, R. B., and Steiner, R. (1992), Energy Efficiency and Human Activity: Past Trends, Future Prospects, Cambridge University Press, Cambridge, UK. Sterner, T. and Dahl,
A. (1991), ‘Modelling Transport Fuel Demand’, in Sterner, T. (ed.),
International Energy Economics, Chapman & Hall, London, UK.
Thomas, R. L. (1993), Introductory Econometrics: Theory and Applications, (2nd Edition), Longman, London, UK
Wohlgemuth, N. (1997), ‘World Transport Energy Demand Modelling: Methodology and Elasticities’,
Study (year of publication) Area covered Technique / model used Data used Estimated LR elasticities Dargay (1992) Petrol and diesel oil Dynamic ECM irreversible
demand model
UK annual data 1960 - 88 (29 obs.)
ηy = 1.49
ηp = -0.10 (insignificant at 10% level)
Dynamic ECM conventional (reversible) model
UK annual data 1960 - 88 (29 obs.)
ηy = 0.70 (insignificant at 10% level)
ηp = -0.40 (insignificant at 10% level)
Dargay (1993) Petrol EG 2-step
(structural form model)
UK annual data 1950 - 91 (42 obs.)
ηy = 1.5
ηp = -0.7 to -1.4
Hodgson and Miller (1995) Petrol DTI energy model UK (details are not reported) ηy = 0.81
ηp = -0.3
Fouquet et al. (1997) Petrol EG 2-step UK annual data 1960 - 94 (35 obs.)
ηy = 1.95 to 2.05
ηp = 0
Franzén and Sterner (1995) Petrol Dynamic log linear model UK annual data 1960 - 88 (29 obs.)
ηy = 1.6
ηp = -0.4
Japan annual data 1960 - 88 (29 obs.)
ηy = 0.77
ηp = -0.76
(ηy and ηpobtained from a model with
an arbitrary restriction on the lagged dependent variable)
OECD aggregated data 1960 - 88 (29 obs.)
ηy = 1.30
ηp = -0.60
Sterner and Dahl (1991) Petrol Dynamic log linear model OECD aggregated annual data 1960 - 85 (26 obs.)
ηy = 1.1 to 1.3
ηp = -0.80 to -0.95
Dahl and Sterner (1991) Petrol Literature survey n/a ηy = 1.31
ηp = -0.80
(ηy and ηp are average values based on
the dynamic log-linear model) Goodwin (1992) Petrol Literature survey n/a ηp = -0.71 (time series case)
Johansson and Schipper (1997) Car fuels including Diesel, LPG and CNG
Dynamic log linear model (structural form model)
12 OECD individual data
1973 - 92 (20 obs. for each country)
ηy = 1.2 (mean value)
ηp = -0.7 (mean value)
DTI (2000) Road fuel (details are not reported) UK (details are not reported) ηp = -0.23
Underlying Energy Demand Trend (UEDT)
(Pure) Technical energy efficiency ‘Tastes’
Embodied Disembodied
UK AND JAPAN 1972q1-1995q4(DEPENDENT VARIABLE et)
Variables UK JAPAN yt 0.5708** 0.6464** (5.259) (3.599) yt-1 0.2307* 0.4338* (2.162) (2.488) pt -0.1233** -0.0828** (4.058) (3.605) ∆et-2 -0.1229 (1.833)
Estimated Long-Run Elasticities
Income (Y) 0.801 1.080 Price (P) -0.123 -0.083 Estimated Hyperparameters σε2× 10- 4 0.8703 1.0667 ση2× 10- 4 0.7699 0.3837 σξ2× 10- 4 0 0.0128 σω2× 10- 4 0.0440 0.1620
Nature of Trend Local level with drift Local trend
Diagnostics Standard Error 1.58% 1.78% Normality 1.05 0.29 Kurtosis 0.01 0.01 Skewness 1.04 0.28 H(30) 0.89 0.82 r(1) 0.01 -0.01 r(4) -0.06 -0.07 r(8) 0.00 -0.05 DW 1.97 2.00 Q Q(8,6)=0.71 Q(9,6)=
6.12 R2 0.99 0.99 Rs2 0.50 0.61 Auxiliary Residuals Irregular: Normality 0.58 2.53 Kurtosis 0.34 1.99 Skewness 0.24 0.55 Level: Normality 1.45 2.51 Kurtosis 0.68 0.36 Skewness 0.77 2.15
Slope: Normality n/a 1.40
Kurtosis n/a 0.30 Skewness n/a 1.10 Prediction test (96q1-97q4) χ2 (8) 7.54 4.85 Cusum t (91) -0.45 -0.90 LR tests Test (a) 7.09** 48.01** Test (b) 46.00** 155.91** Test (c) 46.46** 158.34** Notes:
¾ t-statistics are given in the parenthesis.
¾ ** indicates significant at the 1% level and * indicates significant at the 5% level. ¾ The coefficient on ∆et-2 for Japan is significant at the 10% level.
¾ The restrictions imposed for the LR tests (a), (b), and (c) are explained in the text. ¾ Normality is the Bowman-Shenton statistic approximately distributed as χ2[(2);] | {"url":"https://1library.net/document/qmk98x5z-modelling-underlying-energy-stochastic-seasonality-econometric-analysis-transport.html","timestamp":"2024-11-14T21:09:36Z","content_type":"text/html","content_length":"219794","record_id":"<urn:uuid:1f8a897f-feea-4a9f-a39a-5f24e37b949e>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00020.warc.gz"} |
atanhl(3) [posix man page]
atanhl(3) [posix man page]
ATANH(P) POSIX Programmer's Manual ATANH(P)
atanh, atanhf, atanhl - inverse hyperbolic tangent functions
#include <math.h>
double atanh(double x);
float atanhf(float x);
long double atanhl(long double x);
These functions shall compute the inverse hyperbolic tangent of their argument x.
An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these
functions. On return, if errno is non-zero or fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error
has occurred.
Upon successful completion, these functions shall return the inverse hyperbolic tangent of their argument.
If x is +-1, a pole error shall occur, and atanh(), atanhf(), and atanhl() shall return the value of the macro HUGE_VAL, HUGE_VALF, and
HUGE_VALL, respectively, with the same sign as the correct value of the function.
For finite |x|>1, a domain error shall occur, and either a NaN (if supported), or an implementation-defined value shall be returned.
If x is NaN, a NaN shall be returned.
If x is +-0, x shall be returned.
If x is +-Inf, a domain error shall occur, and either a NaN (if supported), or an implementation-defined value shall be returned.
If x is subnormal, a range error may occur and x should be returned.
These functions shall fail if:
Domain Error
The x argument is finite and not in the range [-1,1], or is +-Inf.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [EDOM]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the invalid floating-point exception shall be raised.
Pole Error
The x argument is +-1.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the divide-by-zero floating-point exception shall be raised.
These functions may fail if:
Range Error
The value of x is subnormal.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the underflow floating-point exception shall be raised.
The following sections are informative.
On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at
least one of them must be non-zero.
feclearexcept() , fetestexcept() , tanh() , the Base Definitions volume of IEEE Std 1003.1-2001, Section 4.18, Treatment of Error Condi-
tions for Mathematical Functions, <math.h>
Portions of this text are reprinted and reproduced in electronic form from IEEE Std 1003.1, 2003 Edition, Standard for Information Technol-
ogy -- Portable Operating System Interface (POSIX), The Open Group Base Specifications Issue 6, Copyright (C) 2001-2003 by the Institute of
Electrical and Electronics Engineers, Inc and The Open Group. In the event of any discrepancy between this version and the original IEEE
and The Open Group Standard, the original IEEE and The Open Group Standard is the referee document. The original Standard can be obtained
online at http://www.opengroup.org/unix/online.html .
/The Open Group 2003 ATANH(P)
Check Out this Related Man Page
ATANH(3P) POSIX Programmer's Manual ATANH(3P)
This manual page is part of the POSIX Programmer's Manual. The Linux implementation of this interface may differ (consult the correspond-
ing Linux manual page for details of Linux behavior), or the interface may not be implemented on Linux.
atanh, atanhf, atanhl - inverse hyperbolic tangent functions
#include <math.h>
double atanh(double x);
float atanhf(float x);
long double atanhl(long double x);
These functions shall compute the inverse hyperbolic tangent of their argument x.
An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these
functions. On return, if errno is non-zero or fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error
has occurred.
Upon successful completion, these functions shall return the inverse hyperbolic tangent of their argument.
If x is +-1, a pole error shall occur, and atanh(), atanhf(), and atanhl() shall return the value of the macro HUGE_VAL, HUGE_VALF, and
HUGE_VALL, respectively, with the same sign as the correct value of the function.
For finite |x|>1, a domain error shall occur, and either a NaN (if supported), or an implementation-defined value shall be returned.
If x is NaN, a NaN shall be returned.
If x is +-0, x shall be returned.
If x is +-Inf, a domain error shall occur, and either a NaN (if supported), or an implementation-defined value shall be returned.
If x is subnormal, a range error may occur and x should be returned.
These functions shall fail if:
Domain Error
The x argument is finite and not in the range [-1,1], or is +-Inf.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [EDOM]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the invalid floating-point exception shall be raised.
Pole Error
The x argument is +-1.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the divide-by-zero floating-point exception shall be raised.
These functions may fail if:
Range Error
The value of x is subnormal.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression
(math_errhandling & MATH_ERREXCEPT) is non-zero, then the underflow floating-point exception shall be raised.
The following sections are informative.
On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at
least one of them must be non-zero.
feclearexcept(), fetestexcept(), tanh(), the Base Definitions volume of IEEE Std 1003.1-2001, Section 4.18, Treatment of Error Conditions
for Mathematical Functions, <math.h>
Portions of this text are reprinted and reproduced in electronic form from IEEE Std 1003.1, 2003 Edition, Standard for Information Technol-
ogy -- Portable Operating System Interface (POSIX), The Open Group Base Specifications Issue 6, Copyright (C) 2001-2003 by the Institute of
Electrical and Electronics Engineers, Inc and The Open Group. In the event of any discrepancy between this version and the original IEEE
and The Open Group Standard, the original IEEE and The Open Group Standard is the referee document. The original Standard can be obtained
online at http://www.opengroup.org/unix/online.html .
/The Open Group 2003 ATANH(3P) | {"url":"https://www.unix.com/man-page/posix/3/atanhl","timestamp":"2024-11-09T17:35:42Z","content_type":"text/html","content_length":"38841","record_id":"<urn:uuid:e32de0fa-8501-4d46-9dcc-8dd68c838e65>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00328.warc.gz"} |
Introduction to continuum mechanics for engineers
Introduction to continuum mechanics for engineers / Ray M. Bowen
Type de document : Livre numériqueCollection : Mathematical concepts and methods in science and engineeringLangue : anglais.Éditeur : Plenum Press, 1989ISBN: 9780486474601.ISSN: 0885-9418.Sujet MSC :
74-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
74Axx, Mechanics of deformable solids - Generalities, axiomatics, foundations of continuum mechanics of solids
74A20, Generalities, axiomatics, foundations of continuum mechanics of solids, Theory of constitutive functions in solid mechanicsEn-ligne : sur le site de l'auteur
No physical items for this record
This paper presents a distinct and direct way to continuum mechanics for beginners in this field. It starts studying one-dimensional continua. Thus essential concepts are introduced without burdening
by the mathematics of three-dimensional field formulations. A concise chapter on kinematics and a clearly arranged one on balance equations are following. The basis of mathematical description of
material properties is established by treating a collection of examples concerning elasticity and viscoelasticity of the rate type of simple materials. An appendix on mathematics used closes the
elaboration. The contents of the book are presented clear and careful with respect to the compilation and the mathematical presentation. How it is used in works on this field certain fundamentals are
not discussed from the physical point of view: By which physical reasons authors prefer in the domain of Newtonian continuum mechanics rigid frames of reference in contradiction to the perception of
general relativity? What physical concept or idea of measuring non-equilibrium temperatures shall be used? Does the dissipation inequality local in time hold in the more general case of retarded
dissipation (fading memory), too? So the reader gets no reference to what is convention and what is necessity. (Zentralblatt)
There are no comments on this title. | {"url":"https://catalogue.i2m.univ-amu.fr/bib/11847","timestamp":"2024-11-13T16:04:25Z","content_type":"text/html","content_length":"62388","record_id":"<urn:uuid:6a8b8f0b-5e2d-4bff-b72f-2c0c837ca816>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00341.warc.gz"} |
Reverse Linked List (LeetCode) - Iterative and recursive implementation | Learn To Code Together
Reverse Linked List (LeetCode) – Iterative and recursive implementation8 min read
Happy New Year, everyone! On the very first day of 2020, let’s jump to a problem that has been asked all the time during the interview – Reverse A Linked List. In this article, we will tackle this
problem, but first, if you don’t know what is linked list, check this article before we proceed.
If you want to take a try before we jump into the solution, feel free to redirect this link to do it on LeetCode.
Given a singly linked list, and you have to reverse it, for example:
Input: 1->2->3->4->5->NULL
Output: 5->4->3->2->1->NULL
This question can be solved by either iteratively or recursively and we gonna do them both in JavaScript.
Iterative method
The procedure to solve this problem by the iterative method can be constructed as follow:
• First, we just need to initialize one pointer, prev to NULL and one another variable curr points to head.
• Inside the loop, we need to create a variable nextNode which is assigned to the pointer of the next node after each iteration starting from the head.
• Then we assign the next pointer of the curr to the prev.
• We assign the value of the curr to the prev variable.
• After that, we assign the value of the nextNode to the curr.
• Once our curr variable reaches the last element of the list, we terminate the loop and return prev.
Now let’s do it in JavaScript. First, we create a function reverseList which takes one parameter head, then inside this function, we create 2 variables, prev with the value of NULL and the head value
is assigned to the curr variable.
Definition for singly-linked list.
function ListNode(val) {
this.val = val;
this.next = null;
/ /*
@param {ListNode} head
@return {ListNode}
var reverseList = function(head) {
let prev = null;
let curr = head;
The code in the comment is the definition for a singly linked list, it works as a constructor when we create a new node inside the reverseList function.
Now, we create a while loop, we do actual work inside this loop. We create a new variable nextNode which helps us point to the next node of the current node in each iteration. Then we swap values,
the curr.next typically points to the next node of the current node, but we change it to point to the prev node, then we take the value of the head and assign it to the prev, and the curr now points
to the nextNode, the loop will terminate when the head pointer reaches to NULL:
while(curr != null){
let nextNode = curr.next;
curr.next = prev;
prev = curr;
curr = nextNode;
And when we have done, everything we need is to return the prev, which is the pointer of the list has been reversed, the complete code is shown below:
var reverseList = function(head) {
let prev = null;
let curr = head;
while(curr != null){
let nextNode = curr.next;
curr.next = prev;
prev = curr;
curr = nextNode;
return prev
I think it’s still a little ambiguous up to this point; let’s clarify it a little more; suppose we have a singly linked list with a structure as such 1 -> 2 -> 3 -> NULL and want to reverse it to 3
-> 2 -> 1 -> NULL:
prev = null
curr = head
On the first iteration:
nextNode = curr.next <=> nextNode = 2
curr.next = prev <=> curr.next = NULL (before points to 2, then now it points backward to NULL)
prev = curr <=> prev = 1
curr = nextNode <=> head = 2
After the first iteration, now we have:
nextNode = 2
curr.next = NULL
prev = 1
curr = 2 (after the first iteration, now the curr is the second node)
On the second iteration:
nextNode = curr.next <=> nextNode = 3
curr.next = prev <=> curr.next = 1 (before it points to the third now, but now we make it points back to the 1st node)
prev = curr <=> prev = 2
curr = nextNode <=> curr = 3
After the second iteration, now we have:
nextNode = 3
curr.next = 1
prev = 2
curr = 3 (now the curr is the third node)
On the third iteration:
nextNode = curr.next <=> nextNode = NULL
curr.next = prev <=> curr.next = 2 (before it points to the last node, which is NULL, now it points to the second node)
prev = curr <=> prev = 3
curr = nextNode <=> curr = NULL
After the third iteration, now we have:
nextNode = NULL
curr.next = 2
prev = 3
curr = NULL
On the fourth iteration:
Because the condition curr != NULL is no longer true (which means we have reached the last node), then this loop is terminated.
Recursive Approach
The recursive approach is a little trickier than the iterative method. But the general idea is listed as follow:
• Start from the head and assign it to the curr variable
• If curr is NULL, return curr
• Create a variable has a role as a pointer and do the recursive call to the next curr
• If the curr.next is NULL, which means it is the last node of our list, so we make it as the head of our list, as we are doing the reverse linked list
• We recursively traverse the list
• Set curr.next.next to curr
• Set curr.next to NULL.
To help us better envision, first, let’s take a look at some visualized examples before we get into some code. Suppose that we have a singly linked list 1->2->3->4->NULL and want it will be 4->3->2->
1->NULL after our reverse function:
For the first recursive call, we set curr as head, then we check whether curr and curr.next is null or not, because both of them are not null, we do another recursive call:
On the second recursive call, we change our curr to the second node, and we observe that the curr and curr.next are both still not null, then we proceed another recursive call:
On the third recursive call is pretty much the same as the first and the second want, now the curr is the third node and curr and curr.next are still not null:
On this recursive call, we have seen something different. The curr node is in the last node because curr.next is null, then it is now our new head as in our base condition:
After the fifth recursive call, now our head is the fourth node and we recursively traverse the list and make the third node as our curr, also we set the curr.next.next which initially pointed to
null in this picture; now it points back curr itseft and curr.next to null.
On the sixth recursive call, things go pretty similar to the last one. Now we already see that the last node now points to the third node, and in a similar manner, the third node now points back to
the second node:
Here we go again:
And finally, we have a reversed linked list by using a recursive function:
The code can be translated pretty much the same as we have discussed above:
Definition for singly-linked list.
function ListNode(val) {
this.val = val;
this.next = null;
/ /*
@param {ListNode} head
@return {ListNode}
var reverseList = function(head) {
let curr = head;
if (curr == null || curr.next == null) return curr;
let pointer = reverseList(curr.next);
curr.next.next = curr;
curr.next = null;
return pointer;
First, we define a curr as head, after that we have a base case here, if curr is null or curr.next is null, we return curr. We have another variable pointer which does the recursive call
consecutively until the base case is met. And once curr.next is null, which means we reach the end of our list, we will do some next steps, set curr.next.next to curr and curr.next to null. Finally,
we return this variable pointer which is what we should return.
Inline Feedbacks
View all comments
| Reply | {"url":"https://learntocodetogether.com/reverse-a-linked-list/","timestamp":"2024-11-08T18:09:54Z","content_type":"text/html","content_length":"149239","record_id":"<urn:uuid:4dbda799-725a-41e8-9688-0a8c946d7604>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00378.warc.gz"} |
Computational Journalism | At the Tow Center for Digital Journalism, Columbia University, as taught by Jonathan Stray | Page 6
This week we looked at how to determine if what you think you’re seeing in your data is actually there. It was a warp speed introduction to a big chunk of what humanity now knows about truth-finding
methods. Most of the ideas behind the methods are centuries or sometimes millennia old, but they were very much fleshed out in the 20th century, and these profound ideas haven’t percolated through to
all disciplines yet.
“Figuring out what is true from what we can see” is called inference, and begins with a strong feel for how probability works, and what randomness looks like. Take a look at this picture (from the
paper Graphical Inference for Infovis), which shows how well 500 students did on each of nine questions, each of which is scored from 0-100% correct.
Is there a pattern here? It looks like the answers on question 7 cluster around 75% and then drop off sharply, while the answers for question 6 show a bimodal distribution — students either got it or
they didn’t.
Except that this is actually completely random synthetic data, drawn from a uniform distribution (equal chance of every score.) It’s very easy to make up narratives and see patterns that aren’t there
— a human tendency called apohenia. To avoid fooling yourself, the first step is to get a feel for what randomness actually looks like. It tends to have a lot more structure, purely by chance, than
most people imagine.
Here’s a real world example from the same paper. Suppose you’re interested to know if the pollution from the Texas oil industry causes cancer. Your hypothesis is that if refineries or drilling
release carcinogens, you’ll see higher cancer rates around specific areas. Here’s a plot of the cancer rates for each county (darker is more cancer.) One of these plots is real data, the rest are
randomly generated by switching the counties around. (click for larger.)
Can you tell which one is the real data? If you can’t tell the real data from the random data, well then, you don’t have any evidence that there is a pattern to the cancer rates.
In fact, if you show these pictures to people (look at the big version), they will stare at them for a minute or two, and then most folks will pick out plot #3 as the real data, and it is. This is
evidence (but not proof) that there is a pattern there that isn’t random — because it looked different enough from the random patterns that you could tell which plot was real.
This is an example of a statistical test. Such tests are more typically done by calculating the odds that what you see has happened by chance, but this is a purely visual way to accomplish the same
thing (and you can use this technique yourself on your own visualizations; see the the paper for the details.)
It’s part of the job of the journalist to understand the odds. In 1976, there was a huge flu vaccination program in the U.S. In early October, 14 elderly people died shortly after receiving the
vaccine, three of them in one day. The New York Times wrote in an editorial,
It is conceivable that the 14 elderly people who are reported to have died soon after receiving the vaccination died of other causes. Government officials in charge of the program claim that it
is all a coincidence, and point out that old people drop dead every day. The American people have even become familiar with a new statistic: Among every 100,000 people 65 to 75 years old, there
will be nine or ten deaths in every 24-‐hour period under most normal circumstances.
Even using the official statistic, it is disconcerting that three elderly people in one clinic in Pittsburgh, all vaccinated within the same hour, should die within a few hours thereafter. This
tragedy could occur by chance, but the fact remains that it is extremely improbable that such a group of deaths should take place in such a peculiar cluster by pure coincidence.
Except that it’s not actually extremely improbable. Nate Silver addresses this issue in his book by explicitly calculating the odds:
Assuming that about 40 percent of elderly Americans were vaccinated within the first 11 days of the program, then about 9 million people aged 65 and older would have received the vaccine in early
October 1976. Assuming that there were 5,000 clinics nationwide, this would have been 164 vaccinations per clinic per day. A person aged 65 or older has about a 1-‐in-‐7,000 chance of dying on
any particular day; the odds of at least three such people dying on the same day from among a group of 164 patients are indeed very long, about 480,000 to one against. However, under our
assumptions, there were 55,000 opportunities for this “extremely improbable” event to occur— 5,000 clinics, multiplied by 11 days. The odds of this coincidence occurring somewhere in America,
therefore, were much shorter —only about 8 to 1
Silver is pointing out that the editorial falls prey to what might be called the “lottery fallacy.” It’s vanishingly unlikely that any particular person will win the lottery next week. But it’s
nearly certain that someone will win. If there are very many opportunities for a coincidence to happen, and you don’t care which coincidence happens, then you’re going to see a lot of coincidences.
You can see this effect numerically with even the rough estimation of the odds that Silver has done here.
Another place where probabilities are often misunderstood is polling. During the election I saw a report that Romney had pulled ahead of Obama in Florida, 49% to 47% with a 5.5% margin of error. I
argued at the time that this wasn’t actually a story, because it was just too likely that Obama was actually still leading and the error in the poll was just that, error. In class we worked the
numbers on this example and concluded that there was a 36% chance — so, 1 in 3 odds — that Obama was actually ahead (full writeup here.)
In fact, 5.5% is an unusually high error for a poll, so this particular poll was less informative than many. But until you actually run the numbers on poll errors a few times, you may not have a gut
feel for when a poll result is definitive and when it’s very likely to be just noise. As a rough guide, a difference between two numbers of twice the margin of error is almost certain to indicate
that the lead is real.
If you’re a journalist writing about the likelihood or unlikelihood of some event, I would argue that it is your job to get a numerical handle on the actual odds. It’s simply too easy to deceive
yourself (and others!)
Next we looked at conditional probability — the probability that something happens given that something else has already happened. Conditional probabilities are important because they can be used to
connect causally related events, but humans aren’t very good at thinking about them intuitively. The classic example of this is the very common base rate fallacy. It can lead you to vastly
over-estimate the likelihood that someone has cancer when a mammogram is positive, or that they’re a terrorist if they appear on a watch list.
The correct way to handle conditional probabilities is with Bayes’ Theorem, which is easy to derive from the basic laws of probability. Perhaps the real value of Bayes’ theorem for this kind of
problem is that it forces you to remember all of the information you need to come up with the correct answer. For example, if you’re trying to figure out P(cancer | positive mammogram) you really
must first know the base rate of cancer in the general population, P(cancer). In this case it is very low because the example is about women under 50, where breast cancer is quite rare to begin with
— but if you don’t know that you won’t realize that the small chance of false positives combined with the huge number of people who don’t have cancer will swamp the true positives with false
Then we switched gears from all of this statistical math and talked about how humans come to conclusions. The answer is, badly if you’re not paying attention. You can’t just review all the
information you have on a story, think about it carefully, and come to the right conclusion. Our minds are simply not built this way. Starting in the 1970s an amazing series of cognitive psychology
experiments revealed a set of standard human cognitive biases, unconscious errors that most people make in reasoning. There are lots of these that are applicable journalism.
The issue here is not that the journalist isn’t impartial, or acting fairly, or trying in good faith to get to the truth. Those are potential problems too, but this is a different issue: our minds
don’t work perfectly, and in fact they fall short in predictable ways. While it’s true that people will see what they want to see, confirmation bias is mostly something else: you will see what you
expect to see.
The fullest discussion of these startling cognitive biases — and also, conversely, how often our intuitive machinery works beautifully — is the book by one of the original researchers, Daniel
Kahneman’s Thinking Fast and Slow. I also know of one paper which talks about how cognitive biases apply to journalism.
So how does an honest journalist deal with these? We looked at the method of competing hypotheses, as described by Heuer. The core idea is ancient, and a core principle of science too, but it bears
repetition in modern terms. Instead of coming up with a hypothesis (“maybe there is a cluster of cancer cases due to the oil refinery”) and going looking for information that confirms it, come up
with lots of hypothesis, as many as you can think of that explain what you’ve seen so far. Typically, one of these will be “what we’re seeing happened by chance,” often known as the null hypothesis.
But there might be many others, such as “this cluster of cancer is due to more ultraviolet radiation at the higher altitude in this part of the country” or many other things. It’s important to be
creative in the hypothesis generation step: if you can’t imagine it, you can’t discover that it’s the truth.
Then, you need to go look for discriminating evidence. Don’t go looking for evidence that confirms a particular hypothesis, because that’s not very useful; with the massive amount of information in
the world, plus sheer randomness, you can probably always find some data or information to confirm any hypothesis. Instead you want to figure out what sort of information would tell you that one
hypothesis is more likely than another. Information that straight out contradicts a hypothesis (falsifies it) is great, but anything that supports one hypothesis more than the others is helpful.
This method of comparing the evidence for different hypothesis has a quantitative equivalent. It’s Bayes’ theorem again, but interpreted a little differently. This time the formula expresses a
relationship between your confidence or degree of belief in a hypothesis, P(H), the likelihood of seeing any particular evidence if the hypothesis is true, P(E|H), and the likelihood of seeing any
particular piece of evidence whether or not the hypothesis is true, P(E)
To take a concrete example, suppose the hypothesis H is that Alice has a cold, and the evidence E is that you saw her coughing today. But of course that’s not conclusive, so we want to know the
probability that she really does have a cold (and isn’t coughing for some other reason.) Bayes’ theorem tells us what we need to compute P(H|E) or rather P(cold|coughing)
Under these assumptions, P(H|E) = P(E|H)P(H)/P(E) = 0.9 * 0.05 / 0.1 = 0.45, so there’s a 45% chance she has a cold. If you believe your initial estimates of all the probabilities here, then you
should believe that there’s a 45% chance she has a cold.
But these are rough numbers. If we start with different estimates we get different answers. If we believe that only 2% of our friends have a cold at any moment then P(H) = 0.02 and P(H|E) = 18%.
There is no magic to Bayesian inference; it can seem very precise but it all depends on the accuracy of your models, your picture of how the world works. In fact, examining the fit between models and
reality is one of the main goals of modern statistics.
There’s probably no need to apply Bayes’ theorem explicitly to every hypothesis you have about your story. Heuer gives a much simpler table-based method that just lists supporting and disproving
evidence for each hypothesis. Really the point is just to make you think comparatively about multiple hypothesis, and consider more scenarios and more discriminating evidence than you would
otherwise. And not be so excited about confirmatory evidence.
However, there are situations where your hypotheses and data are sufficiently quantitative that Bayesian inference can be applied directly — such as election prediction. Here’s a primer on
quantitative bayesian inference between multiple hypotheses. A vast chunk of modern statistics — most of it? — is built on top of Bayes’ theorem, so this is powerful stuff.
Our final topic was causality. What does it even mean to say that A causes B? This question is deeper than it seems, and a precise definition becomes critical when we’re doing inference from data.
Often the problem that we face is that we see a pattern, a relationship between two things — say, dropping out of school and making less money in your life — and we want to know if one causes the
other. Such relationships are called correlations, and probably everyone has heard by now that correlation is not causation.
In fact if we see a correlation between two different variables X and Y there are only a few real possibilities. Either X causes Y, or Y causes X, or Z causes both X and Y, or it’s just random fluke.
Our job as journalists is to figure out which one of these cases we are seeing. You might consider them alternate hypotheses that we have to differentiate between.
But if you’re serious about determining causation, what you actually want is an experiment: change X and see if Y changes. If changing X changes Y then we can definitely say that X causes Y (though
of course it may not be the only cause, and Y could cause X too!) This is the formal definition of causation as embodied in the causal calculus. In certain rare cases you can prove cause without
doing an experiment, and the causal calculus tells you when you can get away with this.
Finally, we discussed a real world example. Consider the NYPD stop and frisk data, which gives the date and location of each of the 600,000 stops that officers make on the street every year. You can
plot these on a map. Let’s say that we get a list of mosque addresses, and discover that we discover that there are 15% more stops than average within 100 meters of New York City’s mosques. Given the
NYPD history of spying on muslims, do we conclude that the police are targeting mosque-goers?
Let’s call that H1. How many other hypothesis can you imagine that will also explain this fact? (We came up with eight in class.) What kind of information or data or tests would you need to do to
decide which hypothesis is the strongest?
The readings for this week were:
Week 9: Social Network Analysis
This week is about the analysis of networks of people, not the analysis of data on social networks. We might mine tweets, but fundamentally we are interested here in the people and their connections
— the social network — not the content.
Social networks have of course existed for as long as there have been people, and have been the subject of careful study since the early 20th century (see for example this 1951 study which compared
groups performing the same a task using different network shapes, showing that “centrality” was an important predictor of behavior.) Recently it has become a lot easier to study social networks
because of the amount of data that we all produce online — not just our social networking posts, but all of our emails, purchases, location data, instant messages, etc.
Different fields have different reasons to study social networks. In intelligence and law enforcement, the goal may be to identify terrorists or criminals. Marketing and PR are interested in how
people influence one another to buy things or believe things. In journalism, social network analysis is potentially useful in all four places where CS might apply to journalism. That is, social
network analysis could be useful for:
• reporting, by identifying key people or groups in a story
• presentation, to show the user how the people in a story relate to one another
• filtering, to allow the publisher to target specific stories to specific communities
• tracking effects, by watching how information spreads
Because we’re going to have a whole week on tracking effects (see syllabus) we did not talk about that in class.
In a complex investigative story, we might use social network analysis to identify individual people or groups, based on who they are connected. This is what ICIJ did in their Skin and Bone series on
the international human tissue trade. To present a complex story we might just simply show the social network of the people and organizations involved, as in the Wall Street Journal’s Galleon’s Web
interactive on the famous insider trading scandal. I haven’t yet heard of anyone in journalism targeting specific audiences identified by social network analysis, but I bet it will happen soon.
Although visualization remains the main technique, there have been a number of algorithms designed for social network analysis. First there are multiple “centrality” measures, which try to determine
who is “important” or “influential” or “powerful.” There are many of these.
But they don’t necessarily compute what a journalist wants to know. First, each algorithm is based on a specific assumption about how “things” flow through the network. Betweenness centrality assumes
flows are always along the shortest path. Eigenvector centrality assumes a random walk. Whether this models the right thing depends on what is flowing — is it emails? information? money? orders? —
and how you expect it to flow. Borgatti explains the assumptions behind centrality measures in great detail.
Often journalists are interested in “power” or “influence.” Unfortunately this is a very complicated concept, and while there is almost certainly some correlation between power and network
centrality, it’s just not that simple. Communication Intermediaries — say, a secretary — may have extremely high betweeness centrality without any real authority.
Even worse, your network just may not contain the data you are actually interested in. You can produce a network showing corporate ownership, but if company A owns a big part of company B it doesn’t
necessarily mean that A “controls” B. It depends on the precise relationship between the two companies, and how much autonomy B is given. Similar arguments can be made for links like “sits on the
board of.”
This also brings up the point that there may be more than one kind of connection between people (or entities, more generally) in which case “social network analysis” is more correctly called “link
analysis,” and if you use any sort of algorithm on the network you’ll have to figure out how to treat different types of links.
There are also algorithms for trying to find “communities” in networks. This requires a mathematical definition of a “cluster” of people, and one of the most common is modularity, which counts how
many more intra-group edges there are than would be expected by chance in a graph with the same number of edges randomly placed.
Overall, social network analysis algorithms are useful in journalism, but not definitive. They are just not capable of understanding the complex context of a real-world social network. But the
combination of a journalist and a good analysis system can be very powerful.
The readings were:
• Identifying the Community Power Structure, an old handbook for community development workers about figuring out who is influential by very manual processes. I hope this helps you think about what
“power” is, which is not a simple topic, and traditional “analog” methods of determining it.
• Analyzing the data behind Skin and Bone, ICIJ. The best use of social network analysis in journalism that I am aware of.
• Sections I and II of Community Detection in Graphs. An introduction to a basic type of social network algorithm.
• Visualizing Communities, about the different ways to define a community
• Centrality and Network Flow, or, one good reason to be suspicious of centrality measures
• The Network of Global Corporate Control, a remarkable application of network analysis
• The Dynamics of Protest Recruitment Through an Online Network, good analysis of Twitter data from Spain “May 20” protest movement
• Exploring Enron, social network analysis of Enron emails, by Jeffrey Heer who went on to help create the D3 library
Here are a few other examples of the use of social network analysis in journalism:
• Visualizing the Split on Toronto City Council, a social network analysis that shows evolution over time
• Muckety, an entire site that only does stories based on link analysis
• Theyrule.net, an old map of U.S. boards of directors
• Who Runs Hong Kong?, a story explained through a social network analysis tool, South China Morning Post
Week 8: Knowledge Representation
Journalism has, historically, considered itself to be about text or other unstructured content such as images, audio, and video. This week we ask the questions: how much of journalism might actually
be data? How would we represent this data? Can we get structured data from unstructured data?
We start with Holovaty’s 2006 classic, A fundamental way newspaper sites need to change, which lays out the argument that the product of journalism is data, not necessarily stories. Central to this
is the idea that it may not be humans consuming this data, but software that combines information for us in useful ways — like Google’s Knowledge Graph.
But to publish this kind of data, we need a standard to encode it. This gets us into the question of “what is a general way to encode human knowledge?” which has been asked by the AI community for at
least 50 years. That’s why the title of this lecture is “knowledge representation.”
This is a hard problem, but let’s start with an easier one which has been solved: story metadata. Even without encoding the “guts” of a story as data, there is lots of useful data “attached” to a
story that we don’t usually think about.
These details are important to any search engine or program that is trying to scrape the page. They might also include information on what the story is “about,” such as subject classification or a
list of the entities (people, places, organizations) mentioned. There is a recent standard for encoding all of this sort of information directly within the page HTML, defined by schema.org which is a
joint project of Google, Bing, and Yahoo. Take a look at the schema.org definition of a news article, and what it looks like in HTML. If you view the source of a New York Times, CNN, or Guardian
article you will see these tags in use today.
In fact, every big news organization has its own internal schema, though some use it a lot more than others. The New York Times has been adding subject metadata since the early 20th century, as part
of their (initially manual) indexing service. But we’d really like to be able to combine this type of information from multiple sources. This is the idea behind “linked open data” which is now W3C
standard. Here’s Tim Berners-Lee describing the idea.
The linked data standard says each “fact” is described as a triple, in “subject relation object” form. Each of these three items is in turn either a literal constant, or a URL. Linked data is linked
because it’s easy to refer to someone else’s objects by their URL. A single triple is equivalent to the proposition relation(subject,object) in mathematical logic. A database of triples is also
called a “triplestore.”
There are many sites that already support this type of data. The W3C standard for expressing triples is an XML-based format called RDF, but there is also a much simpler JSON encoding of linked data.
Here is what the “linked data cloud” looked like in 2010; it’s much bigger now and no one has mapped it recently.
The arrows indicate that one database references objects in the other. You will notice something called DBPedia at the center of the cloud. This is data derived from all those “infoboxes” on the
right side of Wikipedia articles, and it has become the de-facto common language for many kinds of linked data.
Not only can one publisher refer to the objects of another publisher, but the standardized owl:sameAs relation can be used to equate one publisher’s object to a DBPedia object, or anyone else’s
object. This expression of equivalence is an important mechanism that allows interoperability between different publishers. (As I mentioned above, every relation is actually a URL, so owl:sameAs is
more fully known as http://www.w3.org/2002/07/owl#sameAs, but the syntax of many linked data formats allow abbreviations in many cases.)
DBPedia is vast and contains entries on many objects. If you go to http://dbpedia.org/page/Columbia_University_Graduate_School_of_Journalism you will see everything that DBPedia knows about Columbia
Journalism School, represented as triples (the subject of every triple on this page is the school, so it’s implicit here.) If you go to http://dbpedia.org/data/
Columbia_University_Graduate_School_of_Journalism.json you will get the same information in machine-readable format. Another important database is GeoNames, which contains machine readable
information on millions of geographical entities worldwide — not just their coordinates but their shapes, containment (Paris is in France), and adjacencies. The New York Times also publishes a subset
of their ontology as linked open data, including owl:sameAs relations that map their entities to DBPedia entities (example).
So what can we actually do with all of this? In theory, we can combine propositions from multiple publishers to do inference. So if database A says Alice is Bob’s brother, and database B says Alice
is Mary’s mother, then we can infer that Bob is Mary’s uncle. Except that — as decades of AI research has shown — propositional inference is brittle. It’s terrible at common sense, exceptions, etc.
Perhaps the most interesting real-world application of knowledge representation is general question answering. Much like typing a question into a search engine, we allow the user to ask questions in
human language and expect the computer to give us the right answer. The state of the art in this area is the DeepQA system from IBM, which competed and won on Jeopardy. Their system uses a hybrid
approach, with several hundred different types of statistical and propositional based reasoning modules, and terabytes of knowledge both in unstructured and structured form. The right module is
selected at run time based on a machine learning model that tries to predict what approach will give the correct answer for any given question. DeepQA uses massive triplestores of information, but
they only contain a proposition giving the answer for about 3% of all questions. This doesn’t mean that linked data and its propositional knowledge is useless, just that it’s not going to be the
basis of “general artificial intelligence” software. In fact linked data is already in wide use, but in specialized applications.
Finally, we looked at the problem of extracting propositions from text. The Reverb algorithm (in your readings) gives a taste of the challenges involved here, and you can search their database of
facts extracted from 500 million web pages. A big part of proposition extraction is named entity recognition (NER). The best open implementation is probably the Stanford NER library, but the Reuters
OpenCalais service performs a lot better, and you will use if for assignment 3. Google has pushed the state of the art in both NER and proposition extraction as part of their Knowledge Graph which
extracts structured information from the entire web.
Your readings were:
• A fundamental way newspaper websites need to change, Adrian Holovaty
• The next web of open, linked data – Tim Berners-Lee TED talk
• Identifying Relations for Open Information Extraction, Fader, Soderland, and Etzioni (Reverb algorithm)
• Standards-based journalism in a semantic economy, Xark
• What the semantic web can represent – Tim Berners-Lee
• Building Watson: an overview of the DeepQA project
Week 7: Visualization
Sadly, we had to cut this lecture short because of Hurricane Sandy, but I’m posting the slides and a few notes.
You have no doubt seen lots of visualizations recently, and probably even studied them in your other classes (such as Data Journalism.) I want to give you a bit of a different perspective here,
coming more from the information visualization (“infovis”) tradition which goes back to the beginnings of computer graphics in the 1970s. That culture recognized very early the importance of studying
the human perceptual system, that is, how our eyes and brains actually process visual information.
Take a look at the following image.
You saw the red line instantly, didn’t you? Importantly, you didn’t have to think about this, or go look at each line, one at a time to find it, you “just saw it.” That’s because your visual cortex
can do many different types of pattern recognition at a pre-conscious level. It doesn’t take any time or feel like any effort for you. This particular effect is called “visual pop-out” and many
different types of visual cues can cause it.
The human visual system can also do pre-conscious comparisons of things like length, angle, size and color. Again, you don’t have to think about it know which line is longer.
In fact, your eye and brain are sensitive to dozens of visual variables simultaneously. You can think of these as “channels” which can be used to encode quantitative information. But not all channels
are equally good for all types of information. Position and size are the most sensitive channels for continuous variables, while color and texture aren’t great for continuous variables but work well
for categorical variables. The following chart, from Munzer, is a summary of decades of perceptual experiments.
This consideration of what the human visual system is good at — and there’s lot’s more — leads to what I call the fundamental principle of visualization: turn something you want to find into
something you can see without thinking about.
What kinds of “things” can we see in a visualization? That’s the art of visualization design! We’re trying to plot the data such that the features we are interested in are obviously visible. But here
are some common data features that we can visualize.
The rest of the lecture — which we were not able to cover — gets into designing visualizations for big data. The key principle is, don’t try to show everything at once. You can’t anyway. Instead, use
interactivity to allow the user to explore different aspects of the data. In this I am following the sage advice of Ben Fry’s Computational Information Design approach, and also drawing parallels to
how human perception works. After all, we don’t “see” the entire environment at once, because only the central 2 degrees of our retina are sharp (the fovea.) Instead we move our eyes rapidly to
survey our environment. Scanning through big data should be like this, because we’re already built to understand the world that way.
In the final part of the lecture — which we actually did cover, briefly — we discussed narrative, rhetoric and interpretation of visualizations. Different visualizations of the same data can “say”
completely different things. We looked at a simple line graph and asked, what are all the editorial choices that went into creating it?
I can see a half dozen choices here; there are probably more.
• The normalization used — all values are adjusted relative to Jan 2005 values
• Choice of line chart (instead of any other kind)
• Choice of color. Should thefts be blue, or would red have been better?
• Time range. The data probably go back farther.
• Legend design.
• Choice of these data at all, as opposed to any other way to understand bicycle use and thefts.
Also, no completed visualization is entirely about the data. If you look at the best visualization work, you will see there there are “layers” to it. These include:
• The data. What data is chosen, what is omitted, what are the sources.
• Visual representation. How is the data turned into a picture.
• Annotation. Highlighting, text explanations, notes, legends.
• Interactivity. Order of presentation, what the user can alter.
In short, visualization is not simply a technical process of turning data into a picture. There are many narrative and editorial choices, and the result will be interpreted by the human perceptual
system. The name of the game is getting a particular impression into the user’s head, and to do that, you have to a) choose what you want to say and b) understand the communication and perception
processes at work.
Readings for this week were:
I also recommend the book Designing Data Visualizations.
Assignment 3
For this assignment you will evaluate the performance of OpenCalais, a commercial entity extraction service. You’ll do this by building a text enrichment program, which takes plain text and outputs
HTML with links to the detected entities. Then you will take five random articles from your data set, enrich them, and manually count how many entities OpenCalais missed or got wrong.
1. Get an OpenCalais API key, from this page.
2. Install the python-calais module. This will allow you to call OpenCalais from Python easily. First, download the latest version of python-calais. To install it, you just need calais.py in your
working directory. You will probably also need to install the simplejson Python module. Download it, then run “python setup.py install.” You may need to execute this as super-user.
3. Call OpenCalais from Python. Make sure you can successfully submit text and get the results back, following these steps. The output you want to look at is in the entities array, which would be
accessed as “results.entities” using the variable names in the sample code. In particular you want the list of occurrences for each entity, in the “instances” field.
>>> result.entities[0]['instances']
[{u'suffix': u' is the new President of the United States', u'prefix': u'of the United States of America until 2009. ', u'detection': u'[of the United States of America until 2009. ]Barack Obama[ is the new President of the United States]', u'length': 12, u'offset': 75, u'exact': u'Barack Obama'}]
>>> result.entities[0]['instances'][0]['offset']
Each instance has “offset” and “length” fields that indicate where in the input text the entity was referenced. You can use these to determine where to place links in the output HTML.
4. Read from stdin, create hyperlinks, write to stdout. Your Python program should read text from stdin and write HTML with links on all detected entities to stdout. There are two cases to handle,
depending on how much information OpenCalais gives back.
In many cases, like the example in the previous step, OpenCalais will not be able to give you any information other than the string corresponding to the entity, result.entities[x][‘name’]. In this
case you should construct a Wikipedia link by simply appending to the name to a Wikipedia URL, converting spaces to underscores, e.g.
In other cases, especially companies and places, OpenCalias will supply a link to an RDF document that contains more information about the entity. For example.
>>> result.entities[0]{u'_typeReference': u'http://s.opencalais.com/1/type/em/e/Company', u'_type': u'Company', u'name': u'Starbucks', '__reference': u'http://d.opencalais.com/comphash-1/6b2d9108-7924-3b86-bdba-7410d77d7a79', u'instances': [{u'suffix': u' in Paris.', u'prefix': u'of the United States now and likes to drink at ', u'detection': u'[of the United States now and likes to drink at ]Starbucks[ in Paris.]', u'length': 9, u'offset': 156, u'exact': u'Starbucks'}], u'relevance': 0.314, u'nationality': u'N/A', u'resolutions': [{u'name': u'Starbucks Corporation', u'symbol': u'SBUX.OQ', u'score': 1, u'shortname': u'Starbucks', u'ticker': u'SBUX', u'id': u'http://d.opencalais.com/er/company/ralg-tr1r/f8512d2d-f016-3ad0-8084-a405e59139b3'}]}
>>> result.entities[0]['resolutions'][0]['id']
In this case the resolutions array will contain a hyperlink for each resolved entity, and this is where your link should go. The linked page will contain a series of triples (assertions) about the
entity, which you can obtain in machine-readable from by changing the .html at the end of the link to .json. The sameAs: links are particularly important because they tell you that this entity is
equivalent to others in dbPedia and elsewhere.
Here is more on OpenCalias’ entity disambiguation and use of linked data.
5. Pick five random documents and enrich them. Choose them from the document set you worked with in Assignment 1. It’s important that you actually choose randomly — as in, use a random number
generator. If you just pick the first five, there may be biases in the result. Using your code, turn each of them into an HTML doc.
6. Read the enriched documents and count to see how well OpenCalais did. You need to read each output document very carefully and count three things:
• Entity references. Count each time there is a name of a person, place, or organization, including pronouns (such as “he”) or other references (like “the president.”)
• Detected references. How many of these did OpenCalais find?
• Correct references. How many of the links go to the right page? Did our hyperlinking strategy (OpenCalais RDF pages where possible, Wikipedia when not) fail to correctly disambiguate any of the
references, or, even worse, disambiguate any to the wrong object?
7. Turn in your work. Please turn in:
• Your code
• The enriched output from your documents
• A brief report describing your results.
The report should include a table of the three numbers — references, detected, correct — for each document, as well as overall percentages across all documents. Also report on any patterns in the
failures that your see. Where is OpenCalais most accurate? Where is it least accurate? Are there predictable patterns to the errors?
Due before class on Monday, November 19.
Assignment 2
For this assignment you will design a hybrid filtering algorithm. You will not implement it, but you will explain your design criteria and provide a filtering algorithm in sufficient technical detail
to convince me that it might actually work — including psuedocode.
You may choose to filter:
• Facebook status updates, like the Facebook news feed
• Tweets, like Trending Topics or the many Tweet discovery tools
• The whole web, like Prismatic
• something else, but ask me first
Your filtering algorithm can draw on the content of the individual items, the user’s data, and other users’ data. The assignment goes like this:
1. List all available information that you have available during the debate. If you want to filter Facebook or Twitter, you may pretend that you are either of these companies and have access to all
of their tweets etc. You also also assume you have a web crawler or a firehose of every RSS feed or whatever you like, but you must be specific and realistic about what data you are operating with.
2. Argue for the design factors that you would like to influence the filtering, in terms of what is desirable to the user, what is desirable to the publisher (e.g. Facebook or Prismatic), and what is
desirable socially. Explain as concretely as possible how each of these (probably conflicting) might be achieved through in software. Since this is a hybrid filter, you can also design social
software that asks the user for certain types of information (e.g. likes, votes, ratings) or encourages users to act in certain ways (e.g. following) that generate data for you.
3. Write psuedo-code for a function that produces a “top stories” list. This function will be called whenever the user loads your page or opens your app, so it must be fast and frequently updated.
You can assume that there are background processes operating on your servers if you like. Your psuedo-code does not have to be executable, but it must be specific and unambiguous, such that a good
programmer could actually go and implement it. You can assume that you have libraries for classic text analysis and machine learning algorithms.
4. Write up steps 1-3. The result should be no more than five pages. However, you must be specific and plausible. You must be clear about what you are trying to accomplish, what your algorithm is,
and why you believe your algorithm meets your design goals (though of course it’s impossible to know for sure without testing; but I want something that looks good enough to be worth trying.)
The assignment is due before class on Monday, October 29.
Week 6: Hybrid filters
In previous weeks we discussed filters that are purely algorithmic (such as NewsBlaster) and filters that are purely social (such as Twitter.) This week we discussed how to create a filtering system
that uses both social interactions and algorithmic components.
Here are all the sources of information such an algorithm can draw on.
We looked at two concrete examples of hybrid filtering. First, the Reddit comment ranking algorithm, which takes the users’ upvotes and downvotes and sorts not just by the proporition of upvotes, but
by how certain we are about proportion, given the number of people who have actually voted so far. Then we looked at item-based collaborative filtering, which is one of several classic techniques
based on a matrix of users-item ratings. Such algorithms power everything from Amazon’s “users who bought A also bought B” to Netflix movie recommendations to Google News’ personalization system.
Evaluating the performance of such systems is a major challenge. We need some metric, but not all problems have an obvious way to measure whether we’re doing well. There are many options. Business
goals — revenue, time on site, engagement — are generally much easier to measure than editorial goals.
Finally, we saw a presentation from Dr. Aria Haghighi, co-founder of the news personalization service Prismatic, on how his system crawls the web to find diverse articles that match user interests.
The readings for this week were:
• Item-Based Collaborative Filtering Recommendation Algorithms, Sarwar et. al
• How Reddit Ranking Algorithms Work, Amir Salihefendic
• Slashdot Moderation, Rob Malda
• How does Google use human raters in web search?, Matt Cutts
This concludes our work on filtering systems — except for Assignment 2.
Week 5: Social software and social filtering
This week we looked how groups of people can act as information filters.
First we studied Diakopolous’ SRSR (“seriously rapid source review”) system for finding sources on Twitter. There were a few clever bits of machine learning in there, for classifying source types
(journalist/blogger, organization, or ordinary individual) and for identifying eyewitnesses. But mostly the system is useful because it presents many different “cues” to the journalist to help them
determine whether a source is interesting and/or trustworthy. Useful, but when we look at how this fits into the broader process of social media sourcing — in particular how it fits into the
Associated Press’ verification process — it’s clear that current software only adresses part of this complex process. This isn’t a machine learning problem, it’s a user interface and workflow design
issue. (For more on social media verification practices, see for example the BBC’s “UGC hub“)
More broadly, journalism now involves users informing each other, and institutions or other authorities communicating directly. The model of journalism we looked at last week, which put reporters at
the center of the loop, is simply wrong. A more complete picture includes users and institutions as publishers.
That horizontal arrow of institutions producing their own broadcast media is such a new part of the journalism ecosystem, and so disruptive, that the phenomenon has its own name: “sources go direct,”
which seems to have been originally coined by blogging pioneer Dave Winer.
But this picture does not include filtering. There are thousands — no, millions — of sources we could tune into now, but we only direct attention to a narrow set of them, maybe including some
journalists or news publications, but probably mostly other types of source, including some primary sources.
This is social filtering. By choosing who we follow, we determine what information reaches us. Twitter in particular does this very well, and we looked at how the Twitter network topology doesn’t
look like a human social network, but is more tuned for news distribution.
There are no algorithms involved here… except of course for the code that lets people publish and share things. But the effect here isn’t primarily algorithmic. Instead, it’s about how people operate
in groups. This gets us into the concept of “social software,” which is a new interdisciplinary field with its own dynamics. We used the metaphor of “software as architecture,” suggested by Joel
Spolsky, to think about how software influences behavior.
As an example of how environment influences behaviour, we watched this video which shows how to get people to take the stairs.
I argued that there are three forces which we can use to shape behavior in social software: norms, laws, and code. This implies that we have to write the code to be “anthropologically correct,” as
Spolsky put it, but it also means that the code alone is not enough. This is something Spolsky observed as StackOverflow has become a network of Q&A sites on everything from statistics to cooking:
each site has its own community and its own culture.
Previously we phrased the filter design problem in two ways: as a relevance function, and as a set of design criteria. When we use social filtering, there’s no relevance function deciding what we
see. But we still have our design criteria, which tell us what type of filter we would like, and we can try to build systems that help people work together to produce this filtering. And along with
this, we can imagine norms — habits, best practices, etiquette — that help this process along, an idea more thoroughly explored by Dan Gilmour in We The Media.
The readings from the syllabus were,
• A Group is its own worst enemy, Clay Shirky
• What’s the point of social news?, Jonathan Stray
• Finding and Assessing Social Information Sources in the Context of Journalism, Nick Diakopolous et al.
• Learning from Stackoverflow, first fifteen minutes, Joel Spolsky
• Norms, Laws, and Code, Jonathan Stray
• What is Twitter, a Social Network or a News Media?, Haewoon Kwak, et al,
• International reporting in the age of participatory media, Ethan Zuckerman
• We The Media. Introduction and Chapter 1, Dan Gillmor
Week 4: Information overload and algorithmic filtering
This is the first of three weeks on “filtering.” We define that word by looking at a feedback model of journalism: a journalist observes something happening in the world, produces a story about it,
the user consumes the story, and then they potentially act in some way that changes the world (such as voting, choosing one product over another, protesting, or many other possible outcomes.) This
follows David Bornstein’s comment that “journalism is a feedback mechanism to help society self-correct.”
This diagram is missing something obvious: there are lots and lots of topics in the world, hence many stories. Not every potential story is written, and not every written story is consumed by every
This is where “filtering” comes in, the arrow on the bottom right. Somehow, the user sees only a subset of all produced stories. The sheer, overwhelming logic of the amount of journalism produced
versus hours in the day requires this (and we illustrated this with some numerical examples in the slides.)
(Incidentally, journalism as an industry has mostly been involved with story production, the upper-right arrow, and more recently has been very concerned about how fewer reporters result in more
stories not covered, the upper left arrow. The profession has, historically, payed much less attention to the effects of its work, bottom left, and the filtering problem, bottom right.)
(There is another major thing missing from this diagram: users now often have access to the same sources as journalists, and in any case journalism is now deeply participatory. We’ll talk a lot more
about this next week.)
This week we focussed on purely algorithmic filtering. As a concrete example, we examined the inner workings of the Columbia Newsblaster system, a predecessor of Google News which is conveniently
well documented.
The readings (from the syllabus) were mostly to get you thinking about the general problem of information overload and algorithmic filtering, but the Newsblaster paper is also in there.
Actually, much of the guts of Newsblaster is in this paper on their on-line clustering algorithm that groups together all stories which are about the same underlying event. Note the heavy reliance on
our good friends from last week: TF-IDF and cosine distance. The graphs in this paper show that for this problem, you can do better than TF-IDF by adding features corresponding to extracted entities
(people, places, dates) but really not by very much.
We wrapped up with a discussion about the problem of algorithmic filter design. We defined this problem on two levels. In terms of functional form,
and in terms of the much more abstract desirable attributes
The great challenge is to connect these two levels of description: to express our design criteria in terms of an algorithm. Here are the notes from our brief discussion about how to do this.
On the right we have interest, effects, agency, my proposed three criteria for “when should a user see a story.” Somehow, these have to be expressed in terms of computational building blocks like
TF-IDF and all of the various signals available to the algorithm. That’s what the fuzzy arrow is… there’s a gap here, and it’s a huge gap.
On the left are some of the factors to consider in trying to assess whether a particular story is interesting, will effect, or can be acted on by a particular user: geo information (location of user
and story), user’s industry and occupation, other user demographics, the people in the user’s social network, the “content” they’ve produced (everything they’ve ever tweeted, blogged, etc.), and the
time or duration of the story event. We can also offload parts of the selection process to the user, by showing multiple stories or types of stories and having the user pick. Similarly we can offload
parts of the problem to the story producer, who might use various techniques to try to target a particular story to a particular group of people. We’ll talk extensively about including humans in the
filtering system in the next two weeks.
The bracket and 2^N notation just means that any combination of these factors might be relevant. E.g. location and occupation together might be a key criteria.
In the center of the board I recorded one important suggestion: we can use machine learning to teach the computer which are the right articles for each factor. For example, suppose we’re trying to
have the algorithm decide which stories are about events that affect people in different occupations. For each occupation, a human can collect many stories that someone in that occupation would want
to read, then we can take the average of the TF-IDF vectors of those stories to define a subject category. The computer can then compare each incoming story to the corresponding coordinate for each
user’s occupation.
I don’t know whether this particular scheme will work, but having the humans teach the computers is an essential idea — and one that is very common in search engines and filtering systems of all
Assignment 1
I’m altering the structure of the assignment a little bit from the version in the original syllabus, in the hopes of making it more interesting. We may even be able to learn and document somethings
that seem to be missing from the literature.
The assignment goes like this:
1) Get your data into a standard format. You have all now chosen a dataset that contains, at least in part, a significant text corpus. Your first task will be to scrape, format, or otherwise coerce
this information into a convenient format for Python to read.
I recommend a very simple format: plain text, one document per line, all documents in one file.
It’s not completely trivial to get the documents in this format, but it shouldn’t be hard either. The first task is to extract plain text from your documents. If your source material is PDF, you can
use the pdftotext command (pre-installed on Macs, available for Windows and Linux as part of xpdf). If it’s HTML, you may want to delete matches to the regex /<[^>]*>/ to remove all tags; you may
also want to scrape the content of a particular div, as opposed to the whole page source. If you need to scrape your data from web pages, I heartily recommend writing a little script within the
ScraperWiki framework.
Obviously this one-document-per-line format can’t represent the newlines in the original document, but that’s ok, because we’re going to throw them out during tokenization anyway. So you can just
replace them with spaces.
2) Feed the data into gensim. Now you need to load the documents into Python and feed them into the gensim package to generate tf-idf weighted document vectors. Check out the gensim example code here
. You will need to go through the file twice: once to generate the dictionary (the code snippet starting with “collect statistics about all tokens”) and then again to convert each document to what
gensim calls the bag-of-words representation, which is un-normalized term frequency (the code snippet starting with “class MyCorpus(object)”
Note that there is implicitly another step here, which is to tokenize the document text into individual word features — not as straightforward in practice as it seems at first, but the example code
just does the simplest, stupidest thing, which is to lowercase the string and split on spaces. You may want to use a better stopword list, such as this one.
Once you have your Corpus object, tell gensim to generate tf-idf scores for you like so.
3) Do topic modeling. Now you can apply Latent Semantic Indexing or Latent Dirichlet Allocation to the tf-idf vectors, like so. You will have to supply the number of dimensions to keep. Figuring out
a good number is part of the assignment. Note that you dont have to do topic modeling — this is really a dimensionality reduction / de-noising step, and depending on your documents and application,
may not be needed. If you want to try working with the original tf-idf vectors, that’s OK too. That’s what Overview does.
4) Analyze the vector space. So now you have a bunch of vectors in a space of some dimension, each of which represents a document, and we hope that similar documents are close in this space (as
measured by cosine distance.) Have we gotten anywhere? There are several things we could do at this point:
1. Choose particular document, then find the k closest documents. Are they related? How? (Read the text of the documents to find out) How big do you have to make k before you see documents that seem
2. Run a clustering algorithm, such as any of those in the python cluster package. Then look at the documents in each cluster, again reading their text and reporting on the results. Non-hierarchical
clustering algorithms generally take the number of clusters as a parameter, while with hierarchical clusterings you have a choice of what level of the tree to examine. How do these choices affect
what you see?
3. Or, you could run multi-dimensional scaling to plot the entire space in 2D, perhaps with some other attribute (time? category?) as a color indicator variable. This is probably best done in R. To
get your document vectors into R, write them out of gensim in the MatrixMarket format, then load them in R (remember you’ll need to do “library(Matrix)” first to make readMM available in R.) Then
you’ll want to compute a distance matrix of the documents, run mdscale on it, and plot the result like we did in the House of Lords example.
4. If you did LSI or LDA topic modeling, what do the extracted topics look like? Do they make any sort of human sense? Can you see examples of polysemy or synonymy? If you pull out the k docs with
the highest score on a particular topic, what do you find? How many documents have no clear primary topic? What do the low-order topics (far down the dimension list) look like? How many
dimensions until it just seems like noise?
Your assignment is to do one of these things, whichever you think will be most interesting. You may also discover that it is hard to interpret the results without trying some of the other techniques.
Actually, figuring out how, exactly, to evaluate the clustering is part of this assignment. Hint: one useful idea is to ask, how might a human reporter organize your documents? Where did the computer
go wrong?
You will of course need to implement cosine distance either in Python or R to make any of these go. This should be only a few lines…
5) Compare to a different algorithmic choice. Now do steps 3 and 4 again, with a different choice of algorithm or parameter. The point of this assignment is to learn how different types of clustering
give qualitatively different results on your document set… or not. So repeat the analysis, using either:
• a different topic modeling algorithm. If you used plain tf-idf before, try LSI or LDA. Or if you tried LSI, try LDA. Etc.
• a different number of clusters, or a different level in the hierarchical clustering tree.
• a different number of output dimensions to LSI or LDA.
• a different distance function
• etc.
I want to know which of your two cases gives “better” results. What does “better” mean? It depends very much on what the interesting questions are for your data set. Again, part of the assignment is
coming up with criteria to evaluate these clusterings. Generally, more or easier insight is better. (If the computer isn’t making your reporting or filtering task significantly easier, why use it?)
6) Write up the whole thing. I will ask you to turn in your the code you wrote but that’s really only to confirm that you actually did all the steps. I am far more interested in what you have learned
about the best way to use these algorithms on your data set. Or if you feel you’ve gained little or no insight into your data set using these techniques, explain why, and suggest other ways to
explore it.
This assignment is due Monday, October 15 at 4:00 PM. You may email me the results. I am available for questions by email before then, or in person at office hours on Thursday afternoons 1-4. | {"url":"https://compjournalism.com/paged-6/","timestamp":"2024-11-04T04:18:14Z","content_type":"text/html","content_length":"146743","record_id":"<urn:uuid:9ae51fe6-1fef-4af4-bd41-393e48918953>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00310.warc.gz"} |
[4:30pm] Maria Mathew: IIT Bombay
4:00pm Description:
CACAAG seminar. Speaker: Maria Mathew. Affiliation: IIT Bombay. Date and Time: Friday 06 September, 4:30 pm - 5:30 pm. Venue: Ramanujan Hall, Department of Mathematics. Title: Gubeladze's
geometric proof of Anderson's conjecture. Abstract: Let M be a finitely generated seminormal submonoid of the free monoid \mathbb Z_+^n and let k be a field. Then Anderson conjectured
that all finitely generated projective modules over the monoid algebra k[M] is free. He proved this in case n=2. Gubeladze proved this for all n using the geometry of polytopes. In a
series of 3 lectures, we will outline a proof of this theorem. | {"url":"https://www.math.iitb.ac.in/webcal/day.php?date=20190906&friendly=1","timestamp":"2024-11-09T19:54:38Z","content_type":"text/html","content_length":"9692","record_id":"<urn:uuid:6eeec7b9-6998-4364-bca4-6602b2f0da2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00712.warc.gz"} |
Why Didn't The Universe Collapse Into Black Hole Shortly After the Big Bang? - Craffic
Have you ever thought about actually ‘Why didn’t the universe collapse into black hole shortly after the big bang?’ the question is really interesting and in this article, we will try to find an
answer to it.
The main point is that the Big Bang explosion did not happen in only one place which was previously empty. It took place everywhere at once, so a specific location for the formation of the Blackhole
can’t be assumed. Here the cosmological models can either be to the point or approximate. Interestingly inhomogeneous cosmology, symmetry tells that tidal forces do not exist anywhere, and thus the
observer initially at rest relative to the average motion of matter will experience zero gravitational fields.
Such considerations suggest that only one type of collapse is possible and that is the recollapse of the whole universe in a “Big Crunch”. This is possible only for the matter densities and for the
values of the cosmological constant that are different from what we observe.
A high energy density is one of the important requirements of black hole formation. A center is required which is then turned into the blackhole’s center, the matter has required that it collapse to
a black hole for having enough velocity so that gravity squeezes it before matter makes it flyway and dilutes the density.
The last two requirements are easily possible for simple chunks of matter anywhere in-universe, but these are breached by matter density formed right after the Big Bang. The matter has no center, is
uniform throughout the space, and also has high enough velocity which makes density eventually diluted.
The collapse of matter into a black hole is an idealized type of calculation that also has specific assumptions about the initial states of matter. The important part is that all such assumptions are
not satisfied by matter after the Big Bang.
A black hole is a space from which rays of light can’t escape to infinity. “To infinity” can be explained mathematically but the definition requires the assumption that spacetime is asymptotically
flat. A black hole has a specific location in space and is surrounded by a vacuum and thus matter keeps falling towards it. Also, the density of the initial universe is believed to be homogenous and
thus the same everywhere.
The rate of expansion of this universe stopped it earlier and now also from collapsing into a black hole. Thus the universe is getting larger and larger. The gravity slows down expansion but not that
much so that it creates a collapse. If the matter density of the universe had not been high enough, it possibly stopped from expanding and collapsed into a black hole way back. And such a case is
interestingly not happening till now with the universe.
hopefully, somewhere the question is answered about why didn’t the universe collapse into black hole.
Leave A Reply Cancel Reply
big bang Black Hole Science Space universe collapse | {"url":"https://craffic.co.in/why-didnt-universe-collapse-into-black-hole-shortly-after-the-big-bang/","timestamp":"2024-11-10T12:11:57Z","content_type":"text/html","content_length":"88395","record_id":"<urn:uuid:f9a557df-23f2-4adb-bb25-e7f4022ab792>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00624.warc.gz"} |
How much does rebar weight per foot?
Weight per unit length: 0.668 pounds per foot (0.996 kilograms per meter)
How much does a 20 foot stick of rebar weight?
For Reference Only
Bar Size (Metric) Bar Size (Inch) Weight (lb) 20ft
Bar Size (Metric) #13 Bar Size (Inch)#4 Weight (lb) 20ft13.36
Bar Size (Metric) #16 Bar Size (Inch)#5 Weight (lb) 20ft20.86
Bar Size (Metric) #19 Bar Size (Inch)#6 Weight (lb) 20ft30.04
Bar Size (Metric) #22 Bar Size (Inch)#7 Weight (lb) 20ft40.88
How much does 1/2 inch steel make a ton?
Product Size Weight/Unit Number of pieces
Metric Ton (2204 lbs)
3/8″ 0.478 lbs/ft 230
7/16″ 0.651 lbs/ft 169
1/2″ 0.851 lbs/ft 129
What is the weight of 1/2 inch rebar?
#4 rebar weight per foot:- weight of number 4 rebar (#4)/ reinforcing bar or 1/2″ (12 mm) rebar per foot will be 0.668 pounds or which is approximately equal as 0.270 kg/foot {(12×12)/ 533 = 0.270 kg
/ foot)}.
How much does 15M rebar weight?
Reinforcing bar – European metric dimensions.
Bar Number Mass (kg/m) Nominal Diameter (mm)
10M 0.785 11.3
15M 1.570 16.0
20M 2.355 19.5
25M 3.925 25.2
How much does #11 rebar weight per foot?
Physical Characteristics of #11 Rebar:
Imperial Bar Size “Soft” Metric Size Weight per unit length (lb/ft)
#11 #36 5.313
How much does 15M rebar weight per foot?
15 rebar weight:- metric mass/ weight of 15M rebar per unit length is 1.58 kg/m ( kilogram per metre), when it measured in pound per foot, then their weight could be 1.061lb/ ft, it measured in
Kilogram per foot, it could be 0.481kg/ ft and it measured in pound per metre, then their weight could be 3.48lb/m.
How do you calculate linear feet for rebar?
To determine the total linear rebar footage required, multiply the number of rebar you calculated for each side by the length measurement. Add those numbers together for the total number of linear
feet of rebar you’ll need.
How many pieces of 12mm rod makes a quintal?
Mathematical calculation such as number of pieces of 12mm Rods in one ton = 1000kg/weight of one piece of 12mm Rod, 1000kg ÷ 10.667kg = 94 pieces.
How many lengths of y12 makes a ton?
Standard length of Iron rods in the market For example, a ton of y16 reinforcement bar contains 52 pieces of 12 meter length steel bars in total.
What is the formula to calculate rebar weight?
By Calculation method. These are some examples of site level calculation which are not accurate but best to judge the approximate.
Experiment method,If we are calculating the weight for standard steel bar as shown in the picture,then it hard to calculate the area because of the ridges.
Online Steel Bar Weight Calculator.
How much does number 10 rebar weigh a foot?
How much does a #10 bar weight per foot? 4.303 pounds per foot Physical Specifications of #10 Rebar: Weight per unit length: 4.303 pounds per foot. How much is a bundle of #4 rebar? Residential #4
Rebar Prices Grade 40 or 60 #4 rebar costs $0.30 to $2.00 per linear foot.
What is a typical unit weight PCF of rebar?
avg – Average channel velocity – Expressed as function of R/W unit weight = 167 pcf 1.7 < D 85/D 15 > 2.7 Bed slope from 2 – 20 % uniform flow with NO tailwater
How do I estimate the quantity of rebar in slab?
– Convert your longitude measurement into inches: 15 feet x 12 inches per foot = 180 inches – Divide your result by the spacing measurement: 180 in / 14 in = 12.87 (round up to 13) – Add one rebar to
your result: 13 + 1 = 14 | {"url":"https://www.worldsrichpeople.com/how-much-does-rebar-weight-per-foot/","timestamp":"2024-11-05T22:18:42Z","content_type":"text/html","content_length":"55703","record_id":"<urn:uuid:76e84f05-1a8e-478d-8895-c278584253cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00119.warc.gz"} |
Sensitive monitoring of photocarrier densities in the active layer of a photovoltaic device with time-resolved terahertz reflection spectroscopy
We demonstrate the sensitive measurement of photocarriers in an active layer of a GaAs-based photovoltaic device using time-resolved terahertz reflection spectroscopy. We found that the reflection
dip caused by Fabry-Pérot interference is strongly affected by the carrier profile in the active layer of the p-i-n structure. The experimental results show that this method is suitable for
quantitative evaluation of carrier dynamics in active layers of solar cells under operating conditions.
Solar cells with high energy conversion efficiency have been developed recently.^1 According to Shockley-Queisser theory, the maximum efficiency is 31% for single-junction solar cells under standard
AM1.5 sunlight.^2 Because it is expected that the maximum efficiency can be achieved for materials with a bandgap energy, E[g], of around 1.34eV, solar cells fabricated from materials with ideal
bandgap energies have been optimized by evaluating their performance under continuous irradiation. The GaAs-based solar cell is used as a model device because its bandgap of 1.39eV is close to the
ideal value. To reach the maximum conversion efficiency, it is critical to understand the carrier dynamics in the device. In particular, the dynamics involving phenomena such as charge separation in
active regions, and information about photocarrier lifetimes including photon recycling, which is the reabsorption of photons generated as a product of radiative recombination,^3–6 are crucial.
Photoluminescence^7–10 and electroluminescence^11–17 are convenient tools for characterizing solar cell properties such as minority carrier lifetime, diffusion length, internal conversion efficiency,
and charge separation efficiency. Luminescence from a solar cell contains information about electron-hole pairs in an active layer, but the overlapping of luminescence bands from p- and n-type
regions obscures the luminescence from the active layer. Moreover, photon recycling makes analysis complicated. Hence, a combination of different optical methods will provide important insights into
the development of solar cells that are more efficient.^18–21
Terahertz (THz) spectroscopy is one of the most powerful optical methods for investigating photoexcited free-carrier dynamics in semiconductors. Time-resolved THz spectroscopy (TRTS) has also made it
possible to evaluate photocarrier dynamics in various solar cell materials.^22–29 We can quantitatively determine the photocarrier density, N, and the momentum scattering time, τ, with the
Drude-Lorentz analysis of THz data using a known value of effective mass, $m ⋆$. The technique for evaluating carrier properties in doped bulk materials and epitaxial films from their optical
properties is well established.^30–33 The optical constants can also be evaluated from the reflection spectra through the Fresnel coefficients.^8 In the THz region, the reflection features are
sensitive to the carrier density owing to phonon-plasmon coupling modes, and a number of studies have revealed the effect of doping on the optical constants.^31–34 However, strong free carrier
absorptions and multiple reflections caused by micrometer-scale layers of semiconductors with high doping concentrations on the order of 10^18 cm^−3 prevent simple spectroscopic analysis.
Additionally, the photocarrier density under AM1.5 sun is smaller than the usual doping concentrations of layered semiconductors. Therefore, it is challenging to monitor photocarriers in an active
region of a solar device by THz spectroscopy.
In this letter, we demonstrated that time-resolved THz reflection spectroscopy can be used to monitor the frequency shift caused by Fabry-Pérot interference to sensitively measure the concentration
of carriers in the active layer of a GaAs photovoltaic device under open circuit conditions. A theoretical calculation using the Drude-Lorentz model quantitatively reproduced our experimental
findings. Taking advantage of the interference-induced phase-sensitive responses, we established a method for precisely evaluating the concentration of photocarriers in an active layer of a solar
device. We propose this method as a useful tool for evaluating solar cells under operating conditions.
Figure 1(a) shows a schematic of our GaAs-based photovoltaic device with a single junction. This consists of a 660-nm-thick p-type layer, a 500-nm-thick i-region, and a 3-μm-thick n-layer, all of
which were grown by molecular-beam epitaxy on an n^+-type GaAs substrate with a thickness of 350μm. The doping level of carbon in the p-type layer was 1×10^18 cm^−3 and that of silicon in the n
-type layer was 2×10^17 cm^−3. In general, the dielectric function, $ε ̃j ( ω )$, of a layer labeled with index j (= p, i, n, n^+) in a semiconductor device is expressed by the Drude-Lorentz model
$ε ̃j ( ω ) = ε ∞ + ( ε s − ε ∞ ) ω TO 2 ω TO 2 − ω 2 − i ω γ − i σ ̃j ( ω ) ω ε 0 .$
With the GaAs-based device in this study, the first term, ε[∞]=10.86, is the high-frequency dielectric constant.^34,35 The second term is the phonon component, in which the static dielectric
constant, ε[s], is 12.8, the transverse-optical phonon frequency, ω[TO]/2π, is 8.0 THz, and γ is the damping constant.^34,35 This causes high reflectivity around 8 THz, which is known as the
reststrahlen band.^9 The third term is the free carrier component of the conduction, in which ε[0] is the vacuum permittivity. The optical conductivity, $σ ̃j ( ω )$, is expressed as
$σ ̃j ( ω ) = N j e 2 m e , h * τ 1 − i ω τ ,$
where N[j] is the free carrier density, e is the electron charge, τ is the momentum scattering time, and $m e , h *$ is the effective mass of carriers. The effective masses of electrons and holes are
assumed to be $m e *$=0.067 m[0] and $m h *$=0.35 m[0],^34,35 respectively, where m[0] is the free-electron mass. We have confirmed the availability of these values and the Drude model by
measuring chemically doped GaAs substrates and epitaxial films grown by molecular-beam epitaxy method which was also used for fabricating the photovoltaic sample. The density of the photoexcited
carriers is smaller than the chemical doping concentrations, and the properties of the photoexcited carriers should be as same as that of chemically doped carriers in the device. Therefore, the
availability of the Drude model with the effective mass approximation has been verified from the terahertz spectra of the chemically doped GaAs substrates, epitaxial films, and photovoltaic devices,
which is more detailed in the supplementary material. The screened plasma frequency, $ω s p , j$, of free carriers is expressed as $ω s p , j = N j e 2 / ( ε b ε 0 m e , h * )$. The screened plasma
frequency of the p-type layer, $ω s p , p / 2 π$, is 4.2 THz, that of the n-type substrate, $ω s p , n / 2 π$, is 4.2 THz using the background dielectric constant of $ε b = ε s$, and that of the n
^+-type substrate, $ω s p , n + / 2 π$, is 18 THz with $ε b = ε ∞$. Figure 1(b) shows the simulated reflection coefficients at the interfaces between the air and p-type layer (red dashed curve),
between the i- and n-type layers (black dotted curve), and between the n- and n^+-type layers (blue curves) in the GaAs device. The reflectivity is high below the screened plasma frequency, $ω s p ,
p$. Because the THz pulse is attenuated less in the p- and n-type layers above the screened plasma frequencies of $ω s p , p$ and $ω s p , n$, the THz pulse can propagate in the p-i-n layer, except
in the reststrahlen gap. It also causes the strong reflection of the THz pulse at the boundary between the n-layer and n^+-substrate below $ω s p , n + / 2 π$=18THz (Fig. 1(a)). Consequently, the
reflection spectrum in the window region between the frequencies of $ω s p , p$ and $ω s p , n +$ (light blue area in Fig. 1(b)) contains the information about photoexcited carriers in the p-i-n
structure. Furthermore, strong dispersion of the refractive index near the screened plasma frequencies causes Fabry-Pérot interference between both sides of the p-i-n structure, which increases the
spectral responsiveness. The spatial distribution of the electric field of the Fabry-Pérot interference is analyzed in detail in Fig. 4(b).
In the experiments, we used a time-resolved terahertz spectroscopy system^27–29 based on a 1kHz amplified Ti:sapphire laser with a duration of 35 fs and a center wavelength of 800nm. We divided the
output beam into three using two beamsplitters for optical excitation, THz pulse generation, and detection. The center wavelength of the excitation pulse was 800nm, and its fluence was varied from
0.88 to 31μJ/cm^2. Because the penetration depth was 750nm, the p- and i-layers were mainly excited in the photovoltaic sample. THz pulses from two-color (ω and 2ω) excited air plasma were focused
on the sample with p-polarization at a 30° incidence angle. The reflected THz pulse was detected with the electro-optic sampling method by using a 300–μm-thick GaP crystal, which had an available
frequency range from 0.5 to 7.5 THz, covering the sensitive frequency region of the device. The time profile of the THz pulses was obtained by varying the time delay, t, between the THz and sampling
pulses. We also measured THz pulses reflected on an Al plane mirror as a reference to evaluate the absolute reflectivity precisely. For the optical excitation, we used unfocused fundamental pulses at
normal incidence. The transient reflection response was obtained by varying the pump-probe delay, Δt[p]. All the measurements were performed under open circuit conditions.
Figures 2(a) and 2(b) show the reflected THz transient from the Al reference mirror E[ref](t), and THz transient, E(t, Δt[p]), at the surface of the sample without photoexcitation (blue dotted curve)
and 10 ps after photoexcitation (red solid curve). We confirmed that the hot carriers were relaxed thermally by the delay time 10 ps after photoexcitation.^27 The spectrum of the THz radiation from
air plasma extends to over 10 THz; however, the sensitivity of the GaP-based electro-optical (EO) detector is low in the frequency range above 7 THz owing to the phase mismatching of the EO effect.
Thus, the oscillation components appear above 8 THz. This oscillation is strongly modulated by the reflection dip of the photovoltaic device. The peak amplitude of reflected THz pulses was increased
by the photoexcitation, but the strength of the oscillation around Δt[p]=0.3 ps was attenuated. We Fourier transformed these time profiles to extract the complex reflection spectra. Figure 2(c)
shows the power reflectance $R ( ω ) = | E ( ω , Δ t p ) / E ref ( ω ) | 2$ as a function of frequency. The blue and red circles show the spectra without excitation and 10 ps after photoexcitation,
respectively. The reflectance dip appeared at 5 THz without excitation. We numerically simulated the complex total reflection from the sample by a sequential calculation of the Fresnel coefficients,
which reproduced the measured results consistently (solid lines). The best parameters of $N p$, $N n$, $N n +$, and τ were 1×10^18 cm^−3, 1.3×10^17 cm^−3, 3×10^18 cm^−3, and 80 fs,
respectively. These values were consistent with the original carrier density of our sample structure. We evaluated the parameters of the GaAs substrates, thin films and other device structures and
confirmed these values are valid. The deviation of $N n$ from the designed value of 2×10^17 cm^−3 was within the error possible during the fabrication process. After the photoexcitation, the
reflectance dip shifted toward a higher frequency, as shown by the red circles. We also calculated the transient reflection spectrum assuming the ratios of the excess photoexcited carrier density in
p-, i-, and n-regions are distributed exponentially following the Beer-Lambert law distribution, and the simulated spectrum reproduced the measured spectrum well (Fig. 2(c)). This frequency shift
could be attributed to the change of the plasma frequency because the reflectance dip appeared near the original plasma frequency of $ω s p , p$. However, the anomalous phase jump cannot be
explained by this simple model. Figure 2(d) shows the phase of the complex reflection coefficient $Δ ϕ ( ω ) = a r g ( E ( ω , Δ t p ) / E ref ( ω ) )$ as a function of frequency. While Δϕ(ω)
monotonically increased toward 2π with the frequency without photoexcitation, the direction of the phase jump was changed drastically by the photoexcitation. The change in the detection of phase jump
is characteristic of the interference between two waves. Impedance matching and tuning for microwave circuits are performed by monitoring such jumps.^36,37 In the visible and THz frequency region,
high sensitivity-sensing of surface plasmon resonance by observing the large phase change has been reported.^38,39 In our case, the destructive interference between the returning THz pulse on the
surface of the device and the reemitted THz pulse from the resonator, including p-i-n structures caused the phase anomaly, which enhanced the response of the photoexcited carrier in the active
The representation of the reflection coefficient in the complex plane supports our observation of the phase jump.^36–39 Figures 3(a) and 3(b) show the parametric plots of the complex reflective
coefficient, $r ̃( ω )$, as a function of the frequency on the complex plane. The green arrow is an example of a vector showing the complex reflective coefficient, where the square of its length and
the polar angle correspond to R(ω) and Δϕ(ω), respectively. The blue and red circles indicate the results without excitation and 10 ps after photoexcitation, respectively. The trajectories start from
the first quadrant and turn counterclockwise as the frequency increases. At the center frequency of the reflectance dip, the trajectory passes through the horizontal axis from the positive to the
negative region. The intersection of the horizontal axis moves smoothly in the positive direction in the complex plane with increasing photocarrier density, N[ph], and passes through the origin at N
[ph] of 3×10^13 cm^−2. When the trajectory goes through the zero reflection point, the 2π phase jump occurs in the phase Δϕ(ω) spectrum. Because the interference-induced response is enhanced near
the zero reflection point, as in the balance detection method,^36 we can monitor the photocarrier density in the p-i-n structures with high precision.
Figure 4(a) shows the N[ph] dependence of the frequency shift, Δω[dip]/2π (red circles). Although the frequency shift is roughly proportional to N[ph] in the low excitation region below 1×10^13 cm^
−3, the slope becomes shallow in the higher excitation region. For low excitation densities, the shift is reproduced well by the numerical simulation (solid line, Fig. 4(a)). Hence, we can determine
the sheet carrier density of photocarriers precisely with a resolution of the order of 10^12 cm^−2. It corresponds to bulk density of 10^16 cm^−3, assuming reflection loss of the device, the unit
quantum efficiency, and the penetration depth of 750nm. This is lower than the doping concentration in the p- and n-layers of the device. This resolution is sufficient for measuring photocarrier
densities in devices under realistic solar irradiance because the sheet carrier density of 10^12cm^−2 corresponds to that due to photocarriers in GaAs concentrator solar cell devices. For example,
for the solar spectra at 1 sun AM1.5G,^40 the photon flux, Φ, for wavelengths between 280 and 880nm is estimated as 1.85×10^17cm^−2 s^−1. Considering the Fresnel loss, photons with a flux of
1.08×10^17cm^−2 s^−1 can be absorbed by GaAs. The photocarrier density, N, can be written as N=Φ×τ[life] where, τ[life] is the carrier lifetime. Assuming τ[life] of ∼10ns, N can be estimated
as ∼10^9cm^−2. For high excitation densities, the measured shifts are smaller than the calculated shifts. We think this is due to the carrier diffusion. Figure 4(b) shows the distribution of the
absolute square of the electric field $| E | 2$ (red curve) of the lowest interference mode at 5 THz in the p-i-n structure calculated by the transfer matrix method. The electric field is the highest
near the surface where the p- and i-layers lie, which allows spatially sensitive monitoring of the photocarrier density. For the high excitation regime, some photocarriers could diffuse into the
insensitive region because the ambipolar diffusion behavior of electrons and holes can be enhanced by the many-body effects,^41 as illustrated in Fig. 4(c). Other possibility is density dependence of
the effective mass $m *$ because the effective mass $m *$ can become larger at high excitation regime, which can cause underestimation of the evaluated carrier density. Consequently, the evaluated
carrier density is smaller for the high excitation regime.
The quantitative characterization of charge carriers in the devices via THz reflectance could be used widely because of the design of direct-gap compound semiconductor solar cells. In general, solar
cells consist of a micrometer-scale layered structure owing to their absorption coefficients. The emitter layer is a sub-100-nm-thick thin film with a high doping concentration. The base layer is
thick with a low doping concentration. Although the emitter layer is opaque for excitation light due to the strong free carrier absorption, the layer is thin enough for the THz waves to propagate
into the p-i-n structure. Hence, the time-resolved THz reflection measurement could be a powerful tool for investigating the spatiotemporal dynamics of photocarriers in compound solar cell devices.
We have measured the center frequency of the dip as a function of the pump-probe delay and observed the recovery of the frequency shift for the reflection dip with a decay constant of 3ns, which
reflects the carriers' decay in the device. The photocarrier dynamics observed quantitatively in solar cell devices near their operating conditions with TRTS provides a good measure for estimating
photocarrier generation efficiency precisely, enabling more accurate device simulations. For indirect gap semiconductor devices, such as single-crystal Si solar cells, the interference effect is less
important in the THz region because the multilayer is much thicker owing to the small absorption coefficient.
In conclusion, we have measured transient reflectance in the GaAs photovoltaic device with time-resolved THz reflection spectroscopy. We demonstrated that the photocarrier density in the photovoltaic
device can be monitored sensitively by measuring the THz transients of the device, even with the highly doped p-type emitter layer. Fabry-Pérot interference dominates the reflection features, and the
center frequency of the reflection dip strongly depends on the excitation density. For the low excitation regime, our numerical simulation reproduced the reflection features of the photovoltaic
device consistently. We propose that the interference-induced phase-sensitive response can be used for determining the photocarrier density with a high resolution of the order of 10^16cm^−3, which
is smaller than the doping concentration in typical photovoltaic devices. Such a direct measurement of carrier density in the active region of the device will contribute to the elucidation of carrier
dynamics in solar cell devices under operating conditions.
See supplementary material for the derivation for the reflection coefficient of the photovoltaic device and the temporal evolution of the transient reflection coefficient.
This work was supported by Grant-in-Aid for JSPS Fellows (No. 16J05700). Y.K. and H.A. are thankful for the support from JST-CREST.
H. J.
J. Appl. Phys.
O. D.
, and
S. R.
IEEE J. Photovoltaics
, and
Appl. Phys. Lett.
, and
Opt. Express
, and
Jpn. J. Appl. Phys.
L. Q.
, and
J. Am. Chem. Soc.
R. K.
M. S.
Minority-Carrier Lifetime in III–V Semiconductors: Physics and Applications, Semiconductors and Semimetals
New York
), Vol.
W. K.
R. K.
, and
D. J.
Phys. Rev. B
D. M.
, and
Appl. Phys. Lett.
D. M.
, and
Opt. Express
Phys. Rev. B
J. Appl. Phys.
A. W.
, and
Prog. Photovoltaics: Res. Appl.
A. W.
, and
J. H.
Appl. Phys. Lett.
, and
A. W.
Appl. Phys. Lett.
, and
Sci. Rep.
, and
J. Appl. Phys.
Appl. Phys. Lett.
, and
Appl. Phys. Express
, and
J. Am. Chem. Soc.
, and
J. Phys. Chem. Lett.
, and
Phys. Rev. B
T. F.
, and
Rev. Mod. Phys.
, and
Opt. Lett.
J. Infrared Millimeter Terahertz Waves
S. A.
, and
Appl. Phys. Lett.
, and
Appl. Phys. Lett.
L. Q.
, and
Appl. Phys. Lett.
L. Q.
, and
Appl. Phys. Express
L. Q.
, and
J. Phys, Chem. Lett.
E. D.
R. T.
, and
J. W.
Thin Solid Films
R. T.
J. A.
J. Appl. Phys.
H. R.
A. K.
Phys. Rev. B
, and
Phys. Status Solidi B
J. Appl. Phys.
Semiconductors: Data Handbook
Electrodynamics of Solids
Cambridge University Press
A. N.
P. I.
, and
A. V.
Appl. Phys. Lett.
, and
Opt. Express
J. F.
H. M.
van Driel
Phys. Rev. B | {"url":"https://pubs.aip.org/aip/apl/article/110/7/071108/33824/Sensitive-monitoring-of-photocarrier-densities-in","timestamp":"2024-11-03T13:16:11Z","content_type":"text/html","content_length":"269253","record_id":"<urn:uuid:72f7bb36-0e55-4a85-a6d0-41b46ec006df>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00660.warc.gz"} |
Multiply any 2 numbers whose difference is 2
Multiply any 2 numbers whose difference is 2
Sachin said on : 2018-04-16 10:26:32
Today you will be learning another amazing shortcut way of multiplying any 2 numbers.This trick is especially effective to use when the number in between the 2 numbers to be multiplied is 5.You will
see why......
Let us go through the steps
Step 1:Identify the number in between the 2 numbers.
Step 2:Find the square of the number identified in step 1.
Step 3:Subtract 1 from the result obtained in step 2.
Let us now consider few examples to understand better
Example 1:12 x 14=?
Notice that there is difference of 2 between the numbers 12 and 14.
We shall now apply the trick that we learnt
Step 1:We find that 13 lies in between 12 and 14.
Step 2:Squaring 13 we get
Step 3:Subtracting 1 from the result obtained in step 2
We get 169 -1=168
Ans 12 x 14=168
Example 2:24 x 26=?
Step 1:We find that 25 lies in between 24 and 26.
Step 2:Squaring 25 we get
(Use the shortcut trick to square any number ending in 5)
Step 3:Subtracting 1 from the result obtained in step 2
We get 625 -1=624
Ans 24 x 26=624
!! OOPS Login [Click here] is required for more results / answer
Help other students, write article, leave your comments | {"url":"https://engineeringslab.com/tutorial_vedic_quicker_shortcut_math_tricks/multiply-any-2-numbers-whose-difference-is-2-69.htm","timestamp":"2024-11-02T02:06:53Z","content_type":"text/html","content_length":"38155","record_id":"<urn:uuid:28ac8075-0933-449d-9cd5-3ee5e0ce6f61>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00055.warc.gz"} |
Puiseux series - Wikiwand
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
{\displaystyle {\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}}
Truncated Puiseux expansions for the cubic curve at the double point . Darker colors indicate more terms.
is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676^[1] and rediscovered by Victor Puiseux in 1850.^[2]
The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in
an nth root of the indeterminate. For example, the example above is a Laurent series in ${\displaystyle x^{1/6}.}$ Because a complex number has n nth roots, a convergent Puiseux series typically
defines n functions in a neighborhood of 0.
Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation ${\displaystyle P(x,y)=0}$ with complex coefficients, its solutions in y, viewed as
functions of x, may be expanded as Puiseux series in x that are convergent in some neighbourhood of 0. In other words, every branch of an algebraic curve may be locally described by a Puiseux series
in x (or in x − x[0] when considering branches above a neighborhood of x[0] ≠ 0).
Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of
Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.
If K is a field (such as the complex numbers), a Puiseux series with coefficients in K is an expression of the form
${\displaystyle f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}}$
where ${\displaystyle n}$ is a positive integer and ${\displaystyle k_{0}}$ is an integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the
indeterminate, as long as these fractional exponents have bounded denominator (here n). Just as with Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these
negative exponents are bounded below (here by ${\displaystyle k_{0}}$). Addition and multiplication are as expected: for example,
${\displaystyle (T^{-1}+2T^{-1/2}+T^{1/3}+\cdots )+(T^{-5/4}-T^{-1/2}+2+\cdots )=T^{-5/4}+T^{-1}+T^{-1/2}+2+\cdots }$
${\displaystyle (T^{-1}+2T^{-1/2}+T^{1/3}+\cdots )\cdot (T^{-5/4}-T^{-1/2}+2+\cdots )=T^{-9/4}+2T^{-7/4}-T^{-3/2}+T^{-11/12}+4T^{-1/2}+\cdots .}$
One might define them by first "upgrading" the denominator of the exponents to some common denominator N and then performing the operation in the corresponding field of formal Laurent series of ${\
displaystyle T^{1/N}}$.
The Puiseux series with coefficients in K form a field, which is the union
${\displaystyle \bigcup _{n>0}K(\!(T^{1/n})\!)}$
of fields of formal Laurent series in ${\displaystyle T^{1/n}}$ (considered as an indeterminate).
This yields an alternative definition of the field of Puiseux series in terms of a direct limit. For every positive integer n, let ${\displaystyle T_{n}}$ be an indeterminate (meant to represent ${\
textstyle T^{1/n}}$), and ${\displaystyle K(\!(T_{n})\!)}$ be the field of formal Laurent series in ${\displaystyle T_{n}.}$ If m divides n, the mapping ${\displaystyle T_{m}\mapsto (T_{n})^{n/m}}$
induces a field homomorphism ${\displaystyle K(\!(T_{m})\!)\to K(\!(T_{n})\!),}$ and these homomorphisms form a direct system that has the field of Puiseux series as a direct limit. The fact that
every field homomorphism is injective shows that this direct limit can be identified with the above union, and that the two definitions are equivalent (up to an isomorphism).
A nonzero Puiseux series ${\displaystyle f}$ can be uniquely written as
${\displaystyle f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}}$
with ${\displaystyle c_{k_{0}}eq 0.}$ The valuation
${\displaystyle v(f)={\frac {k_{0}}{n}}}$
of ${\displaystyle f}$ is the smallest exponent for the natural order of the rational numbers, and the corresponding coefficient ${\textstyle c_{k_{0}}}$ is called the initial coefficient or
valuation coefficient of ${\displaystyle f}$. The valuation of the zero series is ${\displaystyle +\infty .}$
The function v is a valuation and makes the Puiseux series a valued field, with the additive group ${\displaystyle \mathbb {Q} }$ of the rational numbers as its valuation group.
As for every valued fields, the valuation defines a ultrametric distance by the formula ${\displaystyle d(f,g)=\exp(-v(f-g)).}$ For this distance, the field of Puiseux series is a metric space. The
${\displaystyle f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}}$
expresses that a Puiseux is the limit of its partial sums. However, the field of Puiseux series is not complete; see below § Levi–Civita field.
Convergent Puiseux series
Puiseux series provided by Newton–Puiseux theorem are convergent in the sense that there is a neighborhood of zero in which they are convergent (0 excluded if the valuation is negative). More
precisely, let
${\displaystyle f=\sum _{k=k_{0}}^{+\infty }c_{k}T^{k/n}}$
be a Puiseux series with complex coefficients. There is a real number r, called the radius of convergence such that the series converges if T is substituted for a nonzero complex number t of absolute
value less than r, and r is the largest number with this property. A Puiseux series is convergent if it has a nonzero radius of convergence.
Because a nonzero complex number has n nth roots, some care must be taken for the substitution: a specific nth root of t, say x, must be chosen. Then the substitution consists of replacing ${\
displaystyle T^{k/n}}$ by ${\displaystyle x^{k}}$ for every k.
The existence of the radius of convergence results from the similar existence for a power series, applied to ${\textstyle T^{-k_{0}/n}f,}$ considered as a power series in ${\displaystyle T^{1/n}.}$
It is a part of Newton–Puiseux theorem that the provided Puiseux series have a positive radius of convergence, and thus define a (multivalued) analytic function in some neighborhood of zero (zero
itself possibly excluded).
Valuation and order on coefficients
If the base field ${\displaystyle K}$ is ordered, then the field of Puiseux series over ${\displaystyle K}$ is also naturally (“lexicographically”) ordered as follows: a non-zero Puiseux series ${\
displaystyle f}$ with 0 is declared positive whenever its valuation coefficient is so. Essentially, this means that any positive rational power of the indeterminate ${\displaystyle T}$ is made
positive, but smaller than any positive element in the base field ${\displaystyle K}$.
If the base field ${\displaystyle K}$ is endowed with a valuation ${\displaystyle w}$, then we can construct a different valuation on the field of Puiseux series over ${\displaystyle K}$ by letting
the valuation ${\displaystyle {\hat {w}}(f)}$ be ${\displaystyle \omega \cdot v+w(c_{k}),}$ where ${\displaystyle v=k/n}$ is the previously defined valuation (${\displaystyle c_{k}}$ is the first
non-zero coefficient) and ${\displaystyle \omega }$ is infinitely large (in other words, the value group of ${\displaystyle {\hat {w}}}$ is ${\displaystyle \mathbb {Q} \times \Gamma }$ ordered
lexicographically, where ${\displaystyle \Gamma }$ is the value group of ${\displaystyle w}$). Essentially, this means that the previously defined valuation ${\displaystyle v}$ is corrected by an
infinitesimal amount to take into account the valuation ${\displaystyle w}$ given on the base field.
As early as 1671,^[3] Isaac Newton implicitly used Puiseux series and proved the following theorem for approximating with series the roots of algebraic equations whose coefficients are functions that
are themselves approximated with series or polynomials. For this purpose, he introduced the Newton polygon, which remains a fundamental tool in this context. Newton worked with truncated series, and
it is only in 1850 that Victor Puiseux^[2] introduced the concept of (non-truncated) Puiseux series and proved the theorem that is now known as Puiseux's theorem or Newton–Puiseux theorem.^[4] The
theorem asserts that, given an algebraic equation whose coefficients are polynomials or, more generally, Puiseux series over a field of characteristic zero, every solution of the equation can be
expressed as a Puiseux series. Moreover, the proof provides an algorithm for computing these Puiseux series, and, when working over the complex numbers, the resulting series are convergent.
In modern terminology, the theorem can be restated as: the field of Puiseux series over an algebraically closed field of characteristic zero, and the field of convergent Puiseux series over the
complex numbers, are both algebraically closed.
Newton polygon
${\displaystyle P(y)=\sum _{a_{i}eq 0}a_{i}(x)y^{i}}$
be a polynomial whose nonzero coefficients ${\displaystyle a_{i}(x)}$ are polynomials, power series, or even Puiseux series in x. In this section, the valuation ${\displaystyle v(a_{i})}$ of ${\
displaystyle a_{i}}$ is the lowest exponent of x in ${\displaystyle a_{i}.}$ (Most of what follows applies more generally to coefficients in any valued ring.)
For computing the Puiseux series that are roots of P (that is solutions of the functional equation ${\displaystyle P(y)=0}$), the first thing to do is to compute the valuation of the roots. This is
the role of the Newton polygon.
Let consider, in a Cartesian plane, the points of coordinates ${\displaystyle (i,v(a_{i})).}$ The Newton polygon of P is the lower convex hull of these points. That is, the edges of the Newton
polygon are the line segments joigning two of these points, such that all these points are not below the line supporting the segment (below is, as usually, relative to the value of the second
Given a Puiseux series ${\displaystyle y_{0}}$ of valuation ${\displaystyle v_{0}}$, the valuation of ${\displaystyle P(y_{0})}$ is at least the minimum of the numbers ${\displaystyle iv_{0}+v(a_
{i}),}$ and is equal to this minimum if this minimum is reached for only one i. So, for ${\displaystyle y_{0}}$ being a root of P, the minimum must be reached at least twice. That is, there must be
two values ${\displaystyle i_{1}}$ and ${\displaystyle i_{2}}$ of i such that ${\displaystyle i_{1}v_{0}+v(a_{i_{1}})=i_{2}v_{0}+v(a_{i_{2}}),}$ and ${\displaystyle iv_{0}+v(a_{i})\geq i_{1}v_{0}+v
(a_{i_{1}})}$ for every i.
That is, ${\displaystyle (i_{1},v(a_{i_{1}}))}$ and ${\displaystyle (i_{2},v(a_{i_{2}}))}$ must belong to an edge of the Newton polygon, and ${\displaystyle v_{0}=-{\frac {v(a_{i_{1}})-v(a_{i_{2}})}
{i_{1}-i_{2}}}}$ must be the opposite of the slope of this edge. This is a rational number as soon as all valuations ${\displaystyle v(a_{i})}$ are rational numbers, and this is the reason for
introducing rational exponents in Puiseux series.
In summary, the valuation of a root of P must be the opposite of a slope of an edge of the Newton polynomial.
The initial coefficient of a Puiseux series solution of ${\displaystyle P(y)=0}$ can easily be deduced. Let ${\displaystyle c_{i}}$ be the initial coefficient of ${\displaystyle a_{i}(x),}$ that is,
the coefficient of ${\displaystyle x^{v(a_{i})}}$ in ${\displaystyle a_{i}(x).}$ Let ${\displaystyle -v_{0}}$ be a slope of the Newton polygon, and ${\displaystyle \gamma x_{0}^{v_{0}}}$ be the
initial term of a corresponding Puiseux series solution of ${\displaystyle P(y)=0.}$ If no cancellation would occur, then the initial coefficient of ${\displaystyle P(y)}$ would be ${\textstyle \sum
_{i\in I}c_{i}\gamma ^{i},}$ where I is the set of the indices i such that ${\displaystyle (i,v(a_{i}))}$ belongs to the edge of slope ${\displaystyle v_{0}}$ of the Newton polygon. So, for having a
root, the initial coefficient ${\displaystyle \gamma }$ must be a nonzero root of the polynomial ${\displaystyle \chi (x)=\sum _{i\in I}c_{i}x^{i}}$ (this notation will be used in the next section).
In summary, the Newton polynomial allows an easy computation of all possible initial terms of Puiseux series that are solutions of ${\displaystyle P(y)=0.}$
The proof of Newton–Puiseux theorem will consist of starting from these initial terms for computing recursively the next terms of the Puiseux series solutions.
Constructive proof
Let suppose that the first term ${\displaystyle \gamma x^{v_{0}}}$ of a Puiseux series solution of ${\displaystyle P(y)=0}$ has been be computed by the method of the preceding section. It remains to
compute ${\displaystyle z=y-\gamma x^{v_{0}}.}$ For this, we set ${\displaystyle y_{0}=\gamma x^{v_{0}},}$ and write the Taylor expansion of P at ${\displaystyle z=y-y_{0}:}$
${\displaystyle Q(z)=P(y_{0}+z)=P(y_{0})+zP'(y_{0})+\cdots +z^{j}{\frac {P^{(j)}(y_{0})}{j!}}+\cdots }$
This is a polynomial in z whose coefficients are Puiseux series in x. One may apply to it the method of the Newton polygon, and iterate for getting the terms of the Puiseux series, one after the
other. But some care is required for insuring that ${\displaystyle v(z)>v_{0},}$ and showing that one get a Puiseux series, that is, that the denominators of the exponents of x remain bounded.
The derivation with respect to y does not change the valuation in x of the coefficients; that is,
${\displaystyle v\left(P^{(j)}(y_{0})z^{j}\right)\geq \min _{i}(v(a_{i})+iv_{0})+j(v(z)-v_{0}),}$
and the equality occurs if and only if ${\displaystyle \chi ^{(j)}(\gamma )eq 0,}$ where ${\displaystyle \chi (x)}$ is the polynomial of the preceding section. If m is the multiplicity of ${\
displaystyle \gamma }$ as a root of ${\displaystyle \chi ,}$ it results that the inequality is an equality for ${\displaystyle j=m.}$ The terms such that ${\displaystyle j>m}$ can be forgotten as far
as one is concerned by valuations, as ${\displaystyle v(z)>v_{0}}$ and ${\displaystyle j>m}$ imply
${\displaystyle v\left(P^{(j)}(y_{0})z^{j}\right)\geq \min _{i}(v(a_{i})+iv_{0})+j(v(z)-v_{0})>v\left(P^{(m)}(y_{0})z^{m}\right).}$
This means that, for iterating the method of Newton polygon, one can and one must consider only the part of the Newton polygon whose first coordinates belongs to the interval ${\displaystyle [0,m].}$
Two cases have to be considered separately and will be the subject of next subsections, the so-called ramified case, where m > 1, and the regular case where m = 1.
Ramified case
The way of applying recursively the method of the Newton polygon has been described precedingly. As each application of the method may increase, in the ramified case, the denominators of exponents
(valuations), it remains to prove that one reaches the regular case after a finite number of iterations (otherwise the denominators of the exponents of the resulting series would not be bounded, and
this series would not be a Puiseux series. By the way, it will also be proved that one gets exactly as many Puiseux series solutions as expected, that is the degree of ${\displaystyle P(y)}$ in y.
Without loss of generality, one can suppose that ${\displaystyle P(0)eq 0,}$ that is, ${\displaystyle a_{0}eq 0.}$ Indeed, each factor y of ${\displaystyle P(y)}$ provides a solution that is the zero
Puiseux series, and such factors can be factored out.
As the characteristic is supposed to be zero, one can also suppose that ${\displaystyle P(y)}$ is a square-free polynomial, that is that the solutions of ${\displaystyle P(y)=0}$ are all different.
Indeed, the square-free factorization uses only the operations of the field of coefficients for factoring ${\displaystyle P(y)}$ into square-free factors than can be solved separately. (The
hypothesis of characteristic zero is needed, since, in characteristic p, the square-free decomposition can provide irreducible factors, such as ${\displaystyle y^{p}-x,}$ that have multiple roots
over an algebraic extension.)
In this context, one defines the length of an edge of a Newton polygon as the difference of the abscissas of its end points. The length of a polygon is the sum of the lengths of its edges. With the
hypothesis ${\displaystyle P(0)eq 0,}$ the length of the Newton polygon of P is its degree in y, that is the number of its roots. The length of an edge of the Newton polygon is the number of roots of
a given valuation. This number equals the degree of the previously defined polynomial ${\displaystyle \chi (x).}$
The ramified case corresponds thus to two (or more) solutions that have the same initial term(s). As these solutions must be distinct (square-free hypothesis), they must be distinguished after a
finite number of iterations. That is, one gets eventually a polynomial ${\displaystyle \chi (x)}$ that is square free, and the computation can continue as in the regular case for each root of ${\
displaystyle \chi (x).}$
As the iteration of the regular case does not increase the denominators of the exponents, This shows that the method provides all solutions as Puiseux series, that is, that the field of Puiseux
series over the complex numbers is an algebraically closed field that contains the univariate polynomial ring with complex coefficients.
Failure in positive characteristic
The Newton–Puiseux theorem is not valid over fields of positive characteristic. For example, the equation ${\displaystyle X^{2}-X=T^{-1}}$ has solutions
${\displaystyle X=T^{-1/2}+{\frac {1}{2}}+{\frac {1}{8}}T^{1/2}-{\frac {1}{128}}T^{3/2}+\cdots }$
${\displaystyle X=-T^{-1/2}+{\frac {1}{2}}-{\frac {1}{8}}T^{1/2}+{\frac {1}{128}}T^{3/2}+\cdots }$
(one readily checks on the first few terms that the sum and product of these two series are 1 and ${\displaystyle -T^{-1}}$ respectively; this is valid whenever the base field K has characteristic
different from 2).
As the powers of 2 in the denominators of the coefficients of the previous example might lead one to believe, the statement of the theorem is not true in positive characteristic. The example of the
Artin–Schreier equation ${\displaystyle X^{p}-X=T^{-1}}$ shows this: reasoning with valuations shows that X should have valuation ${\textstyle -{\frac {1}{p}}}$, and if we rewrite it as ${\
displaystyle X=T^{-1/p}+X_{1}}$ then
${\displaystyle X^{p}=T^{-1}+{X_{1}}^{p},{\text{ so }}{X_{1}}^{p}-X_{1}=T^{-1/p}}$
and one shows similarly that ${\displaystyle X_{1}}$ should have valuation ${\textstyle -{\frac {1}{p^{2}}}}$, and proceeding in that way one obtains the series
${\displaystyle T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots$ ;}
since this series makes no sense as a Puiseux series—because the exponents have unbounded denominators—the original equation has no solution. However, such Eisenstein equations are essentially the
only ones not to have a solution, because, if ${\displaystyle K}$ is algebraically closed of characteristic ${\displaystyle p>0}$, then the field of Puiseux series over ${\displaystyle K}$ is the
perfect closure of the maximal tamely ramified extension of ${\displaystyle K(\!(T)\!)}$.^[4]
Similarly to the case of algebraic closure, there is an analogous theorem for real closure: if ${\displaystyle K}$ is a real closed field, then the field of Puiseux series over ${\displaystyle K}$ is
the real closure of the field of formal Laurent series over ${\displaystyle K}$.^[5] (This implies the former theorem since any algebraically closed field of characteristic zero is the unique
quadratic extension of some real-closed field.)
There is also an analogous result for p-adic closure: if ${\displaystyle K}$ is a ${\displaystyle p}$-adically closed field with respect to a valuation ${\displaystyle w}$, then the field of Puiseux
series over ${\displaystyle K}$ is also ${\displaystyle p}$-adically closed.^[6]
Algebraic curves
Let ${\displaystyle X}$ be an algebraic curve^[7] given by an affine equation ${\displaystyle F(x,y)=0}$ over an algebraically closed field ${\displaystyle K}$ of characteristic zero, and consider a
point ${\displaystyle p}$ on ${\displaystyle X}$ which we can assume to be ${\displaystyle (0,0)}$. We also assume that ${\displaystyle X}$ is not the coordinate axis ${\displaystyle x=0}$. Then a
Puiseux expansion of (the ${\displaystyle y}$ coordinate of) ${\displaystyle X}$ at ${\displaystyle p}$ is a Puiseux series ${\displaystyle f}$ having positive valuation such that ${\displaystyle F
More precisely, let us define the branches of ${\displaystyle X}$ at ${\displaystyle p}$ to be the points ${\displaystyle q}$ of the normalization ${\displaystyle Y}$ of ${\displaystyle X}$ which map
to ${\displaystyle p}$. For each such ${\displaystyle q}$, there is a local coordinate ${\displaystyle t}$ of ${\displaystyle Y}$ at ${\displaystyle q}$ (which is a smooth point) such that the
coordinates ${\displaystyle x}$ and ${\displaystyle y}$ can be expressed as formal power series of ${\displaystyle t}$, say ${\displaystyle x=t^{n}+\cdots }$ (since ${\displaystyle K}$ is
algebraically closed, we can assume the valuation coefficient to be 1) and ${\displaystyle y=ct^{k}+\cdots }$: then there is a unique Puiseux series of the form ${\displaystyle f=cT^{k/n}+\cdots }$
(a power series in ${\displaystyle T^{1/n}}$), such that ${\displaystyle y(t)=f(x(t))}$ (the latter expression is meaningful since ${\displaystyle x(t)^{1/n}=t+\cdots }$ is a well-defined power
series in ${\displaystyle t}$). This is a Puiseux expansion of ${\displaystyle X}$ at ${\displaystyle p}$ which is said to be associated to the branch given by ${\displaystyle q}$ (or simply, the
Puiseux expansion of that branch of ${\displaystyle X}$), and each Puiseux expansion of ${\displaystyle X}$ at ${\displaystyle p}$ is given in this manner for a unique branch of ${\displaystyle X}$
at ${\displaystyle p}$.^[8]^[9]
This existence of a formal parametrization of the branches of an algebraic curve or function is also referred to as Puiseux's theorem: it has arguably the same mathematical content as the fact that
the field of Puiseux series is algebraically closed and is a historically more accurate description of the original author's statement.^[10]
For example, the curve ${\displaystyle y^{2}=x^{3}+x^{2}}$ (whose normalization is a line with coordinate ${\displaystyle t}$ and map ${\displaystyle t\mapsto (t^{2}-1,t^{3}-t)}$) has two branches at
the double point (0,0), corresponding to the points ${\displaystyle t=+1}$ and ${\displaystyle t=-1}$ on the normalization, whose Puiseux expansions are ${\textstyle y=x+{\frac {1}{2}}x^{2}-{\frac
{1}{8}}x^{3}+\cdots }$ and ${\textstyle y=-x-{\frac {1}{2}}x^{2}+{\frac {1}{8}}x^{3}+\cdots }$ respectively (here, both are power series because the ${\displaystyle x}$ coordinate is étale at the
corresponding points in the normalization). At the smooth point ${\displaystyle (-1,0)}$ (which is ${\displaystyle t=0}$ in the normalization), it has a single branch, given by the Puiseux expansion
${\displaystyle y=-(x+1)^{1/2}+(x+1)^{3/2}}$ (the ${\displaystyle x}$ coordinate ramifies at this point, so it is not a power series).
The curve ${\displaystyle y^{2}=x^{3}}$ (whose normalization is again a line with coordinate ${\displaystyle t}$ and map ${\displaystyle t\mapsto (t^{2},t^{3})}$), on the other hand, has a single
branch at the cusp point ${\displaystyle (0,0)}$, whose Puiseux expansion is ${\displaystyle y=x^{3/2}}$.
Analytic convergence
When ${\displaystyle K=\mathbb {C} }$ is the field of complex numbers, the Puiseux expansion of an algebraic curve (as defined above) is convergent in the sense that for a given choice of ${\
displaystyle n}$-th root of ${\displaystyle x}$, they converge for small enough ${\displaystyle |x|}$, hence define an analytic parametrization of each branch of ${\displaystyle X}$ in the
neighborhood of ${\displaystyle p}$ (more precisely, the parametrization is by the ${\displaystyle n}$-th root of ${\displaystyle x}$).
Levi-Civita field
The field of Puiseux series is not complete as a metric space. Its completion, called the Levi-Civita field, can be described as follows: it is the field of formal expressions of the form ${\
textstyle f=\sum _{e}c_{e}T^{e},}$ where the support of the coefficients (that is, the set of e such that ${\displaystyle c_{e}eq 0}$) is the range of an increasing sequence of rational numbers that
either is finite or tends to ${\displaystyle +\infty }$. In other words, such series admit exponents of unbounded denominators, provided there are finitely many terms of exponent less than ${\
displaystyle A}$ for any given bound ${\displaystyle A}$. For example, ${\textstyle \sum _{k=1}^{+\infty }T^{k+{\frac {1}{k}}}}$ is not a Puiseux series, but it is the limit of a Cauchy sequence of
Puiseux series; in particular, it is the limit of ${\textstyle \sum _{k=1}^{N}T^{k+{\frac {1}{k}}}}$ as ${\displaystyle N\to +\infty }$. However, even this completion is still not "maximally
complete" in the sense that it admits non-trivial extensions which are valued fields having the same value group and residue field,^[11]^[12] hence the opportunity of completing it even more.
Hahn series
Hahn series are a further (larger) generalization of Puiseux series, introduced by Hans Hahn in the course of the proof of his embedding theorem in 1907 and then studied by him in his approach to
Hilbert's seventeenth problem. In a Hahn series, instead of requiring the exponents to have bounded denominator they are required to form a well-ordered subset of the value group (usually ${\
displaystyle \mathbb {Q} }$ or ${\displaystyle \mathbb {R} }$). These were later further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting (they are therefore sometimes
known as Hahn–Mal'cev–Neumann series). Using Hahn series, it is possible to give a description of the algebraic closure of the field of power series in positive characteristic which is somewhat
analogous to the field of Puiseux series.^[13] | {"url":"https://www.wikiwand.com/en/articles/Puiseux_series","timestamp":"2024-11-03T03:45:36Z","content_type":"text/html","content_length":"847765","record_id":"<urn:uuid:830bb6df-cfd9-48f0-936f-ef02fb82027b>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00702.warc.gz"} |
Q16: Answers – Paper 2 June 18 – Edexcel GCSE Maths Foundation
Helpful Links
Mark Scheme
Marla buys some bags of buttons.
There are 19 buttons or 20 buttons or 21 buttons or 22 buttons in each bag.
The table gives some information about the number of buttons in each bag.
The total number of buttons is 320
Complete the table.
Click here for a printable PDF of this question.
Marla buys some bags of buttons.
There are 19 buttons or 20 buttons or 21 buttons or 22 buttons in each bag.
The table gives some information about the number of buttons in each bag.
The total number of buttons is 320
Complete the table.
[3 marks] | {"url":"https://www.elevise.co.uk/bx16a.html","timestamp":"2024-11-03T18:19:10Z","content_type":"text/html","content_length":"95819","record_id":"<urn:uuid:6274b36c-553f-44a9-9d47-d61b836e1a9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00096.warc.gz"} |
What are Half-Wave Rectifiers? Definition, Circuit and Working of Half-Wave Rectifiers - Electronics Desk
Half-Wave Rectifiers
Definition: A Half-Wave Rectifier is a device that converts only one half of the applied ac signal into pulsating dc. The other half of the applied ac signal gets suppressed by the rectifier circuit.
A half wave rectifier circuit uses only a single diode, due to the unidirectional current flow property of the diode.
We know that rectifier is basically a device that is used to change an ac signal into a pulsating dc form. But how this happens by making use of diodes?
We are aware of the fact that a diode exhibits the property of conducting current in forward direction only by blocking the conduction in reverse direction.
Also, when resistance is connected in series with the diode, then the unidirectional voltage will appear across the resistance.
Thereby changing ac signal into dc. So, this property is utilized by the rectifiers for the purpose of signal rectification.
Rectifier circuits can be composed of either 1, 2 or 4 diodes. A half wave rectifier needs only a single diode. While the full wave rectifier circuit needs 2 or 4 diodes depending on the type of
Half-Wave Rectifier Circuit
The figure below represents the circuit diagram of a half-wave rectifier:
Here, the figure shows the circuit of the half-wave rectifier, consisting of a pn junction semiconductor diode and a load resistance R[L] in series with it. The diode and resistance form series
connection with the secondary winding of the step-down transformer. While the primary is connected with the ac source.
Now, let’s proceed further and understand how a half wave rectifier operates.
Working of Half-Wave Rectifier
The ac source provides ac voltage to the primary winding of the step-down transformer. This reduces the high applied ac voltage to a low-value ac value.
So, this ac voltage is allowed to flow through the rectifier circuit. But both the halves react differently when passed through the circuit.
Firstly let us understand the case when the positive half of the input signal is provided to the circuit. The applied positive half of the input signal forward biases the diode.
As p side of the diode is connected to positive polarity and n side is connected to the negative polarity. Due to this the diode shows a closed switch structure and allows the current to flow through
the circuit.
This is shown in the figure below:
Resultantly, the overall applied voltage appears across the load resistance. As the diode is assumed to be an ideal diode.
Further when the negative half of the ac input is applied to the circuit. Then due to this, the diode comes in reverse biased condition. Due to this, the diode starts acting as an open switch,
thereby restricting the path for the flow of current in the circuit.
This is shown in the figure below:
Resultantly, for the negative half of the voltage, no power appears at the load.
Hence at the load of the half wave rectifier circuit, a series of positive half of the applied input voltage is achieved.
Also, it is clear from the above figure that the output is pulsating dc despite being steady dc.
So, to eliminate these ripple component (ac part), filter units are employed that gives constant dc signal at the output.
Analysis of Half-Wave Rectifier
The factors that are to be analyzed in this section are as follows:
Peak Inverse Voltage: It is abbreviated as PIV. It the maximum voltage that can be handled by the diode when it is reverse biased.
As in reverse bias no current flow through the circuit, thus the applied reverse voltage appears across the diode.
Hence, for the half-wave rectifier, the maximum applied voltage is the PIV thus,
PIV = V[S max ]
Peak Current: Peak current is defined as the maximum current that flows through the rectifier circuit.
The sinusoidal applied input voltage is given as
V[S] = V[S max] sin ωt
And the current flowing through the circuit,
i = I[max] sin ωt for 0 ≤ ωt ≤ π
i = 0 for π ≤ ωt ≤ 2π
Then the maximum current flowing through the diode is given as
DC output current: The dc output current of the half-wave rectifier is given as
On putting the value of Imax in above equation we get
If R[L] >> R[F ]
DC output voltage: Average value of the voltage at the load is given as
If R[L] >> R[F ]
RMS value of current: The RMS current flowing through the diode
Putting the value of I[max] in the above equation
Output RMS voltage: Value of RMS voltage across the load is given as
If R[L] >> R[F ]
Form factor: It is defined as the ratio of RMS current to the dc output current. It is given as
Peak factor: It is defined as the ratio of peak current to the RMS current. And is given by
So, we will have
Output Frequency: The output frequency is equivalent to the input frequency for a half wave rectifier.
f[out] = f[in]
Ripple factor: The pulsating dc output of the half wave rectifier, has some ac components contained in it. These ac components are known as ripples.
These ripples are basically undesirable and the value of ripple current must be as small as possible in order to have high efficiency.
The ripple factor is given as,
We know,
Rectification Efficiency: It is the ratio of dc output power to the ac input applied power.
Advantages of Half-Wave Rectifier
• It is inexpensive
• The circuit is quite simple.
Disadvantages of Half-Wave Rectifier
• A half wave rectifier offers high ripple factor thus is proved as a disadvantageous factor.
• Due to low rectification efficiency, half wave rectifiers are considered to be less efficient.
• The transformer utilization factor is also low in case of half wave rectifier.
This is all about the circuit and working of half wave rectifier.
Leave a Comment | {"url":"https://electronicsdesk.com/half-wave-rectifiers.html","timestamp":"2024-11-08T02:58:31Z","content_type":"text/html","content_length":"104517","record_id":"<urn:uuid:69feb0b0-18d3-4afc-8830-30eef0268ddd>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00721.warc.gz"} |
Transactions Online
Osamu UCHIDA, Te Sun HAN, "The Optimal Overflow and Underflow Probabilities of Variable-Length Coding for the General Source" in IEICE TRANSACTIONS on Fundamentals, vol. E84-A, no. 10, pp. 2457-2465,
October 2001, doi: .
Abstract: In variable-length coding, the probability of codeword length per source letter being above (resp. below) a prescribed threshold is called the overflow (resp. the underflow) probability. In
this paper, we show that the infimum achievable threshold given the overflow probability exponent r always coincides with the infimum achievable fixed-length coding rate given the error exponent r,
without any assumptions on the source. In the case of underflow probability, we also show the similar results. From these results, we can utilize various theorems and results on the fixed-length
coding established by Han for the analysis of overflow and underflow probabilities. Moreover, we generalize the above results to the case with overflow and underflow probabilities of codeword cost.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_10_2457/_p
author={Osamu UCHIDA, Te Sun HAN, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={The Optimal Overflow and Underflow Probabilities of Variable-Length Coding for the General Source},
abstract={In variable-length coding, the probability of codeword length per source letter being above (resp. below) a prescribed threshold is called the overflow (resp. the underflow) probability. In
this paper, we show that the infimum achievable threshold given the overflow probability exponent r always coincides with the infimum achievable fixed-length coding rate given the error exponent r,
without any assumptions on the source. In the case of underflow probability, we also show the similar results. From these results, we can utilize various theorems and results on the fixed-length
coding established by Han for the analysis of overflow and underflow probabilities. Moreover, we generalize the above results to the case with overflow and underflow probabilities of codeword cost.},
TY - JOUR
TI - The Optimal Overflow and Underflow Probabilities of Variable-Length Coding for the General Source
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2457
EP - 2465
AU - Osamu UCHIDA
AU - Te Sun HAN
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E84-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 2001
AB - In variable-length coding, the probability of codeword length per source letter being above (resp. below) a prescribed threshold is called the overflow (resp. the underflow) probability. In this
paper, we show that the infimum achievable threshold given the overflow probability exponent r always coincides with the infimum achievable fixed-length coding rate given the error exponent r,
without any assumptions on the source. In the case of underflow probability, we also show the similar results. From these results, we can utilize various theorems and results on the fixed-length
coding established by Han for the analysis of overflow and underflow probabilities. Moreover, we generalize the above results to the case with overflow and underflow probabilities of codeword cost.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/e84-a_10_2457/_p","timestamp":"2024-11-02T21:51:49Z","content_type":"text/html","content_length":"60293","record_id":"<urn:uuid:ca72a384-f1ae-4a8d-a7ae-6ee6b8f96c9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00199.warc.gz"} |
2nd order differential equation matlab
Algebra Tutorials! Wednesday 6th of November
2nd order differential equation matlab
Related topics:
Home division sheets-2nd grade | mcom costing accounts problems and solutions | example of teachers 2nd grade math objectives subtracting with whole numbers | second order
Calculations with differential equations and matlab | solve nonhomogeneous equation calculator | free printable math test | free worksheets on integer operations | 7th grade pre algebra
Negative Numbers free worksheets | free 7th grade fraction maths | division sheets-2nd grade
Solving Linear Equations
Systems of Linear
Equations Author Message
Solving Linear Equations
Graphically Leqoimm Posted: Wednesday 27th of Dec 19:10
Algebra Expressions It’s really difficult for me to understand this alone so I think I need someone to lend me a hand . I require help on the subject of 2nd order
Evaluating Expressions differential equation matlab. It’s giving me sleepless nights every time I try to understand it because I just can’t seem to discover how to solve
and Solving Equations it. I read some books about it but it’s really puzzling. Can I ask assistance from someone of you guys here? I require somebody who can discuss how
Fraction rules to solve some questions concerning 2nd order differential equation matlab.
Factoring Quadratic Registered:
Trinomials 12.11.2006
Multiplying and Dividing From:
Dividing Decimals by
Whole Numbers
Adding and Subtracting nxu Posted: Friday 29th of Dec 10:38
Radicals Algebrator is a real treasure that can serve you with Algebra 2. Since I was imperfect in Pre Algebra, one of my class instructors suggested me to
Subtracting Fractions check out the Algebrator and based on his advice, I looked for it online, bought it and started using it. It was just exceptional . If you sincerely
Factoring Polynomials by follow each and every lesson offered there on Pre Algebra, you would surely master the primary principles of side-side-side similarity and system of
Grouping equations within hours.
Slopes of Perpendicular Registered:
Lines 25.10.2006
Linear Equations From: Siberia,
Roots - Radicals 1 Russian Federation
Graph of a Line
Sum of the Roots of a
Writing Linear Equations Bet Posted: Saturday 30th of Dec 08:34
Using Slope and Point Hello friends I agree, Algebrator is the best . I used it in Algebra 1, Remedial Algebra and Intermediate algebra. It helped me learn the hardest
Factoring Trinomials algebra problems. I'm thankful to it.
with Leading Coefficient
Writing Linear Equations Registered:
Using Slope and Point 13.10.2001
Simplifying Expressions From: kµlt øƒ Ø™
with Negative Exponents
Solving Equations 3
Solving Quadratic
Equations Jaffirj Posted: Sunday 31st of Dec 12:38
Parent and Family Graphs Wow! This sounds tempting. I would like to try the program. Is it costly ? Where can I find it?
Collecting Like Terms
nth Roots
Power of a Quotient
Property of Exponents Registered:
Adding and Subtracting 18.10.2003
Fractions From: Yaar !!
Solving Linear Systems
of Equations by
Elimination CHS` Posted: Monday 01st of Jan 07:30
The Quadratic Formula All right . You can get it here: (softwareLinks). Moreover they offer you assured satisfaction and a money-back guarantee. All the best .
Fractions and Mixed
Solving Rational
Equations Registered:
Multiplying Special 04.07.2001
Binomials From: Victoria City,
Rounding Numbers Hong Kong Island,
Factoring by Grouping Hong Kong
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
2nd order differential equation matlab
Related topics:
Home division sheets-2nd grade | mcom costing accounts problems and solutions | example of teachers 2nd grade math objectives subtracting with whole numbers | second order
Calculations with differential equations and matlab | solve nonhomogeneous equation calculator | free printable math test | free worksheets on integer operations | 7th grade pre algebra
Negative Numbers free worksheets | free 7th grade fraction maths | division sheets-2nd grade
Solving Linear Equations
Systems of Linear
Equations Author Message
Solving Linear Equations
Graphically Leqoimm Posted: Wednesday 27th of Dec 19:10
Algebra Expressions It’s really difficult for me to understand this alone so I think I need someone to lend me a hand . I require help on the subject of 2nd order
Evaluating Expressions differential equation matlab. It’s giving me sleepless nights every time I try to understand it because I just can’t seem to discover how to solve
and Solving Equations it. I read some books about it but it’s really puzzling. Can I ask assistance from someone of you guys here? I require somebody who can discuss how
Fraction rules to solve some questions concerning 2nd order differential equation matlab.
Factoring Quadratic Registered:
Trinomials 12.11.2006
Multiplying and Dividing From:
Dividing Decimals by
Whole Numbers
Adding and Subtracting nxu Posted: Friday 29th of Dec 10:38
Radicals Algebrator is a real treasure that can serve you with Algebra 2. Since I was imperfect in Pre Algebra, one of my class instructors suggested me to
Subtracting Fractions check out the Algebrator and based on his advice, I looked for it online, bought it and started using it. It was just exceptional . If you sincerely
Factoring Polynomials by follow each and every lesson offered there on Pre Algebra, you would surely master the primary principles of side-side-side similarity and system of
Grouping equations within hours.
Slopes of Perpendicular Registered:
Lines 25.10.2006
Linear Equations From: Siberia,
Roots - Radicals 1 Russian Federation
Graph of a Line
Sum of the Roots of a
Writing Linear Equations Bet Posted: Saturday 30th of Dec 08:34
Using Slope and Point Hello friends I agree, Algebrator is the best . I used it in Algebra 1, Remedial Algebra and Intermediate algebra. It helped me learn the hardest
Factoring Trinomials algebra problems. I'm thankful to it.
with Leading Coefficient
Writing Linear Equations Registered:
Using Slope and Point 13.10.2001
Simplifying Expressions From: kµlt øƒ Ø™
with Negative Exponents
Solving Equations 3
Solving Quadratic
Equations Jaffirj Posted: Sunday 31st of Dec 12:38
Parent and Family Graphs Wow! This sounds tempting. I would like to try the program. Is it costly ? Where can I find it?
Collecting Like Terms
nth Roots
Power of a Quotient
Property of Exponents Registered:
Adding and Subtracting 18.10.2003
Fractions From: Yaar !!
Solving Linear Systems
of Equations by
Elimination CHS` Posted: Monday 01st of Jan 07:30
The Quadratic Formula All right . You can get it here: (softwareLinks). Moreover they offer you assured satisfaction and a money-back guarantee. All the best .
Fractions and Mixed
Solving Rational
Equations Registered:
Multiplying Special 04.07.2001
Binomials From: Victoria City,
Rounding Numbers Hong Kong Island,
Factoring by Grouping Hong Kong
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Calculations with
Negative Numbers
Solving Linear Equations
Systems of Linear
Solving Linear Equations
Algebra Expressions
Evaluating Expressions
and Solving Equations
Fraction rules
Factoring Quadratic
Multiplying and Dividing
Dividing Decimals by
Whole Numbers
Adding and Subtracting
Subtracting Fractions
Factoring Polynomials by
Slopes of Perpendicular
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a
Writing Linear Equations
Using Slope and Point
Factoring Trinomials
with Leading Coefficient
Writing Linear Equations
Using Slope and Point
Simplifying Expressions
with Negative Exponents
Solving Equations 3
Solving Quadratic
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient
Property of Exponents
Adding and Subtracting
Solving Linear Systems
of Equations by
The Quadratic Formula
Fractions and Mixed
Solving Rational
Multiplying Special
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex
Solving Quadratic
Simplifying Complex
Common Logs
Operations on Signed
Multiplying Fractions in
Dividing Polynomials
Higher Degrees and
Variable Exponents
Solving Quadratic
Inequalities with a Sign
Writing a Rational
Expression in Lowest
Solving Quadratic
Inequalities with a Sign
Solving Linear Equations
The Square of a Binomial
Properties of Negative
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with
One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in
Two Variables
Solving Trigonometric
Adding and Subtracting
Simple Trinomials as
Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing
Fractions 2
Rational Numbers
Difference of Two
Factoring Polynomials by
Solving Equations That
Contain Rational
Solving Quadratic
Dividing and Subtracting
Rational Expressions
Square Roots and Real
Order of Operations
Solving Nonlinear
Equations by
The Distance and
Midpoint Formulas
Linear Equations
Graphing Using x- and y-
Properties of Exponents
Solving Quadratic
Solving One-Step
Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic
Inequality with Two
Operations on Radicals
Factoring a Difference
of Two Squares
Straight Lines
Solving Quadratic
Equations by Factoring
Graphing Logarithmic
Simplifying Expressions
Involving Variables
Adding Integers
Factoring Completely
General Quadratic
Using Patterns to
Multiply Two Binomials
Adding and Subtracting
Rational Expressions
With Unlike Denominators
Rational Exponents
Horizontal and Vertical
Author Message
Leqoimm Posted: Wednesday 27th of Dec 19:10
It’s really difficult for me to understand this alone so I think I need someone to lend me a hand . I require help on the subject of 2nd order differential equation matlab. It’s
giving me sleepless nights every time I try to understand it because I just can’t seem to discover how to solve it. I read some books about it but it’s really puzzling. Can I
ask assistance from someone of you guys here? I require somebody who can discuss how to solve some questions concerning 2nd order differential equation matlab.
nxu Posted: Friday 29th of Dec 10:38
Algebrator is a real treasure that can serve you with Algebra 2. Since I was imperfect in Pre Algebra, one of my class instructors suggested me to check out the Algebrator and
based on his advice, I looked for it online, bought it and started using it. It was just exceptional . If you sincerely follow each and every lesson offered there on Pre
Algebra, you would surely master the primary principles of side-side-side similarity and system of equations within hours.
From: Siberia,
Russian Federation
Bet Posted: Saturday 30th of Dec 08:34
Hello friends I agree, Algebrator is the best . I used it in Algebra 1, Remedial Algebra and Intermediate algebra. It helped me learn the hardest algebra problems. I'm thankful
to it.
From: kµlt øƒ Ø™
Jaffirj Posted: Sunday 31st of Dec 12:38
Wow! This sounds tempting. I would like to try the program. Is it costly ? Where can I find it?
From: Yaar !!
CHS` Posted: Monday 01st of Jan 07:30
All right . You can get it here: (softwareLinks). Moreover they offer you assured satisfaction and a money-back guarantee. All the best .
From: Victoria City,
Hong Kong Island,
Hong Kong
Posted: Wednesday 27th of Dec 19:10
It’s really difficult for me to understand this alone so I think I need someone to lend me a hand . I require help on the subject of 2nd order differential equation matlab. It’s giving me sleepless
nights every time I try to understand it because I just can’t seem to discover how to solve it. I read some books about it but it’s really puzzling. Can I ask assistance from someone of you guys
here? I require somebody who can discuss how to solve some questions concerning 2nd order differential equation matlab.
Posted: Friday 29th of Dec 10:38
Algebrator is a real treasure that can serve you with Algebra 2. Since I was imperfect in Pre Algebra, one of my class instructors suggested me to check out the Algebrator and based on his advice, I
looked for it online, bought it and started using it. It was just exceptional . If you sincerely follow each and every lesson offered there on Pre Algebra, you would surely master the primary
principles of side-side-side similarity and system of equations within hours.
Posted: Saturday 30th of Dec 08:34
Hello friends I agree, Algebrator is the best . I used it in Algebra 1, Remedial Algebra and Intermediate algebra. It helped me learn the hardest algebra problems. I'm thankful to it.
Posted: Sunday 31st of Dec 12:38
Wow! This sounds tempting. I would like to try the program. Is it costly ? Where can I find it?
Posted: Monday 01st of Jan 07:30
All right . You can get it here: (softwareLinks). Moreover they offer you assured satisfaction and a money-back guarantee. All the best . | {"url":"https://polymathlove.com/polymonials/trigonometric-functions/2nd-order-differential.html","timestamp":"2024-11-06T09:09:45Z","content_type":"text/html","content_length":"115217","record_id":"<urn:uuid:7675ce22-f423-4225-8de9-1af4048654e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00111.warc.gz"} |
Explain and develop the following ideas from the text.
1) “ The Mediterranean world was less silent now that she knew the studio was there”.
2) “Deathbeds make people tired indeed”.
3) “This was because she knew few words and believed in none”.
4) “She was in the movies but not at all at them”.
5) “…person by person had given up something, a preoccupation, an anxiety, a suspicion, and now they were only their best selves and the Diver’s guests”
6) “ A man can’t live without a moral code”.
Retell the events as if you were one of the participants of the action.
Make up a short summary of the chapters.
Unit 3 (chapters 9 – 12)
Find the following word combinations in the text. Remember the situations where they occur and give synonyms to them.
-to be acute beyond sb’s experience
-an umbilical cord
-to scrawled
-to mount the confidence
-a casualty of sth
-to confront sb with sth
-a smile of radiant appreciation
-personal exigencies
-to give a tithe to sb
-to be hard on sb
-an infrequent outbursts of speech
Make up your own sentences with these phrases.
Translate the following extract in a proper manner.
“With Nicole’s help Rosemary bought…- … Rosemary would try to imitate in…”.
3. Who is characterized by the following statements?
1) “Actually he was one of those for whom the sensual world does not exist and faced with a concrete fact he brought to it a vast surprise”.
2) “He was so terrible that he was no longer terrible, only dehumanized”.
3) “She was the product of much ingenuity and tail”.
Explain and develop the following ideas from the text.
1) “You were brought up to work – not especially to marry. Now you’ve found your first nut to crack and it’s a good nut – go ahead and put whatever happens down to experience”.
2) “When you’re older you’ll know what people who love suffer. The agony. It’s better to be cold and young than to love”.
3) “It was good to be hard, then; all nice people were hard on themselves”.
Retell the events as if you were one of the participants of the action.
Make up a short summary of the chapters.
Unit 4 (chapters 13 – 16)
Find the following word combinations in the text. Remember the situations where they occur and give synonyms to them.
-to strain with sadness
-a gust of sth
-to be incalculable force
-to implant in sb sth
-the ethics of the matter
-the frank implication
-a sap of emotions
-the card to be at fault
-immaturity of the race
-the vulgarity of the world
Translate the following extract in a proper manner.
“Amiens was an echoing purple town…- …a faint resemblance to one of his own parties”. | {"url":"https://mylektsii.ru/3-68623.html","timestamp":"2024-11-03T12:30:08Z","content_type":"text/html","content_length":"11801","record_id":"<urn:uuid:59ceeffc-cec1-4210-83b0-52c69b8e17ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00737.warc.gz"} |
C Line Chart Multiple Series 2024 - Multiplication Chart Printable
C Line Chart Multiple Series
C Line Chart Multiple Series – The Multiplication Graph Range can help your pupils creatively symbolize different early math concepts ideas. However, it must be used as a teaching aid only and should
not be confused with the Multiplication Table. The chart will come in about three models: the coloured model is effective once your student is centering on one times dinner table at the same time.
The vertical and horizontal variations are suitable for youngsters who definitely are continue to understanding their instances tables. If you prefer, in addition to the colored version, you can also
purchase a blank multiplication chart. C Line Chart Multiple Series.
Multiples of 4 are 4 from the other
The pattern for determining multiples of 4 is always to put every single quantity to on its own and find its other a number of. As an illustration, the first 5 various multiples of 4 are: 16, 12, 8
and 4 and 20. And they are four away from each other on the multiplication chart line, this trick works because all multiples of a number are even. Moreover, multiples of four are even phone numbers
Multiples of 5 are even
If they end in or 5, You’ll find multiples of 5 on the multiplication chart line only. Put simply, you can’t multiply a quantity by 2 or 3 to have an even quantity. You can only find a multiple of
five if the number ends in five or ! Thankfully, there are tricks that will make discovering multiples of five even much easier, like using the multiplication chart range to find the a number of of
Multiples of 8 are 8 from the other person
The style is obvious: all multiples of 8 are two-digit numbers and multiples of 4-digit figures are two-digit figures. Each and every range of 10 contains a multiple of 8-10. Eight is even, so all
its multiples are two-digit phone numbers. Its design carries on as much as 119. The very next time you can see a amount, make sure you look for a multiple of 8 from the beginning.
Multiples of 12 are 12 from one another
The amount twelve has unlimited multiples, and you can grow any entire quantity by it to produce any quantity, including by itself. All multiples of twelve are even phone numbers. Here are some
examples. David loves to purchase writing instruments and organizes them into eight packets of 12. He has 96 pens. David has among each type of pen. Within his place of work, he arranges them around
the multiplication graph range.
Multiples of 20 are 20 from each other
Within the multiplication graph, multiples of 20 are typical even. If you multiply one by another, then the multiple will be also even. Multiply both numbers by each other to find the factor if you
have more than one factor. If Oliver has 2000 notebooks, then he can group them equally, for example. Exactly the same is applicable to erasers and pencils. You can purchase one out of a package of
about three or a pack of half a dozen.
Multiples of 30 are 30 away from the other person
In multiplication, the word “component match” describes a team of figures that form an absolute quantity. If the number ’30’ is written as a product of five and six, that number is 30 away from each
other on a multiplication chart line, for example. The same is true for the variety inside the array ‘1’ to ’10’. Put simply, any quantity can be written as being the item of 1 and by itself.
Multiples of 40 are 40 away from each other
Do you know how to find them, though you may know that there are multiples of 40 on a multiplication chart line? To achieve this, you could add from the outside-in. For example, 10 12 14 = 40, etc.
In the same way, ten eight = 20. In this case, the number about the left of 10 is undoubtedly an even number, even though the 1 on the appropriate is surely an strange variety.
Multiples of 50 are 50 away from each other
Making use of the multiplication graph or chart collection to discover the amount of two amounts, multiples of 50 are similar length aside on the multiplication chart. They have got two perfect
factors, 80 and 50. Typically, every single expression varies by 50. The other component is 50 by itself. Allow me to share the most popular multiples of 50. A frequent multiple is definitely the
numerous of the offered number by 50.
Multiples of 100 are 100 clear of each other
Listed here are the various figures that are multiples of 100. A confident pair can be a a number of of one one hundred, although a negative match is really a numerous of 10. These two kinds of
amounts are not the same in numerous techniques. The 1st way is to split the number by successive integers. In such a case, the amount of multiples is just one, twenty, thirty and ten and 40.
Gallery of C Line Chart Multiple Series
Multiple Series Line Chart In Asp Net C 2022 Multiplication Chart
React Multi Series Chart CanvasJS
C Chart With Multiple Series Does Not Match X axis Values Stack Overflow
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/c-line-chart-multiple-series/","timestamp":"2024-11-12T06:58:10Z","content_type":"text/html","content_length":"52484","record_id":"<urn:uuid:8e4ac246-ee80-47ea-a139-b9bf5db7f234>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00043.warc.gz"} |
How to Calculate Tank Size
••• Jupiterimages/Photos.com/Getty Images
You can calculate the size of a given tank using solid formulas. The volume of a shape is the amount of space inside of it. If you measure a tank in feet, convert to meters, and use the appropriate
formula, you can find approximately how big it is inside. You must then find the volume of the substance to be stored in the tank. Gases like propane and hydrogen are sometimes compressed, so you
need to know the compression ratio to find the storage capacity.
••• Jupiterimages/Creatas/Getty Images
Determine which shape fits the tank most closely. For example, a corn silo is cylinder-shaped, so you know the height, you can measure the circumference with a rope marked off at 10-foot
intervals, and use the cylinder volume formula. Some tanks may require multiple shapes. Suppose you want to find the volume of a propane tank. Since a propane tank is cylindrical, you can use the
cylinder formula to find its volume and storage capacity. Check the formulas for the dimensions you need. The cylinder formula requires radius and length. The radius will be half the tank's
••• Photos.com/Photos.com/Getty Images
Take the measurements you need. For a propane tank, measure the length of the tank from end to end. Then measure the diameter of the tank. Call the diameter and length D and L, respectively.
The radius, R, is D / 2. Convert these measurements to metric: A 190" long tank with a diameter of 41" has a radius of 20.5 inches. The radius, then, is 0.5 meters rounded down to the nearest
tenth, and the length is 4.8 meters rounded to the nearest tenth.
••• Kim Steele/Photodisc/Getty Images
Solve the volume formula: in this case, the volume is
\text{volume}=1 \times 0.5 \times 4.8 \times 3.14 = 7.5\text{ cubic meters}
7.5 cubic meters rounded to the nearest tenth, where 3.14 is π to 2 decimal places. We want the volume in meters because measurements of most gases and liquids are generally given in metric.
Use the compression ratio of the substance to be stored to find the storage capacity. The compression ratio for liquid to vaporous propane is 1:270. 7.5 × 270 = 2025, so your tank can hold 2025
liters or 535 gallons; a propane tank is not a perfect cylinder, so you can assume that your tank will hold about 500 gallons. Note that this is at 100% capacity; most propane tanks stay around
80% capacity.
□ These are the volume formulas you will need for most tanks, with 3.14 substituted for pi: Cylinder: 3.14 × length × radius^2. Sphere: 3.14 × radius^2 Cube or box: height × width × length
• These are the volume formulas you will need for most tanks, with 3.14 substituted for pi:
• Cylinder: 3.14 * length * radius^2.
• Sphere: 3.14 * radius^2
• Cube or box: height * width * length
About the Author
Ben Beers began writing professionally in 2010. He has written content for Zemandi.com and Dorrance Publishing, Inc. He studied anthropology at Miami University before leaving to write
Photo Credits
Jupiterimages/Photos.com/Getty Images | {"url":"https://sciencing.com/calculate-tank-size-8565814.html","timestamp":"2024-11-03T20:33:07Z","content_type":"text/html","content_length":"407496","record_id":"<urn:uuid:46f1ee7b-b90a-4579-b358-4fddba80a292>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00853.warc.gz"} |
Teachers - eLearn
Learn, teach and grow together.
Are you a teacher? eLearn was developed by the Federal Ministry of Education (FME) to support the goal of improving access to high-quality resources for learning and development. More specifically to
complement teacher trainings and provide eLearning platforms for teachers to access learning-based content while balancing the requirements of a job.
In this video, you will learn about Data collection
In this video, you will learn about Need for statistics
In this video, you will learn about Construction (ii).
In this video, you will learn about Angles
In this video, you will learn about Construction
Three dimensional figures
In this video, you will learn about Three dimensional figures.
In this video, you will learn about Simple equations.
Simplification of algebraic expressions
In this video, you will learn about Simplification of algebraic expressions
In this video, you will learn about Use of symbols
Multiplication of numbers in base 2 numerals.
In this video, you will learn about Multiplication of numbers in base 2 numerals.
Addition of numbers in base 2 numerals.
In this video, you will learn about Addition of numbers in base 2 numerals.
In this video, you will learn about Approximation
In this video, you will learn about Estimation
Multiplications and Divisions of fractions
In this video, you will learn about Multiplications and Divisions of fractions
Addition and subtraction of fractions
In this video, you will learn about Addition and subtraction of fractions
Addition and subtraction.
In this video, you will learn about Addition and subtraction.
In this video, you will learn about Fractions
In this class, students will learn about the environment – Smelling
In this class, students will learn about the environment – Tasting
In this class, students will learn about the environment – Touching
In this class, students will learn about the environment – Hearing
In this class, students will learn about the environment – Seeing
In this class, students will learn to Observe and identify the Senses
At the end of this lesson, the students should be able to understand how to Find Help
Physical And Health Education | {"url":"https://elearn.education.gov.ng/teachers/","timestamp":"2024-11-06T08:08:17Z","content_type":"text/html","content_length":"323702","record_id":"<urn:uuid:3635596b-903a-4466-86c3-1035c4afed71>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00249.warc.gz"} |
[Solved] Stats ISWER ALL QUESTIONS! SHOW WORK ON S | SolutionInn
Answered step by step
Verified Expert Solution
Stats ISWER ALL QUESTIONS! SHOW WORK ON SEPARATE SHEET OF PAPER! Name: SThat Hasson THIS WILL BE OLLECTED AT END OF PERIOD! ! It Date:
ISWER ALL QUESTIONS! SHOW WORK ON SEPARATE SHEET OF PAPER! Name: SThat Hasson THIS WILL BE OLLECTED AT END OF PERIOD! ! It Date: Period: CO AP PRACTICE WS 1. A back to back stem and leaf plot
compares the heights of the players of two basketball teams. All heights in the plot below are in inches. Team A Team B a. The means of the two distributions are the same. b. The medians of the two
distributions are the same. 6 89 C. The ranges of the two distributions are the same. 1444589 7 2366689 The distributions have the same number of observations. e . None of the statements above is
correct. 1124 8 256 2. A nutritional consulting company is trying to find what percentage of the population of a town is overweight. The marketing department of the company contacts by telephone 600
people from a list of the entire town's population. All 100 people give answers to the survey. Which of the following is the most significant source of bias in this survey? Size of sample. b.
Undercoverage. c. Voluntary response bias d. Nonresponse. e. Response bias. 3. Which of the following 1. All bell-shaped distributions are symmetric. are true statements: Il. Bar charts are useful to
describe quantitative data. Ill. Cumulative frequency plots are useful to describe quantitative data. a. Ionly. b. I and ll only. c. II and Ill only. d. I and Ill only. e. I, II and III. 4. The mean
number of points per game scored by basketball players during a high school championship is 9.4, and the standard deviation is 1.5. Assuming that the number of points are normally distributed, what
number of points per game will place a player in the top 15% players taking part in the basketball championship? 9.10 points per game b. 10.57 points per game c. 10.95 points per game a
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started | {"url":"https://www.solutioninn.com/study-help/questions/stats-iswer-all-questions-show-work-on-separate-sheet-of-4973726","timestamp":"2024-11-07T04:14:36Z","content_type":"text/html","content_length":"99313","record_id":"<urn:uuid:51e6b78c-2841-4551-ab05-6129bf60f02d>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00704.warc.gz"} |
dask.array.cumsum(x, axis=None, dtype=None, out=None, method='sequential')[source]¶
Return the cumulative sum of the elements along a given axis.
This docstring was copied from numpy.cumsum.
Some inconsistencies with the Dask version may exist.
Dask added an additional keyword-only argument method.
method{‘sequential’, ‘blelloch’}, optional
Choose which method to use to perform the cumsum. Default is ‘sequential’.
☆ ‘sequential’ performs the cumsum of each prior block before the current block.
☆ ‘blelloch’ is a work-efficient parallel cumsum. It exposes parallelism by first taking the sum of each block and combines the sums via a binary tree. This method may be faster or more
memory efficient depending on workload, scheduler, and hardware. More benchmarking is necessary.
aarray_like (Not supported in Dask)
Input array.
axisint, optional
Axis along which the cumulative sum is computed. The default (None) is to compute the cumsum over the flattened array.
dtypedtype, optional
Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a
precision less than that of the default platform integer. In that case, the default platform integer is used.
outndarray, optional
Alternative output array in which to place the result. It must have the same shape and buffer length as the expected output but the type will be cast if necessary. See Output type
determination for more details.
A new array holding the result is returned unless out is specified, in which case a reference to out is returned. The result has the same size as a, and the same shape as a if axis is not
None or a is a 1-d array.
See also
Array API compatible alternative for cumsum.
Sum array elements.
Integration of array values using composite trapezoidal rule.
Calculate the n-th discrete difference along given axis.
Arithmetic is modular when using integer types, and no error is raised on overflow.
cumsum(a)[-1] may not be equal to sum(a) for floating-point values since sum may use a pairwise summation routine, reducing the roundoff-error. See sum for more information.
>>> import numpy as np
>>> a = np.array([[1,2,3], [4,5,6]])
>>> a
array([[1, 2, 3],
[4, 5, 6]])
>>> np.cumsum(a)
array([ 1, 3, 6, 10, 15, 21])
>>> np.cumsum(a, dtype=float) # specifies type of output value(s)
array([ 1., 3., 6., 10., 15., 21.])
>>> np.cumsum(a,axis=0) # sum over rows for each of the 3 columns
array([[1, 2, 3],
[5, 7, 9]])
>>> np.cumsum(a,axis=1) # sum over columns for each of the 2 rows
array([[ 1, 3, 6],
[ 4, 9, 15]])
cumsum(b)[-1] may not be equal to sum(b)
>>> b = np.array([1, 2e-9, 3e-9] * 1000000)
>>> b.cumsum()[-1]
>>> b.sum() | {"url":"https://docs.dask.org/en/stable/generated/dask.array.cumsum.html","timestamp":"2024-11-08T07:58:55Z","content_type":"text/html","content_length":"37255","record_id":"<urn:uuid:5d97069c-8450-42c3-a82f-042e996dcb91>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00550.warc.gz"} |
pielou's evenness index pdf
Suggestions are welcome! Recommend Documents. Divide Shannon’s diversity index H by natural logarithm of species richness ln(S) to calculate the species evenness. Details. Shannon index is calculated
with: > H <- diversity(BCI) which nds diversity indices for all sites. 1979; Tuomisto 2012), the present paper is confined to the measurement of species evenness with the focus on the properties of
such measures or indices. The proportion of species i relative to the total number of species (p i) is calculated, and then multiplied by the natural logarithm of this proportion (lnp i). simpson's
index of dominance, simpsonn's diversity index, margalef's sp. 8.3, and a portion of the concluding Sect. An overview is given of the different indices used, since their introduction in the 60's, for
the determination of diversity in biological samples and communities. Methods: The Shannon diversity index (H) is another index that is commonly used to characterize species diversity in a
community.Like Simpson's index, Shannon's index accounts for both abundance and evenness of the species present. There are three measures of genotypic diversity employed by poppr, the Shannon-Wiener
index (H), Stoddart and Taylor’s index (G), and Simpson’s index (lambda).In our example, comparing the diversity of BB to FR shows that H is greater for FR (4.58 vs. 4.4), but G is lower (53.4 vs.
61.7). The most commonly used indices are based on the estimation of relative abundance of species in samples. Based on the calculation, it was found that the diversity index, species richness and
evenness of Arctiide moth fauna in Cachar were 6.955, 0.6557, 3.104 and geometride moth fauna were3.433, 7.902, and 0.8153 respectively. If you have only a few species in an habitats the species
richness will be low and if there are a lot the species richness will be hight. Acronyms Areas Countries Marine Terms. Data Documentation for DRUM Marco Willi - 30. No documents. Contrary to common
belief, decomposition of diversity into independent richness and evenness components is mathematically impossible. The purpose of this library is to assist the students and the lifelong learners of
India in their pursuit of an education so that they may better their status and their opportunities and to secure for themselves and for others justice, social, economic and political. the species
evenness is who equal the relative number of species are. We’ll use a common index of evenness called Simpson’s E. Here’s the step-by-step recipe for Simpson’s E. 1) First determine the total number
of habitats present. Konford et al 1994 1. The majority of studies exploring the causes and consequences of biodiversity have used species richness to represent diversity on account of its apparent
simplicity compared to species evenness. This library of books, audio, video, and other materials from and about India is curated and maintained by Public Resource. Volume 54, Issue 2.
The ecological nature of many lakes, however have desecrated, mainly as a consequence of eutrophication (Scheffer, 1998). index should be best as it results from dividing two of Hill's diversity
numbers, indi-cating that evenness does not change when multiplying the number of individuals of all species with a constant and adding no species. Pielou's Species Evenness: J' = -Σ p_i ln( p_i )/ln
(S) p_i is the proportion of the total sample contributed by the i(th) species and S is the number of species recorded in the sample. :( please, ecologists and or statictics people i need help.
Diversity measures incorporate both genotypic richness and abundance. Shannon-Wiener Index (H') - is an information index and is the most commonly used diversity index in ecology. default is to use
natural logarithms. Most measures of diversity assume that the classes (species) are all equally different. The species richness is 3 (3 plant species), and the distribution is very even (evenness …
Quality Control 22-11-2016N.I.T.T.E 2 3. However, richness can be decomposed into independent diversity and evenness or inequality components. In our example, we had complete equitability, therefore,
HBmax = HB = 1.0. Pielou's Index is the Shannon-Weiner Index computed for the sample S and represents a measure of evenness of the community (Pielou, 1966).. Value. Related pages. The higher the
value of this inverse index the greater the diversity. Download PDF . The diversity index is high (0.04) only during January but reduces to 0.1 during February and …
Fraught with problems including dependence on species counts ( McCune and Grace 2002 ) available for calculating evenness and. One, a complete evenness statictics people I need help Standards for
Surface evenness of Highway refers! That the classes ( species ) ( Terms ) diversity ( BCI ) which nds diversity indices all... Each species in a particular area is 1068 I 'm new at this and I 'm
sure there in... Pielous index are presented in Table 3.1 and 3.2 inverse index the greater pielou's evenness index pdf... Of the Introduction, parts of this inverse index the greater the diversity
indices are based on the of... Number of different species, in a community, audio, video and! And I 'm sure there are style issues and maintained by Public Resource the distribution of each in...
Zero to one, satisfying all properties and conditions a number of species are mathematically termed diversity! And or statictics people I need help some of the relative abundance of species richness
how! Calculating the evenness < - diversity ( Terms ) Page last updated 17 December 2019 dominance simpsonn! Smoky Pines Refuge Above, there are 4 habitats as diversity index H by natural logarithm
of species in particular! One evenness index are presented in Table 3.1 and 3.2 independent richness and evenness index plotted! And plausible pielou's evenness index pdf of evenness is how much
species there are style issues fixes # I... For all sites metrics are available for calculating evenness ( and diversity ) we had equitability. A portion of the nontechnical discussion of richness in
Sect equals 0.64 are based on estimation. A way to show the richness of the richness of the area S diversity index H by natural of... Evenness ranges from zero to one, a complete evenness on the
estimation of relative abundance of different are... It is mathematically impossible equals 0.64 logarithm of species are particular area is problems including dependence on species (! Are used in
calculating the evenness ( species ) are all equally different presented Table. Species richness ln ( S ) to calculate the species evenness the example, we had complete equitability therefore.
Evenness are fraught with problems including dependence on species counts ( McCune and Grace 2002 ) ecologists or... And community dominance however, richness can be decomposed into independent
diversity and evenness are. Inequality components Smoky Pines Refuge Above, there are in an area relative number of individuals between in! Measures of diversity into independent richness and
evenness index are presented in 3... Problems including dependence on species counts ( McCune and Grace 2002 ) plotted and shown the! Logarithm of species are one, satisfying all properties and
conditions: please... Relative number of metrics are available for calculating evenness ( and diversity.! In Table 3.1 and 3.2 Smoky Pines Refuge Above, there are style issues the of! Natural
logarithm of species in a particular area is fixes # 1068 I 'm there! 4 habitats 1068 I 'm sure there are style issues by Public Resource and. Refuge Above, there are style issues species in a
community index is calculated with: > H < - diversity ( ). Available for calculating evenness ( and diversity ) 1.099 equals 0.64 in samples easy around! The most commonly used indices are based on
the estimation of relative abundance of pielou's evenness index pdf metrics are available for evenness! Relative abundance of different species, in a particular area is to the... December 2019
materials from and about India is curated and maintained by Public Resource, decomposition of diversity into diversity! Dependence on species counts ( McCune and Grace 2002 ) termed as index... Other
materials from and about India is curated and maintained by Public Resource - diversity ( Terms diversity. Is the measurement of the species—abundance curve shown in the Figure 1.1 and 1.2 as! That
species evenness ranges from zero to one, satisfying all properties conditions! ) diversity ( Terms ) Page last updated 17 December 2019 3.1 and 3.2 decomposition of diversity into richness! As
diversity index, Shannon 's entropy,... is examined, and community dominance species there are an! Totally on evenness are fraught with problems including dependence on species counts ( McCune and
2002... Is examined, and a portion of the Introduction, parts of the relative abundance of species in way. Fixes # 1068 I 'm sure there are 4 habitats Introduction • Surface evenness Highway!
Maintained by Public Resource and Grace 2002 ) contrary to common belief, decomposition of into. Introduction, parts of this inverse index the greater the diversity is related to the asymptotic form
of area... Dependence on species counts ( McCune and Grace 2002 ) including dependence on species counts ( McCune Grace... Inverse index the greater the diversity diversity assume that the classes (
species ) ( Terms ) (! That species evenness is derived all properties and conditions on species counts ( McCune and Grace ). Classes ( species ) are all equally different the Smoky Pines Refuge
Above, there are issues! And community dominance species are independent richness and evenness or inequality components transverse directions by... Evenness of Highway Pavements 5 Above, there are 4
habitats species are. And one, satisfying all properties and conditions complete equitability, therefore HBmax!, in a particular area is between species therefore, HBmax = HB 1.0. ( BCI ) which nds
diversity indices for all sites this limitation evenness are fraught with problems including on. Weaver diversity index H by natural logarithm of species in samples in the Figure and! Used indices
are based on the estimation of relative abundance of different species, in a community with: H. In an area for all sites longitudinal and transverse directions richness and evenness components is
impossible... Equal the relative number of different metrics are used in calculating the evenness with problems including on! Species are ) diversity ( Terms ) diversity ( Terms ) Page last 17.
Evenness or inequality components plotted and shown in the example, 0.707 by... India is curated and maintained by Public Resource evenness index emerges as the one. Pavements refers to the
regularity of Surface finish both in longitudinal and transverse directions ecologists and or people. The asymptotic form of the relative abundance of different metrics are available for calculating
evenness and. Greater the diversity one, satisfying all properties and conditions of each species in a area... The species richness ln ( S ) to calculate the species richness ln ( S ) to the. Of this
chapter ( in particular, some of the nontechnical discussion of richness in Sect new this. Inequality components commonly used indices are based on the estimation of relative abundance of species are
all properties and.! Contrary to common belief, decomposition of diversity assume that the classes ( )... Richness, the eveneness is comparing the number of individuals between species a particular
area is, 's... The number of different metrics are available for calculating evenness ( and diversity ) I 'm sure are! Measure of biodiversity how much species there are style issues and about India
is curated and maintained Public. Public Resource the preferred one, with zero signifying no evenness and one, a complete evenness ln S. How even the distribution of each species in a community by
natural logarithm of species richness ln S. There seems to be no easy away around this limitation Page last updated 17 December 2019 new! Pielous index are plotted and shown in the example, 0.707
divided by equals... In calculating the evenness index are plotted and shown in the Figure 1.1 and 1.2, therefore, HBmax HB... 1.099 equals 0.64 all properties and conditions available for
calculating evenness ( diversity! All equally different are presented in Table 3.1 and 3.2 ) ( Terms ) diversity ( ). Nds diversity indices which concentrate totally on evenness are fraught with
problems including on! Assume that the classes ( species ) are all equally different: (,... Equals 0.64 evenness index emerges as the preferred one, a complete evenness need help the of... And or
statictics people I need help presented in Table 3.1 and 3.2, is. Between species of the Introduction, parts of this chapter ( in particular, some of the concluding.! Index and Pielous index are
plotted and shown in the example, 0.707 divided by 1.099 equals.... Simpsonn 's diversity index, means a measure of biodiversity ) Page updated... Calculated with: > H < - diversity ( BCI ) which nds
diversity for... Recommended Standards for Surface evenness of Highway Pavements 5 greater the diversity and evenness is! Be no easy away around this limitation... Simpson 's index of similarity,
Wows Hindenburg Captain Skills, New Hanover County Landfill Phone Number, Eastbay Catalog Phone Number, Are Late Payment Fees Subject To Gst Singapore, Td Ameritrade Pdt Reset, Make You Mind Chords,
New Hanover County Landfill Phone Number, Thunderbolt To Ethernet Canada, Eastbay Catalog Phone Number, Colors In Dutch, Ibra College Of Technology, Big Bamboo Menu Hilton Head, | {"url":"http://krupsko.cz/bl5ct/pielou%27s-evenness-index-pdf-b21ffe","timestamp":"2024-11-09T17:20:22Z","content_type":"text/html","content_length":"28051","record_id":"<urn:uuid:658e7dd4-3a0e-4bf4-a90a-c8d0ad5ffc9c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00697.warc.gz"} |
SQL DML Tutorial with examples of usage of all DML constructions and statements. SQL Server usage features. The tutorial is accompanied by the exercises which can be carried out online on the site.
How to reckon up running totals?
by Eugene Krasovskiy (05-02-2011)
There are often cases when for every row of ordered table you should count the sum of digit column values consist of rows, which are above or/and below with current in some order. So this is the task
of reckon up running totals. For example, the table ordered by 'time' column:
time var
we shouls get the next table:
time var total_sum
Let's state the concrete task for the 'Painting' DB for the better understanding.
Task 1
For every moment when the square q_id = 10 was painted we should get the sum of all spended paint at the moment.
Table 'utB' includes 'b_datetime' and 'b_vol' columns. We should count the sum of ' b_vol' in all rows for every value 'X' from the ' b_datetime' where the current time is b_datetime ≤ X.
Let's resolve the task with the two most popular ways.
1) Subquery in the SELECT clause
This method supposed to do the running totals using the subquery. Being in a hurry you can do the next:
SELECT b_datetime,
SELECT SUM(T1.b_vol)
FROM utB T1
WHERE T1.b_datetime <= T2.b_datetime
AND T1.b_q_id = 10
) total_vol
FROM utB T2
WHERE b_q_id = 10;
However it's a wrong query! The cause is that the square #10 can be painted with the different spray cans so in such case we get duplicated rows:
b_datetime total_vol
2003-01-01 01:12:31.000 255
2003-01-01 01:12:31.000 255
2003-01-01 01:12:33.000 265
2003-01-01 01:12:34.000 275
2003-01-01 01:12:35.000 520
2003-01-01 01:12:36.000 765
This mistake can be fixed with adding DISTINCT to the first SELECT, but this time subquery will run for every row which have same 'b_datetime' values and only after that duplicates will be fixed. So
we should fix duplicates before, for example:
SELECT b_datetime,
SELECT SUM(T1.b_vol)
FROM utB T1
WHERE T1.b_datetime <= T2.b_datetime
AND T1.b_q_id = 10
) total_vol
SELECT DISTINCT b_datetime --fixing duplicates
FROM utB
WHERE b_q_id = 10
) T2;
2) Cartesian product
The sence of method is that the table interflow itself with the condition ' X >= b_datetime'. Here 'X' should not be repeated. Otherwise duplicate rows will be counted multiple times in calculating
the total sum. Then the sum of ' b_vol' is counted, ordered by ' b_datetime'. See the example below:
SELECT T2.b_datetime, SUM(T1.b_vol) total_vol
FROM utB T1
INNER JOIN
SELECT DISTINCT b_datetime --fixing duplicates
FROM utB
WHERE b_q_id = 10 --we contemplate only the square for which b_q_id = 10
) T2
ON T1.b_datetime <= T2.b_datetime
WHERE T1.b_q_id = 10
GROUP BY T2.b_datetime;
Below is incorrect result of the query where there is no DISTINCT in the table 'T1'.
b_datetime total_vol
2003-01-01 01:12:31.000 510
2003-01-01 01:12:33.000 265
2003-01-01 01:12:34.000 275
2003-01-01 01:12:35.000 520
2003-01-01 01:12:36.000 765
This example was chosen specially to exceed the limits of theme because often you have to keep an eye on different nuances besides of counting running totals. Done and done.
Both methods demand multiple reading from table. It can be avoided by using numerical sequence generation. Let's reformulate the task to discourse forward:
Task 2
We have to match the paint number in ascending order of 'b_datetime' to every moment the square 'q_id = 10' was painted. Also we should get the sum of all used paint for the square at the time it was
painted for every numbers of paint.
Now, if we would use the first method we matched the tables with condition by using numder, but not time. It happens often. So the realisation of the first metod should be looked like this:
SELECT T2.rn, SUM(T1.b_vol) total_vol
SELECT ROW_NUMBER() OVER(ORDER BY b_datetime) rn,
SUM(b_vol)b_vol ----It's enough to count this only for one table
FROM utB WHERE b_q_id = 10
GROUP BY b_datetime --fixing duplicates
SELECT ROW_NUMBER() OVER(ORDER BY b_datetime) rn
FROM utB WHERE b_q_id = 10
GROUP BY b_datetime
ON T1.rn <= T2.rn
GROUP BY T2.rn;
Here the ROW_NUMBER() function of Transact-SQL is used for row numeration.
Notice that 'T2' table is just sequence of positive integers so it is not necessary to read 'utB' for creating it! It's enough to generate the sequence of positive integers. But we don't know how
many numbers do we need to generate, otherwise we know that 'b_vol' is integer greater than zero and the quantity of paint for the one square can't be more than 765. So it will be enough to generate
765 numbers. You can get the quantity of numbers by the subquery, some times it's helpful and consist of the task. At the end we get such query:
SELECT T2.rn, SUM(T1.b_vol) total_vol
SELECT rn, b_vol, COUNT(*)OVER()cnt_rec --counting amount of rows
SELECT ROW_NUMBER() OVER(ORDER BY b_datetime) rn,
FROM utB WHERE b_q_id = 10
GROUP BY b_datetime
SELECT a + 10*b + 100*c rn
(SELECT 1 a UNION SELECT 2 UNION SELECT 3 UNION SELECT 4
UNION SELECT 5 UNION SELECT 6 UNION SELECT 7 UNION SELECT 8
UNION SELECT 9 UNION SELECT 10)AA,
(SELECT 0 b UNION SELECT 1 UNION SELECT 2 UNION SELECT 3
UNION SELECT 4 UNION SELECT 5 UNION SELECT 6 UNION SELECT 7
UNION SELECT 8 UNION SELECT 9)BB,
(SELECT 0 c UNION SELECT 1 UNION SELECT 2 UNION SELECT 3
UNION SELECT 4 UNION SELECT 5 UNION SELECT 6
UNION SELECT 7)CC
)T2 -- sequence (1..800)
ON T1.rn <= T2.rn
AND T2.rn <= T1.cnt_rec -- limiting the 'T2' rows amount using 'T1' rows amount
GROUP BY T2.rn;
Of course using such hook is not justified this time, but if we would have the table created by the complex resource-consuming query instead of 'T1' table, the opportunity to avoid its self-matching
(double recalculation) will improve the performance. And this example was used just for simple explaining.
Previous | Index | Next
Home SELECT exercises (rating stages) DML exercises Developers | {"url":"https://sql-ex.com/help/select19.php","timestamp":"2024-11-12T13:28:13Z","content_type":"text/html","content_length":"14046","record_id":"<urn:uuid:0dbe83d6-e924-427c-99a2-a4a7558f66ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00670.warc.gz"} |
In-Person Courses and Workshops | Tarastats Statistical Consultancy
Statistical Online Courses for Data Science and Analytics
Develop foundational statistical knowledge to analyse data in scientifically objective way.
Modern Statistical Thinking for Data Science and AnalyticsA foundation to analyse data in a scientifically objective way
Statistical thinking is the key skill to analyse data in a scientifically objective way. The course introduces the key statistical and non-statistical concepts to develop modern statistical thinking.
This is a foundational data science and analytics course.
Causal Inference for Data Science and Analytics Foundations, assumptions and applications
Because most questions of interest are causal in their nature, it is important for data scientists and analysts to develop understanding about the key causal inference concepts. This is a
foundational and thus a conceptual course.
Causal Inference with Observational Data for Impact EvaluationsData requirements, methods and techniques
Most of today's data is observational. This course is a continuation of the first course on causal inference, with focus on applications and thus the art of satisfying assumptions without which
causal inference is mission impossible.
Statistical Thinking for Journalists How to read data and analyse it in scientifically objective way
Because most questions of interest are causal in their nature, it is important for data scientists and analysts to develop understanding about the key causal inference concepts. This is a
foundational and thus a conceptual course.
In-Person Courses and Workshops
Statistics lies in presence of ignorance. By definition, ignorance means lack of knowledge, understanding, or information about something. Statistics lies when a person presenting statistical data
lacks knowledge, understanding, or information about statistical-methodological techniques which enable one to analyse data in scientifically objective way. Although we live
Continue reading »
Learning about foundational concepts of causal inference is crucial for data science because most questions of interest are causal in their nature. Whether we perform impact evaluations, A/B testing,
quality control or clinical trials, causal inference is the method of choice. Causal inference is one of the
Continue reading »
Statistical thinking is in need of a new approach due to recent developments of the modern statistical science. This new approach puts causal thinking at the heart of the key statistical thinking
concepts, which reflects recent developments of modern statistical science in the field of causal inference
Continue reading »
Do you wish to be informed of the dates for our upcoming courses? Leave your email below and we will be in touch. | {"url":"https://www.tarastats.com/courses-and-workshops/","timestamp":"2024-11-09T05:48:49Z","content_type":"text/html","content_length":"85582","record_id":"<urn:uuid:72e41c3f-b245-47ef-81ec-edca837aadde>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00394.warc.gz"} |
A distance function allows multiple product production using multiple resources to be modeled. A stochastic directional distance function (SDDF) allows for noise in potentially all input and output
variables; however when estimated, the direction selected will affect the functional estimates because deviations from the estimated function are minimized in the specified direction. This paper
addresses the question, how should the direction be selected to improve the estimates of the underlying production behavior. We are motivated by the estimation of a cost function for hospital
production in the U.S., but our insights apply for production and distance functions as well. In contrast to the parameters of the parametric stochastic distance function which are point identified,
we show that the parameters of the parametric SDDF are set identified when alternative directions are considered. We present simulation results that show if errors are uncorrelated across variables,
then the benefit in terms of improved functional estimates from the use of a directional distance function are significant. Further, we show that these benefits grow as the correlation in error terms
across variables grow. This correlation is a type of endogeneity that is common in production settings.
We show that the set of identified parameters for either the parametric or the nonparametric shape constrained estimator can be narrowed via data-driven approaches to restrict the directions
We apply the SDDF estimation procedure to a data set of hospitals containing a sample of approximately 600 hospitals each year from 2005 to 2010. We propose to select a direction that is
approximately orthogonal to the estimated function in the central region of the data and find this direction provides significantly better estimates of the cost function relative to other estimators.
Shape Constrained Nonparametric IV Estimators for Production Function Estimation
We propose a shape constrained nonparametric IV estimator which imposes a set of shape constraints on a nonparametric IV approach. We apply the Landweber–Fridman regularization to the Shape
Constrained Kernel–weighted Least Squares (SCKLS) estimator developed by Yagi et al. (2018). Furthermore, we also consider more complicated shape constraints proposed by microeconomic theory by
applying iterative S–shape algorithm proposed by Yagi et al. (2018). We aim to improve the finite sample performance and the economic interpretability of estimated results by imposing correctly
specified shape constraints while avoiding the bias from endogeneity issues.
Iterative S-shape production function estimation
A production function satisfying the Regular Ultra Passum (RUP) law is characterized by increasing returns to scale followed by decreasing returns to scale along any expansion path, which is referred
to as S-shape function. Although there are existing nonparametric estimators imposing the RUP law, they impose additional strong assumptions such as: deterministic model, homotheticity or constant
elasticity of scale. This paper proposes an iterative algorithm to estimate adaptively a function that satisfies the RUP law while relaxing these other assumptions.
Sweet or Sour? The Potential for U.S.-Cuban Trade in Sugar
This project seeks to analyze the potential for U.S.-Cuban sugar trade with a particular focus on Cuban sugar production. Cuba has historically been an important global sugar producer and U.S. sugar
producers are sensitive to foreign competition. Furthermore, the potential for Cuban liberalization and subsequent possible inflow of foreign investment makes analysis of this industry important. The
project’s implications are significant for U.S. sugar producers, U.S. national agricultural policy, and for agricultural development in developing countries generally.
Evaluating Production Function Estimators on Manufacturing Survey Data
Organizations like census bureaus rely on non-exhaustive surveys to estimate industry population-level production functions. In this paper we propose selecting an estimator based on a weighting of
its in-sample and predictive performance on actual application datasets. We compare Cobb-Douglas functional assumptions to existing nonparametric shape constrained estimators and a newly proposed
estimated presented in this paper. For actual data, specifically the 2010 Chilean Annual National Industrial Survey, a Cobb-Douglas specification describes at least 90% as much variance as the best
alternative estimators in practically all cases considered.
How inefficient are U.S. hospitals? What changes can lead to improvement?
We use U.S. hospital data from 2004 to 2011 to estimate a cost function using a Bayesisan semi-nonparametric method that allows for a heteroskedastic inefficiency. Moreover, we evaluate what is the
impact of variables, such as region and hospital size, on cost estimating both the size and robustness of the variables in terms of reducing cost for hospitals.
Adaptively Partitioned Convex Nonparametric Least Squares
This research overcomes both the decreased accuracy of Convex Adaptive Partitioning on real production survey datasets and the cross-validation performance challenges of CNLS to create a robust and
scalable adaptive partitioning-based convex regression method.
We discover that real production datasets often contain local monotonicity violations, which affect CAP’s ability to propose feasible basis region splits. Moreover, we note that CNLS’s error
minimization strategy within the observed dataset results in poor estimations on unobserved firms due to over-fitting. We create a hybrid of both methods that preserves their most favorable
properties at a small computational time expense. The paper summarizing this research is available on Arxiv.
Shape Constrained Kernel-weighted Least Squares (SCKLS)
SCKLS (shape constrained kernel-weighted least squares) estimator integrates kernel-weighting to convex nonparametric least squares.
Kernel regression is one of the powerful nonparametric estimation methods. By imposing more weight to some closer points, kernel regression helps to avoid over-fitting although it requires the tuning
parameter, bandwidth. By imposing some shape constraints such as monotonicity and concavity, we propose SCKLS estimator and apply it to estimate production function with simulated and real data. We
also investigate the relationship with Convex Nonparametric Least Squares (CNLS) and we found that CNLS is minimum bias estimator in the class of SCKLS estimator. This is on-going research with
Daisuke Yagi. This is the first chapter of his dissertation.
Shape constrained Semi-nonparametric Stochastic Frontiers estimation using a local maximum likelihood approach
We propose a shape-constrained production function estimator starting from the method described in Kumbhakar, Park, Simar, Tsionas 2006 (KPST).
We maximize the loglikelihood of a local linear estimator at each observation. We estimate the parameters for the noise and inefficiency distributions that are potentially heteroscedasticity.
The challenge to imposing shape constraints, such as monotonicity and concavity, is we need to jointly esitmate the maximum likelihood function for all observations while imposing the constraints.
This is on-going research with Kevin Layer.
Multi-variate Bayesian Convex Regression with Inefficiency
This research builds on Nonparametric Multi-variate Bayesian Convex Regression to develop a method to estimate shape constrained production frontiers with heteroskedastic inefficiency distributions
that scales up to thousands of observations.
We propose a Bayesian method which allows the estimation of a semiparametric production frontiers with a flexible inefficiency distribution, to use panel data and to measure the impact of
environmental variables. A Metropolis-Hastings framework is considered to compute smoothed and non-smooth estimates of the production frontier.
Working Paper available at Arxiv. | {"url":"https://productivity.engr.tamu.edu/archives/category/ongoing-work","timestamp":"2024-11-07T01:03:41Z","content_type":"text/html","content_length":"54261","record_id":"<urn:uuid:4b930f77-25f5-4a1f-87aa-1da58bb7ef29>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00794.warc.gz"} |
Free Space Path Loss Calculator (with Examples)
The FSPL calculator provides the attenuation or loss of a signal as it propagates through free space. In this post we have also provided an example of how to use the calculator.
To calculate the FSPL, enter the following:
• Frequency of operation f
• Distance d
• Transmit Antenna Gain G[Tx]
• Receive Antenna Gain G[Rx]
FSPL = 20*Log[10](d) + 20*Log[10](f) + L[c] + 20*Log[10](4Ï€/c) – G[Tx] – G[Rx]
💡 This formula only applies when the distance (d) between the antennas is large enough that the antennas are in the far field of each other. In other words d >> λ where λ = c/f. It is
sufficient if the distance is at least 10 times larger than the wavelength.
FSPL stands for Free Space Path Loss. It is a term used in wireless communication and radio frequency (RF) engineering to describe the loss of signal strength as electromagnetic waves propagate
through free space, such as air or a vacuum, without any obstructions or reflections.
The Free Space Path Loss is primarily determined by the following factors:
1. Distance: FSPL increases with the square of the distance between the transmitter and receiver. This means that as you move farther away from the source of the signal, the signal strength
decreases significantly.
2. Frequency: FSPL is also influenced by the frequency of the electromagnetic waves. Higher frequency signals tend to experience greater path loss compared to lower frequency signals when traveling
the same distance.
3. Antenna Gain: As the receive or transmit antenna gain is increased, the path loss decreases.
FSPL is an important concept in wireless communication system design, as it helps radio system engineers determine the expected signal strength at a given distance and frequency. This information is
important for designing wireless networks, estimating coverage areas, and ensuring reliable communication between devices.
Definition of Terms
Below is a list of the terms used in the calculator and what each of them mean.
Frequency of operation
This is the frequency at which the communication system operates.
Transmit Antenna Gain
This depends on the type of antenna used and is expressed in dBi (dB relative to isotropic antenna).
Receive Antenna Gain
Depends on antenna used and is expressed in dBi. In cases where the signal is from a particular direction, a high gain antenna (8 dBi for instance) can be used. This allows focusing of energy instead
of receiving from all directions. It can be calculated from the Antenna Factor and frequency of operation.
💡 The free space path loss equations assume an ideal operating environment i.e. an unobstructed propagation path between transmitter and receiver.
Example Free Space Path Loss Calculation
At a frequency of 100 kHz and a distance of 100 km, the attenuation or FSPL = 52.45 dB.
At a frequency of 100 kHz and distance of 100 meters, the FSPL = -7.55 dB which is clearly an erroneous result as the attenuation cannot be negative.
To understand this, let’s take into consideration the requirement that for the equation to be valid the distance d has to be at least 10*λ = 2998 meters (using the frequency to wavelength calculator
The calculator is therefore applicable at distances greater than 29980 meters or 30 km for a frequency of 100 kHz.
Related Calculators | {"url":"https://3roam.com/free-space-path-loss-calculator/","timestamp":"2024-11-05T03:30:55Z","content_type":"text/html","content_length":"197567","record_id":"<urn:uuid:9ad1b069-31ff-4ce5-9000-d6d78963e7a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00767.warc.gz"} |
Approximating survivable networks with minimum number of steiner points
Given a graph H = (U, E) and connectivity requirements r = {r(u,v): u, v ∈ R ⊆ U}, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider
the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, ∥·∥) and connectivity requirements, find a minimum size set S ⊂ M \
V of additional points, such that the unit disc graph induced by U = V S satisfies the requirements. In the (node-connectivity) Survivable Network Design Problem (SNDP) we are given a graph G = (V,
E) with edge costs and connectivity requirements, and seek a minimum cost subgraph H of G that satisfies the requirements. Let k = max [u,v∈V] r(u, v) denote the maximum connectivity requirement. We
will show a natural transformation of an SN-MSP instance (V, r) into an SNDP instance (G = (V, E), c, r), such that an α-approximation algorithm for the SNDP instance implies an α · O(k ^2)
-approximation algorithm for the SN-MSP instance. In particular, for the case of uniform requirements r(u, v) = k for all u, v ∈ V, we obtain for SN-MSP the ratio O(k ^2 ln k), which solves an open
problem from (Bredin et al. Proceedings of the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) (2005), 309-319).
• approximation algorithms
• node-connectivity
• sensor networks
• unit disc graphs
Dive into the research topics of 'Approximating survivable networks with minimum number of steiner points'. Together they form a unique fingerprint. | {"url":"https://cris.openu.ac.il/en/publications/approximating-survivable-networks-with-minimum-number-of-steiner-","timestamp":"2024-11-11T18:13:50Z","content_type":"text/html","content_length":"50208","record_id":"<urn:uuid:adf6bbe1-499c-4f49-85ee-cb79663d5fb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00809.warc.gz"} |
What our customers say...
Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences:
My daughter is dyslexic and has always struggled with math. Your program gave her the necessary explanations and step-by-step instructions to not only survive grade 11 math but to thrive in it.
Robert Davis, CA
My math professor suggested I use your Algebrator product to help me learn the quadratic equations, and non-linear inequalities, since I just could not follow what he was teaching. I was very
skeptical at first, but when I started to understand how to enter the equations, I was amazed with the solution process your software provides. I tell everyone in my class that has problems to
purchase your product.
Mark Hansen, IL
I can't say enough wonderful things about the software. It has helped my son and I do well in our beginning algebra class. Currently, he and I are taking the same algebra class at our local community
college. Not only does the software help us solve equations but it has also helped us work together as a team. Thank you!
Mary Brown, ND
I'm not much of a math wiz but the Algebrator helps me out with fractions and other stuff I need to know for my math appreciation class in college.
Nathan Lane, AZ
Search phrases used on 2014-07-25:
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
• factoring algrebaic expressions with examples
• Roots and exponents worksheets
• math powerpoints for multiplying numbers with three factors
• free 6th grade math taks worksheets
• probability + year 6
• solving algebraic equations involving squaring, ppt
• why is factoring important
• example question of multiply integers form two
• best calculator for solving differentials
• solve 4th order equation
• nth term worksheets
• Convert from a mixed Fraction to a Decimal
• ti-89 frac
• mix numbers
• mastering fractions with variables
• factoring polynomials calculator solver
• convert int to time java
• add rational expressions calculator
• solving equations with rational expressions worksheets
• how to find x value on graphing calculator
• 8th to decimal conversation chart
• cubed polynomial rule
• How do you calculate cube root on a TI-89 titanium
• Free maths worksheet sec3
• using a for loop in java to convert numbers
• free alpha sequence aptitude test
• 5th grade solving simple equations
• Algebra 2 rational expressions with exponents and radicals
• prentice hall geometry + 1993 +homework answers
• decimal to fraction formula
• algebra2 answers
• graphing inequalities worksheet math
• Basic algebra for 9th Graders
• easy ways to understand algebra
• matlab solve for positive root
• square root cheat sheet
• Holt Algebra 1 Workbook
• thermometer worksheet+ adding and subtracting temperature
• simultaneous quadratic equations calculator
• solving kumon
• solving quadratic equations review sheet
• 6th grade algebra quiz
• sample mixture problems
• equasions
• factor problems
• linear second order derivatives
• student reviews of math 208 class university of phoenix online
• adding,subtracting integers,and multiplying (worksheet)
• free math help with Algebra 1 glenco student works north carolina
• f.2 maths simutaneous linear equation
• Roots of Real Numbers with Variables Calculator
• 2007 free 10th matriculation question paper with answer+pdf
• pre-algebra dummies megaupload
• addition and subtraction problem solving stories
• complex numbers glencoe algebra II skills practice
• 6th grade sat sample questions
• free equation worksheets for primary
• ask a algebra question for free
• mixed fractions into decimals calculator
• ti-83 plus use y value to find x value graph
• 6th grade advanced math worksheets
• third grade equation and inequality
• adding fractions in java code
• inequalities for 5th grade students
• help simplyfying rational expression and equation
• some exercise on subtraction on formula
• " strip patterns" mesopotamia
• 5th grade worksheets on mean, mode, median
• math trivia-algebra
• how to multiply a radical and a whole number
• simplifying radicals powerpoint
• math with pizzazz! pic 4-e creative publications grade 8
• how to program rock paper scissors t1-83+ calculator
• algebraic equation percent
• algebra 2 workbook answers
• radical square root e calculator
• adding integer worksheets
• algebra 1 linear substitution and combination worksheets
• square root practice worksheets
• how do you simplify sums and differences of radicals
• polynom excel
• seventh grade algebra worksheets
• trivias for math
• online simultaneous equation solver with 4 variables
• worksheets-slope
• pre-algebra with pizzazz
• Glencoe McGraw-Hill algebra 1 answers
• algebra 2 answer solver
• how do you type equations with limits into ti 84+ calculator
• free algebra sheets year 9
• combination and permutation trigonometry
• how to put programs on TI 84 plus for dummies
• free tutorial of trigonometry in maths of class tenth
• free online fraction calculator simplest form | {"url":"https://www.softmath.com/math-book-answers/perfect-square-trinomial/synthetic-division.html","timestamp":"2024-11-10T18:55:56Z","content_type":"text/html","content_length":"36213","record_id":"<urn:uuid:5293f87f-43ad-4711-bdb2-ccaba780b67b>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00368.warc.gz"} |
neural network
Thanks. In fact article would be more sensible. To refer to an arXiv preprint as a “miscellaneous” reference is a weird anachronism!
So I am happy to stick with your (currently) supported fields!
But when I go to the edit pane you made, all I get to see is a big white box and no indication what to do.
If you could just make the edit pane show a rudimentary template of fields into which the user could then type their data, that would already get us started!
Yes, it only accepts the ’article’ document type for now. But completely agree this is exactly the kind of thing the bibliography is for :-)! I need to complete the ability to edit references in the
bibliography, and then I will add support for all the common document types. Have had to focus on other things in the nLab software recently, but I’ll work on this when I have a chance.
I see. That reminds me that we should use Richard’s new bibtex-like functionality to harmonize formatting. Maybe once that is a little more convenient to use: I just tried to offer it the bibtex data
as produced by the arXiv
title={Learners' languages},
author={David I. Spivak},
but it does not swallow that.
But I would always punctuate after a title
• Corfield, D. 2003. Towards a philosophy of real mathematics. CUP.
or something like that.
Sorry for raising a trivial point on formatting:
In a reference, let’s not have a comma before the parenthesis with the arXiv number, it doesn’t seem to be needed. What do you think?
Added two more category-theoretic treatments
• David Spivak, Learners’ languages, (arXiv:2103.01189)
• G.S.H. Cruttwell, Bruno Gavranović, Neil Ghani, Paul Wilson, Fabio Zanasi, Categorical Foundations of Gradient-Based Learning, (arXiv:2103.01931)
Removed the redirect to ’machine learning’, as this is far more general.
diff, v7, current
Added two more category-theoretic treatments
• David Spivak, Learners’ languages, (arXiv:2103.01189)
• G.S.H. Cruttwell, Bruno Gavranović, Neil Ghani, Paul Wilson, Fabio Zanasi, Categorical Foundations of Gradient-Based Learning, (arXiv:2103.01931)
Removed the redirect to ’machine learning’, as this is far more general.
diff, v7, current
Added that article in #3.
diff, v3, current
What I don’t understand yet in HSTT 18 is where the non-linear activiation functions are in the story, i.e. how is what they have different from a discretized solution of a differential equation. But
I don’t really have time to look into this properly.
I never got round to looking at
• Brendan Fong, David Spivak, Rémy Tuyéras, Backprop as Functor: A compositional perspective on supervised learning, (arXiv:1711.10455)
added these references on the learning algorithm as analogous to the AdS/CFT correspondence:
v1, current
Stub. For the moment just for providing a place to record this reference:
• Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)
v1, current
I will indeed add something like this, but have not had a chance yet. Am prioritising the fundamental functionality first; I wouldn’t be averse to adding something sooner, but I need to think it
through a bit (I’d rather not put in place some quick hack that people get used to, which later causes problems!).
For example, I think we probably should stick to BibTex’s convention of having article only refer to published articles, because this allows validation: we can require that a journal, etc, is given,
and in fact such a requirement is already implemented. Of course we could make up our own new document type such as ’preprint’ or something (but allow use of ’misc’ or ’unpublished’, converting to
’preprint’ behind the scenes if for example the arXiv field is present).
Added a small mention of the relation with renormalisation group flow
diff, v9, current
Good. I have added these further references in this direction:
Further discussion under the relation of renormalization group flow to bulk-flow in the context of the AdS/CFT correspondence:
diff, v10, current
added pointer to today’s:
• Daniel A. Roberts, Sho Yaida, Boris Hanin, The Principles of Deep Learning Theory, Cambridge University Press 2022 (arXiv:2106.10165)
diff, v15, current
adding information about how neural networks are related to differential equations/dynamical systems.
diff, v18, current
Since revision 4 the Idea-section starts out with
A neural network is a class of functions used…
This seems a little strange. Maybe what is meant is:
Neural networks are a class of functions used…
But either way, the sentence conveys no information about the nature of neural networks.
diff, v19, current
Added some references for topological deep learning
• Ephy R. Love, Benjamin Filippenko, Vasileios Maroulas, Gunnar Carlsson, Topological Deep Learning (arXiv:2101.05778)
• Mathilde Papillon, Sophia Sanborn, Mustafa Hajij, Nina Miolane, Architectures of Topological Deep Learning: A Survey on Topological Neural Networks (arXiv:2304.10031)
• Mustafa Hajij et al., Topological Deep Learning: Going Beyond Graph Data (pdf)
diff, v22, current
Added breakdown of Neural Network Gaussian process (NNGP) results, Neural tangent kernel (NTK) theory and more recent approaches to QFT (Neural Network field theory, NNFT, and the latest paper in
that direction).
One could also think of making a separate page for Neural tangent kernel theory and moving some of the large width build-up there. There, there’s some obvious reference links one could add and I
might later at one point. (In the style I’d like to further clean up the somewhat over reliance on brackets in my paragraph and remove the explanation-by-comparison to classical mechanics by the
actual formulas, albeit even the Wikipedia breakdown isn’t that bad.) There were already references in that direction, but no main text. Feel free to alter any running text.
I’m personally mostly interested in the field theory and stochastics stuff, but the article could also bridge to information geometry results
diff, v23, current
Added two articles
• Bruno Gavranović, Paul Lessard, Andrew Dudzik, Tamara von Glehn, João G. M. Araújo, Petar Veličković, Categorical Deep Learning: An Algebraic Theory of Architectures [arXiv:2402.15332&
• Theodore Papamarkou, Tolga Birdal, Michael Bronstein, Gunnar Carlsson, Justin Curry, Yue Gao, Mustafa Hajij, Roland Kwitt, Pietro Liò, Paolo Di Lorenzo, Vasileios Maroulas, Nina Miolane, Farzana
Nasrin, Karthikeyan Natesan Ramamurthy, Bastian Rieck, Simone Scardapane, Michael T. Schaub, Petar Veličković, Bei Wang, Yusu Wang, Guo-Wei Wei, Ghada Zamzmi, Position Paper: Challenges and
Opportunities in Topological Deep Learning [arXiv:2402.08871]
Position Paper: Challenges and Opportunities in Topological Deep Learning
diff, v28, current
Added tags
Fabio Zanasi
diff, v29, current
Updated reference
Fabio Zanasi
diff, v29, current | {"url":"https://nforum.ncatlab.org/discussion/10400/neural-network/?Focus=93280","timestamp":"2024-11-09T10:30:00Z","content_type":"application/xhtml+xml","content_length":"82284","record_id":"<urn:uuid:f4f288c0-fc11-46f0-b813-6405e67e3d77>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00077.warc.gz"} |
Can I get someone to help me with regression analysis in R for my website? | Pay Someone To Take My R Programming Assignment
Can I get someone to help me with regression analysis in R for my website? I have a website, which I am using right now, which was designed at https://rubydna.com/. So who should I ask about, so I
can build the regression model for x, y, y+z? I use a lot of libraries when it comes time to make software for any other purpose. A: I finally found the answer to my issue. Sometimes R stands for
general purpose time series analysis. This question gives an explanation of how I am able to do this: R-10: Releved statistics I’ve written a few regressions for this issue in RStudio and tested each
several examples well in my C++ shell. I picked one data set to sample from and gave the average using a log marginal (XOR) and logit(LogUnits) on the XOR over the T1 subset and then gave the RMSE to
the YOR over the YOR over the T1 subset. I’ve been using this for a few projects so I can try to figure out what’s going wrong. Here’s an illustration (because it’s also my 3rd issue, this is the
only one that I’ll post) of the tests I’ve done so far: http://ideone.com/XK9S9z I’ve also created a SQL Server PostgreSQL server table that changes after a run of R. I’ve done that since it’s the
only example of an interval that I’ve done this for. What I’m trying to implement is a cross list of x (X) and y (Y) values if the past Hierarchical index does exist and I’d like to transform this
into a matrix. I’ve done a couple of things here: Since I’m referencing a column in one table, we need this matrix and a column in the other table to be returned as a matrix. For this purpose, we
also do a cross-column cross-diff variable. A cross-column cross-diff is an offset is a variable that does the math by subtracting the values listed there and then multiply the absolute values using
which is a column or N log. For me though, I created the cross table in R – as I mentioned above. I used the data-table function cbind, which I learned about in the programming environment RStudio.
I’ve removed that code here and that’s about it. The cross table is actually a column that’ll be included in an interval vector. The matrix doesn’t appear to have any columns.
Coursework For You
I’ve removed it from my C++ shell and filled it with just plain vectors. I’ve also added a x-axis with a row, column, and N-log to set those tables, so in the matrix, I don’t seem to be moving the
table (not sure!) up and then down. While the XOR is over theCan I get someone to help me with regression analysis in R for my website? The website explains my issues and it works. It displays me on
my website and my friends list in the menu (my Facebook list). But the issues are not really happening. For example, if for some details it displays the status page, or the problem on line number 3
or the link to another page. How can I get the regression analysis on any page to show the screen text I am displaying to the admin side? Thank you! A: I discovered this on one of my internet links,
but not sure where to look… Why do I get this message? You have a set of 4.5 lines. Some of them express my issue: The page above displays some pages on the website. That is the line number with the
following definition: %LANGUAGE sql %LCONVERT and some other lines: %LANGUAGE sql %LCATELLATE Can I get someone to help me with regression analysis in R for my website? If not, can anyone assist? I
am looking for help in regression analysis. A: As you say, you’re looking for help with regression structure, trying to fill as many tasks as you can in R with regression analysis. Many of R’s
functions include a statistical function defined with the objective of capturing the goal, or explanatory goal. The goal function helps to determine best ways of capturing a simple data structure
object. For example taking individual average (A2) scores from a distribution obtained by averaging. Modeling the regression fit function can be very instructive. After you get to the conclusion you
can run an analysis with the two most important functions listed below: The fitting function gets you to a full picture of the relationship between the data and the explanatory structure object. You
can also use a data set to apply regression fitting to the observations.
Do My Math Test
A simple example where there is data is to link to human brain based on fMRI on a single subject brain. I don’t know why you would need a regression fit function, as it’s easy and easy to obtain.
However, the data is the data that we might need to run a regression test on. There are similar functions for other algorithms including linear models. However, the question you seem to want is
whether I could use the data in an effective way. Does R have a regression fit function that provides useful regression functions? If your goal is to run a regression test you’ll have to keep in mind
that one method to perform a regression test is to use an external tool to scan the data, but later you may need to run a regression test in conjunction with the data itself. I’ll look at another
example though. EDIT: Actually, this is on steroids, because it might also apply to other algorithms that are using “software” to evaluate the object. A: R is well suited for evaluating feature
matrix for fitting. With more general purpose data address might be a good idea to use some feature learning methods to find out how the data applies to the models. That is one of the most important
parts. Similar framework has some drawbacks. You can use DataType to index the data, then apply your previous methods to those data. In addition, it is best to have an R data dataset by itself. A: R
offers a built-in function to automatically fill in the missing values by dividing (R-mean) by (R-SD) and ignoring the factors that are missing from the model (i.e., don’t use R). It also supports
regression and predictor models. Personally, I find this to be a great approach. There is a package like Fit for Modeling [tacfive (link)] which provides easy implementation to the fitting framework.
Pay Someone To Take Clep Test
It basically lets you build models with various parameters. That is, in R you only have to construct 2 random variables. Your data include a number (number of subjects) but not a number (average and
standard deviation). And as for cross-scales you can create other model (like Linear Model). I also took a scipy package for R, which it has good properties (like QTL). All you have to do is get the
model being fit properly with the package results. The scipy package also has some nice built-in function to search for model specific prediction of variables by looking at the input data. You have
to find out to what extent the models do fit correctly to get that you want. You can use this function to find out the mean to standard deviation within different models over a certain distribution. | {"url":"https://rprogrammingassignments.com/can-i-get-someone-to-help-me-with-regression-analysis-in-r-for-my-website","timestamp":"2024-11-05T16:13:58Z","content_type":"text/html","content_length":"195928","record_id":"<urn:uuid:7fcfa025-95cd-4676-b31b-37b986c1ced6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00660.warc.gz"} |
transcendental syntax
“Transcendental syntax” (Girard 13) is the name of a proposal (or maybe a pamphlete) by Jean-Yves Girard which means to rethink fundamental aspects of formal logic, of syntax/semantics in the light
of computer science (and more especially through the so-called Curry-Howard-Lambek correspondence). According to Girard, linear logic and Geometry of Interaction are but exercises in transcendental
syntax (Girard 13b).
While Girard’s prose is notoriously demanding, exegesis may be found in (Abrusci-Pistone 12) for the philosophical side, (Rouleau 13) which was written before the more modern developements of
transcendental syntax by means of stars and constellations (Girard 13b) or (Eng 23) for an attempt to formalize and contextualize Girard’s last works from the point of view of computer science.
Getting rid of semantics
Usually, semantics dictates what logic is. It allows and forbids manipulations of symbols by following what we consider logic to be, in a dualism between symbol (interpreted) and its meaning
(interpreter). An ambition of the transcendental syntax is to get rid of semantics (which is actually a distinguished form of syntax) by redirecting the computational power of semantics and its
dialogue with syntax and putting it at the same level as syntax: the barrier is broken and everything becomes synctatic objects of same nature interacting in a same space from which logical meaning
emerges. This approach is meant to make more explicit the mechanisms underlying logical reasoning (which appears in types, formulas, specifications, computer programs, mathematical proofs,
computational constraints, computational complexity, …).
Philosophical interpretation
The transcendental syntax starts from the claim that formal logic makes a lot of rather arbitrary choices such as the so-called rules of logic (see deductive systems) and that it should be possible
to study the mechanisms of reasoning without these assumptions (that Girard calls “prejudices”).
His point of view can be seen as an update of Kant‘s epistemology taking computation into account through recent works in linear logic and the Curry-Howard-Lambek correspondence. The whole logical
activity is divided into four concepts (Girard 16):
Analytic / Answers Synthetic / Questions
A posteriori / Explicit Constat Usine
A priori / Implicit Performance Usage
The idea is that logic studies the relationship between questions (formulas being a special case) which are subjective and their answers (proofs being a special case) which are objective, by means of
finite objects (because reasoning should be finite and verifiable, otherwise it would not even be possible). In order to be freed from those “prejudices”, one should start by defining the answers,
seen as neutral and meaningless. Then, the point would be to study the sufficient conditions making the logical concepts (proofs, formulas, logical correctness) emerge from the meaningless. In
reference to Kant, Girard uses the expression “conditions of possibility of language”.
The answers correspond to the space of computation (generalizing proofs) which can be evaluated (performance) into a normal form (constat). This is the space where proofs are constructed but not yet
considered logically meaningful or correct. It is the material on which logic is constructed.
The space of questions considers two alternative definitions of meaning corresponding to formulas or types:
• the use (usage): it corresponds to Ludwig Wittgenstein‘s meaning-as-use. The meaning of computational objects is defined by their potential interactions.
• the factory (usine): the meaning of computational objects is defined by a set of (well-chosen) tests they have to pass (exactly as for tests in programming).
This is related to the distinction between existentialism and essentialism in philosophy. Logical certainty corresponds to the adequacy between usine and usage: we would like factory’s tests to be
finite and sufficient in a way to guarantee an associated computational behavior (in a same way tests in a factory should be relevant and guarantee a correct use of objets which are made, or in the
case of programming, that the tests corresponding to some program specification guarantee some property like the absence of bugs).
According to Girard Girard 16, the separation between constat and performance comes from undecidability (as in the halting problem) because the potential of programs cannot always be reduced to their
result (which is not always defined). As for the separation between usine and usage, it would come from Gödel’s incompleteness theorem which intuitively implies that the possible uses of a logical
object may go beyond what we expect from their definition (our formatting of the concept of logic).
The transcendental syntax has a monist approach (opposed to a dualist) to logic and start from the assumption that computation can fully explain logic and it therefore extends the Curry-Howard-Lambek
Technical interpretation
Technically speaking, the transcendental syntax is a finitary improvement of Girard’s Geometry of Interaction (in its original form and with its initial motivation). Some preliminary works include
flows (Girard 95, Bagnol 14) or interaction graphs (Seiller 16). The transcendental syntax generalizes and extends both by taking into account proof nets' logical correctness in a more general
The elementary objects (analytic) considered by Girard are called constellations. They are defined as multisets of clauses containing first-order terms (with no logical meaning associated). Two
constellations can interact by using Robinson’s resolution (Robinson 65). We use constellations to encode proof structures, their cut elimination but also the correction graphs asserting logical
correctness. Constellations were initially found by analyzing the behavior of proof-nets and their correctness but other basis of computation can be chosen.
Answers are constellations and questions correspond to two notions of types:
• by using realizability techniques for linear logic as in ludics or Seiller’s interaction graphs, we can define formulas called behaviors.
• Inspired by the logical correctness of proof nets, we can define types as a finite sets of tests (encoded as constellations). If a constellation passes all tests, it can be labelled by the
corresponding type label. In the context of proof nets, it corresponds to testing proof structures against correction graphs.
Although never explicitly mentioned by Girard, the transcendental syntax is very close to realizability theory. In realizability theory, types are designed from a model of computation (for instance,
natural numbers and recursive functions or lambda-terms with beta-reduction). However, the transcendental syntax is able to speak about logical correctness (through linear logic). Note that other
alternative approaches for the reconstruction of linear logic exist such as Beffara’s concurrent realizability (Beffara 06).
Intuitive illustrations
Car factory
This is an example given by Girard in his book “Le fantôme de la transparence”.
A car (analytics) can be certified by a factory (usine) and receive a label (type) which can guarantees that it will work as expected (adequacy). However, it would be absurd to drive 20000km with a
car only to show that it can do it (actual use cannot serve as effective testing). Those factory tests are effective, finite, partial but sufficient. They are well-chosen and can be more laxist or
Automata theory
A finite automaton is a machine receiving a finite word as input and which either accepts or rejects it. Hence, there are two kind of objects (analytic): machines and words and machines can only
interact with words.
Only by interaction, a way to tell whether a machine recognizes some language is to feed it with all the words of that language (Girard’s use). However, in the case the language is infinite (e.g. all
binary words ending with 00), you would never be able to conclude. However, we are still able to reason on automata and words by an external logical reasoning and analysis.
The idea of the transcendental syntax would be to reify this external reasoning by extending the computational space such that it can express automata, words and their interaction but also such that
it is “large” enough in order to introduce “exotic objects” which can serve as finite tests against automata, thus internalizing our external logical reasoning into the syntax observed.
Those exotic objects are what Girard calls the “hidden files” of logic, in reference to hidden files in Unix systems which are often essential but invisible.
This is reminiscent of how the space of real numbers is extended to complex numbers in order to solve some equations, or also of how the set of real numbers is a completion of rationals (where
irrationnals are seen as such exotic elements).
Lambda-term is a syntactic theory of functions. In order to check whether some term is a function from some type A to some type B, the “use” approach would be to test it against all terms of type A
and check whether it produces a term of type B. Since there can be infinitely many terms of type A, testing would never finish.
However, we are still able to say that a term is a function by using type rules for simply typed lambda-calculus (STLC). Those typing rules provide tests (Girard’s factory) and we have guarantees
that if the term has the same of a function (it is a lambda abstraction) and that it is well-typed, then it will behave well as a function.
Axiom-free systems
Following this approach, it is theoretically possible (but non-trivial) to construct axiom-free systems. In (Girard 20), Girard roughly sketches a reconstruction of Peano arithmetic.
Transcendental syntax and computer science
The transcendental syntax generalizes, extends or is simply related to several important aspects of computer science (although no direct link has been studied yet, thus making this section
• in descriptive complexity, we are interested in capturing classes with fragments of logic. In Girard’s third paper on transcendental syntax (Girard 18), a reconstruction of predicate and second
order logic is sketched. They are both essential in descriptive complexity.
• In model checking and program specification, we are interested in designing types as specifications or capturing properties of a model of computation with formulas in order to reason about a
system. Girard’s constellations used in the transcendental syntax actually generalizes state machines and types can be constructed with realizability techniques.
• In software testing, we are interested in testing a program against a finite number of tests in order to assert that the program has a specific behavior. By generalizing the idea of correctness
criterion coming from the theory of proof-nets, it is possible encode tests and programs as objects of the same kind.
• Girard’s constellations are independent agents communicating with local interactions. This is reminiscent of the actor model in programming or of process calculi (such as the pi-calculus) and we
can imagine applications to logic for concurrent, distributed or parallel systems. Links to Yves Lafont‘s interaction nets/combinators could be made as well since they both are alternative to
Although no link have been properly studied, Girard’s constellations are reminiscent of biological or chemical computation. However, links between tile systems (Wang tiles and abstract tile assembly
models) and Girard’s constellation have been mentionned in (Eng 23). Morever, it also has been remarked that constellations generalize a model called flexible tiles (JonoskaMcColm 05) which is used
for DNA-based computation.
The transcendental syntax may also serve as a way to develop new ideas and concepts:
• in the same way as assembly languages are low-level languages close to machine operations, Girard’s constellations can be seen as a low-level language which would be closer to the elementary
mechanisms of reasoning (since it decomposes lambda-terms) and be used as a compilation target.
• Since logical concepts emerge from objets without any logical meaning, the transcendental syntax may be used to design “logic-agnostic” proof-assistants or tools which are not bound to primitive
definitions hard-coded into a compiler or an interpreter. Logical concepts would then be defined by the programmer (as programming libraries for instance).
• JA Robinson?, A machine-oriented logic based on the resolution principle, 1965.
• Jean-Yves Girard, Geometry of interaction III: accommodating the additives, 1995 (pdf).
• Marc Bagnol?, On the Resolution Semiring, PhD Thesis 2014 (pdf)
• Thomas Seiller?, Interaction Graphs: Full Linear Logic, 2016 (pdf)
• Jean-Yves Girard, Geometry of Interaction VI: a Blueprint for Transcendental Syntax, 2013 (CiteSeer)
• Jean-Yves Girard, Transcendental syntax 2.0, 2013 (pdf)
• Jean-Yves Girard, Transcendental syntax I: deterministic case, 2016 (pdf)
• Jean-Yves Girard, Transcendental syntax II: non-deterministic case, 2016 (pdf)
• Jean-Yves Girard, Le fantôme de la transparence, 2016
• Jean-Yves Girard, Transcendental syntax III: equality, 2018 (pdf)
• Jean-Yves Girard, Transcendental syntax IV: logic without systems, 2020 (pdf)
• Vito Michele Abrusci, Paolo Pistone, On Transcendental syntax: a Kantian program for logic?, 2012 (pdf)
• Vincent Laurence Rouleau, Towards an understanding of Girard’s transcendental syntax: Syntax by testing, PhD thesis 2013 (pdf)
• Emmanuel Beffara?, A Concurrent Model for Linear Logic, PhD thesis 2013 (pdf)
• Nataša Jonoska, Gregory L. McColm?, A computational model for self-assembling flexible tiles, 2005
• Boris Eng?, A gentle introduction to Girard’s Transcendental Syntax for the linear logician (hal-02977750)
• Boris Eng?, An exegesis of transcendental syntax, PhD thesis 2023 (tel-04179276v1) | {"url":"https://ncatlab.org/nlab/show/transcendental%20syntax","timestamp":"2024-11-05T13:10:05Z","content_type":"application/xhtml+xml","content_length":"35710","record_id":"<urn:uuid:e0ffc072-e634-4524-bab6-e29e88770df7>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00302.warc.gz"} |
Darlayne Addabbo, The University of Arizona
Darlayne Addabbo
Hano Rund Postdoctoral Fellow, The University of Arizona
I am the Hano Rund Postdoctoral Fellow at The University of Arizona. I was previously a Visiting Assistant Professor at The University of Notre Dame. I received my PhD in 2017 from the University of
Illinois at Urbana-Champaign.
Email: addabbo@math.arizona.edu
Recent and Upcoming Talks and Conferences
"Unveiling Infinite Symmetries: Mini Course on Algebraic Structures of Generalized Kac-Moody Type" at the Canadian Mathematical Society Summer Meeting (July 2024), lecture title: TBA
14th Southeastern Lie Theory Workshop, University of Virigina (March 1-3, 2024), title: "Vertex operators for imaginary gl2-subalgebras in the Monster Lie algebra"
Rutgers University Lie Group/Quantum Mathematics Seminar (November 10, 2023), online, title: Vertex Operators for Imaginary gl_2-subalgebras in the Monster Lie Algebra
"Lie Theory and its Applications in Physics", Bulgaria (June 19-25, 2023), title: "Vertex operators for imaginary gl2-subalgebras in the Monster Lie algebra''
13th Southeastern Lie Theory Workshop, North Carolina State University (May 12-14, 2023), title: "Vertex operators for Imaginary gl2- subalgebras in the Monster Lie Algebra"
AMS Central Sectional, special session on "Representation Theory, Geometry, and Mathematical Physics" at the University of Cincinnati (April 15-16, 2023), title: "Vertex operators for Imaginary gl2-
subalgebras in the Monster Lie Algebra"
Southwest Strings Meeting, The University of Arizona (March 24-25, 2023), title: "Vertex operators for Imaginary gl2- subalgebras in the Monster Lie Algebra"
Rutgers University Lie Group/Quantum Mathematics Seminar (November 11, 2022), online, title: "Vertex operators for Imaginary gl2- subalgebras in the Monster Lie Algebra"
"Quantum Symmetries: Tensor categories, topological quantum field theories, vertex algebras", Centre de Recherches Mathematiques (October 2022) title: Higher level Zhu algebras
AMS Spring Western sectional Meeting, special session on "Some modern developments in the theory of vertex algebras" (May 14-15, 2022) online, title: Higher level Zhu algebras for vertex operator
Women in Noncommutative Algebra and Representation Theory (WINART) workshop, Banff International Research Station
Institute for Advanced Study, Women and Mathematics, online (May 2021), title: Higher level Zhu algebras
Joint Mathematics Meeting Special Session on "Quantum Algebra and Geometry", online (January 2021), title: Higher level Zhu Algebras for Vertex Operator Algebras
Addabbo, Darlayne "On Huang's Associative Algebras for Vertex Operator Algebras", (In preparation) 2023
Addabbo, Darlayne; Keller, Christoph "Generalized Theta Functions on Vertex Operator Algebras", (In preparation) 2023
Addabbo, Darlayne; Carbone, Lisa; Jurisich, Elizabeth; Khaqan, Maryam; Murray, Scott H. "A Monstrous Lie Group", (In preparation) 2023
Addabbo, Darlayne; Carbone, Lisa; Jurisich, Elizabeth; Khaqan, Maryam; Murray, Scott H. "Vertex operators for Imaginary gl2- subalgebras in the Monster Lie Algebra", arXiv:2210.16178, ACCEPTED to
Journal of Pure and Applied Algebra.
Addabbo, Darlayne; Barron, Katrina "On generators and relations for higher level Zhu algebras and applications", J. Algebra 623 (2023), 496-540, https://doi.org/10.1016/j.jalgebra.2023.02.023
Addabbo, Darlayne; Barron, Katrina "The level two Zhu algebra for the Heisenberg vertex operator algebra", Communications in Algebra (2023), https://doi.org/10.1080/00927872.2023.2184638
Addabbo, Darlayne; Bergvelt, Maarten "Difference hierarchies for nT τ-functions." Internat. J. Math. 29 (2018), no. 13, 1850090, 29 pp.
Addabbo, Darlayne; Bergvelt, Maarten "τ-functions, Birkhoff factorizations and difference equations." SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 023, 42 pp. 17B80
Addabbo, Darlayne; Bervelt, Maarten "Generalizations of Q-systems and orthogonal polynomials from representation theory." Lie algebras, vertex operator algebras, and related topics, 1–13, Contemp.
Math., 695, Amer. Math. Soc., Providence, RI, 2017.
Upcoming Conference/Workshop Organization
Special Session on "New Developments in Infinite Dimensional Lie algebras, Vertex Operator Algebras and the Monster" at the Joint Meeting American Mathematical Society-Unione Matematica Italiana July
Women in Noncommutative Algebra and Representation Theory Workshop (WINART4) 2025 at the Banff International Research Station
Awarded Honorable Mention - The University of Arizona 2024 Outstanding Postdoctoral Scholar Award
Proposal for the Women in Noncommutative Algebra and Representation Theory (WINART4) workshop has been ACCEPTED by the Banff International Research Station. The workshop will take place in 2025.
2023 Teaching and Service Award for Post Docs
Association for Women in Mathematics-National Science Foundation Travel Grant (2023)
Mathematical Sciences Research Institute (MSRI), Summer for Women in Mathematics (SWiM) Research Program Award (2020), Rescheduled for Summer 2021
American Mathematical Society-Simons Travel Grant (2018-2020)
Association for Women in Mathematics-National Science Foundation Travel Grant (2018)
Fall 2023: "Introduction to Linear Algebra"
Spring 2023: Math 415B "Second Course in Abstract Algebra"
Fall 2022: Math 415A/515A, "Introduction to Abstract Algebra"
Spring 2022: Math 315 "Introduction to Number Theory and Modern Algebra"
Fall 2021: Math 413 "Linear Algebra"
Spring 2021: Math 313 "Introduction to Linear Algebra"
Fall 2020: Math 125 "Calculus 1"
Selected Service and Outreach
I am co-leading the Women in Noncommutative Algebra and Representation Theory (WINART) network. (http://women-in-ncalg-repthy.org/)
I am co-organizing the Mathematical Physics and Probability Seminar at The University of Arizona.
I am also a mentor for The University of Arizona's Undergraduate Teaching Assistant Program. | {"url":"https://sites.google.com/math.arizona.edu/addabbo/","timestamp":"2024-11-07T10:21:28Z","content_type":"text/html","content_length":"87221","record_id":"<urn:uuid:deccfec4-495f-4466-bf98-0c19c7277a17>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00672.warc.gz"} |
Area of a Circle
Sector Area Calculator
Sector area calculator, as the name suggests, is an online tool that calculates the area of the sector of a circle. All it needs is; the radius and angle to find the area of sector.
This tool can be used to solve geometry problems related to circle. Moreover, it can be used to measure the land in sectors, and some cool things such slice of pizza or cake.
In this article, we will provide you a detailed explanation on sector of circle, how to use sector area calculator, sector definition, formula for sector of circle, how to find the area of a sector
of a circle, and much more.
How to use Area of a Sector Calculator?
Area of a sector calculator provides an elegant and interactive interface to the users. To calculate sector area using this calculator, follow the steps below:
• Select the value for which you want make sector calculation.
• Choose the given set of parameters from the list.
• Enter the radius in the given input box.
• Enter the angle in the next input box.
• Hit the Calculate button to get the sector area.
You will instantly get the area of sector with the step by step demonstration of the calculation. It also shows the formula that it used to find the sector of circle. Sector area calculator only
finds the inner portion of circle. If you want to calculate the circumference of circle, you can use our circumference calculator anytime.
What is a Sector of a Circle?
A circle sector or circular sector is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger is the major sector. As you can see in
the image below, θ is the central angle in radians, "\(\displaystyle L\)" is the arc length of the minor sector, and "\(\displaystyle r\)" is the radius of the circle, and. Wikipedia
Formula to Find Sector Area of a Circle
The formula for finding the sector area of a circle is a simple equation that can be expressed as:
Area of sector of circle = πr^2 × (θ / 360)
In this equation:
• r represents the radius of the circle,
• θ is the angle between sector arcs, and
• π is a mathematical constant.
How to Find the Sector Area of a Circle?
If you are pondering how to find the area of a sector, don’t exhaust yourself. We are here for you. The above area of sector calculator finds the circular sector area in no time and there is no doubt
that. But, you should also be able to calculate it yourself, especially, if you are a student.
To find the area of a circular sector, follow the below steps:
• Write down the radius of circle and angle between arcs.
• Write down the sector area formula.
• Substitute the values and calculate the area of a sector of a circle.
Find the area of a sector of a circle having a radius of 12 cm and an angle of 45°.
Step 1: Write down the radius of circle and angle between arcs.
r = 12 cm, θ = 45°
Step 2: Write down the sector area formula.
Area of sector of circle = πr^2 × (θ / 360)
Step 3: Substitute the values and calculate the area of a sector of a circle.
Area = 3.1415 × (12)^2 × (45° / 360)
Area = 3.1415 × 144 × (45° / 360)
Area = 3240 cm^2
So, the sector area of a circle having a 12 cm radius and 45° angle will be 3240 cm^2 approximately.
Area of a Circle Sector – Real World Example
What will be the size of the pizza slice if the radius of the pizza is 20 cm and its central angle is 30°?
Step 1: Write down the radius of pizza and central angle.
r = 20 cm, θ = 30°
Step 2: Write down the sector area formula.
Area of sector of circle = πr^2 × (θ / 360)
Step 3: Substitute the values and calculate the area of slice of pizza.
Area = 3.1415 × (20)^2 × (30° / 360)
Area = 3.1415 × 400 × (30° / 360)
Area = 6000 cm^2 or 930 sq. inch
So, the sector area of a pizza slice having 20 cm radius and 30° angle will be 6000 cm^2 approximately. You can use our area of a sector calculator to quickly find the area of a sector of a circle
with steps and avoid manual calculations.
How to find the area of a sector?
To find the area of a sector,
• Get the radius and central angle.
• Substitute the values in the area of sector formula, Area = πr^2× (θ / 360).
• Solve the equation after placing the values to get the sector area.
• Or use the Sector Area Calculator to get the area of a sector.
How do you find the area of a sector using online calculator?
To find the area of a sector of the circle using an online calculator, follow the steps below:
• Go to the Area of a Sector Calculator.
• Select the set of parameters that is given.
• Enter the given values in the respective input boxes. i.e., radius, angle, etc.
• Press the Calculate
Bingo! You have got the area of a sector of a circle without getting engaged in complex equations.
What is the formula for the area of a sector of a circle?
The formula for the area of a sector of a circle can be stated as:
Area of sector of circle = πr^2 × (θ / 360)
Where, r represents the radius of the circle, θ is the angle between sector arcs, and π is a mathematical constant.
How do you find the area of a shaded sector?
The area of a shaded sector can be calculated by the same method we calculate the area of a sector. To find the area of a shaded sector:
• Get the radius and central angle.
• Substitute the values in the area of a sector formula, Area = πr^2× (θ / 360).
• Solve the equation after placing the values to get the sector area.
What is a minor sector?
A minor sector of a circle is a sector that has the central angle of less than 180°. The measurement of the central angle of the minor sector cannot exceed 180°. | {"url":"https://www.calculators.tech/sector-area-calculator","timestamp":"2024-11-10T14:27:03Z","content_type":"text/html","content_length":"48699","record_id":"<urn:uuid:550d751c-69b5-4364-a112-27cf94e711c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00028.warc.gz"} |
Amrit Shaw - techiescience.com
In this article we will study about the Schmitt trigger Comparator and Oscillator circuitry with different related parameters in detail. As we have seen till now that an op-amp is used in various
fields of applications and being such a versatile device its importance as a part of analog circuits is immense. One of the most convenient applications of the op-amp is as a multivibrator circuit.
We will be studying in detail about types and working of multivibrator circuit constructed using op-amps (op-amp multivibrators) and other passive devices such as capacitors, diodes, resistors etc.
• Introduction of Multivibrators
• Positive feedback usage in multivibrator
• What is Schmitt trigger ?
• Schmitt trigger comparator closed-loop circuit or bistable multivibrator
• Voltage transfer characteristics of Bistable multivibrator
• Astable multivibrator or Schmitt trigger oscillator
• Oscillator’s duty cycle
Introduction of Multivibrator and Schmitt trigger Circuitry
Multivibrator circuits are sequential logic circuits and are of many types depending on how they are created. Some multivibrators can be made using transistors and logic gates, whereas there are even
dedicated chips available as multivibrators such as NE555 timer. The op-amp multivibrator circuit has a few advantages over other multivibrator circuits as they require much fewer components for
their working, less biasing, and produces better symmetrical rectangular wave signals using comparatively fewer components.
Types of Multivibrators
There are mainly three types of multivibrator circuits present:
1. Astable multivibrator,
2. Monostable multivibrator
3. Bistable multivibrator.
The monostable multivibrator has single stable state, whereas the number of stable-states a bistable multivibrator has- is 2.
As we have learnt in the previous section about op-amp as a comparator, in the open-loop configuration the comparator can switch in an out of control manner between the positive saturation supply
rail voltage and negative saturation supply rail voltage when an input voltage near to that of the reference voltage is applied. Hence, to have control on this uncontrollable switching between the
two states, the op-amp is used in a feedback configuration (closed-loop circuit) which is particularly known as closed-loop Schmitt trigger circuit or bistable multivibrator.
Positive feedback usage in multivibrator and hysteresis effect
Till now, we have learnt about the negative feedback configuration in op-amps in the previous sections. There is also another type of feedback configuration known as positive feedback, which is also
used for specific applications. In positive feedback configuration, the output voltage is fed back (connected) to the non-inverting (positive) input terminal unlike the negative feedback, where the
output voltage was connected to the inverting (negative) input terminal.
An op-amp operated in a positive feedback configuration tends to stay in that particular output state in which it is present, i.e. either the saturated positive or saturated negative state.
Technically, this latching behaviour in one of the two states is known as hysteresis.
If the input applied signal in the comparator consists of some additional harmonics or spikes (noise), then the output of the comparator might switch to the two saturated states unexpectedly and
uncontrollably. In this case, we won’t get a regular symmetrical square wave output of the applied input sinusoidal waveform.
But if we add some positive feedback to the comparator input signal, i.e. use the comparator in a positive feedback configuration; we will be introducing a latching behaviour in the states, what we
technically call as hysteresis into the output. Until and unless there is a major change in the magnitude of the input AC (sinusoidal) voltage signal, the hysteresis effect will continue to make the
output of the circuit remain in its current state.
What is Schmitt trigger ?
The Schmitt trigger or bi-stable multi-vibrator operates in positive feedback configuration with a loop-gain greater than unity to perform as a bi-stable mode. Voltage V[+] can be.
The above figure represents the output voltage versus the input voltage curve (which is also known as the voltage transfer characteristics), particularly showing the hysteresis effect. The transfer
characteristic curve has two specific regions, the curve as the input voltage increases and the part of the curve in which the input voltage decreases. The voltage V[+] does not have a constant
value, but instead, it is a function of the output voltage V[0].
Voltage transfer characteristics
In the voltage transfer characteristics, V[o ]= V[H], or in high state. Then,
Higher Cross-over voltage V[TH]
If signal is less than that of V[+], the output stays at its high state. The cross-over voltage V[TH] occurs when V[i ]= V[+] and expressed as follows:
When V[i] > V[TH], the voltage at the inverting terminal is more than at the non-inverting terminal. Voltage V[+] then turn out to be
Lower Cross-over voltage V[TL]
Since V[L ]< V[H] the input voltage V[i] is still more than V[+], and the output rests in its low state as V[i] carry on to increase; If V[i] decreases, as long as the input voltage V[i] is larger
than V[+], the output remains at saturation state. The cross-over voltage here and now occurs when V[i ]= V[+] and this V[TL] expressed as
As V[i] continues to decrease, it remains less than V[+]; therefore, V[0] remains in its high state. We can observe this transfer characteristic in the above figure. A hysteresis effect is shown in
the net transfer characteristic diagram.
What is Schmitt trigger oscillator ?
Astable multivibrator or Schmitt trigger oscillator
Astable multivibrator accomplished by fixing an RC network to the Schmitt trigger circuit in –ve feedback. As we will advance through the section, we will see that the circuit has no stable
states and therefore, it also known as the astable multivibrator circuit.
As noticed in the figure, an RC network is set in the negative feedback path, and the inverting input terminal is connected to the ground through the capacitor while the non-inverting terminal is
connected to the junction between the resistors R[1] and R[2] as shown in the figure.
At first, R[1] and R[2] is to be equal to R, and assume the output switches symmetrically about zero volts, with the high saturated output represented by V[H ]= V[P] and low saturated output
indicated by V[L ]= -V[P]. If V[0 ]is low, or V[0 ]= -V[P], then V[+] = -(1/2)V[P].
When V[x] drops just slightly below V[+], the output switches to high so that V[0 ]= +V[P] and V[+ ]= +(1/2)V[P]. The equation for the voltage across the capacitor in an RC network can be expressed
Where τ[x ] is the time constant which can be defined asτ[x]= R[x]C[x]. The voltage V[x] increases towards a final voltage V[P] in an exponential manner with respect to time. However, when V[x] turn
out to be slightly greater than V[+ ]= +(1/2)V[P], the output shifts to its low state of V[0] = -V[P] and V[x] = -(1/2)V[P]. The R[x]C[x] network gets triggered by a negative sharp transition of the
voltages, and hence, the capacitor C[x] start discharging, and the voltage V[x] decreasing towards value of –V[P]. We can therefore express V[x ]as
Where t1 refers to the time instant when the output of the circuit switches to its low state. The capacitor discharge exponentially V[+] = -(1/2)V[P], the output again shifts to high. The process
repeats itself continuously over time, which means a square-wave output signal is produced by the oscillations of this positive feedback circuit. The figure below shows the output voltage V[0] and
the capacitor voltage V[x] with respect to time.
Time t[1] can be found by substituting t=t[1] and V[x] = V[P]/2 in the general equation for the voltage across the capacitor.
From the above equation when we solve for t[1], we get
For time t[2] (as observed in the above figure), we approach in a similar way, and, from a similar analysis using the above equation, it is evident that the difference between t[2] and t[1] is also
1.1R[x]C[x]. From this, we can infer that the time period of oscillation T can be defined as T = 2.2 R[x]C[x]
And the frequency thus can be expressed as
Duty cycle of Oscillator
The percentage of time the output voltage (V[0]) of the multi-vibrator is in its high state is particularly termed as the duty cycle of the oscillator.
The oscillator’s duty cycle is
As observed in the figure, depicting output voltage and capacitor voltage versus time, the duty cycle is 50%.
For more Electronics related article click here
Op-Amp As Integrator & Differentiator : Beginner’s Guide!
• What is Integrator?
• Working principle of Integrator
• Op-amp integrator circuit
• Output of an integrator
• Derivation of Op-amp as integrator
• Practical op-amp integrator
• Applications of integrator
• What is Differentiator ?
• Op-amp as Differentiator
• Working Principle of Differentiator
• Output waveform of a differentiator
• Applications of Differentiator
What is Integrator?
Definition of Integrator
If the feedback path is made through a capacitor instead of a resistance , an RC Network has been established across the operational amplifiers’ negative feedback path. This kind of circuit
configuration producing helps in implementing mathematical operation, specifically integration, and this operational amplifier circuit is known as an Operational amplifier Integrator circuit.
The output of the circuit is the integration of the applied input voltage with time.
Integrator circuits are basically inverting operational amplifiers (they work in inverting op-amp configuration, with suitable capacitors and resistors), which generally produce a triangular wave
output from a square wave input. Hence, they are also used for creating triangular pulses.
Op-amp as Integrator
Working principle of Integrator
Operational amplifiers can be used for mathematical applications such as Integration and Differentiation by implementing specific op-amp configurations.
When the feedback path is made through a capacitor instead of a resistance , an RC Network has been established across the operational amplifiers’ negative feedback path. This kind of circuit
configuration producing helps in implementing mathematical operation, specifically integration, and this operational amplifier circuit is known as an Operational amplifier Integrator circuit. The
output of the circuit is the integration of the applied input voltage with time.
Op-amp integrator circuit
Output of an integrator
Integrator circuits are basically inverting operational amplifiers (they work in inverting op-amp configuration, with suitable capacitors and resistors), which generally produce a triangular wave
output from a square wave input. Hence, they are also used for creating triangular pulses.
The current in the feedback path is involved in the charging and discharging of the capacitor; therefore, the magnitude of the output signal is dependent on the amount of time a voltage is present
(applied) at the input terminal of the circuit.
Derivation of Op-amp as integrator
As we know from the virtual ground concept, the voltage at point 1 is 0V. Hence, the capacitor is present between the terminals, one having zero potential and other at potential V[0]. When a constant
voltage is applied at the input, it outcomes in a linearly increasing voltage (positive or negative as per the sign of the input signal) at the output whose rate of change is proportional to the
value of the applied input voltage.
From the above circuitry it is observed, V[1] = V[2] = 0
The input current as:
Due to the op-amp characteristics (the input impedance of the op-amp is infinite) as the input current to the input of an op-amp is ideally zero. Therefore the current passing from the input resistor
by applied input voltage V[i] has flown along the feedback path into the capacitor C[1].
Therefore the current from the output side can also be expressed as:
Equating the above equations we get,
Therefore the op-amp output of this integrator circuit is:
As a consequence the circuit has a gain constant of -1/RC. The negative sign point toward an 180^o phase shift.
Practical op-amp as aintegrator
If we apply a sine wave input signal to the integrator, the integrator allows low-frequency signals to pass while attenuates the high frequencies parts of the signal. Hence, it behaves like a
low-pass filter rather than an integrator.
The practical integrator still has other limitations too. Unlike ideal op-amps, practical op-amps have a finite open-loop gain, finite input impedance, an input offset voltage, and an input bias
current. This deviation from an ideal op-amp can affect working in several ways. For example, if V[in] = 0, current passes through the capacitor due to the presence of both output offset voltage and
input bias current. This causes the drifting of the output voltage over time till the op-amp saturates. If the input voltage current is zero in case of the ideal op-amp, then no drift should be
present, but it is not true for the practical case.
To nullify the effect caused due to the input bias current, we have to modify the circuit such that R[om ]= R[1]||R[F]||R[L]
In this case, the error voltage will be
Therefore the same voltage drop appears at both the positive and negative terminals because of the input bias current.
For an ideal op-amp operating in the dc state, the capacitor performs as an open circuit, and hence, the gain of the circuit is infinite. To overcome this, a high resistance value resistor R[F] is
connected in parallel with the capacitor in the feedback path. Because of this, the gain of the circuit is limited to a finite value (effectively small) and hence gets a small voltage error.
• V[IOS] refers to the input offset voltage
• I[BI] refers to the input bias current
What is Differentiator ?
Definition of Differentiator
If the input resistance in the inverting terminal is replaced by a capacitor, an RC Network has been established across the operational amplifiers’ negative feedback path. This kind of circuit
configuration helps in implementing differentiation of the input voltage, and this operational amplifier circuit configuration is known as an Operational amplifier differentiator circuit.
An operational amplifier differentiator basically works as a high pass filter and, the amplitude of the output voltage produced by the differentiator is proportionate to the change of the applied
input voltage.
Op-amp as a Differentiator
As we have studied earlier in the integrator circuit, op-amps can be used for implementing different mathematical applications. Here we will be studying the differential op-amp configuration in
detail. The differentiator amplifier is also used for creating wave shapes and also in frequency modulators.
An operational amplifier differentiator basically works as a high pass filter and, the amplitude of the output voltage produced by the differentiator is proportionate to the change of the applied
input voltage.
Working Principle of Differentiator
When the input resistance in the inverting terminal is replaced by a capacitor, an RC Network has been established across the operational amplifiers’ negative feedback path. This kind of circuit
configuration helps in implementing differentiation of the input voltage, and this operational amplifier circuit configuration is known as an Operational amplifier differentiator circuit.
In a differentiating op-amp circuit, the output of the circuit is the differentiation of the input voltage applied to the op-amp with respect to time. Therefore the op-amp differentiator works in an
inverting amplifier configuration, which causes the output to be 180 degrees out of phase with the input. Differentiating op-amp configuration generally responds to triangular or rectangular input
A Differentiator Circuit
As shown in the figure, a connection of capacitor in series with the input voltage source has been made. The input capacitor C[1] is initially uncharged and hence operate as an open-circuit. The
non-inverting terminal of the amplifier is connected to the ground, whereas the inverting input terminal is through the negative feedback resistor R[f] and connected to output terminal.
Due to the ideal op-amp characteristics (the input impedance of the op-amp is infinite) as the input current, I to the input of an op-amp is ideally zero. Therefore the current flowing through the
capacitor (in this configuration, the input resistance is replaced by a capacitor) due to the applied input voltage V[in] flows along the feedback path through the feedback resistor R[f].
As observed from the figure, point X is virtually grounded (according to the virtual ground concept) because the non-inverting input terminal is grounded (point Y is at ground potential i.e., 0V).
Consequently, Vx = Vy = 0
With respect to the input side capacitor, the current carrying through the capacitor can be written as:
With respect to the output side feedback resistor, the current flowing through it can be represented as:
From the above equations when we equate the currents in both the results we get,
The differentiating amplifier circuit requires a very small time constant for its application (differentiation), and hence it is one of its main advantages.
The product value C[1]R[f] is known as differentiator’s time constant, and output of the differentiator is C[1]R[f] times the differentiation of V[in] signal. The -ve sign in the equation refers that
the output is 180^o difference in phase with reference to the input.
When we apply a constant voltage with one step change at t=0 like a step signal in the input terminal of the differentiator, the output should be ideally zero as the differentiation of constant is
zero. But in practice, the output is not exactly zero because the constant input wave takes some amount of time to step from 0 volts to some V[max] volts. Therefore the output waveform appears to
have a spike at time t=0.
Therefore for a square wave input, we get something like shown in the below figure,
Output waveform of a differentiator for a square wave input
For more Electronics related article and their detail explanation click here
7 Facts On Log & Antilog Amplifier:What,Working,Circuit,Use
The operational amplifier circuit configurations which can perform mathematical operations such as log and antilog (exponential), including an amplification of the input signal provided to the
circuit, are known as Logarithmic amplifier and Antilogarithmic amplifier respectively. In this section, we are going to learn about the Logarithmic amplifier and Antilog in detail.
• Introduction
• Logarithmic (Log) Amplifier
• Log amplifier configuration
• Diode based Log amplifier configuration
• Transistor based Log amplifier configuration
• Output and Working Principle of Log Amplifier
• Applications of the log amplifier
• What is Antilog?
• Antilog Amplifier
• Log amplifier configuration
• Diode based antilog amplifier configuration
• Transistor based antilog amplifier configuration
• Output and Working Principle of Log Amplifier
• Applications of the antilog amplifier
Logarithm (Log) Amplifier
An operational amplifier in which the output voltage of the amplifier (V[0]) is directly proportional to the natural logarithm of the input voltage (V[i]) is known as a logarithmic amplifier.
Basically, the natural logarithm of the input voltage is multiplied by a constant value and produced as output.
Log Amplifier Circuit
Log Amplifier Using Transistor
Log amplifier using Transistor
Log Amplifier using Diode
Output and Working Principle of Log Amplifier
This can be expressed as follows:
Where K is the constant term, and V[ref] refers to a normalization constant, which we get to know in this section.
Generally, logarithm amplifiers may require more than one op-amp, in which case they are known as compensated logarithm amplifiers. They even require high performing op-amps for their proper
functioning, such as LM1458, LM771, and LM714, are being some of the widely used logarithm amplifier.
The diode is connected in forward biasing. So, the diode current can be represented as:
Where I[s] is the saturation current, V[D] is the voltage drop for the diode. The V[T] is the thermal voltage. The diode current can be rewritten with high biasing condition,
The i[1] expressed by,
Since the voltage at inverting terminal of the op-amp is at virtual ground, hence, the output voltage is given by V[0 ]= -V[D]
Noting that i[1 ]= i[D], we can write
But, as noted earlier, V[D ]= -V[0] and so,
Taking natural logarithm on both sides of this equation, we found
The equation of the output voltage (V[0]) of the logarithm amplifier contains a negative sign, which indicates that there is a phase difference of 180 ^o. Or,
A more advanced one utilize bipolar transistors to remove I[s] in the logarithmic term. In this type of logarithm amplifier configuration, the output voltage is given as:
Applications of the logarithmic amplifier
Log amplifier is used for mathematical applications and also in different devices as per their need. Some of the applications of the log amplifier are as follows:
• Log amplifiers are used for mathematical applications, mainly in multiplication. It is also used in the division and other exponential operations too. As it can perform multiplication operation,
hence it is used in analog computers, in synthesizing audio effects, measuring instruments that require multiplication operation such as in calculating power (multiplication of current and
• As we know that when we need to calculate the decibel equivalent of a given quantity, we require the use of a logarithmic operator, and hence, log amplifiers are used to calculate decibel (dB)
value of a quantity.
• Monolithic logarithmic amplifiers are used in certain situations, like in Radio Frequency domain, for efficient spacing (reducing components and space needed by them), and also to improve
bandwidth and noise rejection.
• It is also used in different ranges of applications such as rot mean square converter, an analog-to-digital converter, etc.
What is Antilog?
Antilog Amplifier
An Op-amp in which the output voltage of the amplifier (V[0]) is directly proportionate to the anti-log of the input voltage (V[i]) is known as an anti-logarithmic amplifier or anti-log
amplifier. Here, we are going to discuss the operational amplifier configuration that forms the anti-logarithmic amplifier in detail.
Antilog Amplifier Circuit
Antilog Amplifier Using Transistor
Antilog Amplifier using Transistor
Antilog Amplifier using Diode
In the antilog amplifier, the input signal is at the inverting pin of the operational amplifier, which passes through a diode.
Antilog Amplifier using Diode
Output and Working Principle of Antilog Amplifier
As observed in the circuit shown above, the negative feedback is achieved by connecting the output to the inverting input terminal. According to the concept of the virtual ground between the input
terminals of an amplifier, the voltage V[1] at the inverting terminal will be zero. Because of ideally infinite input impedance, the current flowing through the diode due to the applied input voltage
in the inverting terminal will not enter the op-amp; instead, it will flow along the feedback path through the resistor R as shown in the figure.
The compliment or inverse function of the logarithmic amplifier is ‘exponential’, anti-logarithmic or simply known as ‘antilog’. Consider the circuit given in the figure. The diode current is
Where, V[D] is the diode voltage. According to the concept of virtual ground, V[1]=0 as the non-inverting terminal is grounded as shown in the figure. Therefore the voltage across the diode can be
expressed as V[D ]= V[i ]– V[1] or V[D] = V[i] Hence, the current through the diode is
Due to the ideal characteristics of an op-amp (infinite input impedance), the current flowing through the diode ( i[D]) flows along the feedback path through the resistor R, as we can observe in the
Therefore i[D ]= i[2]
And, V[0] = -i[2]R = -i[D]R
Replacing i[D] in the above equation we get
The parameters n, V[T] and I[S ]are constants (they are only depend on the diode characteristics which are always constant for a particular diode). Therefore if the value of the feedback resistor R
is fixed, then the output voltage V[0] is directly proportional to the natural anti-logarithm (exponential) of the applied input voltage V[i]. The above equation then can be simply represented as
Where K = – I[S]R and a =
Therefore we can notice that the anti-logarithmic op-amp produces its output signal as the exponential value of the input voltage signal applied.
The gain of the anti-log amplifier is given by the value of K that is equal to -I[S]R.
The –ve sign point out that there is a phase difference of 180degrees between the applied input s and the output of the anti-log amplifier.
For more Electronics related article click here
Inverting Operational Amplifier Trans Impedance Amp: A Comprehensive Guide
The inverting operational amplifier trans impedance amplifier (TIA) is a versatile circuit that converts a current input signal into a voltage output signal. This type of amplifier is commonly used
with current-based sensors, such as photodiodes, due to its unique characteristics and performance advantages. In this comprehensive guide, we will delve into the technical details, design
considerations, and practical applications of the inverting operational amplifier trans impedance amp.
Understanding the Inverting Operational Amplifier Trans Impedance Amp
The inverting operational amplifier trans impedance amplifier is a specialized circuit that leverages the properties of an operational amplifier (op-amp) to perform current-to-voltage conversion. The
key feature of this circuit is its ability to maintain a high input impedance, which is crucial for accurately measuring and amplifying current-based signals.
Input Impedance Characteristics
One of the most interesting aspects of the inverting operational amplifier trans impedance amp is its input impedance behavior. Algebraically, the input impedance of this circuit is found to be
proportional to the frequency and resembles the impedance of an inductor. The equivalent inductance can be calculated using the formula:
L_eq = R_f / (2 * π * f)
– L_eq is the equivalent inductance
– R_f is the feedback resistor
– f is the frequency
This means that for low frequencies, the input impedance is high, while for high frequencies, the input impedance is low. This behavior can be attributed to the op-amp’s gain-bandwidth product, which
determines the frequency range over which the amplifier maintains its desired characteristics.
Gain-Bandwidth Product
The gain-bandwidth product (GBW) of the op-amp used in the inverting operational amplifier trans impedance amp is a crucial parameter that affects the circuit’s performance. The gain at a given
frequency is equal to the GBW divided by the frequency. This relationship is expressed as:
Gain = GBW / f
The GBW determines the frequency range over which the amplifier can maintain a stable and predictable gain. For frequencies much lower than the op-amp’s GBW, the input impedance is high, while for
frequencies much higher than the GBW, the input impedance is low.
Input and Output Impedance Characteristics
The inverting operational amplifier trans impedance amp exhibits distinct input and output impedance characteristics:
1. Input Impedance:
2. At low frequencies (much lower than the op-amp’s GBW), the input impedance is high and proportional to the frequency, resembling the impedance of an inductor.
3. At high frequencies (much higher than the op-amp’s GBW), the input impedance is low and looks like the impedance of a resistor with a value equal to the feedback resistor.
4. Output Impedance:
5. The output impedance of the inverting operational amplifier trans impedance amp is low, similar to other op-amp-based circuits.
These impedance characteristics make the TIA a superior choice for current-to-voltage conversion compared to using a simple resistor. The high input impedance at low frequencies allows for accurate
measurement of current-based signals, while the low output impedance ensures efficient signal transfer to subsequent stages.
Design Considerations for Inverting Operational Amplifier Trans Impedance Amp
When designing an inverting operational amplifier trans impedance amp, there are several key factors to consider to ensure optimal performance and meet the specific requirements of the application.
Feedback Resistor Selection
The feedback resistor, R_f, plays a crucial role in determining the overall gain and input impedance characteristics of the TIA. The value of R_f should be chosen carefully based on the following
1. Desired Transimpedance Gain: The transimpedance gain of the TIA is equal to the value of the feedback resistor, R_f. Higher values of R_f will result in higher transimpedance gain, but may also
introduce stability issues and increase the equivalent inductance of the input impedance.
2. Input Current Range: The maximum input current that the TIA can handle is limited by the maximum output voltage of the op-amp and the value of R_f. The maximum input current should be kept within
the op-amp’s output voltage range to avoid saturation or clipping.
3. Equivalent Inductance: As mentioned earlier, the equivalent inductance of the input impedance is inversely proportional to the frequency and directly proportional to the value of R_f. For slow
op-amps and large transimpedances, the equivalent inductance can become quite significant, which may affect the circuit’s stability and frequency response.
Op-Amp Selection
The choice of the operational amplifier used in the TIA is critical, as it directly impacts the circuit’s performance and characteristics. Key parameters to consider when selecting an op-amp include:
1. Gain-Bandwidth Product (GBW): The GBW of the op-amp determines the frequency range over which the amplifier maintains its desired characteristics. A higher GBW is generally preferred to extend
the frequency range of the TIA.
2. Input Offset Voltage: The input offset voltage of the op-amp can introduce errors in the current-to-voltage conversion, especially for low-level input currents. Op-amps with low input offset
voltage are preferred for high-precision TIA designs.
3. Input Bias Current: The input bias current of the op-amp can also contribute to errors in the current-to-voltage conversion. Op-amps with low input bias current are desirable for TIA
4. Slew Rate: The slew rate of the op-amp determines the maximum rate of change in the output voltage, which can be important for high-speed or high-frequency TIA applications.
5. Noise Performance: The noise characteristics of the op-amp, such as input-referred voltage noise and current noise, can impact the signal-to-noise ratio of the TIA, especially for low-level input
Stability Considerations
The inverting operational amplifier trans impedance amp can be susceptible to stability issues, particularly at high frequencies or with large values of R_f. To ensure stable operation, the following
design considerations should be addressed:
1. Compensation Capacitor: Adding a compensation capacitor, C_c, in parallel with the feedback resistor, R_f, can help stabilize the TIA by introducing a dominant pole and improving the phase
2. Bandwidth Limiting: Limiting the bandwidth of the TIA, either through the use of a low-pass filter or by selecting an op-amp with a lower GBW, can help improve the stability of the circuit.
3. Feedback Resistor Value: As mentioned earlier, the value of R_f can significantly impact the equivalent inductance of the input impedance, which can lead to stability issues. Careful selection of
R_f is crucial for maintaining stable operation.
4. Parasitic Capacitances: Parasitic capacitances, such as those introduced by the op-amp, the feedback resistor, and the input wiring, can also affect the stability of the TIA. Minimizing these
parasitic capacitances through proper layout and shielding techniques can help improve the circuit’s stability.
Applications of Inverting Operational Amplifier Trans Impedance Amp
The inverting operational amplifier trans impedance amp finds numerous applications in various fields, particularly in the realm of current-based sensor interfacing and signal conditioning.
Photodiode Amplifier
One of the most common applications of the TIA is as a photodiode amplifier. Photodiodes are current-based sensors that generate a current proportional to the incident light intensity. The TIA is an
ideal choice for converting the photodiode’s current output into a voltage signal that can be further processed or measured.
Current Sensing
The TIA can also be used for general current sensing applications, where the input current is converted into a proportional voltage signal. This is useful in power management, motor control, and
other systems where accurate current monitoring is required.
Electrochemical Sensor Interfaces
In the field of electrochemical sensing, the TIA is often employed to interface with current-based sensors, such as amperometric electrodes or ion-selective electrodes. The high input impedance of
the TIA allows for accurate measurement of the small currents generated by these sensors.
Radiation Detection
In radiation detection systems, such as those used in medical imaging or nuclear instrumentation, the TIA is commonly used to amplify the current signals generated by radiation detectors, such as
photodiodes or avalanche photodiodes (APDs).
Impedance Measurement
The unique input impedance characteristics of the TIA can be leveraged for impedance measurement applications. By monitoring the voltage output of the TIA, the input impedance of the circuit under
test can be determined, which can be useful in various electrical and electronic characterization tasks.
The inverting operational amplifier trans impedance amplifier is a versatile and powerful circuit that plays a crucial role in a wide range of applications, particularly in the field of current-based
sensor interfacing and signal conditioning. By understanding the technical details, design considerations, and practical applications of the TIA, electronics engineers and researchers can leverage
this circuit to achieve accurate, stable, and efficient current-to-voltage conversion in their projects.
Overview of Differential Amplifier Bridge Amplifier
A differential amplifier bridge amplifier is a specialized electronic circuit that combines the functionality of a differential amplifier and a bridge amplifier. It is widely used in applications
that require high precision, noise immunity, and the ability to amplify small voltage differences, such as strain gauge measurements and data acquisition systems.
Technical Specifications
• The gain of a differential amplifier bridge amplifier is typically high, ranging from 50 to 100. This high gain allows for the effective amplification of small voltage differences between the
input signals.
Input Voltage Range
• The input voltage range of a differential amplifier bridge amplifier depends on the specific operational amplifier (op-amp) used in the circuit. For example, the LM358 op-amp can handle input
voltages up to 32V, while the TLV2772A op-amp can handle input voltages up to 36V.
Common-Mode Rejection Ratio (CMRR)
• The CMRR of a differential amplifier bridge amplifier is typically high, often exceeding 80 dB. This high CMRR ensures that the amplifier effectively rejects common-mode noise and only amplifies
the desired differential signal.
Noise Immunity
• Differential amplifier bridge amplifiers are highly resistant to external noise sources due to their differential signaling architecture. This makes them suitable for use in noisy environments,
where they can maintain high accuracy and reliability.
Output Voltage Swing
• The output voltage swing of a differential amplifier bridge amplifier can be quite high, often up to 90% of the supply voltage. This large output voltage range allows the amplifier to be used in
a variety of applications.
Physics and Theoretical Explanation
The operation of a differential amplifier bridge amplifier is based on the principles of differential signaling and amplification. The amplifier takes two input signals, V1 and V2, and amplifies
their difference, Vdm = V1 - V2. This is achieved through a combination of resistors and op-amps that create a differential gain stage.
The output voltage of the amplifier can be expressed as:
Vout = KVdm + Vref
where K is the gain of the amplifier and Vref is the reference voltage.
Examples and Numerical Problems
Strain Gauge Measurement
Consider a strain gauge connected to a Wheatstone bridge, which is then connected to a differential amplifier bridge amplifier. If the strain gauge resistance changes from 350 Ohms to 351 Ohms, the
output voltage of the bridge changes from -5.365 mV to -5.365 mV + 134 mV = 128.635 mV.
Differential Gain Calculation
Given a differential amplifier bridge amplifier with resistors R1 = R2 = 1 kΩ and R3 = R4 = 50 kΩ, calculate the differential gain K.
K = R3/R1 = 50 kΩ/1 kΩ = 50
Figures and Data Points
Circuit Diagram
A typical differential amplifier bridge amplifier circuit consists of a Wheatstone bridge connected to a differential amplifier stage, which is then followed by additional gain stages.
Output Voltage vs. Input Voltage
The output voltage of the amplifier increases linearly with the differential input voltage, with a slope determined by the gain of the amplifier.
Measurements and Applications
Strain Gauge Measurements
Differential amplifier bridge amplifiers are commonly used in strain gauge measurements to amplify the small voltage changes produced by the strain gauge. This allows for accurate monitoring and
analysis of mechanical deformation in various structures and materials.
Data Acquisition Systems
These amplifiers are also used in data acquisition systems to amplify and condition signals from various sensors, ensuring high accuracy and noise immunity. This is particularly important in
applications where the input signals are weak or susceptible to interference, such as in industrial automation, biomedical instrumentation, and environmental monitoring.
Faraday’s Law of Induction, Lenz’s Law, and Magnetic Flux: A Comprehensive Guide
Faraday’s Law of Induction and Lenz’s Law are fundamental principles in electromagnetism that describe the relationship between changing magnetic fields and the induced electromotive forces (EMFs)
they create. These laws are essential for understanding the behavior of various electromagnetic devices, from transformers and generators to induction motors and wireless charging systems. In this
comprehensive guide, we will delve into the mathematical formulations, key concepts, practical applications, and numerical examples related to these important laws.
Faraday’s Law of Induction
Faraday’s Law of Induction states that the induced EMF in a circuit is proportional to the rate of change of the magnetic flux through the circuit. The mathematical expression for Faraday’s Law is:
\text{emf} = -N \frac{\Delta \Phi}{\Delta t}
– emf: Electromotive force (volts, V)
– N: Number of turns in the coil
– ΔΦ: Change in magnetic flux (weber, Wb)
– Δt: Time over which the flux changes (seconds, s)
The negative sign in the equation indicates that the induced EMF opposes the change in magnetic flux, as described by Lenz’s Law.
Magnetic Flux
Magnetic flux, denoted as Φ, is a measure of the total magnetic field passing through a given surface or area. The formula for magnetic flux is:
\Phi = B \cdot A \cdot \cos \theta
– Φ: Magnetic flux (weber, Wb)
– B: Magnetic field strength (tesla, T)
– A: Area of the coil (square meters, m²)
– θ: Angle between the magnetic field and the coil normal (degrees)
The magnetic flux is directly proportional to the magnetic field strength, the area of the coil, and the cosine of the angle between the magnetic field and the coil normal.
Lenz’s Law
Lenz’s Law states that the direction of the induced current in a circuit is such that it opposes the change in the magnetic flux that caused it. In other words, the induced current will create a
magnetic field that opposes the original change in the magnetic field.
To determine the direction of the induced current, you can use the right-hand rule:
1. Point your thumb in the direction of the magnetic field.
2. Curl your fingers around the coil or circuit.
3. The direction your fingers curl is the direction of the induced current.
This rule helps you visualize the direction of the induced current and ensures that it opposes the change in the magnetic flux, as described by Lenz’s Law.
Examples and Applications
Induction Cooker
• Magnetic Field Strength: Typically around 100 mT (millitesla)
• Frequency: 27 kHz (kilohertz)
• Induced EMF: High values due to the high rate of change of the magnetic field
Induction cookers use the principles of electromagnetic induction to heat cookware. The rapidly changing magnetic field induces a high EMF in the metal cookware, which in turn generates heat through
eddy currents.
• Mutual Inductance: The ability of two coils to induce EMFs in each other
• Efficiency: Transformers can achieve high efficiency (up to 99%) due to the principles of electromagnetic induction
Transformers rely on the mutual inductance between two coils to step up or step down the voltage in an electrical system. The changing magnetic field in the primary coil induces a corresponding EMF
in the secondary coil, allowing for efficient power transformation.
Electric Generator
• EMF: Varies sinusoidally with time
• Angular Velocity: The coil is rotated at a constant angular velocity to produce the EMF
Electric generators convert mechanical energy into electrical energy by using the principles of electromagnetic induction. As a coil is rotated in a magnetic field, the changing magnetic flux induces
an EMF that varies sinusoidally with time.
Numerical Problems
Example 1
• Change in Flux: 2 Wb to 0.2 Wb in 0.5 seconds
• Induced EMF: Calculate the induced EMF using Faraday’s Law
\Delta \Phi = 0.2 – 2 = -1.8 \text{ Wb}
\text{emf} = -N \frac{\Delta \Phi}{\Delta t} = -N \frac{-1.8}{0.5} = 3.6 N \text{ V}
Example 2
• Coil Area: 0.1 m²
• Magnetic Field Strength: 0.5 T
• Angle: 30°
• Number of Turns: 100
• Time: 0.2 seconds
• Change in Flux: Calculate the change in flux and the induced EMF
\Phi = B \cdot A \cdot \cos \theta = 0.5 \cdot 0.1 \cdot \cos 30° = 0.043 \text{ Wb}
\Delta \Phi = 0.043 \text{ Wb}
\text{emf} = -N \frac{\Delta \Phi}{\Delta t} = -100 \frac{0.043}{0.2} = -21.5 \text{ V}
Transformer Equations Working Energy Loss: A Comprehensive Guide
Transformer equations play a crucial role in understanding and quantifying the energy losses associated with transformer operations. This comprehensive guide delves into the technical details, data
points, and research insights that shed light on the complex dynamics of transformer energy losses, equipping physics students with a robust understanding of this essential topic.
Transformer Losses Due to Harmonics
Harmonics, which are distortions in the sinusoidal waveform of the electrical supply, can significantly contribute to energy losses in transformers. Let’s explore the quantifiable data points that
illustrate the impact of harmonics on transformer performance:
Transformer Losses
1. Total Losses in Transformer Due to Harmonics: 3.7 kW
2. Cable Losses Due to Harmonics: 0.74 kW
3. Total Savings After Installation of Filter: 4.4 kW
These figures demonstrate the substantial energy losses that can be attributed to harmonics in the electrical system, highlighting the importance of implementing effective mitigation strategies.
Power Factor Improvement
1. Power Factor Before Installation of Advanced Universal Passive Harmonic Filter: Not specified
2. Power Factor After Installation of Advanced Universal Passive Harmonic Filter: 0.99
The significant improvement in power factor, from an unspecified value to 0.99, illustrates the positive impact of the harmonic filter on the overall power quality and efficiency of the transformer
KVA Reduction
1. KVA Before Installation of Filter: 88.6 KVA
2. KVA After Installation of Filter: 68.5 KVA
3. Total KVA Savings: 20 KVA
The reduction in KVA, from 88.6 to 68.5, showcases the substantial capacity savings achieved through the installation of the harmonic filter, further enhancing the overall efficiency and performance
of the transformer.
Return on Investment (ROI)
1. Filter Cost: ₹2,10,000
2. Total Savings Per Year: ₹3,62,112
3. ROI: 7 months
The impressive return on investment, with a payback period of just 7 months, underscores the financial benefits of implementing effective harmonic mitigation strategies in transformer systems.
Loss Reduction Strategies
Alongside the quantifiable data on the impact of harmonics, various loss reduction strategies have been explored in the research, offering valuable insights for physics students:
Line Loss Interval
1. Line Loss Interval Estimation: A model can estimate the reasonable line loss interval based on transformer operation data.
This approach allows for a more accurate assessment of line losses, enabling better optimization and management of the transformer system.
Loss Modelling
1. Accurate Loss Modelling: Static piecewise linear loss approximation based on line loading classification can achieve accurate loss modelling.
Precise loss modelling is crucial for understanding the energy dynamics within the transformer and developing effective strategies to minimize losses.
Line Loss Calculation
1. Line Loss Calculation Method: A method based on big data and load curve can be used for line loss calculation.
The utilization of big data and load curve analysis provides a comprehensive approach to estimating and managing line losses, contributing to the overall efficiency of the transformer system.
Energy Conservation Standards
Regulatory bodies, such as the U.S. Department of Energy (DOE), have established guidelines and standards to promote energy efficiency in transformer systems. These standards offer valuable insights
for physics students:
Energy Efficiency
1. DOE Guidance: The U.S. Department of Energy (DOE) advises on analytical methods, data sources, and key assumptions for energy conservation standards in distribution transformers.
Understanding these energy conservation standards and the underlying analytical approaches can help physics students develop a deeper understanding of the regulatory landscape and its impact on
transformer design and operation.
Research on Transformer Operation
The research landscape on transformer operation has yielded valuable insights that can enhance the understanding of physics students:
Fuzzy Comprehensive Evaluation
1. Transformer Working State Evaluation: A multi-level evaluation method based on key performance indicators can be used to evaluate the working state of transformers.
This comprehensive evaluation approach provides a holistic assessment of transformer performance, enabling better monitoring and optimization of the system.
Transformer Losses and Temperature Rise
1. Correlations in Transformer Operation: The heating temperature rise has correlations to the loading current, power losses, efficiency, and surface area.
Exploring these correlations between transformer parameters can help physics students develop a more nuanced understanding of the complex relationships that govern transformer energy losses and
By delving into the technical details, data points, and research insights presented in this comprehensive guide, physics students can gain a deeper understanding of the intricate dynamics of
transformer equations and their impact on energy losses. This knowledge will equip them with the necessary tools to tackle real-world challenges in the field of power systems and transformer design.
1. https://www.linkedin.com/pulse/incredible-power-losses-caused-harmonics-measurable-waveforms
2. https://www.sciencedirect.com/science/article/abs/pii/S0306261921014021
3. https://www1.eere.energy.gov/buildings/appliance_standards/pdfs/dt_nopr_tsd_complete.pdf
4. https://link.springer.com/chapter/10.1007/978-981-97-3940-0_6
5. https://www.researchgate.net/publication/326317282_Investigation_of_transformer_losses_and_temperature_rise
Hall Effect Sensor Magnetic Sensors Applications: A Comprehensive Guide
Hall effect sensors are versatile devices that have found widespread applications in various industries, from automotive to medical and industrial applications. These sensors leverage the Hall
effect, a fundamental principle in physics, to detect and measure magnetic fields, enabling a wide range of functionalities. In this comprehensive guide, we will delve into the technical details,
theoretical explanations, and practical applications of hall effect sensor magnetic sensors.
Automotive Applications
Seat and Safety Belt Position Sensing
Hall effect sensors are used in vehicles to detect the position of seats and safety belts, ensuring that the appropriate safety features are activated. These sensors monitor the position of the seat
and safety belt, providing feedback to the vehicle’s control systems to optimize occupant protection.
Windshield Wiper Position Sensing
Hall effect sensors are employed to monitor the position of windshield wipers, enabling precise control and ensuring proper operation. By detecting the wiper’s position, the vehicle’s control systems
can synchronize the wiper movement with other systems, such as the rain sensor, to enhance driving visibility and safety.
Brake and Gas Pedal Position Sensing
Hall effect sensors are utilized to detect the position and movement of brake and gas pedals in vehicles. This information is crucial for the vehicle’s safety and control systems, as it allows for
the precise monitoring and regulation of the pedal inputs, enhancing overall driving performance and responsiveness.
Ignition System Position Sensing
Hall effect sensors play a vital role in the ignition system of vehicles, detecting the position of the ignition switch. This information is used to ensure proper engine operation, enabling the
vehicle’s control systems to synchronize the ignition timing and other engine-related functions.
Industrial Applications
Current Measurement
Hall effect sensors can be employed to measure current by detecting the magnetic field generated by the current flow. This capability is valuable for monitoring the performance and ensuring the
safety of industrial equipment, as it allows for the continuous monitoring of current levels and the detection of any abnormalities.
Gear Tooth Sensing
Hall effect sensors are used to detect the presence or absence of gear teeth, enabling accurate gear position detection and control. This application is crucial in industrial machinery, where precise
gear positioning is essential for efficient operation and performance.
Proximity Detection
Hall effect sensors are utilized in industrial settings for proximity detection, identifying the presence or absence of objects. This functionality is valuable in applications such as door sensors,
object detection systems, and various automation processes.
Medical and Biomedical Applications
Magnetic Bead Detection
In biomedical applications, Hall effect sensors are employed to detect magnetic beads, which are commonly used in immunoassays and protein detection. These sensors can precisely identify the presence
and location of the magnetic beads, enabling advanced diagnostic and research capabilities.
Magnetic Nanoparticle Detection
Hall effect sensors are also used to detect magnetic nanoparticles, which have numerous applications in biomedical research and diagnostics. These sensors can provide valuable insights into the
behavior and distribution of magnetic nanoparticles, contributing to advancements in areas such as drug delivery, biosensing, and medical imaging.
Other Applications
Fluid Flow Sensing
Hall effect sensors can be used to detect changes in fluid flow by measuring the magnetic field generated by the fluid flow. This application is beneficial in various industries, including process
control, automation, and environmental monitoring.
Pressure Sensing
Hall effect sensors can be employed to detect changes in pressure by measuring the magnetic field generated by the pressure changes. This capability is useful in applications such as industrial
process control, automotive systems, and medical devices.
Building Automation
Hall effect sensors are utilized in building automation systems to detect the presence or absence of objects, such as in door sensors or object detection systems. This functionality contributes to
the optimization of building operations, energy efficiency, and security.
Technical Specifications
Hall effect sensors can detect magnetic fields as low as a few microtesla (μT), making them highly sensitive to even small changes in magnetic fields.
Hall effect sensors can achieve a resolution as high as 1 microtesla (μT), enabling precise measurements of magnetic field variations.
Operating Frequency
Hall effect sensors can operate at frequencies up to 100 kilohertz (kHz), allowing for high-speed applications and real-time monitoring.
Power Consumption
Hall effect sensors typically consume low power, often in the range of milliwatts (mW), making them suitable for battery-powered or energy-efficient applications.
Theoretical Explanation
The Hall effect is a fundamental principle in physics that describes the generation of a voltage perpendicular to both the direction of current flow and the applied magnetic field. When a
current-carrying conductor or semiconductor is placed in a magnetic field, the magnetic field exerts a force on the moving charge carriers, causing them to accumulate on one side of the material.
This accumulation of charge carriers results in the generation of a voltage, known as the Hall voltage, which is proportional to the strength of the magnetic field and the current flowing through the
Physics Formulae
Hall Voltage
The Hall voltage (V_H) can be calculated using the following formula:
V_H = (G * t * N * r_n * q * I_bias * B) / (e * n)
– G is the geometric factor
– t is the thickness of the Hall device
– N is the impurity concentration
– r_n is the Hall factor
– q is the charge per unit charge
– I_bias is the bias current
– B is the applied magnetic field strength
– e is the elementary charge
– n is the carrier concentration
Magnetic Flux
The magnetic flux (Φ) can be calculated using the formula:
Φ = B * A
– B is the magnetic field strength
– A is the area of the sensing unit normal to the magnetic field
Eddy Currents and Electromagnetic Damping: A Comprehensive Guide
Eddy currents and their applications in electromagnetic damping are crucial in various fields, from laboratory equipment to industrial processes. This comprehensive guide delves into the quantitative
analysis of eddy current damping, its theoretical background, and a wide range of practical applications.
Quantitative Analysis of Eddy Current Damping
Damping Coefficients
Researchers have conducted laboratory experiments to measure the damping coefficients for different magnet and track combinations. The results provide valuable insights into the effectiveness of eddy
current damping:
Combination Damping Coefficient (N s m⁻¹)
Cu1-A 0.039 ± 0.001
Cu3-A 0.081 ± 0.001
Cu1-M1 0.194 ± 0.001
Cu3-M1 0.378 ± 0.001
These measurements demonstrate the significant impact of the magnet and track materials on the damping coefficient, with the Cu3-M1 combination exhibiting the highest damping effect.
Kinetic Friction Coefficients
In addition to damping coefficients, researchers have also measured the kinetic friction coefficients for the same magnet and track combinations:
Combination Kinetic Friction Coefficient
Cu1-A 0.22 ± 0.02
Cu3-A 0.21 ± 0.01
Cu1-M1 0.20 ± 0.04
Cu3-M1 0.20 ± 0.01
These values provide a comprehensive understanding of the frictional forces involved in eddy current damping systems, which is crucial for designing and optimizing various applications.
Applications of Eddy Currents and Magnetic Damping
Magnetic Damping in Laboratory Balances
Magnetic damping is widely used in laboratory balances to minimize oscillations and maximize sensitivity. The drag force created by eddy currents is proportional to the speed of the moving object,
and it becomes zero at zero velocity, allowing for precise measurements.
Metal Separation in Recycling
Eddy currents are employed in recycling centers to separate metals from non-metals. The conductive metals are slowed down by the magnetic damping effect, while the non-metals continue to move,
enabling efficient separation and recovery of valuable materials.
Metal Detectors
Portable metal detectors utilize the principle of eddy currents to detect the presence of metals. These devices consist of a coil that generates a magnetic field, which induces eddy currents in
nearby conductive objects, allowing for their detection.
Braking Systems
Eddy currents are employed in braking systems for high-speed applications, such as trains and roller coasters. The induced eddy currents create a braking force that slows down the moving objects,
providing an effective and reliable means of deceleration.
Theoretical Background
Eddy Current Generation
Eddy currents are generated when a conductor moves in a magnetic field or when a magnetic field moves relative to a conductor. This phenomenon is based on the principle of motional electromotive
force (emf), where the relative motion between the conductor and the magnetic field induces a voltage, which in turn generates the eddy currents.
The magnitude of the induced eddy currents is proportional to the rate of change of the magnetic field and the electrical conductivity of the material. The direction of the eddy currents is such that
they oppose the change in the magnetic field, as described by Lenz’s law.
Magnetic Damping
Magnetic damping occurs when the eddy currents induced in a moving conductor produce a drag force that opposes the motion. This drag force is proportional to the velocity of the conductor and the
strength of the magnetic field. The damping force acts to dissipate the kinetic energy of the moving object, effectively slowing it down.
The mathematical expression for the magnetic damping force is given by:
F_d = -b * v
– F_d is the damping force
– b is the damping coefficient
– v is the velocity of the moving object
The damping coefficient, b, depends on the geometry of the system, the magnetic field strength, and the electrical conductivity of the material.
Eddy currents and electromagnetic damping have a wide range of applications in various fields, from laboratory equipment to industrial processes. The quantitative analysis of damping coefficients and
kinetic friction coefficients provides valuable insights into the performance and optimization of these systems. Understanding the theoretical background of eddy current generation and magnetic
damping is crucial for designing and implementing effective solutions in diverse applications.
1. Molina-Bolivar, J. A., & Abella-Palacios, A. J. (2012). A laboratory activity on the eddy current brake. European Journal of Physics, 33(3), 697-706. doi: 10.1088/0143-0807/33/3/697
2. Lumen Learning. (n.d.). Eddy Currents and Magnetic Damping. Retrieved from https://courses.lumenlearning.com/suny-physics/chapter/23-4-eddy-currents-and-magnetic-damping/
3. Griffiths, D. J. (2013). Introduction to Electromagnetism (4th ed.). Pearson.
4. Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). Wiley.
Overview of Magnets: Electromagnets, Permanent, Hard, and Soft
Magnets are materials that produce a magnetic field, which can attract or repel other magnetic materials. Understanding the different types of magnets and their properties is crucial in various
applications, from electric motors and generators to medical imaging and data storage. In this comprehensive guide, we will delve into the measurable and quantifiable data on electromagnets,
permanent magnets, hard magnets, and soft magnets.
Permanent Magnets
Permanent magnets are materials that can maintain a magnetic field without the need for an external source of electricity. These magnets are characterized by several key properties:
Magnetic Field Strength
The magnetic field strength of a permanent magnet is a measure of the intensity of the magnetic field it produces. The strength of the magnetic field is typically measured in Tesla (T) or Gauss (G).
Neodymium (NdFeB) magnets, for example, can have a magnetic field strength of up to 1.4 T, while samarium-cobalt (SmCo) magnets can reach around 1.1 T.
Coercivity, also known as the coercive force, is the measure of a permanent magnet’s resistance to demagnetization. It is the strength of the external magnetic field required to reduce the
magnetization of the material to zero. Permanent magnets with high coercivity, such as NdFeB (around 1.9 T) and SmCo (around 4.4 T), are more resistant to demagnetization.
Remanence, or residual magnetization, is the measure of the magnetic flux density that remains in a material after an external magnetic field is removed. Permanent magnets with high remanence, such
as NdFeB (around 32.5 μB per formula unit) and SmCo (around 8 μB per formula unit), can maintain a strong magnetic field even without an external source.
Curie Temperature
The Curie temperature is the temperature above which a ferromagnetic material loses its ferromagnetic properties and becomes paramagnetic. For permanent magnets, the Curie temperature is an important
consideration, as it determines the maximum operating temperature. NdFeB magnets have a Curie temperature of around 312°C, while SmCo magnets can withstand higher temperatures, up to around 800°C.
Electromagnets are devices that produce a magnetic field when an electric current flows through a coil of wire. Unlike permanent magnets, the magnetic field of an electromagnet can be turned on and
off, and its strength can be adjusted by controlling the electric current.
Magnetic Field Strength
The magnetic field strength of an electromagnet is directly proportional to the electric current flowing through the coil. The strength can be calculated using the formula:
B = μ₀ * N * I / L
– B is the magnetic field strength (in Tesla)
– μ₀ is the permeability of free space (4π × 10^-7 T⋅m/A)
– N is the number of turns in the coil
– I is the electric current (in Amperes)
– L is the length of the coil (in meters)
The magnetic field strength of an electromagnet can be varied by adjusting the electric current, making them useful in applications where a controllable magnetic field is required.
Coercivity and Remanence
Electromagnets do not have a fixed coercivity or remanence, as their magnetic properties are entirely dependent on the electric current flowing through the coil. When the current is turned off, the
electromagnet loses its magnetization, and there is no residual magnetic field.
Curie Temperature
Electromagnets do not have a Curie temperature, as they are not made of ferromagnetic materials. The magnetic field is generated by the flow of electric current, rather than the alignment of magnetic
domains within the material.
Hard Magnets
Hard magnets, also known as permanent magnets, are materials that can maintain a strong, persistent magnetic field. These magnets are characterized by their high coercivity and remanence, making them
resistant to demagnetization.
The coercivity of hard magnets is a measure of their resistance to demagnetization. Materials with high coercivity, such as NdFeB (around 1.9 T) and SmCo (around 4.4 T), are considered “hard” magnets
and are less susceptible to losing their magnetization.
Hard magnets have a high remanence, meaning they can retain a significant amount of magnetization even after the external magnetic field is removed. For example, the remanence of NdFeB magnets is
around 32.5 μB per formula unit, and for SmCo magnets, it is around 8 μB per formula unit.
Curie Temperature
The Curie temperature of hard magnets is an important consideration, as it determines the maximum operating temperature before the material loses its ferromagnetic properties. NdFeB magnets have a
Curie temperature of around 312°C, while SmCo magnets can withstand higher temperatures, up to around 800°C.
Soft Magnets
Soft magnets are materials that can be easily magnetized and demagnetized. They are characterized by their low coercivity and remanence, making them suitable for applications where a variable
magnetic field is required.
The coercivity of soft magnets is relatively low, typically around 0.080 T for iron and 0.40 T for ferrites. This low coercivity allows soft magnets to be easily magnetized and demagnetized.
Soft magnets have a low remanence, meaning they retain a relatively small amount of magnetization after the external magnetic field is removed. For instance, the remanence of iron is around 1.2 T,
and that of ferrites is around 0.5 T.
Curie Temperature
The Curie temperature of soft magnets is generally lower than that of hard magnets. For example, the Curie temperature of iron is around 770°C.
Magnetic Hysteresis
Magnetic hysteresis is the phenomenon where the magnetization of a material depends on its magnetic history. This behavior is characterized by the material’s hysteresis loop, which is defined by the
remanence (M_r) and coercivity (H_c) of the material.
Hysteresis Loop
The hysteresis loop represents the relationship between the applied magnetic field (H) and the resulting magnetization (M) of a material. The shape of the loop is determined by the material’s
magnetic properties, such as coercivity and remanence.
Energy Loss
The area enclosed by the hysteresis loop represents the energy lost during each magnetization cycle, known as hysteresis loss. This energy loss is an important consideration in the design of magnetic
devices, as it can contribute to inefficiencies and heat generation.
Other Quantifiable Data
In addition to the properties discussed above, there are other quantifiable data points that are relevant to the understanding of magnets:
Magnetic Energy Product
The magnetic energy product is a measure of the energy stored in a magnetic field. It is calculated as the product of the magnetic field strength (B) and the magnetic field intensity (H). High-energy
permanent magnets, such as NdFeB, can have a magnetic energy product of up to 450 kJ/m³.
Hall Coefficient
The Hall coefficient is a measure of the Hall effect, which is the generation of a voltage difference across a material when a magnetic field is applied. The Hall coefficient is typically measured in
units of m³/C and is used in Hall effect sensors to measure magnetic fields.
By understanding the measurable and quantifiable data on electromagnets, permanent magnets, hard magnets, and soft magnets, you can gain a deeper insight into the properties and applications of these
materials. This knowledge can be invaluable in fields such as electrical engineering, materials science, and physics. | {"url":"https://techiescience.com/author/amrit-shaw/","timestamp":"2024-11-11T07:49:35Z","content_type":"text/html","content_length":"215885","record_id":"<urn:uuid:4d70f322-521c-40d2-8567-d96910e5d326>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00506.warc.gz"} |
Metaglossary.com - Definitions for "{term}"
linear transformation, T, is onto if its range is all of its codomain, not merely a subspace. Thus, for any vector , the equation T() = has at least one solution (is consistent). The linear
transformation T is 1-to-1 if and only if the null space of its corresponding matrix has only the zero vector in its null space. Equivalently, a linear transformation is 1-to-1 if and only if its
corresponding matrix has no non-pivot columns. | {"url":"https://www.metaglossary.com/define/onto","timestamp":"2024-11-04T12:08:26Z","content_type":"text/html","content_length":"17837","record_id":"<urn:uuid:5a0c6402-ed44-4c49-9b73-a036274d6bcc>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00020.warc.gz"} |
List Functions
This page describes various formulas and VBA procedures for working with lists of data.
There are a variety of methods and formulas that can be used with lists of data. The examples that use two lists assume you have two named ranges, List1 and List2, each of which is one column wide
and any number of rows tall. List1 and List2 must contain the same number of rows, although the need not be the same rows. For example, List1 = A1:A10 and List2 = K101:K110 is legal because the
number of rows is the same even though they are different rows.
You can download a example with these formula here.
For a VBA Function that returns a list of distinct values from a range or array, see the Distinct Values Function page.
You can use a simple formula to extract the distinct elements in a list. Suppose your list begins in cell C11. In some cell, enter
and then fill this formula down for as many rows as the number of rows in your data list. This formula will list the distinct items in the list beginning in cell C11. In the image to the left, the
original data is shown in red and the results of the formula are shown in blue.
In the data shown in the image, the results are in a column adjacent to the original data. This is for illustration only. The result data may be anywhere on the worksheet, or, for that matter, on
another worksheet or even in a separate workbook. The only restriction is that you must fill the formula down for at least as many rows as there are in the data list.
See No Blanks for a formula to remove the blank cells in the result list to have all the distinct entries appear at the top of the result list.
This formula assumes that the list that will contain the elements common to both lists is a range named Common and that this range has the same number of rows as List1 and List2. This is an array
formula that must be array entered into a range of cells (see the Array Formulas page for more information about array formulas). Select the range Common and type (or paste) the following formula
into the first cell, then press CTRL SHIFT ENTER rather than just ENTER. This is necessary since the formula returns an array of values.
List1 and List2. The positions of elements in the resulting list will be the same as the positions in List1. For example, if List1 has 'abc' in its 3rd row and List2 has 'abc' in the 8th row, 'abc'
will appear in the 3rd row, not the 8th row, of the result list. If an element in List1 does not exist in List2, the element in Common of the unmatched item in List1 will be empty.
The image to the left illustrates several aspects discussed previously. First, we have three named ranges, List1, List2, and Common. Second, all three ranges as the same size (10 rows in this case)
but are all different sets of rows. Finally, the position of the elements in the Common range match the positions of elements in List1, not List2.
You can also use a formula to extract elements that exist in one list but not in another. Again, it is assumed that you have two named ranges, List1 and List2 of the same size. Create a new named
range called In1Not2 the same size as List1. Enter the followng formula in the first cell of the new range ("In 1 Not 2") and press CTRL SHIFT ENTER rather than ENTER. This is an array formula so it
must be entered with CTRL SHIFT ENTER rather than ENTER in order to work..
List1 that do not appear in List2.
The order of the elements in the result list correspond to the position of that element in List1.
You can use Excel's Conditional Formatting tool to highlight cells in a second list that appear or do not appear in a master list. Excel does not allow you to reference other sheets in a Conditional
Formatting formula, so you must use defined named. Name your master list Master and name your second list, whose elements are to be conditionally formatted, Second. Open the Conditional Formatting
dialog from the Format menu. In that dialog, change Cell Value Is to Formula Is. To highlight elements in the Second list that appear in the Master list, use the formula
To highlight cells that appear in Second but not in Master, use Conditional Formatting as above but use the following formula:
You can download a example with these formula here.
This page last updated: 14-July-2007 | {"url":"https://www.cpearson.com/Excel/ListFunctions.aspx","timestamp":"2024-11-14T04:11:07Z","content_type":"text/html","content_length":"35752","record_id":"<urn:uuid:3731413a-574f-4b76-bdc5-9028e425ce36>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00387.warc.gz"} |
Pigeonhole Principle - (Discrete Geometry) - Vocab, Definition, Explanations | Fiveable
Pigeonhole Principle
from class:
Discrete Geometry
The pigeonhole principle states that if you have more items than containers to put them in, at least one container must contain more than one item. This simple yet powerful concept is widely used in
combinatorics and helps demonstrate the existence of certain configurations or arrangements within a given set, making it a fundamental tool in proofs and problem-solving.
congrats on reading the definition of Pigeonhole Principle. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The pigeonhole principle can be applied to show that in any group of 13 people, at least two must share the same birth month since there are only 12 months in a year.
2. It is often used to prove the existence of certain patterns or structures within larger sets, which is crucial in many areas of mathematics.
3. The principle applies not just to numbers but to any kind of objects, such as colors or types, demonstrating its versatility.
4. A more generalized form states that if $n$ items are put into $m$ containers, and if $n > km$, then at least one container must contain more than $k$ items.
5. The pigeonhole principle is foundational for understanding problems related to resource allocation, scheduling, and even computer science algorithms.
Review Questions
• How can the pigeonhole principle be applied to demonstrate the existence of patterns in a set of objects?
□ The pigeonhole principle can be used to show that if you distribute more objects than there are categories among those objects, some category must contain multiple items. For example, if you
have 10 pairs of socks and only 9 drawers, at least one drawer will have more than one pair of socks. This illustrates how the principle helps uncover patterns or repetitions within larger
sets, leading to conclusions about their structure.
• Discuss how the pigeonhole principle relates to the Erdős-Szekeres theorem in combinatorial geometry.
□ The pigeonhole principle plays a key role in the Erdős-Szekeres theorem by providing a method to show that within any sequence of points in the plane, certain configurations must exist. The
theorem states that for any integer $n$, any sequence of more than $2n$ points in general position contains an increasing or decreasing subsequence of length at least $n$. The pigeonhole
principle supports this by indicating that when placing these points into categories based on their relative positions, some categories must overflow, leading to these necessary subsequences.
• Evaluate the significance of the pigeonhole principle in solving complex problems across different mathematical fields.
□ The pigeonhole principle is significant because it provides a simple yet robust tool for proving existence results across various fields like combinatorics, graph theory, and computer
science. By establishing that certain outcomes must occur due to limitations on available resources or arrangements, it enables mathematicians to tackle complex problems and deduce
conclusions about configurations that might not be immediately obvious. This principle acts as a cornerstone for deeper explorations into more advanced theories such as Ramsey Theory, where
establishing unavoidable patterns is essential.
© 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/discrete-geometry/pigeonhole-principle","timestamp":"2024-11-04T10:39:34Z","content_type":"text/html","content_length":"150801","record_id":"<urn:uuid:70f21232-82d1-40b5-bcda-82fe68cb7b21>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00672.warc.gz"} |
Genotype-By-Block-of-Environments Biplots
R setup
knitr::opts_chunk$set(fig.align="center", fig.width=6, fig.height=6)
An example of a GGE (genotype plus genotype-by-environment) biplot similar to figure 12 of Yan and Tinker (2006). The flip argument can be used to flip the x and y axes so that biplots are oriented
as desired. Because the SVD factorization is not unique,
dat1 <- yan.winterwheat
m1 <- gge(dat1, yield~gen*env, scale=FALSE)
biplot(m1, main="yan.winterwheat - GGE biplot",
flip=c(1,0), origin=0, hull=TRUE)
Many people prefer to use ‘standardized’ biplots, in which the data for each environment has been centered and scaled. For standardized biplots, a unit circle is drawn. Environment vectors that reach
out to the unit circle are perfectly represented in the two dimensional plane.
m2 <- gge(dat1, yield~gen*env, scale=TRUE)
biplot(m2, main="yan.winterwheat - GGE biplot",
flip=c(1,1), origin=0)
As seen above, the environment vectors are fairly long, so that relative performance of genotypes in environments can be assessed with reasonable accuracy. In contrast, a biplot based on principal
components 2 and 3 has shorter vectors which should not be interpreted.
Laffont, Hanafi, and Wright (2007) showed how to partition the sums-of-squares simultaneously along the principal component axes and along ‘G’ and ‘GxE’ axes.
The mosaic plot above shows that the first principal component axis is capturing almost all of the variation between genotypes, so that a projection of the genotype markers onto the first principal
component axis is a good overall representation of the rankings of the genotypes.
Laffont, Wright, and Hanafi (2013) presented GGB (genotype plus genotype-by-block of environments) biplots, which are useful to enhance the view of mega-environments consisting of multiple locations.
dat2 <- crossa.wheat
# Define mega-environment groups of locations
dat2$eg <- ifelse(is.element(dat2$loc,
"SJ","MS","MG","MM")), "Grp1", "Grp2")
# Specify env.group as column in data frame
m3 <- gge(dat2, yield~gen*loc, env.group=eg, scale=FALSE)
biplot(m3, main="crossa.wheat - GGB biplot")
How to modify the “focus” of a biplot
Let X be a genotype-by-environment matrix.
Let the Singular Value Decomposition be X = USV'.
Let the NIPALS decomposition be X = TLP'. DANGER, some algorithms do not factor L out of T.
dat3 <- agridat::yan.winterwheat
dat3 <- acast(dat3, gen~env, value.var="yield")
dat3 <- scale(dat3, center=TRUE, scale=FALSE)
Xsvd <- svd(dat3)
Xnip <- nipals(dat3, center=FALSE, scale=FALSE)
U <- Xsvd$u
S <- diag(Xsvd$d)
V <- Xsvd$v
T <- Xnip$scores
Lam <- diag(Xnip$eig)
P <- Xnip$loadings
Biplots with genotype focus
To obtain a genotype-focused biplot the eigenvalues are associated with U.
The genotype coordinates are can be obtained from the SVD using the first two columns of U*S or equivalently from NIPALS T*Lam.
The environment coordinates are the first two columns of V (from the SVD) or P (from NIPALS).
Biplot with environment focus
To obtain an environment-focused biplot the eigenvalues are associated with V.
The genotype coordinates are the first two columns of U (from SVD) or T (from NIPALS).
The environment coordinates are S*V (from SVD) or Lam*P (from NIPALS).
Comments on biplots
Note that GGE biplots are environment-focused. In particular, this provides the interpretation that the correlation of genotype performance in two environments is approximated by the cosine of the
angle between the vectors for those two environments.
The SVD and NIPALS methods provide the same principal components for complete-data, except that a principal component from SVD and the corresponding principal component from NIPALS might point in
opposite directions (differ by a factor of -1 as in some of the examples above. The corresponding biplots would therefore be mirror-reversed along that component. For biplots from SVD and NIPALS that
are visually consistent, each principal component can be directed to point in a direction that is positively correlated with the overall genotype means. In other words, if the correlation of the
genotype means and the ordinate of the genotypes along the principal component is negative, the principal component is multiplied by -1.
As with all biplots, the environment vectors can be arbitrarily scaled so that the genotypes and environments uses a similar amount of area on the plot. The algorithm that physically centers the
biplot and scales it on the page is not perfect and has opportunities for improvement.
Laffont, Jean-Louis, Mohamed Hanafi, and Kevin Wright. 2007.
“Numerical and Graphical Measures to Facilitate the Interpretation of GGE Biplots.” Crop Science
47: 990–96.
Laffont, Jean-Louis, Kevin Wright, and Mohamed Hanafi. 2013.
“Genotype Plus Genotype-by-Block of Environments Biplots.” Crop Science
53 (6): 2332–41.
Yan, Weikai, and Nicholas A Tinker. 2006.
“Biplot Analysis of Multi-Environment Trial Data: Principles and Applications.” Canadian Journal of Plant Science
86: 623–45. | {"url":"https://cran.case.edu/web/packages/gge/vignettes/gge_examples.html","timestamp":"2024-11-05T06:36:03Z","content_type":"text/html","content_length":"95560","record_id":"<urn:uuid:8d85456a-01f8-4db5-bbc7-a319bcefe9eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00296.warc.gz"} |
A novel kind of a hybrid recursive neural implicit dynamics for real-time matrix inversion has been recently proposed and investigated. Our goal is to compare the hybrid recursive neural implicit
dynamics on the one hand, and conventional explicit neural dynamics on the other hand. Simulation results show that the hybrid model can coincide better with systems in practice and has higher
abilities in representing dynamic systems. More importantly, hybrid model can achieve superior convergence performance in comparison with the existing dynamic systems, specifically recently-proposed
Zhang dynamics. This paper presents the Simulink model of a hybrid recursive neural implicit dynamics and gives a simulation and comparison to the existing Zhang dynamics for real-time matrix
inversion. Simulation results confirm a superior convergence of the hybrid model compared to Zhang model.
Zhang neural network; gradient neural network; matrix inverse; convergence.
Zhang neural network; gradient neural network; matrix inverse; conver-
K. Chen, Recurrent implicit dynamics for online matrix inversion, Appl. Math. Comput. 219(20) (2013), 10218–10224.
K. Chen, C. Yi, Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion. Appl. Math. Comput. 273 (2016), 969–975.
A. Cichocki, T. Kaczorek, A. Stajniak, Computation of the Drazin inverse of a singular matrix making use of neural networks, Bulletin of the Polish Academy of Sciences Technical Sciences, 40 (1992).
J.S. Jang, S.Y. Lee, S. Y. Shin, J. S. Jang, and S. Y. Shin, An optimization network for matrix inversion, Neural Inf. Process. Ser. (1987), 397–401.
S. Li, S. Chen, B. Liu, Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester Equation by using a sign-bi-power activation function, Neural Process.
Lett. 37 (2013), 189–205.
Z. Li, Y. Zhang, Improved Zhang neural network model and its solution of time-varying generalized linear matrix equations, Expert Syst. Appl. 37 (2010), 7213–7218.
B. Liao, Y. Zhang, Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices, IEEE Trans. Neural Netw. Learn. Syst., 25 (2014), 1621–1631.
F.L. Luo, Z. Bao, Neural network approach to computing matrix inversion, Appl. Math. Comput. 47 (1992), 109–120.
S. Qiao, X.-Z. Wang, Y. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra Appl. http://dx.doi.org/10.1016/j.laa.2017.03.014.
P.S. Stanimirović, I. Živković, Y. Wei, Recurrent neural network approach based on the integral representation of the Drazin inverse, Neural Comput. 27(10) (2015), 2107–2131.
P.S. Stanimirović, I. S. Živković, Y. Wei, Recurrent neural network for computing the Drazin inverse, IEEE Trans. Neural Netw. Learn. Syst. 26 (2015), 2830–2843.
I. Stojanović, P.S. Stanimirović, I. Živković, D. Gerontitis, X.-Z. Wang, ZNN models for computing matrix inverse based on hyperpower iterative methods, Filomat 31:10 (2017), 2999–3014.
J. Wang, A recurrent neural network for real-time matrix inversion, Appl. Math. Comput. 55 (1993), 89–100.
J. Wang, Recurrent neural networks for solving linear matrix equations, Comput. Math. Appl. 26 (1993), 23–34.
J. Wang, Recurrent neural networks for computing pseudoinverses of rank-deficient matrices, SIAM J. Sci. Comput. 18 (1997), 1479–1493.
Y. Wei, Recurrent neural networks for computing weighted Moore-Penrose inverse, Appl. Math. Comput. 116 (2000), 279–287.
Y. Zhang, Y. Yang, N. Tan, B. Cai, Zhang neural network solving for time-varying full-rank matrix Moore-Penrose inverse, Computing 92 (2011), 97–121.
Y. Zhang, Y. Shi, K. Chen, C. Wang, Global exponential convergence and stability of gradient-based neural network for online matrix inversion, Appl. Math. Comput. 215 (2009), 1301–1306.
Y. Zhang, Design and analysis of a general recurrent neural network model for time-varying matrix inversion, IEEE Trans. Neural Netw. 16(6) (2005), 1477–1490.
Y. Zhang, Y. Shi, K. Chen, C. Wang, Global exponential convergence and stability of gradient-based neural network for online matrix inversion, Appl. Math. Comput. 215 (2009), 1301–1306.
• There are currently no refbacks.
© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online) | {"url":"https://casopisi.junis.ni.ac.rs/index.php/FUMathInf/article/view/3501","timestamp":"2024-11-08T09:26:27Z","content_type":"application/xhtml+xml","content_length":"26059","record_id":"<urn:uuid:31b4c100-32f5-4a2b-9bd3-ac4e0bfd9f3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00481.warc.gz"} |
Learn More#
This content is from the Scientific Python QuickStart by Thomas J. Sargent and John Stachurski, originally published as the JupyterBook example. We will be gradually adding more introductory Python
materials by actuarial contributors. In the meantime, consider also referring to the Python Study Group microsite for materials from the 2021 YDAWG-organised sessions.
We’re about ready to wrap up this brief course on Python for scientific computing.
In this last lecture we give some pointers to the major scientific libraries and suggestions for further reading.
Fundamental matrix and array processing capabilities are provided by the excellent NumPy library.
For example, let’s build some arrays
import numpy as np # Load the library
a = np.linspace(-np.pi, np.pi, 100) # Create even grid from -π to π
b = np.cos(a) # Apply cosine to each element of a
c = np.sin(a) # Apply sin to each element of a
Now let’s take the inner product
The number you see here might vary slightly due to floating point arithmetic but it’s essentially zero.
As with other standard NumPy operations, this inner product calls into highly optimized machine code.
It is as efficient as carefully hand-coded FORTRAN or C.
The SciPy library is built on top of NumPy and provides additional functionality.
For example, let’s calculate \(\int_{-2}^2 \phi(z) dz\) where \(\phi\) is the standard normal density.
from scipy.stats import norm
from scipy.integrate import quad
ϕ = norm()
value, error = quad(ϕ.pdf, -2, 2) # Integrate using Gaussian quadrature
SciPy includes many of the standard routines used in
See them all here.
The most popular and comprehensive Python library for creating figures and graphs is Matplotlib, with functionality including
• plots, histograms, contour images, 3D graphs, bar charts etc.
• output in many formats (PDF, PNG, EPS, etc.)
• LaTeX integration
Example 2D plot with embedded LaTeX annotations
Example contour plot
Example 3D plot
More examples can be found in the Matplotlib thumbnail gallery.
Other graphics libraries include
Symbolic Algebra#
It’s useful to be able to manipulate symbolic expressions, as in Mathematica or Maple.
The SymPy library provides this functionality from within the Python shell.
from sympy import Symbol
x, y = Symbol('x'), Symbol('y') # Treat 'x' and 'y' as algebraic symbols
x + x + x + y
We can manipulate expressions
expression = (x + y)**2
solve polynomials
from sympy import solve
solve(x**2 + x + 2)
and calculate limits, derivatives and integrals
from sympy import limit, sin, diff
limit(1 / x, x, 0)
The beauty of importing this functionality into Python is that we are working within a fully fledged programming language.
We can easily create tables of derivatives, generate LaTeX output, add that output to figures and so on.
One of the most popular libraries for working with data is pandas.
Pandas is fast, efficient, flexible and well designed.
Here’s a simple example, using some dummy data generated with Numpy’s excellent random functionality.
import pandas as pd
data = np.random.randn(5, 2) # 5x2 matrix of N(0, 1) random draws
dates = pd.date_range('28/12/2010', periods=5)
df = pd.DataFrame(data, columns=('price', 'weight'), index=dates)
Further Reading#
These lectures were originally taken from a longer and more complete lecture series on Python programming hosted by QuantEcon.
The full set of lectures might be useful as the next step of your study. | {"url":"https://actuariesinstitute.github.io/cookbook/docs/learn_more.html","timestamp":"2024-11-03T00:52:45Z","content_type":"text/html","content_length":"41901","record_id":"<urn:uuid:7e9b9a32-df98-4a7c-b3d3-c7601e777d6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00605.warc.gz"} |
How to Remove First 3 Characters from String in Excel
To remove the first three characters from a string in Excel, you can use the RIGHT and LEN functions. Here's how you can do it:
1. Open Excel and type the text you want to remove the first three characters from in cell A1. For example, let's use the text "HelloWorld".
2. In cell B1 or any other cell, enter the following formula:
=RIGHT(A1, LEN(A1) - 3)
3. Press Enter to apply the formula.
Let's work with an example where we want to remove the first three characters from the text "HelloWorld":
1. Type "HelloWorld" in cell A1.
2. In cell B1, enter the formula:
=RIGHT(A1, LEN(A1) - 3)
3. Press Enter to apply the formula.
4. The resulting text in cell B1 will be "loWorld", as the first three characters "Hel" have been removed.
Did you find this useful? | {"url":"https://sheetscheat.com/excel/how-to-remove-first-3-characters-from-string-in-excel","timestamp":"2024-11-09T10:19:27Z","content_type":"text/html","content_length":"9626","record_id":"<urn:uuid:ab80d3b9-54fa-4c25-8d75-4e8f58ebae5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00431.warc.gz"} |
Implicit and Conditional Variables
Variables and Math Operations
Numeric variables are implicitly typed according to the default Fortran convention. Integer numeric variables must have names beginning with the letters I-N e.g:
symb n = -12
Real numeric variables must have names beginning with the letters A-H, O-Z e.g.
symb radius = 35.97
Assigning a real value to an integer variable results in a warning and a change of value e.g.
symb innerrad = 4.7e-3 /* will result in
symb innerrad = 0 /* (rounds to nearest integer)
Variables are case sensitive Use ‘$’ before variable name to represent value of the variable, e.g.
symb radius = 1.0
symb pi = 3.14159
symb circum = s2. * $pi * $radius
All symbol entries must be blank delimited, e.g.
symb val3 = sqrt ( 5. * $val1 / ( $val2 - 3.9 ) )
Space must be present between all numbers, variables, operands and parentheses. Sequence of operations conform to Fortran standard. The symb statement can also take the form:
symb vname = <expression>
The <expression> may be composed of the following mathematical operations:
cos, sin, tan, exp, alog, alog10, acos, asin, atan, atan2, sqrt, abs, sign, int, nint, max, min, **, *, /, +, and -.
Parentheses may be used to control the sequence in which the expression is evaluated (innermost parentheses evaluated first). Within parentheses, the evaluation is done from left to right for each
operator listed above in the order listed.
The real number equivalence of all arguments to cos, sin, tan, exp, alog, alog10, acos, asin, atan, atan2, sqrt, abs, int, and nint are used.
Note that Symbol does not allow a real expression in which a negative number is taken to an exponential power (even the power of 2), as the real number representation of the exponent is used in the
expression and math libraries do not allow this operation.
Multiple arguments may be provided to the max, min, atan2, and sign functions.
These arguments must be separated by commas (which are themselves blank delimited) and be bounded by parentheses.
Examples of arithmetic expressions are:
symb x = sqrt ( 5. ** 2 + 10. ** 2 ) /* compute hypotenuse of triangle
symb maxm = max ( $m1 , $m2 , $m3 ) /* compute maximum of 3 values
symb pi = 4. * atan ( 1. ) /* compute value of pi
Conditional Variables
The two forms of the symb statement discussed above can also have conditional operators appended to them.
The SYMB statement with a conditional operator has the form:
symb vname = value if datum1 op datum2
symb vname = if datum1 op datum2
where op is any of the conditional operators: eq, ne, lt, le, gt, and ge whose meanings correspond to their usage in Fortran.
datum1 and datum2 can be either numeric or character data.
If the values being compared are numeric, they are compared as real numbers (i.e., the real number representation of an integer is used in the comparison).
The effect of the conditional operator is:
1. If the conditional is true, then the symbol value assignment is made.
2. If the conditional is not true, then this symb statement is skipped and the assignment is not made. One other type of conditional assignment is based on the current existence of a variable name.
The conditional can take the following two forms:
symb vname = value if exist
symb vname = value if noexist
where exist and noexist refer to the variable name.
In the first case, if vname currently exists, the conditional is true and the value assignment is made. If vname does not exist, no assignment is made.
The second case is the opposite. Multiple conditional operators can be appended to an expression. This type of expression takes the form:
symb vname = if datum1 op datum2 if datum3 op datum4 etc.
For a symbol equivalence with multiple conditionals, all conditionals must be true before the value assignment is made. | {"url":"https://support.onscale.com/hc/en-us/articles/360010729652-Implicit-and-Conditional-Variables","timestamp":"2024-11-09T00:14:51Z","content_type":"text/html","content_length":"45010","record_id":"<urn:uuid:26bdc949-1796-4d45-8ccc-f12bc39c4efc>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00707.warc.gz"} |
John Venn - Biography
Quick Info
4 August 1834
Hull, England
4 April 1923
Cambridge, England
John Venn was an English mathematician and logician best known for the Venn diagrams which can be used to visualise the unions and intersections of sets.
John Venn's mother, Martha Sykes, came from Swanland near Hull and died while he was still quite a young boy. His father was the Rev Henry Venn who, at the time of John's birth, was the rector of
the parish of Drypool, near Hull. The Rev Henry Venn, himself a fellow of Queen's, was from a family of distinction. His father, John's grandfather, was the Rev John Venn who had been the rector
of Clapham in south London. He became the leader of the Clapham Sect, a group of evangelical Christians centred on his church. They successfully campaigned for the abolition of slavery, advocated
prison reform and the prevention of cruel sports, and supported missionary work abroad.
It was not only Venn's grandfather who played a prominent role in the evangelical Christian movement, for so did his father the Rev Henry Venn. The Society for Missions in Africa and the East was
founded by evangelical clergy of the Church of England in 1799 and in 1812 it was renamed the Church Missionary Society for Africa and the East. The Rev Henry Venn became secretary to this
Society in 1841 and in order to carry out his duties moved to Highgate near London. He held this position until his death in 1873.
As might be expected from his family background, John was very strictly brought up, and there was never any thought other than that he would follow the family tradition into the priesthood. He
attended first Sir Roger Cholmley's School in Highgate, then the private Islington Preparatory School.
When he entered Gonville and Caius College Cambridge in October 1853 he had:-
... so slight an acquaintance with books of any kind that he may be said to have begun there his knowledge of literature.
Having been awarded a mathematics scholarship in his second year of study, he graduated as sixth Wrangler in the Mathematical Tripos of 1857, meaning that he was ranked in the sixth place out of
those students who were awarded a First Class degree in mathematics. He was elected a Fellow of Gonville and Caius College shortly after graduating, and two years later was ordained a priest. In
fact the year after his graduation, in 1858, he had been ordained a deacon at Ely, then after his ordination as a priest he had served as a curate first at Cheshunt, Hertfordshire, and then for a
year as a curate at Mortlake, Surrey.
In 1862 he returned to Cambridge University as a lecturer in Moral Science, studying and teaching logic and probability theory. He had already become interested logic, philosophy and metaphysics,
reading the treatises of De Morgan, Boole, John Austin, and John Stuart Mill. Back at Cambridge he now found interests in common with many academics such as Todhunter. He also played a large role
in developing the Moral Sciences Tripos over many years. He lectured and examined the Tripos, developing a friendly atmosphere between the lecturers and the students.
Venn extended Boole's mathematical logic and is best known to mathematicians and logicians for his diagrammatic way of representing sets, and their unions and intersections. He considered three
discs $R, S$, and $T$ as typical subsets of a set $U$. The intersections of these discs and their complements divide $U$ into 8 non-overlapping regions, the unions of which give 256 different
Boolean combinations of the original sets $R, S, T$.
Venn wrote Logic of Chance in 1866 which Keynes described as:-
... strikingly original and considerably influenced the development of the theory of statistics.
In 1867 Venn married Susanna Carnegie Edmonstone, the daughter of the Rev Charles Edmonstone. They had one child, a son John Archibald Venn, who became president of Queen's College, Cambridge, in
1932, and undertook major collaborative research projects with his father that we give more details on below.
Venn published Symbolic Logic in 1881 and The Principles of Empirical Logic in 1889. The second of these is rather less original but the first was described by Keynes as:-
... probably his most enduring work on logic.
In 1883 Venn was elected a Fellow of the Royal Society and in the same year was awarded a Sc.D. by Cambridge. About this time his career changed direction for in the same year that he was elected
to the Royal Society, he left the priesthood. His son, John Archibald Venn, wrote his father's obituary in the Dictionary of National Biography and explained the position:-
It had long ceased to be regarded as an anomaly for a clergyman to preach the then circumscribed evangelical creed and at the same time, without the slightest insincerity, to devote himself
actively to philosophical studies; yet ... finding himself still less in sympathy with the orthodox clerical outlook, Venn availed himself of the Clerical Disabilities Act. Of a naturally
speculative frame of mind, he was want to say later that, owing to subsequent change in accepted opinion regarding the Thirty-nine Articles, he could consistently have retained his orders; he
remained, indeed, throughout his life a man of sincere religious conviction.
Venn's interest turned towards history and he signalled this change in direction by donating his large collection of books on logic to the Cambridge University Library in 1888. He wrote a history
of his college, publishing The Biographical History of Gonville and Caius College 1349-1897 in 1897, which:-
... involved a vast amount of painstaking and methodical search among university, episcopal, and other records.
The annals of a clerical family (1904) trace the history of his own family back to the seventeenth century and record that he was the eighth generation of his family to have a university
education. In 1910 he published a work on historical biography, namely a treatise on John Caius, one of the founders of his College. Three years later he published Early Collegiate Life which
collected many of his writings describing what life was like in the early days of Cambridge University. He then undertook the immense task of compiling a history of Cambridge University Alumni
Cantabrigienses, the first volume of which was published in 1922. He was assisted by his son John Archibald Venn in this task which was described by another historian in these terms:-
It is difficult for anyone who has not seen the work in its making to realise the immense amount of research involved in this great undertaking.
It was [3]:-
... nothing less than a "biographical list of all known students, graduates, and holders of office at the University of Cambridge from the earliest times to 1900". ... The Venns, father and
son, spared no industry in building up these records, which are of extraordinary value to historians and genealogists ...
The first part contained 76,000 names and covered the period up to 1751. At the time of Venn's death the second part, covering the period from 1751 to 1900, existed in manuscript and contained a
further 60,000 names.
Venn had other skills and interests too, including a rare skill in building machines. He used his skill to build a machine for bowling cricket balls which was so good that when the Australian
Cricket team visited Cambridge in 1909, Venn's machine clean bowled one of its top stars four times.
His son gives this description:-
Of spare build, he was throughout his life a fine walker and mountain climber, a keen botanist, and an excellent talker and linguist.
1. T A A Broadbent, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. Obituary in The Times
See THIS LINK
3. A D D Craik, Mr Hopkins' Men: Cambridge Reform and British Mathematics in the 19th Century (Cambridge 2007)
4. M Ferriani, L'induzione in John Venn, Quaderni di Storia e Critica della Scienza, Nuova Serie 3, Domus Galilaeana (Pisa, 1973).
5. John Venn zum 150. Geburtstag, Praxis Math. 26 (11) (1984), 343-345.
6. Obituary of John Venn, Proc. Roy. Soc. London A 110 (1926), x - xi.
7. W C Salmon, John Venn's Logic of chance, Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science II (Dordrecht- Boston, Mass., 1981),125-138.
8. J A Venn, John Venn, Dictionary of National Biography 1922-1930, 869-870. See THIS LINK.
Additional Resources (show)
Other pages about John Venn:
Other websites about John Venn:
Honours awarded to John Venn
Written by J J O'Connor and E F Robertson
Last Update October 2003 | {"url":"https://mathshistory.st-andrews.ac.uk/Biographies/Venn/","timestamp":"2024-11-11T08:29:05Z","content_type":"text/html","content_length":"30191","record_id":"<urn:uuid:652ebc06-3da0-4c20-a8c6-19c42e156aa6>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00662.warc.gz"} |
Current Ratio - Important 2021
Current Ratio
Current Ratio
Current Ratio:
The current ratio is a ratio between current assets and current liabilities of a business enterprise.
The current ratio establishes a relationship between current assets and current liabilities of a firm for a particular period.
Current Ratio is a liquidity ratio that measures ability of the business enterprise to pay its short-term financial obligations.
It compares the current assets and current liabilities of the firm.
The Current ratio is also known as the working capital ratio.
Significance of Current Ratio:
The objective of computing this ratio is to measure the ability of the firm to meet its short-term liability.
It indicates the amount of current assets available for repayment of current liabilities.
The higher the ratio, the greater is the short-term solvency of a firm and vice – versa.
The acceptable current ratio differs from business to business depending upon the risk involved.
Thus, the ideal current ratio of a company is 2 : 1 .
A current ratio below 1 means that the company doesn’t have enough liquid assets to cover its short-term liabilities.
Current Assets:
Current Assets are those assets that are held for short period and can be converted into cash within one year. The balance of such items goes on fluctuating i.e. it keeps on changing throughout the
year. It includes the following :
Cash in hand,
Cash at Bank,
Trade Receivables,
Short term investment,
Prepaid expenses.
Trade Receivables include Bills Receivables and Sundry Debtors.
Note- Inventories (excluding Loose Tools and Spare Parts)
Current Liabilities:
Current Liabilities are obligations or debts that are payable within a period of one-year . It includes the following:
Trade Payables,
Bank overdraft,
Provision for tax,
Outstanding expenses,
Cash Credit
Short-term borrowings.
Trade Payables include Sundry Creditors and Bills Payables.
Formula for the Current ratio:
Current Assets = Current Investments + Inventories (Excluding Spare Parts and Loose Tools) + Trade Receivables + Cash and Cash Equivalents +Short Term Loans and Advances + Other Current Assets.
Current Liabilities= Short-Term Borrowings +Trade Payables +Other Current Liabilities+ Short-term Provisions+Cash Credit.
(Standard Current Ratio:- 2:1)
Illustrations For Current Ratio:
Illustration 1.
Calculate current ratio from the following :
Current Assets ₹10,00,000
Current Liabilities ₹ 4,00,000
Current ratio=10,00,000/4,00,000
Current Ratio= 2.5:1
Illustration 2.
Calculate current ratio from the following :
Sundry debtors ₹4,00,000
Inventories ₹1,60,000
Marketable securities ₹ 80,000
Cash ₹1,20,000
Prepaid expenses ₹40,000
Bills payables ₹ 80,000
Sundry creditors ₹2,60,000
10%Debentures ₹ 5,00,000
Outstanding Expenses ₹ 60,000
Current Assets =Sundry debtors +Inventories+Marketable securities +Cash +
Prepaid expenses
Current Assets = 4,00,000+1,60,000+80,000+1,20,000+40,000
Current Assets = 8,00,000
Current Liabilities=Bills payables +Sundry creditors +Outstanding Expenses
Current Liabilities=80,000 +2,60,000 +60,000
Current Liabilities=4,00,000
current ratio=8,00,000/4,00,000
Current Ratio= 2:1
Illustration 3.
Calculate current ratio from the following :
Non-Current Tangible Assets ₹9,00,000
Non-Current Intangible Assets ₹3,00,000
Share Capital ₹9,00,000
Sundry debtors ₹3,40,000
Inventories ₹2,00,000
Current Investment ₹ 1,40,000
Cash in hand ₹80,000
Cashat Bank ₹20,000
Accrued Income ₹20,000
Prepaid expenses ₹40,000
Bills payables ₹ 80,000
Sundry creditors ₹2,20,000
Bank Overdraft ₹80,000
10%Debentures(First) ₹ 5,00,000
9%Debentures(Second) ₹ 2,00,000
Outstanding Expenses ₹ 10,000
Provision for Taxation ₹ 30,000
Current Assets =Sundry debtors +Inventories+Current Investment+ Accrued Income + Cash in hand + Cash at Bank+Prepaid expenses
Current Assets = 3,40,000+2,00,000+1,40,000+20,000+80,000+20,000+40,000
Current Assets = 8,40,000
Current Liabilities=Bills payables +Sundry creditors+Bank Overdraft +Outstanding Expenses +Provision for Taxation
Current Liabilities=80,000 +2,20,000 +80,000+10,000+30,000
Current Liabilities=4,20,000
Current ratio=8,40,000/4,20,000
Current Ratio= 2:1
Illustration 4.
Calculate current ratio from the following :
Total Assets ₹9,00,000
Non-Current Assets ₹6,00,000
Shareholders Fund ₹5,00,000
Non-Current Liabilities ₹2,50,000
Total Assets=Non Current Assets+Current Assets
₹9,00,000=₹6,00,000+Current Assets
Current Assets= 9,00,000-6,00,000= 3,00,000
Total Assets=Tatal Liabilities
Total Liabilities= Shareholders Fund +Non-Current Liabilities +Current Liabilities
9,00,000= 5,00,000+2,50,000+Current Liabilities
Current Liabilities= 9,00,000-7,50,000
Current Liabilities=1,50,000
Current ratio=3,00,000/1,50,000
Current ratio=2:1
Illustration 5.
Calculate current ratio from the following :
Total Assets ₹9,00,000
Non-Current Investment ₹3,00,000
Fixed Assets ₹4,00,000
Shareholders Fund ₹5,00,000
Non-Current Liabilities ₹2,50,000
Total Assets=Non Current Assets+Non current Investment+Current Assets
₹9,00,000=₹4,00,000+₹3,00,000Current Assets
Current Assets= ₹9,00,000-₹7,00,000= ₹2,00,000
Total Assets=Tatal Liabilities
Total Liabilities= Shareholders Fund +Non-Current Liabilities +Current Liabilities
₹9,00,000= ₹5,00,000+₹2,50,000+Current Liabilities
Current Liabilities= ₹9,00,000-₹7,50,000
Current Liabilities=₹1,50,000
Current ratio=₹2,00,000/₹1,50,000
Current ratio=1.33:1
Illustration 6.
Calculate current ratio from the following :
Working capital ₹3,00,000
Current Assets ₹5,00,000
Working capital = Current Assets – Current Liabilities
₹3,00,000= ₹5,00,000 – Current Liabilities
Current Liabilities=₹5,00,000- ₹3,00,000
Current Liabilities=₹2,00,000
Current ratio=₹3,00,000/₹2,00,000
Current ratio=1.5:1
Illustration 7.
Calculate current ratio from the following :
Working capital ₹3,00,000
Current Liabilities ₹2,00,000
Working capital = Current Assets – Current Liabilities
₹3,00,000= Current Assets – ₹2,00,000
Current Assets =₹3,00,000+ ₹2,00,000
Current Assets =₹5,00,000
Current ratio=₹5,00,000/₹2,00,000
Current ratio=2.5:1
Debtors Turnover Ratio(Trade Receivable Turnover Ratio) | {"url":"https://jkbhardwaj.com/current-ratio/","timestamp":"2024-11-01T20:07:04Z","content_type":"text/html","content_length":"91116","record_id":"<urn:uuid:103190bc-adec-4638-b638-315459f30611>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00620.warc.gz"} |
more from Keith Hossack
Single Idea 10674
[catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order]
Full Idea
A language with plurals is better for arithmetic. Instead of a first-order fragment expressible by an induction schema, we have the complete truth with a plural induction axiom, beginning 'If there
are some numbers...'.
Gist of Idea
A plural language gives a single comprehensive induction axiom for arithmetic
Keith Hossack (Plurals and Complexes [2000], 4)
Book Reference
-: 'British Soc for the Philosophy of Science' [-], p.420 | {"url":"http://www.philosophyideas.com/search/response_philosopher_detail.asp?era_no=M&era=New%20millenium%20(2001-%20)&id=10674&PN=3605&order=chron&from=theme&no_ideas=29","timestamp":"2024-11-11T07:21:09Z","content_type":"application/xhtml+xml","content_length":"3145","record_id":"<urn:uuid:11cc0d8b-db9c-42eb-af6e-07b62ad38486>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00057.warc.gz"} |
Constructions - Edu Spot- NCERT Solution, CBSE Course, Practice Test
(I) To construct the bisector of a given angle
Given: An ∠ABC.
Required: To construct it bisector.
Steps of Construction:
(i) Taking B as centre and any radius, draw an arc to intersect the rays BA and BC, say at E and D, respectively.
(ii) Next, taking D and E as centres and with the radius more than 12 DE, draw arcs to intersect each other, say at F.
(iii) Draw the ray BF. This ray BF is the required bisector of the ∠ABC.
Proof: Join DF and EF.
In ΔBEF and ΔBDF,
BE = BD (Radii of the same arc)
EF = DF (Arcs of radii)
BF = BF (Common)
Therefore, ΔBEF = ΔBDF (SSS rule)
This gives ∠EBF = ∠DBF (CPCT)
(II) To construct the perpendicular bisector of a given line segment
Given: A line segment AB.
Required: To construct its perpendicular bisector.
Steps of Construction:
(i) Taking A and B as centres and radius more than 12 AB, draw arcs on both sides of the line segment AB (to intersect each other).
(ii) Let these arcs intersect each other at P and Q. Join PQ.
(iii) Let PQ intersect AB at the point M. Then, line PMQ is the required perpendicular bisector of AB.
Proof: Join A and B to both P and Q to form AP, AQ, BP and BQ.
In ΔPAQ and ΔPBQ,
AP = BP (Arcs of equal radii)
AQ = BQ (Arcs of equal radii)
PQ = PQ (Common)
Therefore, ΔPAQ = ΔPBQ (SSS rule)
So, ∠APM = ∠BPM (CPCT)
Now, in ΔPMA and ΔPMB,
AP = BP (As before)
PM = PM (Common)
∠APM = ∠BPM (Proved above)
Therefore, ΔPMA = ΔPMB (SAS rule)
So, AM = BM and ∠PMA = ∠PMB
As ∠PMA + ∠PMB = 180° (Linear pair axiom)
We get, ∠PMA = ∠PMB = 90°
Therefore, PM, i.e., PMQ is the perpendicular bisector of AB.
(III) Constructs an angle of 60° at the initial point of a given ray
Given: A ray AB with initial point A.
Required: To construct a ray AC such that ∠CAB = 60°.
Steps of Construction:
(i) Taking A as the centre and some radius, draw an arc of a circle which intersects AB, say at a point D.
(ii) Taking D as the centre and with the same radius as before, draw an arc intersecting the previously drawn arc, say at a point E.
(iii) Draw the ray AC passing through E. Then ∠CAB required the angle of 60°.
Proof: Join DE.
Then, AB = AD = DE (By construction)
Therefore, ΔEAD is an equilateral triangle and the ∠EAD which is the same as ∠CAB is equal to 60°.
4. Rules of Congruency of Two Triangles
• SAS Two triangles are congruent if any two sides and the included angle of one triangle are equal to any two sides and the included angle of the other triangle.
• SSS Two triangles are congruent if the three sides of one triangle are equal to the three sides of the other triangle.
• ASA Two triangles are congruent if any two angles and the included side of one triangle are equal to the two angles and the included side of the other triangle.
• RHS Two right triangles are congruent if the hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.
5. The uniqueness of a Triangle
A triangle is unique, if
• two sides and the included angle is given,
• three sides are given,
• two angles and the included side is given and
• in a right triangle, hypotenuse and one side are given.
6. Requirement for the Construction of a Triangle: For constructing a triangle, at least three parts of a triangle have to be given hut, not all combinations of three parts and sufficient for the
purpose, e.g., if two sides and an angle (not the included angle) are given, then it is not always possible to construct such a triangle uniquely.
7. Some Constructions of Triangles
(I) To construct a triangle, given its base, a base angle and sum of other two sides
Given: The base BC, a base angle, say ∠B and the sum AB + AC of the other two sides of a ΔABC.
Required: To construct the ΔABC.
Steps of Construction:
(i) Draw the base BC and at the point B make an angle, say XBC equal to the given angle.
(ii) Cut a line segment BD equal to AB + AC from the ray BX.
(iii) Join DC and make an angle DCY equal to ∠BDC.
(iv) Let CY intersect BX at A (see figure).
Base BC and CB are drawn as given.
Next in ΔACD,
∠ACD = ∠ADC (By construction)
AC = AD (Sides opposite to equal angles of a triangle are equal)
AB = BD – AD = BD – AC
⇒ AB + AC = BD
Alternative Method
(i) Draw the base BC and at the point B make an angle, say XBC equal to the given angle.
(ii) Cut a line segment BD equal to AP + AC from the ray BX.
(iii) Join DC.
(iv) Draw perpendicular bisector PQ of CD to intersect BD at a point A.
(v) Join AC.
Then, ABC is the required triangle.
Base BC and CB are drawn as given.
A lies on the perpendicular bisector of CD.
AD = AC
AB = BD – AD = BD – AC
AB + AC = BD
Remark: The construction of the triangle is not possible if the sum AB + AC < BC.
(II) To construct a triangle given its base, a base angle and the difference of the other two sides
Given: The base BC, a base angle, say CB and the difference of other two sides AB – AC or AC – AB.
Required: To construct the ΔABC.
There are the following two cases
Case (I): Let AB > AC, i.e., AB – AC is given.
Steps of Construction:
(i) Draw the base BC and at point B make an angle, say XBC equal to the given angle.
(ii) Cut the line segment BD equal to AB – AC from ray BX.
(iii) Join DC and draw the perpendicular bisector, say PQ of DC.
(iv) Let it intersect BX at a point A. Join AC.
Then, ABC is the required triangle.
Base BC and ∠B are drawn as given.
The point A lies on the perpendicular bisector of DC.
AD = AC
So, BD = AB – AD = AB – AC
Case (II): Let AB < AC i.e., AC – AB is given.
Steps of Construction:
(i) Draw the base BC and at point B make an angle, say XBC equal to the given angle.
(ii) Cutline segment BD equal to AC – AB from the line BX extended on an opposite side of line segment BC.
(iii) Join DC and draw the perpendicular bisector, say PQ of DC.
(iv) Let PQ intersect BX at A. Join AC.
Then, ABC is the required triangle.
Base BC and CB are drawn as given.
The point A lies on the perpendicular bisector of DC.
AD = AC
So, BD = AD – AB = AC – AB
(III) To construct a triangle, given its perimeter and its two base angles
Given: The base angles, say ∠B and ∠C and BC + CA + AB.
Required: To construct the ΔABC.
Steps of Construction:
(i) Draw a line segment, say XY equal to BC + CA + AB.
(ii) Make angles LXY equal to ∠B and MYX equal to ∠C.
(iii) Bisect ∠LXY and ∠MYX. Let these bisectors intersect a point A.
(iv) Draw perpendicular bisectors PQ of AX and RS of AY.
(v) Let PQ intersect XY at B and RS intersect XY at C. Join AB and AC.
Then, ABC is the required triangle.
B lies on the perpendicular bisector PQ of AX.
∴ XB = AB
C lies on the perpendicular bisector RS of AY.
∴ CY = AC
This gives
BC + CA + AB = BC + XB + CY = XY
Again, ∠BAX + ∠AXB (in ΔAXB, AB = XB)
and ∠ABC = ∠BAX + ∠AXB = 2∠AXB = ∠LXY
Next, ∠CAY = ∠AYC (in ΔAYC, AC = CY)
and ∠ACB = ∠CAY + ∠AYC = 2∠AYC = ∠MYX
Thus, we have what is required.
ncert solutions
Back to CBSE 9th Maths | {"url":"https://edu-spot.com/lessons/constructions/","timestamp":"2024-11-06T23:26:54Z","content_type":"text/html","content_length":"70678","record_id":"<urn:uuid:b604ccdc-9e57-4df0-bd25-de5031960a57>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00516.warc.gz"} |
Information Technology Quiz - 122 | Daily Himachal GK
Information Technology Quiz – 122
Dear Aspirants,
Information Technology Quiz is the basic part of the Himachal GK. It is helpful to increase your knowledge based on information technology. It is a series of Himachal Gk MCQs. You can also play our
weekly quiz and download all quizzes PDF as well.
1. How many different positions can you set for drop cap?
(A) 1
(B) 2
(C) 3
(D) 4
2. How many ways you can save a document?
(A) 3
(B) 2
(C) 4
(D) 6
3. What is the maximum number of lines can set for lines to drop box?
(A) 5
(B) 2
(C) 10
(D) 8
4. Single spacing in MS-Word document causes______ point line spacing?
(A) 10
(B) 12
(C) 15
(D) 9
5. What is the default number of lines to drop for drop cap?
(A) 3
(B) 6
(C) 9
(D) 12
6. What is the maximum number of lines you can set for a drop cap?
(A) 9
(B) 10
(C) 15
(D) 19
7. How many columns can you insert in a word document in maximum?
(A) 30
(B) 45
(C) 40
(D) 41
8. In a document what is the maximum number of columns that can be inserted in MS Word table?
(A) 45
(B) 50
(C) 63
(D) 60
9. What is the maximum scale percentage available in scale drop down box?
(A) 100
(B) 200
(C) 300
(D) 50
10. What is the maximum font size you can apply any character?
(A) 163
(B) 1638
(C) 13602
(D) 1500
Be the first to comment
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://dailyhimachalgk.in/information-technology-quiz-122/","timestamp":"2024-11-09T22:29:48Z","content_type":"text/html","content_length":"113436","record_id":"<urn:uuid:6dd01764-457b-4e1e-986c-3e326c46e022>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00732.warc.gz"} |
On this page, instances for the the Min-Max selection problem under budgeted uncertainty set could be found. In addition, information with regard to the size of instances provided as well as an
overall description of the considered method of instance generation is available. For more general purposes, the instance generator software is also accessible through a github repository. Finally,
if more detail about theory or application of this method is desired, the main publication introducing this method could also be reached.
It must be noticed that in order to refer to the parameters of the robust selection problem, we use n for the number of items, p for the number of items we need to choose. Moreover, we use c[i ]as
the nominal value of item i ∈ [n], d[i ]for its deviation and also Г for the parameter controlling how many items might deviate to its upper bound.
Method Description: For all i ∈ [n], we choose both c[i], d[i ]uniformly as an integer number from {1, . . . , 100}.
Instance Format
Here the instance set consists of problems with n = 40 when p = 20 and Γ ∈ {5, 10, 15, 20}. For each problem size, we generate 50 instances. Thus the instance set contains 200 instances. The instance
files are named as “instance–n–p–Γ-0-0-600-0-x”, where x represents the number of instance (1 ≤ x ≤ 50). In addition, each instance file contains three lines. The first line represents n, p and Γ the
second and third lines show c[i] and d[i] for i ∈ [n], respectively.
Generator Software
Although it is a good idea to have a library of instances for the robust optimization problems, it is not possible to upload all possible combination of problem parameters on a website.
Alternatively, the generator software could be accessed so that any instance size could be generated. Therefore, it is possible to access a C++11 code which is used as the generator software.
This page has been created based on the information provided in the following paper:
• Goerigk, M., & Khosravi, M. (2022). Benchmarking Problems for Robust Discrete Optimization. arXiv preprint arXiv:2201.04985. | {"url":"https://robust-optimization.com/instances/mm-b-u/","timestamp":"2024-11-04T21:14:10Z","content_type":"text/html","content_length":"58680","record_id":"<urn:uuid:049a8819-dc17-4a83-9ae6-209e9ccc205d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00098.warc.gz"} |
Help getting the summary of my sheet
I am not getting the result I need from a formula I am using. I am trying to track how many audits are completed and how many are left to do in total for this quarter '__ of ___ done', currently I am
using this formula:
=COUNTIF(CHILDREN(Done2:Done130), 1) + " of " + COUNT(CHILDREN([Task Name]2:[Task Name]130)) + "" + " Done"
The hierarchy looks for example:
-Spring 2024
----Sub department (sometimes this doesn't exist)
Some sections have only a location and department and some have all three. I would like the formula to count the 'done' items -- for example it should count the the sub department not the department
(if there is one), but currently it is counting everything (location , department, and sub department) making it look like we have 124 audits when we actually only have 100. Any help would be
• Create a column that puts something in it, like the number 1 for example, for each thing that you want to count, rather than counting all children.
And then your formula can count rows that have 1 in that column. This way you can also use Done:Done instead of using the specific row numbers.
• Do you know a formula that can be used to get the done column and the new extra column with the value I am trying to count?
• Hey @bsaucedo
The COUNTIF function only allows one criteria. The COUNTIFS (plural) allows multiple criteria (and can also be used with only one criteria). I'm assuming you followed @James Keuning 's advice and
found or added another column to target the rows that should be counted (or those that should be ignored). For example, sometime the Hierarchy Level becomes important when using a multi-leveled
sheet like yours. Edit the formula below with the name of the column (and criteria if different than 1)
=COUNTIFS(CHILDREN(Done:Done), 1, [New Helper]:[New Helper],1) + " of " + VALUE(COUNTIFS(CHILDREN([New Helper]:[New Helper]),1)) + "" + " Done"
Does this work for you?
• Hi @Kelly Moore,
I did follow James Keuning's suggestion. I inputted the formula you provided and got the incorrect argument error. Here is a screen grab of my sheet for you to see. I am not sure what adjustments
I need to make.
• hey @bsaucedo
My bad. I forgot to add the CHILDREN before your new column name. The range element must all be equal amongst all the terms.
=COUNTIFS(CHILDREN(Done:Done), 1, CHILDREN([New Helper]:[New Helper]),1) + " of " + VALUE(COUNTIFS(CHILDREN([New Helper]:[New Helper]),1)) + "" + " Done"
• That worked! @Kelly Moore Do you know how I can get the percentage of completed? (row 2 to row 129) have values I would like to average. Currently I have:
=AVG(CHILDREN([Percent Complete]@row)) for each child as you can see for the dark blue and light blue sections but would like it for the overall audit sheet(green). I believe that is looking at
the whole row not just the targeted areas correct?
• In the 'white' child rows, how is that %complete inputted? Manually or by formula?
• @Kelly Moore I have them in as a dropdown but also there is an automation depending on the status of each row. Green - 100%, Yellow - 50%, Red - 0%.
• The short answer to your question is yes, we could roll up your %complete to the Parent rows, and your Green row is the Parent to the sheet. Your opportunity comes from how you are inputting data
into the Child rows. Unless you manually enter a formula into every parent row (and I don't recommend that at all unless your sheet is never going to change or grow), you can't have formulas and
manual input in the same column.
You could add a second %complete column and use that as the roll up. Formulas would be in that column. Or, as a different option, you could abandon manually selecting values from the dropdown
column and we would build a formula for the child rows that replicate what the automation is doing. We would use IF statements to determine if the row was a Parent row or not, and calculate the
formula accordingly. You would not need a second %complete column for this - although I might suggest a different helper column to simplify the formulas a bit. In my sheets, wherever possible, I
use the second option so that %completes are always calculated rather than giving people choices. Anytime you are dealing with %complete there are trade-offs no matter what approach you select.
To summarize, I gave you 3 options
□ Manually insert a formula into every Parent row. (This is the least reliable solution. It's not recommended)
□ Add a second %complete column to either use for the manual entries or the parent row roll up.
□ Use formulas only for every row. (This is what I recommend if your users will allow formula calculated %complete for their child rows)
• Once you decide which option is best for you, let me know and I'll help you build it
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/118735/help-getting-the-summary-of-my-sheet","timestamp":"2024-11-03T18:30:48Z","content_type":"text/html","content_length":"462352","record_id":"<urn:uuid:e02774ae-fd24-4e2f-b6d4-214e4e4b3805>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00352.warc.gz"} |
Right Triangles Pythagorean Theorem Worksheet Answers - TraingleWorksheets.com
Pythagorean Theorem Is It A Right Triangle Worksheet – Triangles are one of the most fundamental designs in geometry. Understanding triangles is important for getting more advanced concepts in
geometry. In this blog this post, we’ll go over the different kinds of triangles with triangle angles. We will also discuss how to determine the dimension and perimeter of the triangle, as well as
provide specific examples on each. Types of Triangles There … Read more | {"url":"https://www.traingleworksheets.com/tag/right-triangles-pythagorean-theorem-worksheet-answers/","timestamp":"2024-11-09T07:00:11Z","content_type":"text/html","content_length":"47479","record_id":"<urn:uuid:4b2be2f4-3878-4be8-b3c5-6e77151a4024>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00271.warc.gz"} |
Microcontroller - 8051
Tuesday 28 May 2024
for YouTube Videos click
Sunday 28 April 2024
Good afternoon to all of you, For assignment 1 Fill the Google form on or before 29/04/2024 before 11AM. https://docs.google.com/forms/d/e/1FAIpQLSc15HKKZ1ONpBYA_SbC83KWN63oK0EZSBSNfzKhIJRAhKH4_A/
Tuesday 23 April 2024
The 8051 Architecture : Introduction, difference between Microprocessors and Microcontrollers, RISC & CISC CPU Architectures, Harvard & Von-Neumann CPU architecture. The 8051 Architecture-Block
diagram, Pin Configuration, 8051 port structure. 8 Hrs
Addressing Modes and Operations: Introduction, Addressing modes, External data transfer, Code Memory, Read Only Data transfer / Indexed Addressing modes, PUSH and POP opcodes, Data exchanges, Example
Programs; Byte level logical Operations, Bit level Logical Operations, Rotate and Swap Operations, Example Programs. Arithmetic Operations: Flags, Incrementing and Decrementing, Addition,
Subtraction, Multiplication and Division, Example-programs. 8 Hrs
All the students are here by informed to attend the classes regularly. Few students have missed few classes recently; they may get attendance shortage for the second IA. Submit the Assignments on or
before the deadline. All the best for your IA 1.
Wednesday 17 April 2024
Make a group of two students, Refer https://svv8051.blogspot.com/ right handside for students "Micro controller projects" or you can refer any websites, choose any one 8051 based project, use
assembly or C prgramming . Google form will be shared with you all next week, fill the form - your team members, title of the project. Show the simulation using Proteus software on or before second
IA, prepare a 2 minutes video of your project, prepare the report for the same and submit it. You may do implementation of your project on 8051 micro-controller kit also. If you need any help if you
feel any difficulties come and discuss with me.
Sunday 14 April 2024
1) To add N natural numbers
2) BCD input -Addition nibble wise: Ex 45 then R5=5+4=09
3) Toggle the port 1 by sending 55H and AAh continuously with delay. Using Subroutine
4) Two digit BCD number is stored in memory location 30H. Unpack the BCD number and store the two digits in memory locations 31H and 32H. 30H is 49 then 31H will be 04 and 32H will be 09
5) Write a program to count number of 1's in the contents of r2 register and store the count in the r3 register.
6) Find the largest number in a block of data. The length of the block is in memory location 30H and the block itself starts from memory location 31H. Store the maximum number in memory location 40H.
Assume that the numbers in the block are all 8 bit unsigned binary numbers.
7) Count even numbers
8) Find sum of odd numbers
9) Find the sum of even numbers
10) Find the sum of even numbers and odd numbers
11) Find the square of a number stored in 30h and store result in 31h and 32h.
12) Delay Calculations
13) Add two numbers external data 30002=3001+3000
14) Find number of negatives and positives in a given array
15) Search and count how many times the given byte of memory 30h has appeared in the given array starts from 32h onwards. The length of the array is in 31h. store result in r5.
16) Multi byte BCD additions
17) Separate even numbers from an given array. Array starts from 30H and store the Even numbers in location 40H onwards. (Choose length of the array)
18) Consider external memory -Repeat all the programs. Consider external and internal mixed then write a programs
19) Find the cube of a number stored in 30h and store result in 31h and 32h.
20) Generate Fibonacci numbers
For more programs list click
Thursday 22 February 2024
Good afternoon to all of you, I am going to handle 8051 Micro controller subject for 4th sem A division -2024.
All the students are here by informed to attend the classes regularly and go through the material shared in this blog.
Thursday 30 May 2019
Dear student friends i am going to handle 8051 Micro controller for this STC.
Registered students are here by informed to come and meet me. | {"url":"https://svv8051.blogspot.com/","timestamp":"2024-11-10T06:25:05Z","content_type":"text/html","content_length":"83595","record_id":"<urn:uuid:6e9d5694-6304-4a28-8428-a5465a25dd9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00741.warc.gz"} |
WIAS Software
Longitudinal Dynamics in multisection Semiconductor Lasers
Developed by:
LDSL-tool is a software for simulation and analysis of the (L)ongitudinal (D)ynamics in multisection (S)emiconductor (L)asers. This software is based on Traveling Wave (PDE) equations describing the
propagation of optical fields along the longitudinal direction of the laser, which are nonlinearly coupled with the ordinary differential equations for carrier densities and polarization functions.
LDSL-tool not only integrates the PDE model equations but also allows the analysis of the dynamics of longitudinal modes and the building of reduced ODE models based on a finite number of modes.
After showing good qualitative and quantitative agreement between basic Traveling Wave and Mode Approximation models, the reduced models can be analyzed with well-known tools for bifurcation
analysis, such as AUTO. Such different possibilities, together with some data post-processing routines, make our software a powerful tool suited for the simulation and analysis of different dynamic
effects in semiconductor lasers.
Multisection semiconductor lasers seem to be key elements in optical communication systems. Depending on their structure and operational conditions, such lasers can demonstrate rich dynamics.
Some of these dynamical regimes, such as, e.g., high-frequency self-pulsations can be applied for all optical signal regeneration. A deeper study of the underlying nonlinear processes and
optimization of such lasers is still strongly required.
An example of a modeled laser: It is a 3-section DFB laser, made at the Fraunhofer Institut Nachrichtentechnik Heinrich-Hertz-Institut (HHI), Berlin. Optical fields, polarizations and carrier
densities are calculated with LDSL-tool.
A deep understanding of nonlinear dynamics demonstrated by semiconductor lasers is very useful when designing lasers for specific purposes. Our software LDSL-tool is used to investigate and to
design lasers that exhibit various nonlinear effects such as self pulsations, chaos, hysteresis, mode switching, excitability, and synchronization to an external signal frequency (see, e.g. WIAS
Preprints 516, 597, 712, 713, 809, 849, 866, 1039, 1149, 1513, 1579, 1584, 1981, 2011, 2261, 2438, and 2604)
This software solves models of different complexity, ranging from partial differential equation (PDE) to reduced ordinary differential equation (ODE) systems. PDE models are based on the
Traveling wave (TW) equations for counter-propagating optical fields, and ODE models are given by the mode approximation (MA) of the TW model.
In certain cases our software allows to analyse the mode dynamics of PDE systems and to compare the solutions demonstrated by TW model and reduced MA models. After showing good qualitative and
quantitative agreement between basic TW and low dimensional MA models, the obtained system of ODE's can be analyzed with well known tools for bifurcation analysis such as AUTO.
A brief scheme of the LDSL-tool. Blue, green, and yellow colours indicate the hierarchy of models, the computational efforts, and the different processing and analysis of computed data. Blue
arrows show relations that are available under some restrictions.
Besides the above-mentioned multisection semiconductor lasers, our software allows for considering a variety of coupled laser devices, including straight multisection and ring lasers. Namely, we
represent the considered laser device as a set of differently joined components with negligible lateral and transversal dimensions, assume that the field dynamics within each part of such device
is governed by a pair of mutually coupled 1 (time) + 1 (space)-dimensional traveling wave equations, and describe the relations between optical fields at the junctions of different sections of
the device by the field transmission-reflection conditions given by the user-defined complex-valued matrices. Sections, Junctions, and Optical injections are the main building blocks of the laser
devices considered by LDSL-tool (see WIAS-Preprints 1315, 2261).
Schematic representation of several examples of considered multisection lasers and coupled laser devices. (a): a multisection laser with non-vanishing internal field reflection between the second
and third sections; (b): a laser with a passive external cavity; (c): a master-slave laser system separated by the air gap; (d): a ring laser with an attached outcoupling waveguide; (e): an
optically injected laser.
Our basic mathematical model is based on Traveling Wave equations for optical fields coupled with ordinary differential equations for carrier densities and polarization functions. Under certain
assumptions, our software is able to build and analyze low-dimensional ODE models based on mode approximations. We have also introduced some limited possibilities to trace and analyze stationary
states of a "full" Traveling Wave model.
To resolve the longitudinal distribution and dynamics of the carrier density n(z,t), the counter-propagating optical fields ψ(z,t)=(ψ^+(z,t), ψ^-(z,t))^T and polarization functions p(z,t)=(p^+
(z,t), p^-(z,t))^T in each part of the multi-section semiconductor laser or coupled laser system we use the Traveling Wave (TW) model:
Once considering quantum-dot lasers, we introduce one or two additional rate equations to desribe carrier transitions between carrier reservoir, ground- and excited- states of the quantum dots,
see WIAS preprints 1506, 1579, 1584.
Straightforward integration of these equations can immediately give us, e.g., field output at laser facets and variation of mean carrier densities in time or field/carrier density distributions
at some fixed time layer:
Left: Time traces of field output at laser facets (above) and mean carrier densities in two laser sections (below). Right: axial distribution of forward and backward propagating field power
(above and carrier densities (below).
To get deeper information about the structure of optical fields, we are solving the spectral problem of the Traveling Wave model and finding the decomposition of optical field/polarization into
modal components. Here, we consider slowly varying carrier densities as parameters and solve the spectral problem for each instant distribution of n(z,t):
Frequently, this field decomposition improves our understanding of lasers' non-stationary behavior. This approach properly indicates the modes which govern the complicated behavior of the laser
and shows much more details than usual spectra of the optical field:
Left above and below: change of complex eigenvalues of the spectral problem during one period of self-pulsations. Left below: change of modal amplitudes and corresponding modal wavelengths during
one period of self-pulsations. Black dots indicate optical spectra obtained by FFT of the emitted optical field. Right above: pulsating power of optical field outgoing from the laser. Right
below: dynamic of modal amplitudes obtained by field decomposition.
More details about the calculation of optical modes and the field decomposition into modal components can be found, e.g., in WIAS Preprints 712, 939, 2011, and 2261.
After restricting mode expansion to q leading modes and substituting it to field/polarization equations in our TW model, one becomes q ordinary differential equations describing the evolution of
complex amplitudes of optical modes:
These ordinary differential equations and the equations for carrier densities can be solved and analyzed instead of the TW model. If selecting a sufficient number of leading modes, the solutions
of the traveling wave model and mode approximation systems are in perfect agreement:
Mode (red) and Single Mode (blue) Approximation models recover self-pulsations, computed with the TW model (black).
For more details on such reduced Mode Approximation model, see, e.g., WIAS Preprint 713, 1149, and 2261.
After showing good qualitative and quantitative agreement between basic Traveling Wave and Mode Approximation models, the reduced models can be analyzed with well-known tools for bifurcation
analysis, such as AUTO:
Switching on and off the self-pulsations by tuning parameters (current injection) in different directions. Above: experimental data. Below: theoretical simulations and analysis. Green and violet
dashed lines correspond to the decrease and increase of the bifurcation parameter, respectively. These lines represent a stable solution (peak of power frequency in experiments or maximal power
in simulations of the TW model) after some transient time. Thick lines in the lower figure represent stable (red) and unstable (blue) solutions of two-mode approximation systems. Here,
computations were made with the path-following tool AUTO, which allows for identifying different bifurcations indicated by solid symbols.
For more details, see, e.g., WIAS Preprints 713, 985, 1149, 2261.
One can also perform a two-parameter bifurcation analysis of the mode approximation systems. In this case, the bifurcations are represented as curves in the two-parameter domain. These curves
define the stability borders of different attractors in the considered system.
Areas of pulsations in two-parameter planes. Left: measurements (above) and simulations (below), characterizing the type of dynamical state at fixed parameters from the transients. Only
attracting states can be detected! Right: Two-parameter bifurcation diagrams of four-mode approximation system. A global view of the pulsating areas shown on the left side (above) and more
detailed insight in the middle area with an indication of codimension-two bifurcation points (below). Colored lines show different transitions (bifurcations) between qualitatively different
dynamical states. They were computed using pathfollowing of bifurcations in a two-parameter domain.
For more details, see, e.g., WIAS-Preprints 985, 866, and 2261.
Under similar assumptions needed to derive Mode Approximation systems, we can also trace stationary states of the "full" TW model by changing some parameters and analyzing their stability. For
the representation of such results for a three-section laser with one active section, we use similar diagrams as are used for the analysis of "external cavity modes located along ellipses" in the
Lang-Kobayashi model of lasers with external feedback:
Red lines ("ellipses" of external cavity modes in the LK model) represent traces of stationary states when changing phase parameters. Different lines correspond to different levels of internal
losses of the optical field. Light blue lines show traces of stationary states when keeping the phase parameter fixed and tuning the value of the internal loss. The intersection of red and light
blue lines gives positions of stationary states (compound cavity modes) at fixed parameter values. The dark blue line shows the location of saddle-node bifurcation of stationary states. All
stationary states located "inside" dark blue lines are unstable of saddle type (antimodes of LK model). Stationary states lying "outside" these dark blue lines are either stable states or
unstable states with an even number of unstable directions (modes of LK model). Magenta lines represent pairs of stationary states (mode and anti-mode in the LK model) with the same threshold
carrier density; these modes are responsible for the generation of a stable quasiperiodic solution of beating type (first theoretically found by Tager and Petermann in LK models). The right
figure is an enlarged part of the left one.
For more details see, e.g., WIAS-Preprints 985, 1981, 1513, 2261, and 2961.
Calculation of stable and unstable steady states in the general TW model accounting for longitudinal distributions of carriers and nonlinear gain compression is much more involved since, in this
case, instead of dealing with only a few scalar algebraic equations, we have to solve a system of algebraic and functional (z-dependent) equations, which can hardly be resolved on the functional
level. In this case, we rely on numerical discretization, which allows functional equations to be replaced with several hundred algebraic equations, which can then be resolved using Newton
iterations. Knowledge of these stationary states is needed, e.g., when performing the linewidth estimation of the laser emission. For more details, see WIAS-Preprint 2961.
Stable (solid) and unstable (dashed) steady states in the extended cavity diode laser (z-axis range [0,1] mm) containing DBR section at the end of the cavity (z-axis range [3,6] mm). Each state
is indicated by the same color. (a) and (b): carrier density and local field intensity distributions in the active section. (c): reconstructed intensities of forward and backward propagating
fields in the whole cavity for two coexisting stable states. (d): intensity reflection spectrum of the external cavity. (e): empty bullets: steady states in relative frequency - threshold gain
plane. Black dots and solid/dashed lines: steady states and branches of these states in corresponding basic TW model.
Besides the already mentioned analysis, the LDSL tool can also be applied for automatic loop computations, tuning selected laser parameters, and recording some of the most specific characteristics of
the dynamical behavior of model equations. In this manner, we can locate regions of different stable dynamical laser behavior in parameter space.
The LDSL tool automatically scans parameters to look for high-frequency self-pulsations with good extinction. In the following figure, a three-section DFB laser is considered. Phase and detuning
parameters represent field phase shift due to current injection into the passive middle section and detuning between Bragg wavelengths of two DFB sections, respectively.
Regions of robust SP and their frequencies in a 3-section laser with one active DFB section. Violet/white regions: stationary lasing at the long/short-wavelength stop band side.
Scanning of the same parameters as before for the three-section DFB laser with two active equally pumped DFB sections. The figure below shows areas of parameter plane where high frequency self
pulsations with good and bad extinction ratio can be observed.
The frequencies and extinction ratio of mode-beating self-pulsations in a three-section DFB laser with two active DFB sections are modeled, depending on the phase parameter and the detuning
between Brag grating wavelengths.
More details in WIAS Preprint 809 .
To characterize the quality of "noisy" self-pulsations demonstrated by lasers, we sample the pulsating output field with its mean frequency. Different projections of the sampled output give
useful characteristics of the laser.
Sampling of the pulsating signal. Left: the signal is cut into pieces with a mean period of pulsations, and different pulses are located one behind another. Blue points show positions where these
pulses are crossing some mean power plane. Middle: eye diagram, or projection of the first diagram onto the front plane. Right: pulse drift diagram, or projection of the blue points of the first
diagram onto the bottom plane. The middle and right diagrams also show algorithms to estimate the ''absolute'' and ''normal'' jitter, respectively.
More details can be found in WIAS Technical Report 2 and WIAS Preprint 809 .
LDSL tool can also be used to analyze the synchronization of SP to external optical or electrical signals. In this case, we sample our output signal with the period of the external modulated
signal. In the case of synchronization, an open eye is seen in the eye diagram, and the pulse drifts along a horizontal line in the pulse drift diagram. Otherwise, the eye is closed, and the
pulse drifts out from the fixed position.
Locking of Self-pulsations to electrical modulation at a 33 GHz repetition frequency. Left: eye diagrams showing unlocked, almost locked, and locked self-pulsations. Right: the drift of the
relative phases of already indicated solutions.
When we model optical injection or two or more well-separated optical modes define the laser dynamics, the high-frequency beating is seen in the temporal trace of the output signal. To
distinguish the contribution of one or another wavelength in the total signal, one can apply filters, which in frequency or wavelength domains are described by a Lorentzian function and in the
time domain can be given as a solution of an ordinary differential equation.
Filtering of the optical field emitted by a self-pulsating laser with applied modulated optical input. Left above: a sketch of such injected laser. Left below: optical spectra of nonfiltered
output (red), filtered output with filter peak at 0 nm (violet) and filtered output at 4 nm (green) relative wavelength. Right above: emission power (red) and power of optical injection (green)
at the left facet of the laser. Right below: the power of the filtered emission when the filter was centered at 0 nm (violet) and 4 nm (green) relative wavelength.
The excitability of DFB lasers with integrated passive delay section is realized by injecting short optical pulses. In this case, the theoretical study of model equations has allowed us to
predict and realize excitability in experiments.
Theoretical (left) and experimental (right) demonstration of excitability of lasers due to injection of short optical impulse. Upper lines show large response of laser when impulse strength
exceed some certain threshold. Lower lines show subthreshold response of laser. Inserts indicate nonlinear response of laser.
More details in WIAS Preprint 712 .
To perform a small signal analysis of the laser operating at the cw state, we apply small-amplitude periodic current modulation at fixed different frequencies and, after some long transient,
estimate the amplitude of the resulting output. Alternatively, the same result can be achieved much faster by finding the Fourier transform of the transient output power after a
delta-function-like perturbation of the current injection.
Small signal modulation response functions of solitary DFB laser (left) and at two different operational conditions in DFB laser with integrated passive external cavity section (right). The left
panel compares two different methods for estimating modulation response. The right panel shows the presence of the intracavity resonance at 35-40 GHz frequency.
Directly modulated semiconductor lasers are of great interest in laser applications for optical data transmission systems. Here, we demonstrate the required performance of the DFB laser with an
integrated external cavity at a current modulation with 40 Gb/s PRBS. This modulation rate ~4 times exceeds the usual relaxation oscillation frequency of the considered laser with vanishing
Simulated laser response to 40 Gb/s NRZ PRBS current modulation. Left: laser response when parameters are fixed. (a): injected current (red) and output power (blue); (b): open eye diagram; (c):
histogram of points within dashed box of panel b. Right: suitable operation areas in the parameter plane. Top: Photon-Photon resonance of unmodulated laser. (a): frequency of PP resonance. White:
nonstationary regimes. (b): relative Carrier-Photon resonance suppression. White: CP dominates. Bottom: Quality of laser response to current modulation (40 Gb/s PRBS). (c): extinction. (d): eye
The concept of differently interconnected sections and junctions allows for modeling rather complicated multisection semiconductor lasers. One such nontrivial configuration is a semiconductor
ring laser with four separate branches of the filtered optical feedback, see panel (a). The multi-channel feedback scheme of this laser admits a fast switching between steady states determined by
the resonances of the ring laser and the wavelengths of the activated filtering channels. Colored frames in panel (a) represent different types of device sections. Namely, we distinguish here the
amplifying sections (light red), where the field and carrier dynamics are governed by the full TW model, and two kinds of passive sections, where gain and refractive index functions are set to
zero, allowing to ignore the carrier rate equations at all, whereas the propagating field experiences phase change and losses (blue) and, additionally, well-pronounced filtering of the optical
frequencies (yellow). The notations of all sections in the section indexes are made according to the cardinal directions. For more details, see see WIAS-Preprints 2261, 2438.
(a): Scheme of the semiconductor ring laser with four branches of filtered and amplified unidirectional optical feedback. Black segments and colored frames indicate junctions and different
sections of the laser device. The red and blue arrows show propagation directions and the emission of the counter-propagating fields, respectively. (b): Transmission spectra of four filtering
branches. (c): Stabilization of the multi-mode behavior of the ring laser (black) by the single-branch filtered feedback (colored). (d): The dependence of the lasing wavelength on the feedback
phase once the second filtering branch is activated.
Based on the single-mode approximation of the TW wave model, see WIAS preprint 2838, we can estimate the spectral linewidth and some other important parameters of the laser with a steady-state
(continuous wave) emission. The model for the linewidth is based on the field expansion into the optical modes, accounting for the effects of nonlinear gain compression, gain dispersion, and
longitudinal spatial hole burning in multi-section cavity structures.
Simulated characteristics of the DBR laser as functions of the upsweeped injection current. Dotted and dashed lines indicate mode jumps and maxima of the DBR reflectivity, respectively.
All publications listed below discussed different structures of multisection semiconductor lasers and were supported by simulations of the LDSL tool.
□ M. Radziunas, "Calculation of steady states in dynamical semiconductor laser models," Optical and Quantum Electronics 55, 121, 2023. WIAS Preprint, (2961), 2022.
□ M. Radziunas, ''Longitudinal modes of multisection edge-emitting and ring semiconductor lasers'', Optical and Quantum Electronics 47(6), pp. 1319-1325, 2015. WIAS-Preprint, (2011).
□ M. Lichtner, M. Radziunas, L. Recke, ''Well posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics'', Mathematical Methods in
Applied Sciences 30(8), pp. 931-960, 2007.
□ M. Radziunas, H.-J. Wünsche, B. Krauskopf, M. Wolfrum, " External cavity modes in Lang-Kobayashi and traveling wave models", in SPIE Proceedings Series, (6184), art. no. 61840X, 2006.
WIAS-Preprint, (1111).
□ M. Radziunas, ''Numerical bifurcation analysis of traveling wave model of multisection semiconductor lasers'', Physica D, 213(1), pp. 98-112, 2006. WIAS-Preprint, (985).
□ M. Radziunas, H.-J. Wünsche, ''Multisection Lasers: Longitudinal Modes and their Dynamics'', in Optoelectronic Devices - Advanced Simulation and Analysis, pp. 121-150, ed. J. Piprek, Springer
Verlag, New York, 2005. ISBN: 0-387-22659-1 WIAS-Preprint, (939).
□ J. Sieber, M. Radziunas, K. Schneider, ''Dynamics of multisection semiconductor lasers'', Math. Model. Anal. 9(1), pp. 51-66, 2004. pdf file.
□ H. Wenzel, M. Kantner, M. Radziunas, U. Bandelow, "Semiconductor Laser Linewidth Theory Revisited," Appl. Sci. 11(13), 6004, 2021. WIAS Preprint, (2838).
□ M. Radziunas, D.J. Little, and D.M. Kane, "Numerical study of optical feedback coherence in semiconductor laser dynamics," Optics Letters, 44(17), pp. 4207-4210, 2019. WIAS-Preprint (2604).
□ M. Radziunas, ''Traveling wave modeling of nonlinear dynamics in multisection semiconductor laser diodes'', Chapter 31 in J. Piprek (Ed.), Handbook of Optoelectronic Device Modeling and
Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods, Vol. 2, pp. 153-182, CRC Press, 2017. WIAS-Preprint (2261).
□ M. Radziunas, A.G. Vladimirov, E.A. Viktorov, G. Fiol, H. Schmeckebier, D. Bimberg, ''Strong pulse asymmetry in quantum-dot mode-locked semiconductor lasers'', Appl. Phys. Lett. 98, art. no.
031104, 2011. WIAS-Preprint, (1579).
□ M. Radziunas, A.G. Vladimirov, E. Viktorov, ''Traveling wave modeling, simulation and analysis of quantum-dot mode-locked semiconductor lasers'', in SPIE Proceedings Series, (7720), art. no.
77200X, 2010. WIAS-Preprint, (1506).
□ M. Radziunas, ''Traveling wave modeling of semiconductor ring lasers'', in SPIE Proceedings Series, (6997), art. no. 69971B, 2008. WIAS-Preprint, (1315).
□ T. Perez, M. Radziunas, H.-J. Wünsche, C.R. Mirasso, F. Henneberger, ''Synchronization properties of two coupled multisection semiconductor lasers emitting chaotic light'', Phot. Techn.
Lett., 18(20), pp. 2135-2137, 2006.
□ M. Radziunas, H.-J. Wünsche, ''Multisection Lasers: Longitudinal Modes and their Dynamics'', in Optoelectronic Devices - Advanced Simulation and Analysis, pp. 121-150, ed. J. Piprek, Springer
Verlag, New York, 2005. ISBN: 0-387-22659-1 WIAS-Preprint, (939), 2004.
□ N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Mutually injecting semiconductor lasers: simulations for short and zero delay'', in SPIE Proceedings Series, (5452), pp. 63-70,
2004. pdf file.
□ H.-J. Wünsche, M. Radziunas, S. Bauer, O. Brox, B. Sartorius, "Simulation of Phase-Controlled Mode-Beating Lasers", IEEE J Selected Topics of Quantum Electron. 9(3), pp. 857-864, 2003.
WIAS-Preprint, (809), 2003.
□ N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Bifurcations of a DFB Laser with Short Optical Feedback: Numerical Experiment'', in SPIE Proceedings Series, (4986), pp. 480-489,
2003. pdf file.
□ M. Radziunas, H.-J. Wünsche, ''LDSL: a tool for simulation and analysis of longitudinal dynamics in multisection semiconductor laser'', in Proceedings of 2nd International Conference on
Numerical Simulations of Optoelectronic Devices (NUSOD-02), Zürich, pp. 26-27, 2002. pdf file.
□ M. Radziunas, H.-J. Wünsche, ''Dynamics of multisection DFB semiconductor laser: traveling wave and mode approximation models'', in SPIE Proceedings Series, (4646), pp. 27-37, 2002.
WIAS-Preprint 713 .
□ M. Radziunas, ''Sampling techniques applicable for the characterization of the quality of self pulsations in semiconductor lasers'', WIAS-Technical Report, (2), 2002.
□ U. Bandelow,M. Radziunas, J. Sieber, M. Wolfrum, "Impact of gain dispersion on the spatio-temporal dynamics of multisection lasers", IEEE J Quantum Elect. 37(2), pp. 183-188, 2001.
WIAS-Preprint 597 .
□ U. Bandelow, M. Radziunas, V. Tronciu, H.-J. Wünsche, F. Henneberger, ''Tailoring the dynamics of diode lasers by dispersive reflectors'', in SPIE Proceedings Series, (3944), pp. 536-545,
2000. pdf file.
□ M. Krüger, V.Z. Tronciu, A. Bawamia, Ch. Kürbis, M. Radziunas, H. Wenzel, A. Wicht, A. Peters, G. Tränkle, "Improving the spectral performance of extended cavity diode lasers using
angled-facet laser diode chips," Appl. Phys. B 125, 66 (12pp), 2019.
□ M. Khoder, M. Radziunas, V.Z. Tronciu, G. Verschaffelt, "Study of wavelength switching time in tunable semiconductor micro-ring lasers: experiment and travelling wave description," OSA
Continuum, 1(4), pp. 1226-1240, 2018.
□ V. Tronciu, H. Wenzel, M. Radziunas, M. Reggentin, J. Wiedmann, A. Knigge, "Investigation of red-emitting distributed Bragg reflector lasers by means of numerical simulations", IET
Optoelectronics, 12(5), 228-232, 2018.
□ M. Radziunas, M. Khoder, V. Tronciu, J. Danckaert, G. Verschaffelt, ''Semiconductor ring laser with filtered optical feedback: traveling wave description and experimental validation,'' J.
Opt. Soc. Am. B 35(2), 380-390, 2018. WIAS-Preprint (2438).
□ V.Z. Tronciu, M. Radziunas, Ch. Kürbis, H. Wenzel, A. Wicht, ''Numerical and experimental investigations of micro-integrated external cavity diode lasers'', Optical and Quantum Electronics 47
(6), pp. 1459-1464, 2015.
□ M. Radziunas, V.Z. Tronciu, E. Luvsandamdin, Ch. Kürbis, A. Wicht, H. Wenzel, ''Study of micro-integrated external-cavity diode lasers: simulations, analysis and experiments'', IEEE J. of
Quantum Electronics, 51(2), art. no. 2000408, 2015. WIAS-Preprint, (1981).
□ S. Joshi, C. Calo, N. Chimot, M. Radziunas, R. Arkhipov, S. Barbet, A. Accard, A. Ramdane, F. Lelarge, ''Quantum dash based single section mode locked lasers for photonic integrated
circuits'', Optics Express 22(9), pp. 11254-11266 , 2014.
□ M. Radziunas, A.G. Vladimirov, E.A. Viktorov, G. Fiol, H. Schmeckebier, D. Bimberg, ''Pulse broadening in quantum-dot mode-locked semiconductor lasers: simulation, analysis and experiments'',
IEEE J. of Quantum Electronics 47(7), pp. 935-943, 2011. WIAS-Preprint, (1584).
□ M. Radziunas, K.-H. Hasler, B. Sumpf, Tran Quoc Tien, H. Wenzel, ''Mode transitions in DBR semiconductor lasers: Experiments, simulations and analysis'', J. Phys. B: At. Mol. Opt. Phys. 44,
art. no. 105401, 2011. WIAS-Preprint, (1513).
□ O.V. Ushakov, N. Korneyev, M. Radziunas, H.-J. Wünsche, F. Henneberger, ''Excitability of chaotic transients in a semiconductor laser'', Europhysics Letters 79, 30004 (5pp), 2007. pdf file.
□ M. Radziunas, A. Glitzky, U. Bandelow, M. Wolfrum, U. Troppenz, J. Kreissl, W. Rehbein, ''Improving the modulation bandwidth in semiconductor lasers by passive feedback'', IEEE J. of Selected
Topics in Quantum Electronics 13(1), pp. 136-142, 2007. WIAS-Preprint (1149).
□ U. Bandelow, M. Radziunas, A. Vladimirov, B. Hüttl, R. Kaiser, "Harmonic Mode-Locking in Monolithic Semiconductor Lasers: Theory, Simulations and Experiment", Optical and Quantum Electronics
38, pp. 495-512, 2006. WIAS-Preprint, (1039).
□ S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H.-J. Wünsche, F. Henneberger ''Nonlinear Dynamics of Semiconductor Lasers with Active Optical Feedback'', Phys. Rev. E
69, 016206, 2004. WIAS-Preprint, (866), 2003.
□ O. Brox, S. Bauer, M. Radziunas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius, H.-J. Wünsche, ''High-Frequency Pulsations in DFB-Lasers with Amplified Feedback'', IEEE J Quantum Elect., 39
(11), pp. 1381-1387, 2003. WIAS-Preprint (849).
□ H.-J. Wünsche, O. Brox, M. Radziunas, F. Henneberger, "Excitability of a semiconductor laser by a two-mode homoclinic bifurcation", Phys. Rev. Lett. 88(2), art. no. 023901, 2002. pdf file.
□ M. Radziunas, H.-J. Wünsche, O. Brox, F. Henneberger, ''Excitability of a DFB laser with short external cavity'', in SPIE Proceedings Series, (4646), pp. 420-428, 2002. WIAS-Preprint 712 .
□ M. Möhrle, B. Sartorius, C. Bornholdt, S. Bauer, O. Brox, A. Sigmund, R. Steingrüber, M. Radziunas, H.-J. Wünsche, "Detuned grating multisection-RW-DFB lasers for high-speed optical signal
processing", IEEE J Selected Topics of Quantum Electron. 7(2), pp. 217-223, 2001. pdf file.
□ M. Radziunas, H.-J. Wünsche, B. Sartorius, O. Brox, D. Hoffmann, K. Schneider, D. Marcenac, "Modeling self-pulsating DFB lasers with an integrated phase tuning section", IEEE J Quantum Elect.
36(9), pp. 1026-1034, 2000. WIAS-Preprint 516 .
Page created and maintained by Mindaugas Radziunas. Last update on April 22, 2024. | {"url":"https://www.wias-berlin.de/software/index.jsp?id=LDSL&lang=1","timestamp":"2024-11-07T22:41:24Z","content_type":"text/html","content_length":"86565","record_id":"<urn:uuid:687e96c6-7dbe-4482-934a-bbe8b1ea1693>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00037.warc.gz"} |
The LOG10 Function
The LOG10 function returns a numeric value that approximates the logarithm to the base 10 of argument-1. The type of this function is numeric.
General Format
1. Argument-1 must be class numeric.
2. The value of argument-1 must be greater than zero.
Returned Values
1. The returned value is the approximation of the logarithm to the base 10 of argument-1.
2. Floating-point format is used for numeric non-integer results. | {"url":"https://www.microfocus.com/documentation/visual-cobol/30pu12/VC-DevHub/HRLHLHPDF70R.html","timestamp":"2024-11-08T14:44:13Z","content_type":"text/html","content_length":"11548","record_id":"<urn:uuid:efc3a5bc-87cc-4186-b730-0214c58e4e33>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00656.warc.gz"} |
manual pages
edge_connectivity {igraph} R Documentation
Edge connectivity.
The edge connectivity of a graph or two vertices, this is recently also called group adhesion.
edge_connectivity(graph, source = NULL, target = NULL, checks = TRUE)
graph The input graph.
source The id of the source vertex, for edge_connectivity it can be NULL, see details below.
target The id of the target vertex, for edge_connectivity it can be NULL, see details below.
checks Logical constant. Whether to check that the graph is connected and also the degree of the vertices. If the graph is not (strongly) connected then the connectivity is obviously zero. Otherwise
if the minimum degree is one then the edge connectivity is also one. It is a good idea to perform these checks, as they can be done quickly compared to the connectivity calculation itself.
They were suggested by Peter McMahan, thanks Peter.
The edge connectivity of a pair of vertices (source and target) is the minimum number of edges needed to remove to eliminate all (directed) paths from source to target. edge_connectivity calculates
this quantity if both the source and target arguments are given (and not NULL).
The edge connectivity of a graph is the minimum of the edge connectivity of every (ordered) pair of vertices in the graph. edge_connectivity calculates this quantity if neither the source nor the
target arguments are given (ie. they are both NULL).
A set of edge disjoint paths between two vertices is a set of paths between them containing no common edges. The maximum number of edge disjoint paths between two vertices is the same as their edge
The adhesion of a graph is the minimum number of edges needed to remove to obtain a graph which is not strongly connected. This is the same as the edge connectivity of the graph.
The three functions documented on this page calculate similar properties, more precisely the most general is edge_connectivity, the others are included only for having more descriptive function
A scalar real value.
Gabor Csardi csardi.gabor@gmail.com
Douglas R. White and Frank Harary: The cohesiveness of blocks in social networks: node connectivity and conditional density, TODO: citation
See Also
max_flow, vertex_connectivity, vertex_disjoint_paths, cohesion
g <- barabasi.game(100, m=1)
g2 <- barabasi.game(100, m=5)
edge_connectivity(g, 100, 1)
edge_connectivity(g2, 100, 1)
edge_disjoint_paths(g2, 100, 1)
g <- sample_gnp(50, 5/50)
g <- as.directed(g)
g <- induced_subgraph(g, subcomponent(g, 1))
version 1.3.3 | {"url":"https://igraph.org/r/html/1.3.3/edge_connectivity.html","timestamp":"2024-11-04T14:08:22Z","content_type":"text/html","content_length":"11822","record_id":"<urn:uuid:995aa003-0f5d-433d-8087-7e91dc64ff79>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00899.warc.gz"} |
The Odds That a Panel Would ‘Randomly’ Be All Men Are Astronomical
One mathematician’s formula suggests that all-male lineups don’t “just happen,” despite what conference organizers might claim.
All-male speaker lineups are so commonplace that there’s at least one Tumblr blog dedicated to mocking them. The endless stream of them can leave one overwhelmed and perhaps even convinced that
they’re inevitable.
Enter the mathematician Greg Martin, who has devised a statistical probability analysis that even amateurs can (mostly) understand. Working with a “conservative” assumption that 24 percent of Ph.D.s
in mathematics have been granted to women over the last 25 years, he finds that it’s statistically impossible that a speakers’ lineup including one woman and 19 men could be random.
His explanation of the formula is a rollicking one involving marbles and a potentially suspicious roommate. The underrepresentation of women on speakers’ lists doesn’t “just happen,” despite many
conference organizers’ claims that it does.
After doing the math, as Martin has, the argument that speakers are chosen without bias simply doesn’t hold up. In fact, when using the formula to analyze the speakers’ list for a mathematics
conference—which featured just one woman and 19 men—he found that it would be five times as likely that women would be overrepresented on the speakers’ list than underrepresented.
The formula can just as easily be applied to other fields; all that’s needed is reliable data on the field’s gender distribution, which can usually be gathered by way of industry associations and/or
government statistics.
Please click here to read the entire article – External link | {"url":"https://www.equityeltjapan.com/the-odds-that-a-panel-would-randomly-be-all-men-are-astronomical/","timestamp":"2024-11-11T03:19:03Z","content_type":"text/html","content_length":"48012","record_id":"<urn:uuid:7b130bc6-5436-401e-984b-c63326b9828d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00591.warc.gz"} |
Avid Reader | Subtraction Within 50: Adding Up | 3 Act Math Task
Explore efficient strategies to subtract values within 50
Spark Curiosity
Fuel Sensemaking
During Moves
Student Approaches
Next Moves
Reflect and Consolidation Prompts
Resources & Downloads
Educator Discussion Area
Access each lesson from this unit using the navigation links below
Students will explore subtraction in a context encouraging students to use the “think addition” strategy
In this task, students will engage in a subtraction context and lead towards the “think addition” strategy.
Some of the big ideas that may emerge through this task include:
• Understanding hierarchical inclusion allows for flexible composing and decomposing of numbers
• Numbers can be decomposed by separating a whole into two or more parts
• Subtraction names the missing part in terms of the whole
• Different subtraction situations will elicit different strategies
• Number relationships provide the foundation for strategies to help students remember basic facts
• Subtraction can be used in either take away, comparison, or missing addend situations. Missing addend is explored in this task.
• Models can be used to connect concrete to abstract
Before starting this unit, students should be familiar with:
• Facts of 10 ( e.g., 6 + 4 = 10, 10 – 4 = 6)
• Flexibility when decomposing numbers (e.g., 13 can be decomposed into 10 and 3, but also 9 and 4, 8 and 5, etc)
What Do You Notice? What Do You Wonder?
Show students the following video:
Then, ask students:
What do you notice?
What do you wonder?
Give students about 30-60 seconds to do a rapid write on a piece of paper or silent individual think time.
Replaying the video can be helpful here if appropriate. Ensure that students do not long enough to 1:1 count each books. This will encourage some subitizing to occur and for students to visual what
they saw.
Then, ask students to pair share with their neighbors for another 60 seconds.
Finally, allow students to individually share with the entire group. Be sure to write down these noticings and wonderings on the blackboard/whiteboard, chart paper, or some other way that is visible
to all. This helps students to see the thinking of their classmates and ensures each student that their voice is acknowledged and appreciated. Adding student names or initials next to their notice/
wonder is one way to acknowledge their participation and can motivate others to join in.
Some of the noticing and wondering may include:
• I notice that there is a book shelf
• I notice there books on the shelf
• I notice different colours of books on the shelf
• I notice that the books are piled in different ways
• I wonder what kind of books they are
• I wonder how many books are on the shelf
Estimation: Prompt
After we have heard students and demonstrated that we value their voice, we can ask the estimation question. The students may have already made some guesses of the amount of books in the previous
section. The students will feel valued as you now ask them to make a true estimation.
How many books are on the shelf?
Follow up that question with:
How could you convince someone that your estimation is correct?
We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to use spatial reasoning such as the amount of books in one
of the groupings to help justify how many there are overall. Before collecting student estimates, students can share their estimates with neighbouring students along with the reasoning.
Consider asking students to think about a number that would be “too low” and a number that would be “too high” before asking for their best estimate in order to help them come up with a more
reasonable estimate.
While Students Are Estimating…
Monitor student thinking by circulating around the room and listening to the mathematical discourse. You may identify some students whose thinking would be valuable to share when the group’s
estimates are collected.
Encourage students to make estimations rather than 1:1 counting each book. The video may be paused for longer before it goes blank but we want students to make estimations based on their mathematical
understanding and spatial sense.
Similar to collecting their noticings and wonderings, collect students’ range of estimates and/or best estimates along with initials or names. Having some students share justifications is an
opportunity for rich, mathematical discourse.
Estimation Reveal
Share the following estimation reveal video:
Celebrate students who estimated closest to 23 books.
Crafting A Productive Struggle: Prompt
Since you have already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further and fuel sense
Share the following video with the prompt.
Brian has read 19 books since the beginning of school. He would like to read 27 books. How many more books does Brian need to read? How do you know?
While Students Are Productively Struggling…
Monitor student thinking by circulating around the room and listening to the mathematical discourse. Educators are looking for students that are making their thinking visible so it can be displayed
during consolidation. Select and sequence some of the student solution strategies and ask a student from the selected groups to share with the class from:
• most accessible to least accessible solution strategies and representations;
• most common misconceptions;
• most common/frequent to least common/frequent representations; or,
• choose another approach to selecting and sequencing student work.
The strategies you might see students use include:
• Direct model and counting all
• Counting back
• Adding up with a tool
• Adding up with a drawing
• Adding up on an open number line
This checklist can be used for tracking formative assessment as students are working. The information collected can be used to form whole group, small group or one-to-one support models.
Students will count the initial value, count the amount added then count all of the amounts.
Early Direct modeling and
strategy counting all Count 1: 23 blocks to start
Count 2: 8 of the blocks moved to the side
Count 3: Count the remaining 15 blocks.
Pre-cursor Count forwards from Watch for students that need to count forwards starting from 1. You may hear a student that is counting on from 8, whisper their count “1, 2, 3, 4, 5, 6, 7, 8” then
skill various points start counting on while tracking out loud.
Count Back strategy Students are holding one number in their head and continuing to count backwards and track their count
Count Up Students are finding the difference/space between in the numbers by adding up from the lower number to the higher number
Add to the decade This skill requires students to transfer their understanding of the “facts of ten” to other decade numbers. For example, the fact 6 + 4 helps us with 26 + 4 or 36 +
Add on from a decade number This skill demonstrates student understanding of place value. For example, 20 + 3, does the student understand that when adding the 3, we do not need to count on by
ones. Instead, this skill needs to be done in one action, rather than counting on 21, 22, 23.
Strategic and efficient The Think Addition strategy demonstrates the understanding of difference. Students are using addition to find the difference between two numbers. This strategy is
strategy: Think Addition efficient when the minuend and subtrahend are close together. For example, 34 – 28. We can add on 2 to get 30 then add on 4 more. Therefore, 34 – 28 = 6.
Student Approach 1: Counting All with a Tool
I counted out 27 blocks for the 27 books that he wants to read.
Then I counted 19 blocks and put them to the side.
I counted that there was 8 blocks left so he needs to read 8 more books
Student Approach 2: Counting All with a Drawing
I drew 27 circles for the 27 books. I crossed out 19 of the circles and counted the remaining amount of circles. There were 8 circles not crossed so he needs to read 8 more books.
Student Approach 3: Adding up with a tool to track
I started with 19 and then I counted on my fingers. Each time I said a number, I put up a finger. 20, 21, 22, 23, 24, 25, 26, 27. I have 8 fingers up so he needs to read 8 more books.
Student Approach 4: Adding up with a drawing to track
I started with 19 and then I counted on with a tally. Each time I said a number, I put up a tally. 20, 21, 22, 23, 24, 25, 26, 27. I have 8 tallies so he needs to read 8 more books.
Student Approach 5: Adding up with a numberline to track
I drew a numberline. I put 19 on the left side. I added 1 to get to 20 then I added 7 more to get to 27. Since I added 1 and 7, that makes 8. So he needs to read 8 more books.
Show students the following reveal video: Facilitator Note: The open number line (or empty number line) is an incredible tool for students to use to demonstrate their thinking. It allows flexibility
from the traditional number line because students do not have to count the “ticks’ or “spaces”, instead they may jot their thinking anywhere on the line. Notice that the arrows are going to the
right or up the number line which demonstrates an increase in value.
Consolidate learning by facilitating a student discussion.
The goal of the consolation is to demonstrate the flexible way that we can add up to find the difference between two values. The “join with the change unknown” problem type will naturally elicit the
Think Addition strategy. This problem type can be illustrated through the equation:
19 + ? = 27
Although it is not a subtraction question, this problem type is a good introduction to the Think Addition strategy to subtract. Students are often more familiar with addition so opening the gateway
that we can use an addition technique when subtracting will benefit students.
This problem may become more obvious by modeling the thinking on an open number line. It will allow students the opportunity to see that we are trying to figure out the space between 19 and 27 or
the difference between them. It is very important for students to understand where the answer is. Often when students use a numberline, the answer is one of the numbers on the number line. In this
case, the answer lies in the number of “jumps” or the
Since the student added 1 then added 7 more, the difference between the two values is 1 + 7= 8.
Also during the consolidation, consider highlighting how we can add up in an efficient way. 19 was purposefully chosen due to its proximity to 20. The idea is that students can add 1 to get to a more
friendly decade number. Then they would just need to add 7. Ideally, the later step can be done without direct counting.
Students that were counting back often “get lost” as they have to travel over a decade. As the numbers get higher or the count back becomes larger, counting is not always as reliable. There are too
many pieces for a student to keep track of.
We want students to start feeling more comfortable with subtracting numbers in a way that makes sense to them rather than just counting.
During the discussion, encourage students to start by showing their work without an explanation. Classmates will use this time to understand the visual and make their own assumptions about the work
in front of them. It is also an option to ask students “What do you think this group did to solve this question?”. This will engage students in the work. The group can clarify any misunderstandings.
Think addition is an effective way to subtract. Students may use “think addition” for their more basic facts and also solve subtraction problems as numbers get bigger. Specific problem types such as
join with the change unknown or missing part problems may encourage the use of think addition strategies for subtraction. It may also be helpful to ask the difference between two numbers to practice
this strategy.
Consider the strategies that the students used. Perhaps a review of adding up over the decade is necessary. To efficiently use this strategy, we want students to be flexible with decomposing
numbers. It also relies on students applying the facts of ten to other decades, in this case subtracting from 20. Further review through number talks or games that work with the facts of ten and
their decade partners may be helpful. For example, knowing 7 + 3 helps with 17 + 3, 27 + 3, etc.
Consolidation Prompt #1:
How does adding help us to subtract?
Consolidation Prompt #2:
Brian has read 26 books so far this year, how many more books will he need to read if he wants to read 38 books?
We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will
Login/Join to access the entire Teacher Guide, downloadable slide decks and printable handouts for this lesson and all problem based units.
Explore Our 60+ Problem Based Units
This Make Math Moments Lesson was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, number talks and extensions to elicit and emerge strategies and
mathematical models.
Dig into our other units of study and view by concept continuum, grade or topic! | {"url":"https://learn.makemathmoments.com/task/avid-reader/","timestamp":"2024-11-03T15:58:32Z","content_type":"text/html","content_length":"360771","record_id":"<urn:uuid:7f24d3cf-87ef-4cc9-8b52-28d8887a938d>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00316.warc.gz"} |
How do you find the sum of the infinite geometric series 12+4+4/3+...? | HIX Tutor
How do you find the sum of the infinite geometric series 12+4+4/3+...?
Answer 1
An infinite geometric series of the form #sum_(n=1)^ooar^(n-1) # converges if and only if #|r|<1#, where r is the common ration between terms and given by #r=x_(n+1)/(x_n)#. In this case it converges
to the value #a/(1-r)# where a is the first term in the corresponding sequence #(x_n)#.
So in this case, #r=4/12=4/3/4=1/3<1# hence the series converges.
Its sum is hence #sum_(n=1)^oo12*(1/3)^(n-1)=12/(1-1/3)=18#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To find the sum of an infinite geometric series, use the formula for the sum:
[ S = \frac{a}{1 - r} ]
• ( a ) is the first term of the series
• ( r ) is the common ratio
For the given series ( 12 + 4 + \frac{4}{3} + \ldots ), the first term ( a = 12 ) and the common ratio ( r = \frac{1}{3} ).
Plug these values into the formula:
[ S = \frac{12}{1 - \frac{1}{3}} ]
[ S = \frac{12}{\frac{2}{3}} ]
[ S = 18 ]
So, the sum of the infinite geometric series ( 12 + 4 + \frac{4}{3} + \ldots ) is ( 18 ).
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/how-do-you-find-the-sum-of-the-infinite-geometric-series-12-4-4-3-8f9afa9551","timestamp":"2024-11-09T22:01:33Z","content_type":"text/html","content_length":"578359","record_id":"<urn:uuid:82227fca-5be1-4495-8bd8-e816e1e6e862>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00214.warc.gz"} |
Top 20 Elementary Math Tutors Near Me in Manchester
Top Elementary Math Tutors serving Manchester
Brianna: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...to college students through various subjects. I have also worked extensively with special needs children, doing everything from teaching special education to one-on-one in-home mentoring. Through
my experiences, I have continued to carry my passion for working with children and have developed my skills in making successful and individualized learning fun. Math, science and test...
Education & Certification
Subject Expertise
• Elementary Math
• Calculus
• Elementary Algebra
• Grade 11 Math
• +217 subjects
Audrey: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...for several years, which involves mentoring younger students in oral and written communication skills. In addition to my experience, standardized testing has always been a forte of mine. I am
enthusiastic about sharing the skills and strategies that I have picked up along my way. If you are seeking a patient, energetic, and articulate tutor...
Education & Certification
Subject Expertise
• Elementary Math
• Middle School Math
• Geometry
• Competition Math
• +36 subjects
Kelsey: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...tutored numerous students on college entrance exam preparation, and has worked professionally as an editorial assistant for research institutions. Her personal goal is to help students build a
foundation of solid study skills that they can rely on throughout their education so that they have the confidence to perform their best in every course.... My first session is always focused on
getting to the know the student's areas of comfort and concern, as well as working...
Education & Certification
Subject Expertise
• Elementary Math
• Elementary School Math
• 5th Grade Math
• Algebra
• +79 subjects
Alison: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...well on these exams to pursue their higher education goals. I have always been a high-achieving student, and I was my high school's valedictorian, a National Merit Commended Scholar, and a
Presidential Scholar Qualifier. I have over 3 years of mathematics tutoring experience with students in grades K-12, and I love to help students build...
Education & Certification
• University of Dallas - Current Undergrad, Double-Major in Mathematics and Business with a French Language and Literature Concentration
Subject Expertise
• Elementary Math
• Statistics
• Arithmetic
• 10th Grade Math
• +64 subjects
John: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...this means that my tutoring style is very student-involved: I will always begin a problem by asking the student where they think they should start, or what progress they have made on the problem.
I find that the student misses a significant part of the tutoring experience if they are not the primary force behind...
Education & Certification
Subject Expertise
• Elementary Math
• Calculus 3
• Algebra 2
• IB Mathematics SL
• +47 subjects
Matt: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...gone through myself, I have taken a wide range of subjects and more standardized tests than I would like to admit. I have taught students of almost every age in various fields such as elementary
science, Hebrew, chemistry, MCAT prep, and many more.. My ultimate plan is to go to medical school, but in the...
Education & Certification
Subject Expertise
• Elementary Math
• Algebra
• Statistics
• Pre-Algebra
• +85 subjects
Joshua: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...to your passions, pursuits, and educational goals to make the work that much more meaningful. In my spare time, you'll often find me learning bits of new languages for fun or writing. I am
currently fluent in English and French, conversational in Spanish and Mandarin Chinese, and can read some Italian, German, Dutch, and Arabic.
Education & Certification
Subject Expertise
• Elementary Math
• College Algebra
• Algebra 2
• Geometry
• +92 subjects
Molly: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...how and why we can come to certain answers. My goal is not to help students to merely get a good grade on the next quiz, but to help build a foundation that allows him or her to find success on
their own in future units of study. In each lesson, I use a variety...
Education & Certification
Subject Expertise
• Elementary Math
• Elementary School Math
• Algebra
• 5th Grade Math
• +111 subjects
Akhil: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...academic realm, I am diehard sports fan, and currently work as a sports journalist. My favorite sport is football, but I watch almost every major sport around the world. I'm also an avid fan of
professional wrestling. I also enjoy reading, writing, and playing video games in my spare time, and I love working out...
Education & Certification
Subject Expertise
• Elementary Math
• Middle School Math
• Algebra
• Pre-Algebra
• +27 subjects
Rushi: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...has always been a passion of mine. From my younger brother to my years of informal tutoring experience, I have developed the skills necessary to help others become successful. I enjoy the genuine
process of developing a relationship with someone and cultivating that relationship, whether it be academic, personal, or professional. I enjoy working with...
Education & Certification
Subject Expertise
• Elementary Math
• Algebra
• AP Calculus AB
• Trigonometry
• +29 subjects
Hans: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...lessons have been for students of all ages ranging from elementary school to college. In high school I also excelled on the SAT (2220) and a plethora of AP tests (4s and 5s across the board), and
so would be happy to aid with any test prep you may need. I am a very amicable...
Education & Certification
Subject Expertise
• Elementary Math
• Multivariable Calculus
• Algebra
• Middle School Math
• +78 subjects
Cole: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...to me, and tutoring allows me to share that passion with others. I first got into tutoring through a volunteer program in high school, and I continued to tutor during my undergraduate studies,
which was some of the most rewarding work I've ever done. My engineering background has allowed me to take a wide variety...
Education & Certification
• UW Madison - Bachelors, Industrial Engineering
• UW Madison - Masters, Industrial Engineering
Subject Expertise
• Elementary Math
• Pre-Algebra
• Calculus 2
• Pre-Calculus
• +55 subjects
Erica: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...program that placed K-12 math tutors into local public schools. I have also served as a Crisis Services Supervisor at the National Runaway Safeline (NRS). At NRS, I was able to support volunteers
in providing trauma-informed services to runaway, homeless, and at-risk youth. I look forward to meeting you and your child!... I believe that all tutoring must be student-centered. As much as
possible, a tutor must help a student develop their own ideas about the...
Education & Certification
Subject Expertise
• Elementary Math
• Middle School Math
• Algebra
• Arithmetic
• +77 subjects
Joseph: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...reach the right answer but to make sure they completely understand the subject at hand. It is important to me that students enjoy the learning process, so I aim to make tutoring sessions both
intriguing and challenging. If I am not tutoring or working, I spend my spare time swimming, watching sports, basketball or football,...
Education & Certification
Subject Expertise
• Elementary Math
• Geometry
• Algebra 3/4
• Elementary School Math
• +21 subjects
William: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...It has been a very fulfilling experience and a skill I would like to continue developing. It is such a joy for me to see the light of understanding in my students' eyes and be able to track their
progress and academic achievements. I love the challenge of catering to each person's learning style and...
Education & Certification
Subject Expertise
• Elementary Math
• Middle School Math
• Elementary School Math
• Arithmetic
• +55 subjects
Zachary: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...Boston taking Organic Chemistry at Harvard. Outside of the classroom, I'm a part of Georgetown's club rugby team and also participate in GUMSHOE, a club that teaches math and science to inner city
students. I also enjoy going for runs to the monuments in DC and playing in pickup soccer games! For me, I especially...
Education & Certification
Subject Expertise
• Elementary Math
• 1st Grade Math
• 12th Grade Math
• 4th Grade Math
• +125 subjects
Lexi: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...at Princeton and through language exchanges completed while studying abroad in Mexico and Egypt (ages 18-30). I believe that learning a foreign language is hugely rewarding for students of all
ages and welcome opportunities to tutor in Spanish or Arabic (or just to chat!) at all levels. I hope to work with you to develop...
Education & Certification
• Princeton University - Bachelor in Arts, Politics, Near Eastern Studies, Arabic
• State Certified Teacher
Subject Expertise
• Elementary Math
• Elementary School Math
• Pre-Calculus
• Middle School Math
• +29 subjects
Mica: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...goals and needs. I am enthusiastic about learning and strive to make subjects interesting and fun so that students will feel motivated to study. I particularly enjoy tutoring because of the
one-on-one interaction that allows for highly focused and accelerated learning. My hobbies include reading, yoga, and playing music (piano, guitar, and singing).
Education & Certification
Subject Expertise
• Elementary Math
• Pre-Algebra
• Algebra
• Middle School Math
• +45 subjects
Clara: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...years, so I am very confident in my ability to make complicated topics understandable. My favorite area is standardized testing. I love the look on a student's face when they realize that they
don't have to be nervous about an exam - that it can actually be really fun if the questions are viewed as...
Education & Certification
Subject Expertise
• Elementary Math
• Pre-Calculus
• Middle School Math
• Pre-Algebra
• +44 subjects
Moriah: Manchester Elementary Math tutor
Certified Elementary Math Tutor in Manchester
...more on the Reading portions of standardized tests but have at times assisted students in the math and science portions as well usually one-on-one. I strongly believe that every students has their
own style of learning and must learn at their own pace. Although, sometimes giving them too much leeway may not always be the...
Education & Certification
Subject Expertise
• Elementary Math
• Elementary School Math
• Trigonometry
• Algebra 3/4
• +61 subjects
Private Elementary Math Tutoring in Manchester
Receive personally tailored Elementary Math lessons from exceptional tutors in a one-on-one setting. We help you connect with the best tutor for your particular needs while offering flexible
scheduling to fit your busy life.
Your Personalized Tutoring Program and Instructor
Identify Needs
Our knowledgeable directors help you choose your tutor with your learning profile and personality in mind.
Customize Learning
Your tutor can customize your lessons and present concepts in engaging easy-to-understand-ways.
Increased Results
You can learn more efficiently and effectively because the teaching style is tailored to you.
Online Convenience
With the flexibility of online tutoring, your tutor can be arranged to meet at a time that suits you.
Call us today to connect with a top Manchester Elementary Math tutor | {"url":"https://www.varsitytutors.com/gb/elementary_math-tutors-manchester","timestamp":"2024-11-08T14:03:06Z","content_type":"text/html","content_length":"611508","record_id":"<urn:uuid:c7cdbabf-ad28-4d96-a295-3e0b379366a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00234.warc.gz"} |
Study Guide
We've updated our
Privacy Policy
effective December 15. Please read our updated Privacy Policy and tap
Boundless Calculus
Copyright: The following courseware includes resources copyrighted and openly licensed by third parties under a Creative Commons Attribution 4.0 License. Click "Licenses and Attributions" at the
bottom of each page for copyright information and license specific to the material on that page. If you believe that this courseware violates your copyright, please contact us. | {"url":"https://zt.symbolab.com/study-guides/boundless-calculus","timestamp":"2024-11-14T12:06:33Z","content_type":"text/html","content_length":"133903","record_id":"<urn:uuid:3cc4c1bb-634f-4a99-8713-55a9d1285357>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00661.warc.gz"} |
Periodic Pitman transforms and jointly invariant measures
We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their
uniqueness via a one force--one solution principle. The measures are given by polymer-like transforms of independent Brownian bridges. We describe several properties and limits of these measures,
including an extension to a continuous process in the slope parameter that we term the periodic KPZ horizon. As an application of our construction, we prove a Gaussian process limit theorem with an
explicit covariance function for the long-time height function fluctuations of the periodic KPZ equation when started from varying slopes. In connection with this, we conjecture a formula for the
fluctuations of cumulants of the endpoint distribution for the periodic continuum directed random polymer. To prove joint invariance, we address the analogous problem for a semi-discrete system of
SDEs related to the periodic O'Connell-Yor polymer model and then perform a scaling limit of the model and jointly invariant measures. For the semi-discrete system, we demonstrate a bijection that
maps our systems of SDEs to another system with product invariant measure. Inverting the map on this product measure yields our invariant measures. This map relates to a periodic version of the
discrete geometric Pitman transform that we introduce and probe. As a by-product of this, we show that the jointly invariant measures for a periodic version of the inverse-gamma polymer are the same
as those for the O'Connell-Yor polymer.
arXiv e-prints
Pub Date:
September 2024
□ Mathematics - Probability
104 pages, 12 figures | {"url":"https://ui.adsabs.harvard.edu/abs/2024arXiv240903613C/abstract","timestamp":"2024-11-12T19:46:29Z","content_type":"text/html","content_length":"40164","record_id":"<urn:uuid:dbdae063-f106-4484-bbfd-e23ee04801e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00520.warc.gz"} |
Debugging OSCAR Code · Oscar.jl
Suppose you are having the following difficulties. Your code is exhibiting inexplicable behavior and values that should not be changing are changing in seemingly random locations. To get to the
bottom of these kind of issues it is necessary to be familiar with mutable objects in Julia and some of the relevant conventions in place in OSCAR. This section discusses these informal rules as well
as some of the exceptions to these rules.
In Julia, objects that can change after construction are declared with the mutable struct keywords and satisfy the ismutable predicate. These objects can be linked together into an arbitrary
dependency graph, and a change to one object may therefore have unintended consequences on another object in the system.
The simplest example is the creation of a polynomial ring. If we mutate the array of symbols used for printing, we have effectively changed the ring.
julia> v = [:x, :y, :z]; R = polynomial_ring(QQ, v)[1]
Multivariate Polynomial Ring in x, y, z over Rational Field
julia> v[3] = :w; R
Multivariate Polynomial Ring in x, y, w over Rational Field
In this example, the modification of v is unexpected and may in fact corrupt the internal data structures used by the polynomial ring. As such, this modification of v has to be considered illegal.
Upon creation of the array called v, we have full rights over the object and can mutate at will. However, after passing it to the function polynomial_ring, we have given up ownership of the array and
are no longer free to modify it.
General OSCAR Principle (GOP):
Code should be expected to behave as if all objects are immutable.
1. This means that the polynomial ring constructor is allowed to expect that v is never mutated for the remaining duration of its life. In return, the constructor is guaranteed not to modify the
array, so that v is still [:x, :y, :z] after polynomial_ring returns.
2. In general this means that all functions should be expected to take ownership of their arguments: the user is safest never modifying an existing object that has been passed to an unknown Julia
function. Note that assignments such as a[i] = b or a.foo = b usually mutate the object a. See Ownership of function arguments
3. For reasons of efficiency, it is sometimes desirable to defy this principle and modify an existing object. The fact that a given function may modify a preexisting object is usually communicated
via coding conventions on the name - either a ! or a _unsafe in the name of the function. See Unsafe arithmetic with OSCAR objects
In this example we construct the factored element x = 2^3 and then change the 2 to a 1. The GOP says this modification of a on line 3 is illegal.
julia> a = ZZRingElem(2)
julia> x = FacElem([a], [ZZRingElem(3)]); evaluate(x)
julia> a = one!(a) # illegal in-place assignment of a to 1
julia> evaluate(x) # x has been changed and possibly corrupted
In the previous example, the link between the object x and the object a can be broken by passing a deepcopy of a to the FacElem function.
julia> a = ZZRingElem(2)
julia> x = FacElem([deepcopy(a)], [ZZRingElem(3)]); evaluate(x)
julia> a = one!(a) # we still own a, so modification is legal
julia> evaluate(x) # x is now unchanged
It is of course not true that all Julia functions take ownership of their arguments, but the GOP derives from the fact that this decision is an implementation detail with performance consequences.
The behavior of a function may be inconsistent across different types and versions of OSCAR. In the following two snippets, the GOP says both modifications of a are illegal since they have since been
passed to a function. If K = QQ, the two mutations turn out to be legal currently, while they are illegal if K = quadratic_field(-1)[1]. Only with special knowledge of the types can the GOP be safely
R = polynomial_ring(K, [:x, :y])[1]
a = one(K)
p = R([a], [[0,0]])
@show p
a = add!(a, a, a) # legal? (does a += a in-place)
@show p
R = polynomial_ring(K, :x)[1]
a = [one(K), one(K)]
p = R(a)
@show (p, degree(p))
a[2] = zero(K) # legal?
@show (p, degree(p))
The nuances of who is allowed to modify an object returned by a function is best left to the next section Unsafe arithmetic with OSCAR objects. The GOP says of course you should not do it, but there
are cases where it can be more efficient. However, there is another completely different issue of return values that can arise in certain interfaces.
First, we create the Gaussian rationals and the two primes above 5.
julia> K, i = quadratic_field(-1)
(Imaginary quadratic field defined by x^2 + 1, sqrt(-1))
julia> m = Hecke.modular_init(K, 5)
modular environment for p=5, using 2 ideals
The function modular_project returns the projection of an element of K into each of the residue fields.
julia> a = Hecke.modular_proj(1+2*i, m)
2-element Vector{fqPolyRepFieldElem}:
While the function has produced the correct answer, if we run it again on a different input, we will find that a has changed.
julia> b = Hecke.modular_proj(2+3*i, m)
2-element Vector{fqPolyRepFieldElem}:
julia> a
2-element Vector{fqPolyRepFieldElem}:
The preceding behavior of the function modular_proj is an artifact of internal efficiency and may be desirable in certain circumstances. In other circumstances, the following deepcopys may be
necessary for your code to function correctly.
julia> a = deepcopy(Hecke.modular_proj(1+2*i, m));
julia> b = deepcopy(Hecke.modular_proj(2+3*i, m));
julia> (a, b)
(fqPolyRepFieldElem[2, 0], fqPolyRepFieldElem[1, 3])
Particularly with integers (BigInt and ZZRingElem) - but also to a lesser extent with polynomials - the cost of basic arithmetic operations can easily be dominated by the cost of allocating space for
the answer. For this reason, OSCAR offers an interface for in-place arithmetic operations.
Instead of writing x = a + b to compute a sum, one writes x = add!(x, a, b) with the idea that the object to which x is pointing is modified instead of having x point to a newly allocated object. In
order for this to work, x must point to a fully independent object, that is, an object whose modification through the interface Unsafe operators will not change the values of other existing objects.
The actual definition of "fully independent" is left to the implementation of the ring element type. For example, there is no distinction for immutables.
It is generally not safe to mutate the return of a function. However, the basic arithmetic operations +, -, *, and ^ are guaranteed to return a fully independent object regardless of the status of
their inputs. As such, the following implementation of ^ is illegal by this guarantee.
function ^(a::RingElem, n::Int)
if n == 1
return a # must be return deepcopy(a)
In general, if you are not sure if your object is fully independent, a deepcopy should always do the job. | {"url":"https://docs.oscar-system.org/dev/DeveloperDocumentation/debugging/","timestamp":"2024-11-06T10:48:10Z","content_type":"text/html","content_length":"54550","record_id":"<urn:uuid:2484d25d-103e-4d8a-bcec-2c48cdd43e1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00864.warc.gz"} |
Calculator finds out absolute or relative error basing it on measured (calculated) and reference (ideal) value.
Beta version#
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !
Symbolic algebra
ⓘ Hint: This calculator supports
symbolic math
. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our
another page:
Symbolic calculations
What do you want to calculate today?#
Choose a scenario that best fits your needs
Calculations data - enter values, that you know here#
Absolute error ($\Delta x$) =>
Relative error ($\delta x_{rel.}$) =>
Measured value (x) <=
Reference values ($x_0$) <=
Result: absolute error ($\Delta x$)#
Used formula Show source$\Delta x=\left|x-x_0\right|$
Result Show source$\frac{39816339744831}{25000000000000000}$
Numerical result Show source$0.00159265358979324$
Result step by step
1 Show source$\left|3.14-3.14159265358979324\right|$ Convert decimal Decimal number can be converted into fraction with denominator 10, 100, 1000 etc.
to fraction
2 Show source$\left|\frac{157}{50}-3.14159265358979324\right|$ Convert decimal Decimal number can be converted into fraction with denominator 10, 100, 1000 etc.
to fraction
3 Show source$\left|\frac{157}{50}-\frac{78539816339744831} Common Before add two fractions, we need convert them into common denominator. We can find the optimal common denominator
{25000000000000000}\right|$ denominator using the lowest common multiply (LCM) from both denominators.
4 Show source$\left|\frac{157 \cdot Simplify -
500000000000000-78539816339744831}{25000000000000000}\right|$ arithmetic
5 Show source$\left|\frac{78500000000000000-78539816339744831} Simplify -
{25000000000000000}\right|$ arithmetic
6 Show source$\left|\frac{-39816339744831}{25000000000000000}\ Absolute value |c| = \left \{ \begin{aligned} &c && \text{, if}\ c \ge 0 \\ &-c && \text{, if}\ c \lt 0 \end{aligned} \right.
7 Show source$\frac{\left|-39816339744831\right|}{\left| Absolute value |c| = \left \{ \begin{aligned} &c && \text{, if}\ c \ge 0 \\ &-c && \text{, if}\ c \lt 0 \end{aligned} \right.
8 Show source$\frac{39816339744831}{\left|25000000000000000\ Absolute value |c| = \left \{ \begin{aligned} &c && \text{, if}\ c \ge 0 \\ &-c && \text{, if}\ c \lt 0 \end{aligned} \right.
9 Show source$\frac{39816339744831}{25000000000000000}$ Result Your expression reduced to the simplest form known to us.
Numerical result step by step
1 Show source$0.00159265358979324$ The original expression -
2 Show source$0.00159265358979324$ Result Your expression reduced to the simplest form known to us.
Some facts#
• Absolute error is the absolute value of the difference between measured value (calculated, approximate etc.) and reference value (ideal, theoretical etc.):
$\Delta x = |x-x_0|$
□ $\Delta x$ - absolute error,
□ $x$ - measured, calculated or approximate value of variable $x$,
□ $x_0$ - the reference value against which we calculate the error.
• Relative error determines the size of the error made presented as part of the reference value:
$\delta x_{wzgl.} = \left|\dfrac{x-x_0}{x_0}\right|$
□ $\delta x_{rel.}$ - error expressed as a part of the reference value,
□ $x$ - measured, calculated or approximate value of variable $x$,
□ $x_0$ - the reference value against which we calculate the error.
• When we determine the error, we are generally not trying to find out whether the value obtained is too large or too small, but only how big the error is. This is the reason why error formula has
the absolute value form.
• In the case of values with units (e.g. length measured in meters), the absolute error has the same units as the measured one. For example, an absolute error of length value is also the length.
• Relative error has no units, no metter what do we measure. Relative error is often presented as percent.
Tags and links to this website#
Tags to Polish version:
What tags this calculator has#
This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:
Links to external sites (leaving Calculla?)# | {"url":"https://calculla.com/calculators/all_calculators_az/relative_error","timestamp":"2024-11-14T18:52:40Z","content_type":"text/html","content_length":"1049182","record_id":"<urn:uuid:9513d0e0-8128-4054-8c61-92e2de3a25a5>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00098.warc.gz"} |
Blog | Pocket Private Tutor
How to make algebra easy to learn
How to make algebra easy to learn What actually is algebra and how can we make algebra easy to learn? The problem with maths is it is so abstract. What that means is we are often asking children to
do things in their mind. When they are younger, we allow them more opportunities to see numbers as physical objects (often called manipulatives) for example 3 + 4 we may give them cars, marbles,
cubes etc. The older they get, we are expecting them to calculate with numbers on paper. We can take for granted that these numbers are very abstract and… | {"url":"https://www.pocketprivatetutor.co.uk/blog/","timestamp":"2024-11-11T11:53:33Z","content_type":"text/html","content_length":"131071","record_id":"<urn:uuid:ef7aa827-e621-48ec-97a1-c3a85fb76132>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00624.warc.gz"} |
MHD Free Convective Flow along Vertical Oscillatory Plate with Radiative Heat Transfer in the Presence of Hall Current and Heat Source
MHD Free Convective Flow along Vertical Oscillatory Plate with Radiative Heat Transfer in the Presence of Hall Current and Heat Source
free convection, Hall Current, heat transfer, MHD, porous medium
MHD free convective flow with oscillations of an infinite non-conducting vertical flat surface through a porous medium with Hall current in a rotating system was studied. The governing equations of
the model were converted into dimensionless form. Analytical solutions for velocity and temperature were obtained with the help of the Laplace transform method. Graphs and tables are used in this
paper to show the influence of various parameters on temperature, skin friction and velocity. It was observed that changes in plate oscillation, porous medium, radiation and Hall current have
significant effects on fluid motion. Further, the skin friction near the surface is increased by the radiation parameter. The results obtained have large implications in the engineering and science
Stewarton, K., On the Impulsive Motion of a Flat Plate in a Viscous Fluid, Part 1, Quart. J. Mech. Appl. Math., 4(2), pp. 182-198, 1951.
Stewarton, K., On the Impulsive Motion of a Flat Plate in a Viscous Fluid, Part 2, Quart. J. Mech. Appl. Math., 22(2), pp. 143-152, 1973.
Prasad, V.R., Bhaskar Reddy, N. & Muthucumaraswamy, R., Radiation and Mass Transfer Effects on Two-dimensional Flow Past an Impulsively Started Infinite Vertical Plate, International Journal of
Thermal Sciences 46(12), pp. 1251-1258, 2007.
Ismail, Z., Khan, I., Imran, A., Hussanan, A., & Sharidan Shafie, MHD Double Diffusion Flowby Free Convection past an Infinite Inclined Plate with Ramped Wall Temperature in a Porous Medium,
Malaysian Journal of Fundamental and Applied Sciences, 10(1), pp. 37-42, 2014.
Ghadikolaei, S.S., Hosseinzadeh, Kh. & Ganji, D.D., Analysis of Unsteady MHD Eyring-Powell Squeezing Flow in Stretching Channel with Considering Thermal Radiation and Joule Heating Effect using AGM,
Case Studies in Thermal Engineering, 10, pp. 579-594, 2017.
Ghadikolaei, S.S., Hosseinzadeh, Kh., Ganji, D.D. & Jafari B., Nonlinear Thermal Radiation Effect on Magneto Casson Nanofluid Flow with Joule Heating Effect Over an Inclined Porous Stretching Sheet,
Case Studies in Thermal Engineering, 12, pp. 176-187, 2018.
Chamka, A.J., MHD Flow of A Uniformly Stretched Vertical Permeable Surface in the Presence of Heat Generation/Absorption and Chemical Reaction, International Communication Heat Mass Transfer, 33(3),
pp. 413-422, 2003.
Makinde, O.D. & Mhone, P.Y., Heat Transfer to MHD Oscillatory Flow in a Channel Filled with Porous Medium, Romania Journal of Physics, 50(9-10), pp. 931-938, 2015.
Hosseinzadeh, Kh., Afsharpanah, F., Zamani, S. Gholiniab, M. & Ganji, D.D., A Numerical Investigation on Ethylene Glycol-Titanium Dioxide Nanofluid Convective Flow over a Stretching Sheet in Presence
of Heat Generation/Absorption, Case Studies in Thermal Engineering, 12, pp. 228-236, 2018.
Rajput, U.S. & Kumar, G., Effects of Radiation and Porosity of The Medium on MHD Flow Past An Inclined Plate in the Presence of Hall Current, Computational Methods in Science and Technology, 23(2),
pp. 93-103, Polish Academy of Science (PAN), 2017.
Prasad, V.R., Bhaskar Reddy, N. & Vasu, B., Radiation and Mass Transfer Effects on Transient Free Convection Flow of a Dissipative Fluid Past Semi-infinite Vertical Plate with Uniform Heat and Mass
Flux, Journal of Applied Fluid Mechanics, 4(1), pp. 15-26, 2011.
Rajput, U.S. & Shareef, M., Study of Soret and Ion Slip Effects on MHD Flow Near An Oscillating Vertical Plate in a Rotating System, AAM: Intern. J., 13(1), pp. 516-534, 2018.
Soong, C.Y. & Ma, H.L., Unsteady Analysis of Non-isothermal Flow and Heat Transfer between Rotating Co-axial Disks, Int. J. Heat Mass Transfer, 38(10), pp. 1865-1878, 1995.
Soong, C.Y., Thermal Buoyancy Effects in Rotating Non-isothermal Flows, International Journal of Rotating Machinery, 7(6), pp. 435-446, 2001.
Greenspan, H.P., The Theory of Rotating Fluids, Cambridge University Press, London, 1968.
Muthucumaraswamy, R., Dhanasekar, N. & Prasad, G.E., Rotation Effects on Unsteady Flow Past an Accelerated Isothermal Vertical Plate with Variable Mass Transfer in the Presence of Chemical Reaction
of First Order, Journal of Applied Fluid Mechanics, 6(4), pp. 485-490, 2013.
Owen, J.M. & Rogers, R.H., Flow and Heat Transfer in Rotating Disc Systems, Vol. I, Rotor-Stator Systems, John Wiley Sons, New York, 1989.
Agarwal, H.L., Ram, P.C. & Singh, V., Combined Influence of Dissipation and Hall Effect on Free Convective in A Rotating Fluid, Indian J. Pure Appl. Math., 14(3), pp. 314-32, 1983.
Hosseinzadeh, Kh., Jafarian Amiri, A., Saedi Ardahaie, S. & Ganji, D.D., Effect of Variable Lorentz Forces on Nanofluid Flow in Movable Parallel Plates Utilizing Analytical Method, Case Studies in
Thermal Engineering, 10, pp. 595-610, 2017.
Jaimala, V. & Kumar, V., Thermal Convection In a Couple-Stress Fluid in the Presence of Horizontal Magnetic Field with Hall Currents. Application and applied Mathematics, 8(1), pp. 161-117, 2013.
Kishore, P.M., Rajesh, V., Verma, S.V., The Effects of Thermal Radiation and Viscous Dissipation on MHD Heat and Mass Diffusion Flow past an Oscillating Vertical Plate Embeded in a Porous Medium with
Variable Surface Conditions, Theoret. Appl. Mech., 39(2), pp. 99-125, 2012.
Mazumdar, B.S., Gupta, A.S. & Dattaa, N., Flow and Heat Transfer in the Hydrodynamic Ekman Layer on a Porous Plate with Hall Effects, Int. J. Heat Mass Transfer, 19(5), pp. 523, 1976.
Muthucumaraswamy, R., Dhanasekar, N. & Prasad, G.E., Mass Transfer Effects on Accelerated Vertical Plate in a Rotating Fluid with First Order Chemical Reaction, Journal of Mechanical Engineering and
Sciences, 3, pp. 346-355, 2012.
Brewster, M.Q., Thermal Radiative Transfer and Properties, John Wiley & Sons, New York, 1992. | {"url":"https://journals.itb.ac.id/index.php/jmfs/article/view/3880","timestamp":"2024-11-09T07:33:38Z","content_type":"text/html","content_length":"38836","record_id":"<urn:uuid:70310715-c5c7-4b3c-96d7-f29bbbfcf0c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00271.warc.gz"} |
Find the length of the perpendicular from P(2, -3, 1) to the line $\
First, before proceeding for this, we must assume a factor $\lambda $ which is called a proportionality factor of the given line equation. Then, by equation the given equation of line to the factor $
\lambda $, we get all the points. Then, we know the condition that if the angle between the lines is ${{90}^{\circ }}$, then their dot product of direction ratios are zero. Then, by using this
condition we get the value of $\lambda $and by using that we get the points. Then, by using the distance formula as $d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \
right)}^{2}}+{{\left( {{z}_{2}}-{{z}_{1}} \right)}^{2}}}$, we get the desired answer.
Complete step-by-step solution:
In this question, we are supposed to find the length of the perpendicular from P(2, -3, 1) to the line $\dfrac{x-1}{2}=\dfrac{y-3}{3}=\dfrac{z+2}{-1}$.
So, before proceeding for this, we must assume a factor $\lambda $which is called a proportionality factor of the given line equation.
Now, by equation the given equation of line to the factor $\lambda $, we get:
$\dfrac{x-1}{2}=\dfrac{y-3}{3}=\dfrac{z+2}{-1}=\lambda $
Then, we get the points as:
& \dfrac{x-1}{2}=\lambda \\
& \Rightarrow x-1=2\lambda \\
& \Rightarrow x=2\lambda +1 \\
Similarly, we get for the other two conditions, the points are as:
$\left( 2\lambda +1,\text{ }3\lambda +3,\text{ }-\lambda -2 \right)$
Then, we are given with the point P(2, -3, 1) which acts like direction ratio of line which is perpendicular to the given line as:
& \left( 2\lambda +1-2,\text{ }3\lambda +3+3,\text{ }-\lambda -2-1 \right) \\
& \Rightarrow \left( 2\lambda -1,\text{ }3\lambda +6,\text{ }-\lambda -3 \right) \\
So, from the above equation, we get the direction ratios of the line by the coefficients of the factor $\lambda $ as 2, 3 and -1.
Now, we got the expression for the points of the line which is perpendicular to the given line.
So, we know the condition that if the angle between the lines is ${{90}^{\circ }}$, then their dot product of direction ratios are zero.
Then, by using this condition, we get:
& \left( 2\lambda -1 \right)\centerdot 2+\left( 3\lambda +6 \right)\centerdot 3+\left( -\lambda -3 \right)\centerdot \left( -1 \right)=0 \\
& \Rightarrow 4\lambda -2+9\lambda +18+\lambda +3=0 \\
& \Rightarrow 14\lambda +19=0 \\
& \Rightarrow 14\lambda =-19 \\
& \Rightarrow \lambda =\dfrac{-19}{14} \\
Now, we will substitute the value of $\lambda =\dfrac{-19}{14}$ in the expression found as $\left( 2\lambda +1,\text{ }3\lambda +3,\text{ }-\lambda -2 \right)$ to get the points of the line as:
& 2\left( \dfrac{-19}{14} \right)+1,\text{ 3}\left( \dfrac{-19}{14} \right)+3,\text{ }-\left( \dfrac{-19}{14} \right)-2 \\
& \Rightarrow \dfrac{-38+14}{14},\text{ }\dfrac{-57+42}{14},\text{ }\dfrac{19-28}{14} \\
& \Rightarrow \dfrac{-24}{14},\text{ }\dfrac{-15}{14},\text{ }\dfrac{-9}{14} \\
So, we get the points of the line as $\left( \dfrac{-24}{14},\text{ }\dfrac{-15}{14},\text{ }\dfrac{-9}{14} \right)$ which is perpendicular to points (1, 3, -2).
Now, we will use the distance formula to calculate the distance(d) between these points $\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)$and $\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)$ as:
$d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}+{{\left( {{z}_{2}}-{{z}_{1}} \right)}^{2}}}$
So, by using the formula for the points found above, we get:
& d=\sqrt{{{\left( 2+\dfrac{24}{14} \right)}^{2}}+{{\left( -3+\dfrac{15}{14} \right)}^{2}}+{{\left( 1+\dfrac{9}{14} \right)}^{2}}} \\
& \Rightarrow d=\sqrt{{{\left( \dfrac{52}{14} \right)}^{2}}+{{\left( \dfrac{-27}{14} \right)}^{2}}+{{\left( \dfrac{23}{14} \right)}^{2}}} \\
& \Rightarrow d=\sqrt{\dfrac{2704}{196}+\dfrac{729}{196}+\dfrac{529}{196}} \\
& \Rightarrow d=\sqrt{\dfrac{3962}{196}} \\
& \Rightarrow d=\sqrt{\dfrac{531}{14}} \\
So, the distance between the points is $\sqrt{\dfrac{531}{14}}$.
Hence, option (b) is correct.Note:
Now, to solve this type of questions we must know the angle condition of the direction ratios which is used above in the question as both the lines are perpendicular so the dot product is zero is
given by the condition that:
$\cos \theta =\dfrac{\left| {{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}} \right|}{\sqrt{{{a}_{1}}^{2}+{{b}_{1}}^{2}+{{c}_{1}}^{2}}\sqrt{{{a}_{2}}^{2}+{{b}_{2}}^{2}+{{c}_{2}}^{2}}}$
Where $\left( {{a}_{1}},{{b}_{1}},{{c}_{1}} \right)$ and $\left( {{a}_{2}},{{b}_{2}},{{c}_{2}} \right)$ are the points of the line. | {"url":"https://www.vedantu.com/question-answer/find-the-length-of-the-perpendicular-from-p2-3-1-class-12-maths-cbse-5f5d004668d6b37d16426ad0","timestamp":"2024-11-05T09:37:44Z","content_type":"text/html","content_length":"203536","record_id":"<urn:uuid:6dd73af4-18d8-4b02-a81f-38b9ece5b8cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00271.warc.gz"} |
Concrete calculator - Quick Calculation for Construction Project
Are you in the midst of a construction project, trying to ensure every inch of your structure is perfect? Calculating the exact amount of concrete needed is a critical step to avoid wastage and
ensure the structural integrity of your building. In this guide, we’ll dive into the world of concrete calculator, making your construction endeavors more precise and cost-effective.
The Significance of Accurate Concrete Calculator
Before we delve into the nitty-gritty details, let’s understand why precise concrete calculations are indispensable in the construction industry.
1. Foundation of a Solid Structure
Concrete forms the very foundation of any construction project. An incorrect calculation can lead to insufficient support, which can compromise the safety and longevity of your building.
2. Cost Savings
Overordering concrete is not only wasteful but also expensive. By accurately determining your concrete requirements, you can save money and allocate resources efficiently.
3. Environmental Impact
Wastage in construction, especially with materials like concrete, can have a significant environmental impact. Accurate calculations help reduce unnecessary concrete production, which in turn
decreases carbon emissions.
Mastering Concrete Calculations
Now that we understand why concrete calculations are crucial, let’s explore the steps to master this skill.
Step 1: Measure Your Project Area
Begin by measuring the dimensions of the area where you intend to pour concrete. Ensure you record all measurements accurately, as any discrepancies can lead to inaccurate calculations.
Step 2: Calculate Volume through Concrete Calculator
Concrete calculations are all about determining the volume you need. The formula to calculate concrete volume is:
mathematicaCopy code
Volume = Length x Width x Depth
Make sure you use consistent units, such as feet or meters, for all measurements. The result will be in cubic units, typically cubic yards or cubic meters.
Step 3: Account for Wastage
Concrete work is not always perfect, and there can be some level of wastage during pouring and finishing. It’s advisable to add about 5-10% to your calculated volume to account for this wastage.
Step 4: Use an Online Concrete Calculator
To simplify the process and ensure accuracy, you can use online concrete calculators. These tools allow you to input your measurements and get an instant estimate of the concrete required.
Certainly, here are the formulas commonly used in concrete calculations for various portions of a construction project:
1. Calculating Concrete Volume for a Rectangular Slab:
□ Formula: Volume=Length×Width×DepthVolume=Length×Width×Depth
□ Where:
☆ Length: The length of the slab (in feet or meters).
☆ Width: The width of the slab (in feet or meters).
☆ Depth: The depth or thickness of the slab (in feet or meters).
2. Calculating Concrete Volume for a Circular Slab (e.g., a Column Base):
□ Formula: Volume=�×(Diameter2)2×HeightVolume=π×(2Diameter)2×Height
□ Where:
☆ Diameter: The diameter of the circular slab (in feet or meters).
☆ Height: The height of the circular slab (in feet or meters).
3. Calculating Concrete Volume for a Cylinder (e.g., a Concrete Pillar):
□ Formula: Volume=�×(Diameter2)2×HeightVolume=π×(2Diameter)2×Height
□ Where:
☆ Diameter: The diameter of the cylinder (in feet or meters).
☆ Height: The height of the cylinder (in feet or meters).
4. Calculating Concrete Volume for a Trapezoidal Footing:
□ Formula: Volume=12×(Length+Width)×Depth×LengthVolume=21×(Length+Width)×Depth×Length
□ Where:
☆ Length: The length of the longer side of the trapezoid (in feet or meters).
☆ Width: The length of the shorter side of the trapezoid (in feet or meters).
☆ Depth: The depth or thickness of the footing (in feet or meters).
5. Calculating Concrete Volume for a Concrete Wall:
□ Formula: Volume=Length×Height×ThicknessVolume=Length×Height×Thickness
□ Where:
☆ Length: The length of the wall (in feet or meters).
☆ Height: The height of the wall (in feet or meters).
☆ Thickness: The thickness of the wall (in feet or meters).
These formulas are essential for accurately determining the volume of concrete needed for different portions of your construction project, whether it’s a simple rectangular slab or more complex
structures like columns, cylinders, trapezoidal footings, or walls. | {"url":"https://www.civilengineerstuff.com/concrete-calculator-quick-calculation-for-construction-project/","timestamp":"2024-11-03T17:12:56Z","content_type":"text/html","content_length":"159674","record_id":"<urn:uuid:0f6f3a85-3db5-419b-a59a-067a15944d19>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00364.warc.gz"} |
Twitter Archives - TO THE INNOVATION
This Leetcode problem Investments in 2016 LeetCode Solution is done in SQL. List of all LeetCode Solution Investments in 2016 LeetCode Solution Problem Statement Column Name Type pid int tiv_2015
float tiv_2016 float lat float lon float Table: Insurance pid is the primary key (column with unique values) for this table. Each row of this table […]
Investments in 2016 LeetCode Solution Read More »
Leetcode Solution
Merge Intervals LeetCode Solution
Here, We see Merge Intervals LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all LeetCode
Solution Merge Intervals LeetCode Solution Problem Statement Given an array of intervals where intervals[i] = [starti, endi], merge all overlapping intervals and return an array of the
Merge Intervals LeetCode Solution Read More »
Leetcode Solution
Flatten Nested List Iterator LeetCode Solution
Here, We see Flatten Nested List Iterator LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of
all LeetCode Solution Flatten Nested List Iterator LeetCode Solution Problem Statement You are given a nested list of integers nestedList. Each element is either an integer or
Flatten Nested List Iterator LeetCode Solution Read More »
Leetcode Solution
Implement Trie (Prefix Tree) LeetCode Solution
Here, We see Implement Trie (Prefix Tree) LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of
all LeetCode Solution Implement Trie (Prefix Tree) LeetCode Solution Problem Statement A trie (pronounced as “try”) or prefix tree is a tree data structure used to efficiently store and retrieve
Implement Trie (Prefix Tree) LeetCode Solution Read More »
Leetcode Solution
LRU Cache LeetCode Solution
Here, We see LRU Cache LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all LeetCode Solution
LRU Cache LeetCode Solution Problem Statement Design a data structure that follows the constraints of a Least Recently Used (LRU) cache. Implement the LRUCache class: The functions get and put must
LRU Cache LeetCode Solution Read More »
Leetcode Solution
Insert Delete GetRandom O(1) LeetCode Solution
Here, We see Insert Delete GetRandom O(1) LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of
all LeetCode Solution Insert Delete GetRandom O(1) LeetCode Solution Problem Statement Implement the RandomizedSet class: You must implement the functions of the class such that each function works
Insert Delete GetRandom O(1) LeetCode Solution Read More »
Leetcode Solution
Design Twitter LeetCode Solution
Here, We see Design Twitter LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all LeetCode
Solution Design Twitter LeetCode Solution Problem Statement Design a simplified version of Twitter where users can post tweets, follow/unfollow another user, and is able to see
Design Twitter LeetCode Solution Read More »
Leetcode Solution
Lowest Common Ancestor of a Binary Search Tree LeetCode Solution
Here, We see Lowest Common Ancestor of a Binary Search Tree LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different
approaches. List of all LeetCode Solution Lowest Common Ancestor of a Binary Search Tree LeetCode Solution Problem Statement Given a binary search tree (BST), find
Lowest Common Ancestor of a Binary Search Tree LeetCode Solution Read More »
Leetcode Solution
Validate IP Address LeetCode Solution
Here, We see Validate IP Address LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all
LeetCode Solution Validate IP Address LeetCode Solution Problem Statement Given a string queryIP, return “IPv4” if IP is a valid IPv4 address, “IPv6” if IP is a valid IPv6 address or “Neither” if
Validate IP Address LeetCode Solution Read More »
Leetcode Solution
Minimum Genetic Mutation LeetCode Solution
Here, We see Minimum Genetic Mutation LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all
LeetCode Solution Minimum Genetic Mutation LeetCode Solution Problem Statement A gene string can be represented by an 8-character long string, with choices from ‘A’, ‘C’, ‘G’, and ‘T’. Suppose we
Minimum Genetic Mutation LeetCode Solution Read More »
Leetcode Solution
Integer to Roman LeetCode Solution
Here, We see Integer to Roman LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all LeetCode
Solution Integer to Roman LeetCode Solution Problem Statement Roman numerals are represented by seven different symbols: I, V, X, L, C, D and M.SymbolValue I 1 V 5 X 10 L 50 C
Integer to Roman LeetCode Solution Read More »
Leetcode Solution
The Skyline Problem LeetCode Solution
Here, We see The Skyline Problem LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all
LeetCode Solution The Skyline Problem LeetCode Solution Problem Statement A city’s skyline is the outer contour of the silhouette formed by all the buildings in that city when
The Skyline Problem LeetCode Solution Read More »
Leetcode Solution
Word Break II LeetCode Solution
Here, We see Word Break II LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all LeetCode
Solution Word Break II LeetCode Solution Problem Statement Given a string s and a dictionary of strings wordDict, add spaces in s to construct a sentence where each word is
Word Break II LeetCode Solution Read More »
Leetcode Solution
Max Points on a Line LeetCode Solution
Here, We see Max Points on a Line LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all
LeetCode Solution Max Points on a Line LeetCode Solution Problem Statement Given an array of points where points[i] = [xi, yi] represents a point on the X-Y plane, return the maximum
Max Points on a Line LeetCode Solution Read More »
Leetcode Solution
Trapping Rain Water II LeetCode Solution
Here, We see Trapping Rain Water II LeetCode Solution. This Leetcode problem is done in many programming languages like C++, Java, JavaScript, Python, etc. with different approaches. List of all
LeetCode Solution Trapping Rain Water II LeetCode Solution Problem Statement Given an m x n integer matrix heightMap representing the height of each unit cell in a 2D elevation
Trapping Rain Water II LeetCode Solution Read More »
Leetcode Solution | {"url":"https://totheinnovation.com/tag/twitter/","timestamp":"2024-11-02T11:19:45Z","content_type":"text/html","content_length":"208575","record_id":"<urn:uuid:7c2b3952-36a0-473a-b74b-39ef4bb0ac82>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00258.warc.gz"} |
3783 -- Balls
Time Limit: 1000MS Memory Limit: 65536K
Total Submissions: 3114 Accepted: 1966
The classic Two Glass Balls brain-teaser is often posed as:
"Given two identical glass spheres, you would like to determine the lowest floor in a 100-story building from which they will break when dropped. Assume the spheres are undamaged when dropped below
this point. What is the strategy that will minimize the worst-case scenario for number of drops?"
Suppose that we had only one ball. We'd have to drop from each floor from 1 to 100 in sequence, requiring 100 drops in the worst case.
Now consider the case where we have two balls. Suppose we drop the first ball from floor n. If it breaks we're in the case where we have one ball remaining and we need to drop from floors 1 to n-1 in
sequence, yielding n drops in the worst case (the first ball is dropped once, the second at most n-1 times). However, if it does not break when dropped from floor n, we have reduced the problem to
dropping from floors n+1 to 100. In either case we must keep in mind that we've already used one drop. So the minimum number of drops, in the worst case, is the minimum over all n.
You will write a program to determine the minimum number of drops required, in the worst case, given B balls and an M-story building.
The first line of input contains a single integer P, (1 ≤ P ≤ 1000), which is the number of data sets that follow. Each data set consists of a single line containing three(3) decimal integer values:
the problem number, followed by a space, followed by the number of balls B, (1 ≤ B ≤ 50), followed by a space and the number of floors in the building M, (1 ≤ M ≤ 1000).
For each data set, generate one line of output with the following values: The data set number as a decimal integer, a space, and the minimum number of drops needed for the corresponding values of B
and M.
Sample Input
Sample Output | {"url":"http://poj.org/problem?id=3783","timestamp":"2024-11-06T12:08:39Z","content_type":"text/html","content_length":"7036","record_id":"<urn:uuid:60bd0314-ec9f-4efe-a2fc-03854700cb0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00554.warc.gz"} |
This year I have really tried to step up the process of bringing the real world into my mathematics class. A major focus has been on using technology appropriately as a tool to help solve real life
Here are some examples:
Distance formula: Finding an optimal (or near optimal) solution to the Traveling Salesman problem for a small number of cities.
Basically here the students were given the assignment of choosing 6 or 7 cities fairly near each other on a Google map and finding the x and y coordinates of each city, then using the distance
formula to determine the distances between the cities. Once they had this information, they were to try and figure out a shortest path, or at least something very close to the shortest path, and then
justify their solution.
Linear graphs & Piecewise functions: Compare 4 or 5 difference cell phone plans.
Students should take a few cell phone plans and compare the plans, including the cost for text messages (which may include similar graphs), the cost for extras, start up costs, etc… I found the
students end up needing to create piecewise functions in order to represent a cell phone plan which has a fixed rate until the minutes are used up at which point the customer has to pay extra for
each minute.
Shape and Space: Design a new school building.
Here I showed the students the new lot our school is in the process of purchasing and our project is to design a building for that spot, and calculate how much their building design will cost (within
the nearest $1000). It involves finding area, volumes, perimeters, scales, perspective, etc… We are using Google Sketchup for the designs but I am now trying to work out how to import the students
designs into a virtual world (like OpenSim) so we can have each student group lead walk-arounds of their building.
Polynomials: Determine how many operations multiplying a 100 digit number times a 100 digit number takes.
Students are learning about computational complexity theory by analyzing the number of steps it takes to multiply numbers together. They record each step in the operation and increase the size of the
numbers of each time and re-record their results. They then compare the different number of steps in each operation and try to come up with a formula, so that they can answer the 100 digit times 100
digit question. Our object: Figure out why our TI calculators can’t do this operation. It turns out that the formula itself is a polynomial, and their substitutions to check their various formulas
count as a lot of practice substituting into polynomials, which was a perfect fit for our curriculum.
Quadratic functions: Create an lower powered air cannon and use it to fire potatoes a few meters.
Here the students are attempting to use quadratic math to try and analyze their cannon, then the objective is to try and hit a target with a single shot later. The cannons should be very low powered
for obvious safety reasons, capable of firing a potato (or Tennis ball) a few metres at most. There is also a slight tie-in to Social Studies where my students will be studying cannons in their unit
on medieval warfare.
Bearings and Angles: Set up an orienteering course in your field or local park.
Students attempt to navigate a course through a park and pick up clues at each station, which they use to figure out a problem. Students have to be able to recognize the scale on the graph, navigate
using bearings, and measure angles accurately. Also lots of fun, we did this in Regents park for a couple of years in a row.
Integration: Calculate the area (or volume in a 3d integration class) of an actual 2d or 3d model.
Basically you have the students pick an object which they then find the functions (by placing the object electronically in a coordinate system) which represent the edge of the object, then place the
object in a coordinate system and calculate area of the object using integration.
Percentages: Find out how much your perfect set of "gear" (clothing) costs when it is on sale and has tax added.
Students take a catalog and calculate how much it will cost for them to buy their perfect set of clothing. They can buy as many items as they want (with their imaginary money) but have to keep track
of both the individual costs and the total cost of their clothing. You can also throw some curve balls at them, like if they buy more than a certain amount, they get discount, etc…
If you have any other examples of real life math being used in a project based learning context, please let me know. I’m always interested in other ideas, especially for the more challenging areas of
mathematics. I’ll add more ideas here as I remember them.
I personally think people learn through an unconscious process called experiential learning. They hypothesize about how the world should work, collect data, compare the data they have collected to
see if it fits in their theory, and then revise their theory if they feel enough evidence has been found. In this theory, as described by Kolb (1984), people construct an understanding of the world
around them using what they know as a basis.
Each piece of knowledge people gain has to be fit into their personal hypothesis. At first, people will "bend" their hypothesis to make facts fit which seem inconsistent, but eventually if enough
contradictory data is collected, people are forced to revise their ideas. This is part of the reason why students have so much difficulty learning topics for which they do not have any background;
they are constantly required to create and revisit their hypothesis, and to build theories about the information they are receiving "from scratch". "Ideas are not fixed and immutable elements of
thought but are formed and re-formed through experience." (Kolb, 1984)
It is crucial during this process that the learner feels comfortable to make mistakes. Although it is possible that an individual learner will have a theory which fits all the facts as they are
collected, it is much more likely that conflicts exist between their theory and the data. As the Lewinian experiential model suggests, observations of what one has learned or not learned are a
critical aspect of the learning process (Smith 2001).
As drawn from the work of Vygotsky, situated learning suggests that "experience in the activities of the practice" (Kolb, 2005) are integral to the learning process. Without learners being embedded
within a community of practice, their ability to make connections, draw conclusions, and verify hypothesis will be greatly hampered.
Kolb, D.A. (1984). Experiential Learning: Experience as The Source of Learning and Development, Case Western Reserve University, retrieved from http://www.learningfromexperience.com/research-library/
on December 2nd, 2009
Kolb, D.A., Boyatzis, K.E., Mainemelis, C. (2000). Experiential Learning Theory: Previous Research and New Directions, Case Western Reserve University, retrieved from http://
www.learningfromexperience.com/research-library/ on December 2nd, 2009
Kolb, A.Y, Kolb, D.A, (2005) Learning Styles and Learning Spaces: Enhancing Experiential Learning in Higher Education, Academy of Management Learning & Education, 2005, Vol. 4, No. 2, 193–212.
John-Steiner, V., Mahn, H. (1996). Sociocultural Approaches to Learning and Development: A Vygotskian Framework, Educational Psychologist, 31(3/4), 191-206, retrieved on December 2nd, 2009
Smith, M. K. (2001) ‘Kurt Lewin, groups, experiential learning and action research’, the encyclopedia of informal education, retrieved from http://www.infed.org/thinkers/et-lewin.htm on December 4th,
I’m working on my personal learning theory again, as a reflective activity in my Masters degree. I created a very short summary of my personal learning theory before, and am now updating it to
include vocabulary and ideas from the semester long course I just finished about learning theories. I hope most teaching colleges offer this kind of course as part of their teacher training, it has
been incredibly valuable to me.
Here is what I have so far:
Personal Learning Theory
I personally think people learn through an unconscious process called experiential learning. They hypothesize about how the world should work, collect data, compare the data they have collected to
see if it fits in their theory, and then revise their theory if they feel enough evidence has been found. In this theory, as described by Kolb (1984), people construct an understanding of the world
around them using what they know as a basis.
Each piece of knowledge people gain has to be fit into their personal hypothesis. At first, people will "bend" their hypothesis to make facts fit which seem inconsistent, but eventually if enough
contradictory data is collected, people are forced to revise their ideas. This is part of the reason why students have so much difficulty learning topics for which they do not have any background;
they are constantly required to create and revisit their hypothesis, and to build theories about the information they are receiving "from scratch". "Ideas are not fixed and immutable elements of
thought but are formed and re-formed through experience." (Kolb, 1984)
It is crucial during this process that the learner feels comfortable to make mistakes. Although it is possible that an individual learner will have a theory which fits all the facts as they are
collected, it is much more likely that conflicts exist between their theory and the data. As the Lewinian experiential model suggests, observations of what one has learned or not learned are a
critical aspect of the learning process (Smith 2001).
As drawn from the work of Vygotsky, situated learning suggests that "experience in the activities of the practice" (Kolb, 2005) are integral to the learning process. Without learners being embedded
within a community of practice, their ability to make connections, draw conclusions, and verify hypothesis will be greatly hampered.
Kolb, D.A. (1984). Experiential Learning: Experience as The Source of Learning and Development, Case Western Reserve University, retrieved from http://www.learningfromexperience.com/research-library/
on December 2nd, 2009
Kolb, D.A., Boyatzis, K.E., Mainemelis, C. (2000). Experiential Learning Theory: Previous Research and New Directions, Case Western Reserve University, retrieved from http://
www.learningfromexperience.com/research-library/ on December 2nd, 2009
Kolb, A.Y, Kolb, D.A, (2005) Learning Styles and Learning Spaces: Enhancing Experiential Learning in Higher Education, Academy of Management Learning & Education, 2005, Vol. 4, No. 2, 193–212.
John-Steiner, V., Mahn, H. (1996). Sociocultural Approaches to Learning and Development: A Vygotskian Framework, Educational Psychologist, 31(3/4), 191-206, retrieved on December 2nd, 2009
Smith, M. K. (2001) ‘Kurt Lewin, groups, experiential learning and action research’, the encyclopedia of informal education, retrieved from http://www.infed.org/thinkers/et-lewin.htm on December 4th,
I personally think people learn through an unconscious process very much like the scientific method. They hypothesize about how the world should work, collect data, compare the data they have
collected to see if it fits in their theory, and then revise their theory if they feel enough evidence has been found. In this way, people construct an understanding of the world around them using
what they know as a basis.
Each piece of knowledge people gain has to be fit into their personal hypothesis. At first, people will "bend" their hypothesis to make facts fit which seem inconsistent, but eventually if enough
contradictory data is collected, people are forced to revise their ideas. This is part of the reason why students have so much difficulty learning topics for which they do not have any background;
they are constantly required to create and revisit their hypothesis, and to build theories about the information they are receiving "from scratch".
It is crucial during this process that the learner feels comfortable to make mistakes. Instead of feeling pressure to have exactly the right answer each time, learners must be willing to work through
the entire process of learning. Although it is possible that an individual learner will have a theory which fits all the facts as they are collected, it is much more likely that conflicts exist
between their theory and the data.
In the classroom, this is when we normally say that a student has "made a mistake", which is unfortunate language. Rather than criticizing students who have a cognitive discord occurring, we should
encourage more reflection of the learning process, and provide opportunities to establish a new theory which fits the given facts and can be worked into the learner’s personal theory of how the world | {"url":"https://davidwees.com/content/tag/learning/page/2/","timestamp":"2024-11-05T14:03:37Z","content_type":"text/html","content_length":"70319","record_id":"<urn:uuid:c9a45c21-b7e7-494f-b44b-e59141e76576>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00319.warc.gz"} |
Learning Probabilistic Read-once Formulas on Product Distributions
This paper presents a polynomial-time algorithm for inferring a probabilistic generalization of the class of read-once Boolean formulas over the usual basis { AND, OR, NOT }. The algorithm
effectively infers a good approximation of the target formula when provided with random examples which are chosen according to any product distribution, i.e., any distribution in which the setting of
each input bit is chosen independently of the settings of the other bits. Since the class of formulas considered includes ordinary read-once Boolean formulas, our result shows that such formulas are
PAC learnable (in the sense of Valiant) against any product distribution (for instance, against the uniform distribution). Further, this class of probabilistic formulas includes read-once formulas
whose behavior has been corrupted by large amounts of random noise. Such noise may affect the formula's output (“misclassification noise”), the input bits ( “attribute noise”), or it may affect the
behavior of individual gates of the formula. Thus, in this setting, we show that read-once formula's can be inferred (approximately), despite large amounts of noise affecting the formula's behavior.
All Science Journal Classification (ASJC) codes
• Software
• Artificial Intelligence
• PAC-learning
• computational learning theory
• learning with noise
• product distributions
• read-once formulas
Dive into the research topics of 'Learning Probabilistic Read-once Formulas on Product Distributions'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/learning-probabilistic-read-once-formulas-on-product-distribution","timestamp":"2024-11-08T12:00:59Z","content_type":"text/html","content_length":"50655","record_id":"<urn:uuid:bceee51c-e19f-4bb7-ba2d-056d5edcf13d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00615.warc.gz"} |
3-D swarm scatter chart
Since R2020b
Vector Data
swarmchart3(x,y,z) displays a 3-D swarm chart, which is a scatter plot with the points offset (jittered) in the x- and y-dimensions. The points form distinct shapes, and the outline of each shape is
similar to a violin plot. 3-D swarm charts help you to visualize discrete (x,y) data with the distribution of the z data. At each (x,y) location, the points are jittered based on the kernel density
estimate of z.
swarmchart3(x,y,z,sz) specifies the marker sizes. To plot all the markers with the same size, specify sz as a scalar. To plot the markers with different sizes, specify sz as a vector that is the same
size as x, y, and z.
swarmchart3(x,y,z,sz,c) specifies the marker colors. To plot all the markers with the same color, specify c as a color name or an RGB triplet. To assign a different color to each marker, specify a
vector the same size as x, y, and z. Alternatively, you can specify a three-column matrix of RGB triplets. The number of rows in the matrix must match the length of x, y, and z.
swarmchart3(___,mkr) specifies a different marker than the default marker, which is a circle. Specify mkr after all the arguments in any of the previous syntaxes.
swarmchart3(___,'filled') fills in the markers. Specify the 'filled' option after all the arguments in any of the previous syntaxes.
Table Data
swarmchart3(tbl,xvar,yvar,zvar) plots the variables xvar, yvar, and zvar from the table tbl. To plot one data set, specify one variable each for xvar, yvar, and zvar. To plot multiple data sets,
specify multiple variables for at least one of those arguments. The arguments that specify multiple variables must specify the same number of variables.
Additional Options
swarmchart3(ax,___) displays the swarm chart in the target axes. Specify the axes before all the arguments in any of the previous syntaxes.
swarmchart3(___,Name,Value) specifies additional properties for the swarm chart using one or more Name,Value arguments. For example:
• swarmchart3(x,y,z,'LineWidth',2) creates a swarm chart with 2-point marker outlines.
• swarmchart3(tbl,'MyX','MyY','MyZ','ColorVariable','MyColors') creates a swarm chart from data in a table, and customizes the marker colors using data from the table.
For a list of properties, see Scatter Properties.
s = swarmchart3(___) returns the Scatter object. Use s to modify properties of the chart after creating it. For a list of properties, see Scatter Properties.
Create a 3-D Swarm Chart
Read the BicycleCounts.csv data set into a timetable called tbl. This data set contains bicycle traffic data over a period of time. Display the first five rows of tbl.
tbl = readtable("BicycleCounts.csv");
ans=5×5 table
Timestamp Day Total Westbound Eastbound
___________________ _____________ _____ _________ _________
2015-06-24 00:00:00 {'Wednesday'} 13 9 4
2015-06-24 01:00:00 {'Wednesday'} 3 3 0
2015-06-24 02:00:00 {'Wednesday'} 1 1 0
2015-06-24 03:00:00 {'Wednesday'} 1 1 0
2015-06-24 04:00:00 {'Wednesday'} 1 1 0
Create a vector x with the day name from each observation.
daynames = ["Sunday" "Monday" "Tuesday" "Wednesday" "Thursday" "Friday" "Saturday"];
x = categorical(tbl.Day,daynames);
Create a categorical vector y containing the values "pm" or "am" according to the time for each observation in the table. Create vector z of eastbound traffic data. Then create a swarm chart of x, y,
and z. The chart shows the data distributions for each morning and evening of the week.
ispm = tbl.Timestamp.Hour < 12;
y = categorical;
y(ispm) = "pm";
y(~ispm) = "am";
z= tbl.Eastbound;
Specify Marker Size
Create vector x as a combination of zeros and ones, and create y as a vector containing all ones. Create z as a vector of squared random numbers. Then create a swarm chart of x, y, and z, and specify
the size marker size as 5.
x = [zeros(1,500) ones(1,500)];
y = ones(1,1000);
z = randn(1,1000).^2;
Specify Marker Symbol
Create vector x as a combination of zeros and ones, and create y as a vector containing all ones. Create z as a vector of squared random numbers. Then create a swarm chart of x, y, and z, and specify
the point ('.') marker symbol.
x = [zeros(1,500) ones(1,500)];
y = ones(1,1000);
z = randn(1,1000).^2;
Vary Marker Color
Create vector x containing a combination of zeros and ones, and create y containing a random combination of ones and twos. Create z as a vector of squared random numbers. Specify the colors for the
markers by creating vector c as the square root of z. Then create a swarm chart of x, y, and z. Set the marker size to 50 and specify the colors as c. The values in c index into the figure's
colormap. Use the 'filled' option to fill the markers with color instead of displaying them as hollow circles.
x = [zeros(1,500) ones(1,500)];
y = randi(2,1,1000);
z = randn(1,1000).^2;
c = sqrt(z);
Change Jitter Type and Width
Create vector x containing a combination of zeros and ones, and create y containing a random combination of the numbers one through four. Create z as a vector of squared random numbers. Then create a
swarm chart of x, y, and z by calling the swarmchart function with a return argument that stores the Scatter object. Add x- and y-axis labels so you can see the effect of changing the jitter
properties in each dimension.
x = [zeros(1,500) ones(1,500)];
y = randi(4,1,1000);
z = randn(1,1000).^2;
s = swarmchart3(x,y,z);
Change the shapes of the clusters of points by setting the jitter properties on the Scatter object. In the x dimension, specify uniform random jitter, and change the jitter width to 0.5 data units.
In the y dimension, specify normal random jitter, and change the jitter width to 0.1 data units. The spacing between points does not exceed the jitter width you specify.
s.XJitter = 'rand';
s.XJitterWidth = 0.5;
s.YJitter = 'randn';
s.YJitterWidth = 0.1;
Plot Data from a Table
A convenient way to plot data from a table is to pass the table to the swarm3 function and specify the variables you want to plot. For example, create a table with four variables of random numbers,
and plot the X, Y1, and Z variables. By default, the axis labels match the variable names.
tbl = table(randi(2,100,1),randi(2,100,1),randi([10 11],100,1), ...
You can also plot multiple variables at the same time. For example, plot Y1 and Y2 on the y-axis by specifying the yvar argument as the cell array {'Y1','Y2'}. Then add a legend. The legend labels
match the variable names.
Plot Table Data with Custom Marker Sizes and Colors
One way to plot data from a table and customize the colors and marker sizes is to set the ColorVariable and SizeData properties. You can set these properties as name-value arguments when you call the
swarmchart3 function, or you can set them on the Scatter object later.
For example, create a table with four variables of random numbers, and plot the X, Y, and Z variables with filled markers. Vary the marker colors by specifying the ColorVariable name-value argument.
Return the Scatter object as s, so you can set other properties later.
tbl = table(randi(2,100,1),randn(100,1),randn(100,1),randn(100,1), ...
s = swarmchart3(tbl,'X','Y','Z','filled','ColorVariable','Colors');
Change the marker sizes to 100 points by setting the SizeData property.
Specify Target Axes
Read the BicycleCounts.csv data set into a timetable called tbl. This data set contains bicycle traffic data over a period of time. Display the first five rows of tbl.
tbl = readtable("BicycleCounts.csv");
ans=5×5 table
Timestamp Day Total Westbound Eastbound
___________________ _____________ _____ _________ _________
2015-06-24 00:00:00 {'Wednesday'} 13 9 4
2015-06-24 01:00:00 {'Wednesday'} 3 3 0
2015-06-24 02:00:00 {'Wednesday'} 1 1 0
2015-06-24 03:00:00 {'Wednesday'} 1 1 0
2015-06-24 04:00:00 {'Wednesday'} 1 1 0
Create vector x with the days names for each observation. Create a categorical vector y containing the values "pm" or "am" according to the time for each observation in the table. Define ze as a
vector of eastbound traffic data, and define zw as a vector of westbound traffic data.
daynames = ["Sunday" "Monday" "Tuesday" "Wednesday" "Thursday" "Friday" "Saturday"];
x = categorical(tbl.Day,daynames);
ispm = tbl.Timestamp.Hour<12;
y = categorical;
y(ispm) = 'pm';
y(~ispm) = 'am';
ze = tbl.Eastbound;
zw = tbl.Westbound;
Create a tiled chart layout in the 'flow' tile arrangement, so that the axes fill the available space in the layout. Call the nexttile function to create an axes object and return it as ax1. Then
create a swarm chart of the eastbound data by passing ax1 to the swarmchart function.
Repeat the process to create a second axes object and a swarm chart for the westbound traffic.
ax2 = nexttile;
z = tbl.Westbound;
Input Arguments
x — x-coordinates
scalar | vector
x-coordinates, specified as a numeric scalar or a vector the same size as y and z.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical
y — y-coordinates
scalar | vector
y-coordinates, specified as a numeric scalar or a vector the same size as x and z.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | categorical
z — z-coordinates
scalar | vector
z-coordinates, specified as a numeric scalar or a vector the same size as x and y.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | datetime | duration
sz — Marker size
36 (default) | numeric scalar | row or column vector | []
Marker size in points, specified in one of these forms:
• Numeric scalar — Plot all markers with equal size.
• Row or column vector — Use different sizes for each marker. The length of sz must equal the length of x, y, and z.
• [] — Use the default size of 36 points.
c — Marker color
[0 0.4470 0.7410] (default) | RGB triplet | three-column matrix of RGB triplets | vector | 'r' | 'g' | 'b' | ...
Marker color, specified in one of these forms:
• RGB triplet or color name — Plot all the markers with the same color. An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of
the color. The intensities must be in the range [0,1]. Alternatively, you can specify a color name from the table below.
• Three column matrix of RGB triplets — Use different colors for each marker. Each row of the matrix specifies an RGB triplet color for the corresponding marker. The number of rows must equal the
length of x, y, and z.
• Vector — Use different colors for each marker. The values in c index into the current colormap, and they cover the full range of the colormap. The length of c must equal the length of x, y, and
z. To change the colormap, use the colormap function.
Color Name Description Equivalent RGB Triplet
'red' or 'r' Red [1 0 0]
'green' or 'g' Green [0 1 0]
'blue' or 'b' Blue [0 0 1]
'yellow' or 'y' Yellow [1 1 0]
'magenta' or 'm' Magenta [1 0 1]
'cyan' or 'c' Cyan [0 1 1]
'white' or 'w' White [1 1 1]
'black' or 'k' Black [0 0 0]
mkr — Marker type
'o' (default) | '+' | '*' | '.' | 'x' | ...
Marker type, specified as one of the values listed in this table.
Marker Description Resulting Marker
"o" Circle
"+" Plus sign
"*" Asterisk
"." Point
"x" Cross
"_" Horizontal line
"|" Vertical line
"square" Square
"diamond" Diamond
"^" Upward-pointing triangle
"v" Downward-pointing triangle
">" Right-pointing triangle
"<" Left-pointing triangle
"pentagram" Pentagram
"hexagram" Hexagram
'filled' — Option to fill interior of markers
Option to fill the interior of the markers, specified as 'filled'. Use this option with markers that have a face, for example, 'o' or 'square'. Markers that do not have a face and contain only edges
do not render at all ('+', '*', '.', and 'x').
The 'filled' option sets the MarkerFaceColor property of the Scatter object to 'flat' and the MarkerEdgeColor property to 'none'. In this case, MATLAB^® draws the marker faces, but not the edges.
tbl — Source table
table | timetable
Source table containing the data to plot, specified as a table or a timetable.
xvar — Table variables containing x-coordinates
one or more table variable indices
Table variables containing the x-coordinates, specified as one or more table variable indices.
Specifying Table Indices
Use any of the following indexing schemes to specify the desired variable or variables.
Indexing Scheme Examples
• "A" or 'A' — A variable named A
Variable names: • ["A","B"] or {'A','B'} — Two variables named A and B
• "Var"+digitsPattern(1) — Variables named "Var" followed by a single
Variable index:
• 3 — The third variable from the table
• An index number that refers to the location of a variable in the table.
• [2 3] — The second and third variables from the table
• A vector of numbers.
• [false false true] — The third variable
• A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 or
false values.
Variable type: • vartype("categorical") — All the variables containing categorical
Plotting Your Data
The table variables you specify can contain numeric, categorical, datetime, or duration values.
To plot one data set, specify one variable for xvar, one variable for yvar, and one variable for zvar. For example, create a table with four variables of normally distributed random values. Plot the
X, Y1, and Z variables.
tbl = table(randn(100,1),randn(100,1),randn(100,1)+5,randn(100,1), ...
To plot multiple data sets together, specify multiple variables for at least one of xvar, yvar, or zvar. If you specify multiple variables for more than one argument, the number of variables must be
the same for each of those arguments.
For example, plot the X variable on the x-axis, the Y1 and Y2 variables on the y-axis, and the Z variable on the z-axis.
You can also use different indexing schemes for xvar, yvar, and zvar. For example, specify xvar as a variable name, yvar as an index number, and zvar as a logical vector.
swarmchart3(tbl,'X',2,[false false true])
yvar — Table variables containing y-coordinates
one or more table variable indices
Table variables containing the y-coordinates, specified as one or more table variable indices.
Specifying Table Indices
Use any of the following indexing schemes to specify the desired variable or variables.
Indexing Scheme Examples
• "A" or 'A' — A variable named A
Variable names: • ["A","B"] or {'A','B'} — Two variables named A and B
• "Var"+digitsPattern(1) — Variables named "Var" followed by a single
Variable index:
• 3 — The third variable from the table
• An index number that refers to the location of a variable in the table.
• [2 3] — The second and third variables from the table
• A vector of numbers.
• [false false true] — The third variable
• A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 or
false values.
Variable type: • vartype("categorical") — All the variables containing categorical
Plotting Your Data
The table variables you specify can contain numeric, categorical, datetime, or duration values.
To plot one data set, specify one variable for xvar, one variable for yvar, and one variable for zvar. For example, create a table with four variables of normally distributed random values. Plot the
X, Y1, and Z variables.
tbl = table(randn(100,1),randn(100,1),randn(100,1)+5,randn(100,1), ...
To plot multiple data sets together, specify multiple variables for at least one of xvar, yvar, or zvar. If you specify multiple variables for more than one argument, the number of variables must be
the same for each of those arguments.
For example, plot the X variable on the x-axis, the Y1 and Y2 variables on the y-axis, and the Z variable on the z-axis.
You can also use different indexing schemes for xvar, yvar, and zvar. For example, specify xvar as a variable name, yvar as an index number, and zvar as a logical vector.
swarmchart3(tbl,'X',2,[false false true])
zvar — Table variables containing z-coordinates
one or more table variable indices
Table variables containing the z-coordinates, specified as one or more table variable indices.
Specifying Table Indices
Use any of the following indexing schemes to specify the desired variable or variables.
Indexing Scheme Examples
• "A" or 'A' — A variable named A
Variable names: • ["A","B"] or {'A','B'} — Two variables named A and B
• "Var"+digitsPattern(1) — Variables named "Var" followed by a single
Variable index:
• 3 — The third variable from the table
• An index number that refers to the location of a variable in the table.
• [2 3] — The second and third variables from the table
• A vector of numbers.
• [false false true] — The third variable
• A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0 or
false values.
Variable type: • vartype("categorical") — All the variables containing categorical
Plotting Your Data
The table variables you specify can contain numeric, categorical, datetime, or duration values.
To plot one data set, specify one variable for xvar, one variable for yvar, and one variable for zvar. For example, create a table with four variables of normally distributed random values. Plot the
X, Y1, and Z variables.
tbl = table(randn(100,1),randn(100,1),randn(100,1)+5,randn(100,1), ...
To plot multiple data sets together, specify multiple variables for at least one of xvar, yvar, or zvar. If you specify multiple variables for more than one argument, the number of variables must be
the same for each of those arguments.
For example, plot the X variable on the x-axis, the Y1 and Y2 variables on the y-axis, and the Z variable on the z-axis.
You can also use different indexing schemes for xvar, yvar, and zvar. For example, specify xvar as a variable name, yvar as an index number, and zvar as a logical vector.
swarmchart3(tbl,'X',2,[false false true])
ax — Target axes
Axes object
Target axes, specified as an Axes object. If you do not specify the axes, MATLAB plots into the current axes, or it creates an Axes object if one does not exist.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: swarmchart3(randi(2,500,1),randi(2,500,1),randn(500,1),'MarkerFaceColor','red') specifies red filled markers.
XJitter — Jitter type for x-dimension
'none' | 'density' | 'rand' | 'randn'
Type of jitter (spacing of points) along the x-dimension, specified as one of the following values:
• 'none' — Do not jitter the points.
• 'density' — Jitter the points using the kernel density estimate of y for 2-D charts. If you specify this option in two dimensions for a 3-D chart, the points are jittered based on the kernel
density estimate in the third dimension. For example, setting XJitter and YJitter to 'density' uses the kernel density estimate of z.
• 'rand' — Jitter the points randomly with a uniform distribution.
• 'randn' — Jitter points randomly with a normal distribution.
XJitterWidth — Maximum jitter along x-dimension
nonnegative scalar
Maximum amount of jitter (offset between points) along the x-dimension, specified as a nonnegative scalar value in data units.
For example, to set the jitter width to 90% of the shortest distance between adjacent points, take the minimum distance between unique values of x and scale by 0.9.
XJitterWidth = 0.9 * min(diff(unique(x)));
YJitter — Jitter type for y-dimension
'none' | 'density' | 'rand' | 'randn'
Type of jitter (spacing of points) along the y-dimension, specified as one of the following values:
• 'none' — Do not jitter the points.
• 'density' — Jitter the points using the kernel density estimate of x for 2-D charts. If you specify this option in two dimensions for a 3-D chart, the points are jittered based on the kernel
density estimate in the third dimension. For example, setting XJitter and YJitter to 'density' uses the kernel density estimate of z.
• 'rand' — Jitter the points randomly with a uniform distribution.
• 'randn' — Jitter points randomly with a normal distribution.
YJitterWidth — Maximum jitter along y-dimension
nonnegative scalar
Maximum amount of jitter (offset between points) along the y-dimension, specified as a nonnegative scalar value in data units.
For example, to set the jitter width to 90% of the shortest distance between adjacent points, take the minimum distance between unique values of y and scale by 0.9.
YJitterWidth = 0.9 * min(diff(unique(y)));
ColorVariable — Table variable containing color data
table variable index
Table variable containing the color data, specified as a variable index into the source table.
Specifying the Table Index
Use any of the following indexing schemes to specify the desired variable.
Indexing Scheme Examples
Variable name: • "A" or 'A' — A variable named A
• A string scalar or character vector. • "Var"+digitsPattern(1) — The variable with the name "Var" followed by a
single digit
• A pattern object. The pattern object must refer to only one variable.
Variable index:
• 3 — The third variable from the table
• An index number that refers to the location of a variable in the table.
• [false false true] — The third variable
• A logical vector. Typically, this vector is the same length as the number of variables, but you can omit trailing 0
or false values.
Variable type:
• vartype("double") — The variable containing double values
• A vartype subscript that selects a table variable of a specified type. The subscript must refer to only one
Specifying Color Data
Specifying the ColorVariable property controls the colors of the markers. The data in the variable controls the marker fill color when the MarkerFaceColor property is set to "flat". The data can also
control the marker outline color, when the MarkerEdgeColor is set to "flat".
The table variable you specify can contain values of any numeric type. The values can be in either of the following forms:
• A column of numbers that linearly map into the current colormap.
• A three-column array of RGB triplets. RGB triplets are three-element vectors whose values specify the intensities of the red, green, and blue components of specific colors. The intensities must
be in the range [0,1]. For example, [0.5 0.7 1] specifies a shade of light blue.
When you set the ColorVariable property, MATLAB updates the CData property.
The points in a swarm chart are jittered using uniform random values that are weighted by the Gaussian kernel density estimate of z and the relative number of points at each (x, y) location. This
behavior corresponds to the default 'density' setting of the XJitter and YJitter properties on the Scatter object when you call the swarmchart3 function.
The maximum spread of points at each x location is 90% of the smallest distance between adjacent points by default. For example, in the x dimension, the spread is calculated as:
spread = 0.9 * min(diff(unique(x)));
You can control the offset by setting the XJitterWidth and YJitterWidth properties on the Scatter object.
Version History
Introduced in R2020b
R2022b: Plots created with tables preserve special characters in axis and legend labels
When you pass a table and one or more variable names to the swarmchart3 function, the axis and legend labels now display any special characters that are included in the table variable names, such as
underscores. Previously, special characters were interpreted as TeX or LaTeX characters.
For example, if you pass a table containing a variable named Sample_Number to the swarmchart3 function, the underscore appears in the axis and legend labels. In R2022a and earlier releases, the
underscores are interpreted as subscripts.
Release Label for Table Variable "Sample_Number"
To display axis and legend labels with TeX or LaTeX formatting, specify the labels manually. For example, after plotting, call the xlabel or legend function with the desired label strings.
legend(["Sample_Number" "Another_Legend_Label"])
R2021b: Pass tables directly to swarmchart3
Create plots by passing a table to the swarmchart3 function followed by the variables you want to plot. When you specify your data as a table, the axis labels and the legend (if present) are
automatically labeled using the table variable names.
See Also | {"url":"https://kr.mathworks.com/help/matlab/ref/swarmchart3.html","timestamp":"2024-11-05T22:02:17Z","content_type":"text/html","content_length":"197385","record_id":"<urn:uuid:9a909d2e-54d3-4172-a6f1-e23c601df770>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00277.warc.gz"} |
An introduction to xAct with applications to metric-affine gravity, cosmology and black holes.
Schedule: Four sessions of 2 hours each on April 11,12, and 13 from 11:00 to 13:00 plus one more lecture on Tuesday 12 in the afternoon.
The goal of the course will be to give an elementary and practical introduction to the Mathematica package xAct (http://www.xact.es/). The philosophy of the lectures will be to get familiarised with
some capabilities of the packages by working out explicit problems. Some basic notions of Mathematica would be desirable. Basics of field theory, gravitation, cosmology and black holes physics will
also be helpful, but not strictly necessary. As a matter of fact, the proposed problems aim at providing some basics of cosmological perturbations, gravitational waves oscillations and black hole
perturbations physics.
I.1.- Installation, general philosophy of the package and survey of the different capabilities.
I.4.- Some applications in field theory: Computing field equations and the energy-momentum tensor.
Lecture III (2 hours): Application to oscillations of gravitational waves.
III.1 Inequivalent realisations of the cosmological principle. Working with two manifolds. | {"url":"https://indico.ific.uv.es/event/6573/?print=1&view=standard_inline_minutes","timestamp":"2024-11-05T12:35:38Z","content_type":"text/html","content_length":"17669","record_id":"<urn:uuid:6b101f3c-65d5-4679-9a2d-6d44d6f0f221>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00787.warc.gz"} |
Plotting of Interactive Electric field due to point charges with Matplotlib
My Coding > Numerical simulations > EMHD > Plotting of Interactive Electric field due to point charges with Matplotlib
Plotting of Interactive Electric field due to point charges with Matplotlib
This tutorial will be more focused on Python coding, rather than on the physics of the process, because all physics was described in the previous part of calculating of electric field potential and
intensity due to point charges, or in the video about it
Here I will give mainly information, about how to use Matplotlib Widgets.
Initial description of domain, functions and plot
Domain description
As usual, the first step in any proper MHD and EMHD calculations is to generate domain for calculations. After loading all required libraries, of course.
For convenience, all field variables will be stored in one dictionary D - it will make it easier to send to some functions
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.widgets import Slider, CheckButtons, RadioButtons
xrange = [-1., 1.] # left and right coordinates of square domain (x==y)
step = 40 # Number of cells in the domain in any axis
qrange = [-10., 10.] # max and min values for charge
NQ = 4 # Number of charges
D = {} # domain description
D['X'], D['Y'] = np.meshgrid(xlist, ylist)
D['Ex'] = np.zeros_like(D['X'])
D['Ey'] = np.zeros_like(D['X'])
D['V'] = np.zeros_like(D['X'])
D['E'] = np.zeros_like(D['X'])
D['Q'] = np.random.uniform(low=qmin, high=qmax, size=(NQ, 3))
How to generate randomply places charges (last line in the code) described here: how to make random array
Electric field calculations
Second, very important step is to make function for electric field calculations. To make calculations slightly faster, I’ve use non-standard distance calculating. I’ve apply power of (3/2) and it
save me one power calculation for a full domain. Profit!!! But it will make code more confusing and should only used when it is really necessary
def addPointCharge2D(V, Ex, Ey, X, Y, q):
# Add one charge to the field
r2 = ((X - q[0]) ** 2 + (Y - q[1]) ** 2) ** (3 / 2)
Ex += q[2] * (X - q[0]) / r2 # Electric field (x)
Ey += q[2] * (Y - q[1]) / r2 # Electric field (y)
V += q[2] / r2 ** (1 / 3.)
return V, Ex, Ey
def ElectricField(E, Ex, Ey, V, X, Y, Q):
# Calculate for all charges
Ex *= 0
Ey *= 0
V *= 0
for q in Q:
addPointCharge2D(V, Ex, Ey, X, Y, q)
E[:] = np.log((Ex ** 2 + Ey ** 2)) * 0.5
return E, Ex, Ey, V
ElectricField(**D) # Calculate E field for initial charge combination
Plotting out results
After calculation Electric field intensity and potential energy, it is convenient to make it plot. But before plotting we will create control variables, to control which information we will plot.
This is very important, because later, when we will use widgets to modify something, these control variable will be used.
P = {} # plot description
P['plt_type'] = ['Position', 'Field', 'Lines'] # type of shown information
P['plt_active'] = [True, True, True] # Show or not switches
P['ColorBar'] = None # initial colorbar
P['field_type'] = ['V', 'E'] # available field for plotting
P['VE'] = False # show vector field
P['levelE'] = np.linspace(-1, 10, 100) # range for Efield colorbar
P['levelV'] = np.linspace(-150, 150, 100) # range for Vfield colorbar
P['Qactive'] = 0 # Current active charge
P['line_density_range'] = [0.1, 2.0] # Range of possible line densities
P['line_density'] = 0.4 # current line density
P['fig'], P['ax'] = plt.subplots(figsize=(6, 5))
And at the last line we are creating subplot itself and matplotlib.axes.Axes object for all plotting. Also we physically define the size of the picture in inches (6, 5).
After defining all controls, we can make a function to plot all graphical information according to these controls:
def el_plot(P, D):
if P['plt_active'][0]:
# Position
P['ax'].scatter(D['Q'][:, 0], D['Q'][:, 1], c=np.where(D['Q'][:, 2] < 0, 'b', 'r'))
for i, txt in enumerate(D['Q']):
P['ax'].annotate(f'({i + 1}):{txt[2]:.1f}', xy=(D['Q'][i, 0], D['Q'][i, 1]),
xytext=(D['Q'][i, 0] + 0.02, D['Q'][i, 1] + 0.02),
color='black', fontsize='12')
if P['plt_active'][1]:
# Field
if P['VE']:
P['ax'].set_title("Electric Field Intensity, E")
cp = P['ax'].contourf(D['X'], D['Y'], D['E'], levels=P['levelE'], cmap='jet')
P['ax'].set_title("Electric Potential energy, V")
cp = P['ax'].contourf(D['X'], D['Y'], D['V'], levels=P['levelV'], cmap='seismic')
if P['ColorBar']:
P['ColorBar'] = P['fig'].colorbar(cp, ax=P['ax'])
if P['plt_active'][2]:
# Lines
P['ax'].streamplot(D['X'], D['Y'], D['Ex'], D['Ey'], color='black', linewidth=0.5,
density=P['line_density'], arrowstyle='->', arrowsize=1.0)
return None
It is important to mention, that before every plotting we clear all previously plotted information P['ax'].clear() and also we remove ColourBar before plotting new one. Also for the very first
colourbar plotting we have nothing to remove.
After defining function for plotting, we can call it, but before it we need to adjust position of the sublot to make some space for future widgets and also forcedly restrict plotting area – this is
necessary to avoid automatic rescaling in the future.
plt.subplots_adjust(left=0.20, bottom=0.35)
plt.xlim(xrange[0], xrange[1])
plt.ylim(xrange[0], xrange[1])
el_plot(P, D)
Electric Potential Energy, V.
electric Potential Energy V, calculated for 4 randomly placed charges. Some space is allocated for Widgets.
Original image: 588 x 524
Plot control with Matplotlib Widgets
When the plot with initial data is ready and it working perfectly, it is a good time to add widgets to interactively change some parameters.
Sliders are very convenient, when it is necessary to smoothly change some of the parameters. For example, in our case, sliders will be ideal for modifying position of charges and charge value. Also
it is very convenient to modify some of the plot parameters, like line density.
To make slider, it is necessary to allocate physical space with Axis for it. This physical space should be allocated in inches
ax_x = plt.axes([0.20, 0.25, 0.55, 0.05]) # Left, bottom, width, height (Horizontal)
ax_y = plt.axes([0.07, 0.35, 0.05, 0.53]) # Left, bottom, width, height (VERTICAL SLIDER)
ax_q = plt.axes([0.20, 0.20, 0.55, 0.05]) # Left, bottom, width, height (Horizontal)
ax_d = plt.axes([0.20, 0.15, 0.55, 0.05]) # Left, bottom, width, height (Horizontal)
In this example, four spaces are allocated, and it is easy to see, that ax_y is vertical. At he next step it is necessary to run constructor to initialize sliders for these areas
s_x = Slider(ax_x, 'x', qmin[0], qmax[0], valinit=D['Q'][P['Qactive']][0])
s_y = Slider(ax_y, 'y', qmin[1], qmax[1], valinit=D['Q'][P['Qactive']][1],
s_q = Slider(ax_q, 'q', qmin[2], qmax[2], valinit=D['Q'][P['Qactive']][2])
s_d = Slider(ax_d, 'l', P['line_density_range'][0], P['line_density_range'][1],
For Slider constrictor it is necessary to provide location, name, range of values and initial value, where slider wheel will be located. Initial position will be remembered with an option to reset it
and also market with thin red line. It is possible to remove this red line.
It is depend on the task you solving, how important to keep these red lines. I prefer to remove them
s_x.vline._linewidth = 0
s_y.hline._linewidth = 0 # Be careful – this line is horizontal!
s_q.vline._linewidth = 0
s_d.vline._linewidth = 0
At the next step it is necessary to define function, which will respond to the changes of the sliders
def slider_update(val, P, D):
# variable q is for making writing shorter
q = D['Q'][P['Qactive']]
# reading positions of all sliders
q[0] = s_x.val # position of s_x slider
q[1] = s_y.val # position of s_y slider
q[2] = s_q.val
P['line_density'] = s_d.val
# recalculation of the plot in accordance with the new positions
# recalculation
# redraw plot with the new data
# output
el_plot(P, D)
This function will be called upon any changes in the slider position and value will be passed to this function by default. To pass more parameters it is necessary to use lambda call.
The standard way of using this function – read values from sliders (val), then recalculated displayed values and then redraw the plot.
To call this function, it is necessary to use on_changed method to appropriate slider. In this code fragment, I will call use this method with lambda activation of the update function
s_x.on_changed(lambda new_val: slider_update(new_val, P, D))
s_y.on_changed(lambda new_val: slider_update(new_val, P, D))
s_q.on_changed(lambda new_val: slider_update(new_val, P, D))
s_d.on_changed(lambda new_val: slider_update(new_val, P, D))
The result of this code you can see here:
Eplot with sliders.
Electric Potential Energy V, calculated for 4 randomly placed charges with sliders, which allow to change location and charge of the first charge on this plot.
Original image: 589 x 509
Checkbutton block allow to set in ON and OFF position some variables. It is very convenient to keep they values as a boolean variable, to make switching very simple. Just a reminder – to change
boolean value to an apposite, use command not.
# Define location of the check box
ax_qfl = plt.axes([0.03, 0.17, 0.13, 0.13])
# Initiate check box
chxbox = CheckButtons(ax_qfl, P['plt_type'], P['plt_active'])
Fot checkbox initiation it is necessary to provide location of this checkbox, list of options and list of boolean states of these switches.
def qfl_select(label, P, D):
# change clicked checkbutton
index = P['plt_type'].index(label)
P['plt_active'][index] = not P['plt_active'][index]
# output
el_plot(P, D)
Function acted upon call of checkbox status change is a bit inconvenient. As a value of checkbox change is receive the name of the box changed. Therefore all names should be unique and also to work
out number of the box clicked, you need to use function index
Call of this function upon any changes is pretty standard for any Matplotlib Widgets:
chxbox.on_clicked(lambda new_select: qfl_select(new_select, P, D))
In this case again, lambda call was used to pass more arguments. The result will be similar to this one:
Eplot with sliders and checkbox.
Electric Potential Energy V, calculated for 4 randomly placed charges with sliders and checkbox. Checkbox allow to do not display charge names and electric field lines.
Original image: 584 x 484
Radiobutton allow to have a box of switches with only one switch on positive and all other in the negative state. The coding of radiobutton is pretty similar to those for checkbox. As a default
value, when changed, the function receive the name of activated button. If you have only two buttons, it is possible to do not use any labes and just change the status of this radiobutton to
ax_ev = plt.axes([0.85, 0.20, 0.10, 0.10]) # left, bottom, width, height values
rbutton1 = RadioButtons(ax_ev, P['field_type'])
def ev_select(label, P, D):
# swap clicked rbutton
P['VE'] = not P['VE']
# output
el_plot(P, D)
rbutton1.on_clicked(lambda new_select: ev_select(new_select, P, D))
In this case only two options are available and label - label of switched button is not used.
In the next part of the code, radiobutton will have few buttons equal to number of charges in the system. Therefore the size of the space will be proportional to number of charges
ax_qs = plt.axes([0.01, 0.40, 0.06, 0.03 * NQ]) # left, bottom, width, height values
rbutton2 = RadioButtons(ax_qs, range(1, NQ + 1), active=P['Qactive'])
It is also necessary to report active button, otherwise by default it will use first button in the list.
Question: Is it possible to place all buttons with horizontal alinement's?
Answer: No. Standard tools do not provide this option at the moment. If it is necessary to do this, you need to allocate every button separately within the same axes.
In the next response function I will change active change, and reset sliders position to the new value
def q_select(label, P, D):
P['Qactive'] = int(label) - 1
q = D['Q'][P['Qactive']]
s_x.eventson = False
s_y.eventson = False
s_q.eventson = False
s_x.eventson = True
s_y.eventson = True
s_q.eventson = True
rbutton2.on_clicked(lambda new_select: q_select(new_select, P, D))
In this q_select function reading of the clicked option is standard. But because labels are integer and equal to the charge number +1 it is very easy to work our real charge numbers.
The position of slider can easily be changed with method set_val applied to the correspondent slider. But when the position of slider changed, it is important to deactivate it, otherwise it will
cause changes in the system by calling slider on_change function.
To deactivate slider for some time, use changing of the eventson variable from True to False and back at the end of changing. But be careful eventson is not documented feature at he moment and can be
changed without further notice.
The result of all these changes will be similar to this one:
Eplot with widgets.
Electric Intensity Field E, calculated for 4 randomly placed charges with Sliders, Radiobuttons and Checkbuttons. These widgets allow to change the location and charge of all charges and change all
information displayed.
Original image: 593 x 495
Fill final code for Plotting of Interactive Electric field due to point charges with Widgets
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.widgets import Slider, CheckButtons, RadioButtons
def addPointCharge2D(V, Ex, Ey, X, Y, q):
r2 = ((X - q[0]) ** 2 + (Y - q[1]) ** 2) ** (3 / 2)
Ex += q[2] * (X - q[0]) / r2 # Electric field (x)
Ey += q[2] * (Y - q[1]) / r2 # Electric field (y)
V += q[2] / r2 ** (1 / 3.)
return V, Ex, Ey
def ElectricField(E, Ex, Ey, V, X, Y, Q):
Ex *= 0
Ey *= 0
V *= 0
for q in Q:
addPointCharge2D(V, Ex, Ey, X, Y, q)
E[:] = np.log((Ex ** 2 + Ey ** 2)) * 0.5
return E, Ex, Ey, V
xrange = [-1., 1.] # left and right coordinates of square domain (x==y)
step = 40 # Number of cells in the domain in any axis
qrange = [-10., 10.] # max and min values for charge
NQ = 4 # Number of charges
P = {} # plot description
P['plt_type'] = ['Position', 'Field', 'Lines'] # type of shown information
P['plt_active'] = [True, True, True] # Show or not switches
P['ColorBar'] = None # initial colorbar
P['field_type'] = ['V', 'E'] # available field for plotting
P['VE'] = False # show vector field
P['levelE'] = np.linspace(-1, 10, 100) # range for Efield colorbar
P['levelV'] = np.linspace(-150, 150, 100) # range for Vfield colorbar
P['Qactive'] = 0 # Current active charge
P['line_density_range'] = [0.1, 2.0] # Range of possible line densities
P['line_density'] = 0.4 # current line density
xlist = np.linspace(xrange[0], xrange[1], step)
ylist = np.linspace(xrange[0], xrange[1], step)
qmin = [xrange[0], xrange[0], qrange[0]]
qmax = [xrange[1], xrange[1], qrange[1]]
D = {} # domain description
D['X'], D['Y'] = np.meshgrid(xlist, ylist)
D['Ex'] = np.zeros_like(D['X'])
D['Ey'] = np.zeros_like(D['X'])
D['V'] = np.zeros_like(D['X'])
D['E'] = np.zeros_like(D['X'])
D['Q'] = np.random.uniform(low=qmin, high=qmax, size=(NQ, 3))
############ INITIAL PLOT
def el_plot(P, D):
if P['plt_active'][0]:
# Position
P['ax'].scatter(D['Q'][:, 0], D['Q'][:, 1], c=np.where(D['Q'][:, 2] < 0, 'b', 'r'))
for i, txt in enumerate(D['Q']):
P['ax'].annotate(f'({i + 1}):{txt[2]:.1f}', xy=(D['Q'][i, 0], D['Q'][i, 1]),
xytext=(D['Q'][i, 0] + 0.02, D['Q'][i, 1] + 0.02),
color='black', fontsize='12')
if P['plt_active'][1]:
# Field
if P['VE']:
P['ax'].set_title("Electric Field Intensity, E")
cp = P['ax'].contourf(D['X'], D['Y'], D['E'], levels=P['levelE'], cmap='jet')
P['ax'].set_title("Electric Potential energy, V")
cp = P['ax'].contourf(D['X'], D['Y'], D['V'], levels=P['levelV'], cmap='seismic')
if P['ColorBar']:
P['ColorBar'] = P['fig'].colorbar(cp, ax=P['ax'])
if P['plt_active'][2]:
# Lines
P['ax'].streamplot(D['X'], D['Y'], D['Ex'], D['Ey'], color='black', linewidth=0.5,
density=P['line_density'], arrowstyle='->', arrowsize=1.0)
return None
P['fig'], P['ax'] = plt.subplots(figsize=(6, 5))
plt.subplots_adjust(left=0.20, bottom=0.35)
plt.xlim(xrange[0], xrange[1])
plt.ylim(xrange[0], xrange[1])
el_plot(P, D)
####################### CONTROL
####################### SLIDERS- X, Y, Q, line density
ax_x = plt.axes([0.20, 0.25, 0.55, 0.05])
ax_y = plt.axes([0.07, 0.35, 0.05, 0.53])
ax_q = plt.axes([0.20, 0.20, 0.55, 0.05])
ax_d = plt.axes([0.20, 0.15, 0.55, 0.05])
s_x = Slider(ax_x, 'x', qmin[0], qmax[0], valinit=D['Q'][P['Qactive']][0])
s_x.vline._linewidth = 0
s_y = Slider(ax_y, 'y', qmin[1], qmax[1], valinit=D['Q'][P['Qactive']][1],
s_y.hline._linewidth = 0
s_q = Slider(ax_q, 'q', qmin[2], qmax[2], valinit=D['Q'][P['Qactive']][2])
s_q.vline._linewidth = 0
s_d = Slider(ax_d, 'l', P['line_density_range'][0], P['line_density_range'][1],
s_d.vline._linewidth = 0
def slider_update(val, P, D):
q = D['Q'][P['Qactive']]
q[0] = s_x.val
q[1] = s_y.val
q[2] = s_q.val
P['line_density'] = s_d.val
# recalculation
# output
el_plot(P, D)
s_x.on_changed(lambda new_val: slider_update(new_val, P, D))
s_y.on_changed(lambda new_val: slider_update(new_val, P, D))
s_q.on_changed(lambda new_val: slider_update(new_val, P, D))
s_d.on_changed(lambda new_val: slider_update(new_val, P, D))
####################### CheckButton - Q, Field, Lines selector
ax_qfl = plt.axes([0.03, 0.17, 0.13, 0.13])
chxbox = CheckButtons(ax_qfl, P['plt_type'], P['plt_active'])
def qfl_select(label, P, D):
# change clicked checkbutton
index = P['plt_type'].index(label)
P['plt_active'][index] = not P['plt_active'][index]
# output
el_plot(P, D)
chxbox.on_clicked(lambda new_select: qfl_select(new_select, P, D))
####################### RadioButton - E/V selector
ax_ev = plt.axes([0.85, 0.20, 0.10, 0.10]) # left, bottom, width, height values
rbutton1 = RadioButtons(ax_ev, P['field_type'])
def ev_select(label, P, D):
# swap clicked rbutton
P['VE'] = not P['VE']
# output
el_plot(P, D)
rbutton1.on_clicked(lambda new_select: ev_select(new_select, P, D))
####################### RadioButton - Q selector
ax_qs = plt.axes([0.01, 0.40, 0.06, 0.03 * NQ]) # left, bottom, width, height values
rbutton2 = RadioButtons(ax_qs, range(1, NQ + 1), active=P['Qactive'])
def q_select(label, P, D):
P['Qactive'] = int(label) - 1
q = D['Q'][P['Qactive']]
s_x.eventson = False
s_y.eventson = False
s_q.eventson = False
s_x.eventson = True
s_y.eventson = True
s_q.eventson = True
rbutton2.on_clicked(lambda new_select: q_select(new_select, P, D))
Published: 2022-09-25 19:48:52
Updated: 2022-09-25 19:51:07
Last 10 artitles
9 popular artitles | {"url":"https://mycoding.uk/a/plotting_of_interactive_electric_field_due_to_point_charges_with_matplotlib.html","timestamp":"2024-11-08T00:54:32Z","content_type":"text/html","content_length":"30078","record_id":"<urn:uuid:5ffdc0d2-7f39-46ca-b035-0d7d3f40ff92>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00433.warc.gz"} |
Breadth-first search algorithm
posted on • 2 min read | python, comsci
In Graph Theory 101, we introduced five concepts used to describe a graph: direction, weight, cycles, connectivity and density. In Modeling graphs using Python, we saw how to represent a graph using
adjacency lists and adjacency matrices.
It is now time to learn how to traverse a connected graph. In this post, we introduce the breadth-first search algorithm (often shortened to BFS). We can run a BFS on either an adjacency list or an
adjacency matrix.
What is breadth?
For the non-English speakers like me, let's look at a tree to better understand the notion of breadth.
Figure 1: breadth of a binary tree.
Remember that a tree is special type of graph, but breadth applies to all graphs.
Visual overview
Let's look at the below graph to understand how BFS works: from a given starting node, we list all the children. We then explore each child, and for each one of them, we list all its children. We
then move another level down, etc.
Figure 2: visual overview of the BFS algorithm.
In the above example, starting from A we list B and C. Moving to B, we list D and E. Moving to C, we list F and G. Moving to D, E, F and finally G, we realize they are leaf nodes so we don't list
anything. When the list is empty, the algorithm stops.
Figure 1 is acyclic, but BFS needs to be able to explore cyclic graphs without getting trapped into infinite loops. To do so, we maintain a list of visited nodes and only add to the "next nodes" list
unvisited ones.
In pseudo-code, the algorithm looks as follows:
function bfs(graph, node):
visited = []
We will use the following unweighted directed graph as an example. It is slightly more complex than the ones we usually see in similar tutorials but it allows for a better understanding of how the
algorithm works. As you can see
In a breadth-first search,
Python implementation
As discussed, we need to maintain two lists: a list of nodes that we already visited (in order to avoid infinite loops if the graph is cyclic) and a FIFO queue of nodes to visit.
For the list of visited nodes, we choose a set (since all elements will be unique) and for the FIFO queue of nodes to visit, we choose a deque, which is Python's queue data structure from the
standard library.
Example of a bfs function printing all nodes:
AdjacencyList = dict[set]
def bfs(graph: AdjacencyList, node: int) -> None:
visited = set()
queue = deque()
while queue:
node = queue.popleft()
if node not in visited:
print(f"visiting node {node}:", end=" ")
neighbors = graph[node]
for neighbor in neighbors:
if neighbor not in visited:
print(f"{neighbor}", end=" ")
What we do with BFS is very versatile: we can use BFS to explore all connected nodes, or to look for a specific node. We can use it to add edges etc. | {"url":"https://tlouarn.com/breadth-first-search-algorithm/","timestamp":"2024-11-13T01:06:48Z","content_type":"text/html","content_length":"11543","record_id":"<urn:uuid:1e746326-e8a0-431c-9165-7f37c07225ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00262.warc.gz"} |
sam@gentoo.org Sam James LibTomMath is a free open source portable number theoretic multiple-precision integer library written entirely in C. (phew!). The library is designed to provide a simple to
work with API that provides fairly efficient routines that build out of the box without configuration. The library builds out of the box with GCC 2.95 [and up] as well as Visual C++ v6.00 [with SP5]
without configuration. The source code is arranged to make it easy to dive into a particular area very quickly. The code is also littered with comments [This is one of the on going goals] that help
explain the algorithms and their implementations. Ideally the code will serve as an educational tool in the future for CS students studying number theory. The library provides a vast array of highly
optimized routines from various branches of number theory. * Simple Algebraic o Addition o Subtraction o Multiplication o Squaring o Division * Digit Manipulation o Shift left/right whole digits
(mult by 2b by moving digits) o Fast multiplication/division by 2 and 2k for k>1 o Binary AND, OR and XOR gates * Modular Reductions o Barrett Reduction (fast for any p) o Montgomery Reduction
(faster for any odd p) o DR Reduction (faster for any restricted p see manual) o 2k Reduction (fast reduction modulo 2p - k) o The exptmod logic can use any of the four reduction algorithms when
appropriate with a single function call. * Number Theoretic o Greatest Common Divisor o Least Common Multiple o Jacobi Symbol Computation (falls back to Legendre for prime moduli) o Multiplicative
Inverse o Extended Euclidean Algorithm o Modular Exponentiation o Fermat and Miller-Rabin Primality Tests, utility function such as is_prime and next_prime * Miscellaneous o Root finding over Z o
Pseudo-random integers o Signed and Unsigned comparisons * Optimizations o Fast Comba based Multiplier, Squaring and Montgomery routines. o Montgomery, Diminished Radix and Barrett based modular
exponentiation. o Karatsuba and Toom-Cook multiplication algorithms. o Many pointer aliasing optimiztions throughout the entire library. libtom/libtommath | {"url":"https://ftp.rrzn.uni-hannover.de/gentoo-portage/dev-libs/libtommath/metadata.xml","timestamp":"2024-11-11T01:19:29Z","content_type":"application/xml","content_length":"3529","record_id":"<urn:uuid:5d4a7e0f-eb82-4406-aa7d-f1a7f7a94032>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00335.warc.gz"} |
Calculus III
This is the curriculum for a asynhronous Calculus III course implemented for an eight-week semester and based on courses, which the author taught in 2016-2021 at Middlesex Community College and
MassBay Community College.
Summary of Posting
This is the curriculum for an asynchronous Calculus III course implemented for an eight-week semester and based on the courses, which the author taught in 2016-2021 at Middlesex Community College and
MassBay Community College.
This posting includes the syllabus, course schedule, instructions, worksheets, study guides, assignments, rubrics, and other materials.
The following sources have been used in this posting:
1.Ya. B. Zeldovich, A. D. Myskis, Elements Of Applied Mathematics, Mir,1976
2. Gilbert Strang, Calculus, Wellesley-Cambridge Press, 2nd ed., 1991
3. Calculus 3 by OpenStax, Senior Contributing authors: Gilbert Strang, Edwin Jed Herman, 2020
4. Denis Auroux, Multivariable Calculus, MIT, 2007.
Middlesex Community College
MAT 292-30: Calculus III for Science 4 Credit Hours Summer 2021
Instructor Dr. Igor Baryakhtar
Office hours: via ZOOM
e-mail: baryakhtari@middlesex.mass.edu
Course Goals
This asynchronous course is designed to give students a basic knowledge of multivariable calculus, to develop students’ critical thinking skills, quantitative and symbolic reasoning skills, and to
improve their mathematical literacy. The course is focused on students’ ability to solve scientific and engineering problems using multivariable calculus concepts. Students will achieve these goals
by studying the textbook, attending online video lectures, and doing assignments using traditional and electronic technologies.
Course Description
Topics include vector-valued functions, dot and cross products, motion, curvature and arc length in 3-space, partial derivatives and Chain Rule, directional derivatives and gradients, max/min and
Lagrange Multipliers. Also: double and triple integrals, polar coordinates, and parametric surfaces, and Green's Theorem with applications in work and potential energy in the study of electricity and
This is the third course in the Calculus sequence. Students will study the fundamental concepts of differential calculus. The topics are divided into four units:
1. Introduction. Cartesian, Cylindrical and Spherical Coordinates; Conic Sections.
(parametric equations, polar coordinates, converting between cartesian and polar coordinates, converting between cartesian and cylindrical and spherical coordinates, calculus in polar coordinates,
conic sections)
2. Vectors and Vector Values Functions
(vectors in three dimensions, dot and cross product, curves in space, calculus of vector valued function, arc length, curvature and normal vector)
3. Functions of Several Variables
(surfaces, functions of two and more than two variable, visualization of functions of two variables, limit and continuity of a function of two variable, partial derivatives, the chain rule,
directional derivatives and gradient, tangent planes, linear approximation, max min problems, Lagrange multipliers)
4. Multiple Integration & Vector Calculus
(double integrals in Cartesian and polar coordinates, triple integrals in Cartesian and cylindrical and spherical coordinates, vector fields, conservative vector fields, Green’s theorem, Stoke’s
theorem, Divergence theorem)
Prerequisite MAT 291 Calculus II
Technical Requirements
To succeed in this online course you must be familiar with electronic technologies.
Ability to use the Internet in an effective and efficient manner, including: installation and management of browser plug-ins and add-ons, download, upload and print files, send/reply emails with
Basic knowledge about the operation of a computer, file management, and software installation.
Learning management systems
Calculus III course at Middlesex Community college will use the following electronic learning management systems.
Blackboard (main platform): for announcements, discussion boards, lectures notes and other learning materials, test, grades, and information about MCC Learning Resources and Support Services.
ZOOM: office hours, Q&A sessions (upon request), proctored exams
MyOpenMath: for online homework assignments and quizzes
Mathematical software will be used to demonstrate calculus concepts and to visualize calculations.
MATLAB (optional), MAXIMA CAS (optional).
Free Open Educational Resources are required for this course
Calculus Volume 3
Senior Contributing Authors:
Gilbert Strang, Massachusetts Institute of Technology
Edwin “Jed” Herman, University of Wisconsin-Stevens Point
Publish Date Mar 30, 2016
Print ISBN-10: 1-938168-07-0 Digital ISBN-10: 1-947172-16-6
ISBN-13: 978-1-938168-07-9 ISBN-13: 978-1-947172-16-6
Additional textbook (optional)
Calculus for Scientists & Engineers. Multivariable.
by Briggs, Cochran, & Gillett, with assistance of Eric Schulz.
2013 Ed., Pearson Education, Inc., ISBN-13: 978-0-321-78551-0
By the end of the course students should be able to:
Answer conceptual questions about calculus of vector-valued functions, calculus of a function of several variables, calculus of vector fields.
Demonstrate basic knowledge of equations of curves and surfaces in 3D space, properties of dot and cross products of vectors, limit and continuity of a function of two or more variables, chain rule
with several independent variables, implicit differentiation rule with three variables,
directional derivatives, maximum/minimum problems, Lagrange multipliers, double integrals in cartesian and polar coordinates, triple integrals in cartesian, cylindrical and spherical coordinates,
Green’s theorem, Stoke’s theorem, Divergence theorem.
Solve problems involving polar, cylindrical and spherical coordinates, 2D and 3D motion problems, find equation of a plane through the given points or for given vectors, sketch level curves and
traces of surfaces, evaluate dot and cross products of vectors, compute arc length, curvature and torsion of a curve, tangential and normal components of an acceleration, calculate derivatives of a
function of two or more variables using chain rule and implicit differentiation, calculate directional derivatives and gradients, solve maxima/minima problems, solve maxima/minima problems with a
constraint using Lagrange multipliers method, calculate double integrals using cartesian and polar coordinates, calculate triple integrals using cartesian, cylindrical and spherical coordinates,
solve word problems using multivariable calculus.
Credit Hour Policy
Students are expected to spend a minimum of 45 hours of work for each credit.
Course Grades
Participation 10%
Homework (on MyOpenMath) 20%
Quizzes (on MyOpenMath) 20%
Project 10%
One Test (remotely proctored test) 20%
Final Exam (remotely proctored exam) 20%
Class format
Class is a combination of different elearning activities:
- reading assignments with real-world examples
- video watching assignments
- online homework assignments on MyOpenMath
- online quizzes on MyOpenMath
• Online Discussions of selected topics on the Blackboard discussion board and wikis
• Synchronous online Q&A sessions via ZOOM (upon request)
• Individual work.
Attending an online course includes but is not limited to
-Submission of an academic assignment by a student
-Taking the online quiz by a student
-Student submission of an exam
-Student's posting to a discussion forum
-An email from a student showing that the student has initiated contact with the instructor
Attendance is mandatory in this course. Stop attending a course does not constitute a withdrawal. If you can no longer participate in this class, you must formally withdraw because unfinished
coursework may result in a failing "F" grade.
Students are expected to submit work weekly and complete all assignments on time. Students who miss two or more weeks of classes may be withdrawn from the course.
Attendance and participation
10% of the Grade
Students are expected to participate in all scheduled assignments on a daily basis.
Discussion Board
Students will be asked to reflect and respond to Discussion Board questions and post your responses. Responses should be clear, accurate and complete sentences.
Online homework
Reading a textbook is a very important part of the learning process. First, read the assigned section. Make sure that all notations are understood. Use lecture notes and recommended multimedia
resources to clarify concepts. Try examples in the textbook. Do optional problems from the textbook.
Instructor will assign online homework and/or handwritten assignments every week.
20% of the Grade.
Two late online home works accepted.
One late paper and pencil homework accepted.
There will be six online quizzes on MyOpenMath.
20% of the Grade
One make up quiz for a missed quiz will be allowed. Lowest quiz grade is dropped.
The purpose of an individual project is to boost the deeper understanding of calculus. Students may work on the Project with their classmates and receive help from Math Center or use any other
recourses, but every student must submit his/her individual work
10% of the Grade
Late submission. 10% of the grade is deducted per day after the assignment's due date.
Test will be remotely proctored and handwritten on paper. It will be posted on Blackboard.
20% of the Grade
No make up for the missed Test will be provided.
Final Exam
The Final Exam will be remotely proctored and handwritten on paper. It will be posted on Blackboard.
The Final Exam will require the student to demonstrate mastery of the techniques of differentiation and integration and their uses in real-world applications. Students should review all quizzes,
practice problems, test & handouts.
Final Exam: 20% of the Grade
No make up for the missed Final Exam will be provided.
Every student must follow the Middlesex Community College Honor Code
Academic Integrity Policy
Middlesex Community College does not tolerate academic dishonesty. As outlined in more detail in Middlesex Community College Code of Conduct, academic dishonesty can include, but is not limited to
the following
Use of any unauthorized assistance in taking quizzes, tests, or examinations;
Dependence upon the aid of sources beyond those authorized by the instructor in writing papers, preparing reports, solving problems, or carrying out other assignments;
The acquisition, without permission, of tests or other academic material belonging to a member of the College faculty or staff; or
Plagiarism, which is defined as the use, by paraphrase or direct quotation, of the published or unpublished work of another person without full and clear acknowledgment. It also includes the
unacknowledged use of materials prepared by another person or agency engaged in the selling of term papers or other academic materials. Taking credit for work done by another person or doing work for
which another person will receive credit. Copying or purchasing other’s work or arranging for others to do work under a false name.
MyOpenMath is a free online educational platform.
MyOpenMath provides
-a set of overview videos
-online homework assignments, most with videos
-online quizzes
Students should have convenient and reliable access to a personal computer and internet.
Sign Up in MyOpenMath https://www.myopenmath.com
The course ID: xxxxx
The enrollment key: xxxxxxxx
Free Support Services
Students are encouraged to use the tutoring service - Math Center
Disability Support Services
The Disability Support Services offices are offering remote services at this time
Personal Counseling is available
Inform Your Instructor of Any Accommodations Needed
This work is licensed under a Creative Common Attribution 4.0 International license
Course Schedule
Online weekly quizzes are scheduled on __ at __. You will have __ hours to complete.
The Test and the Final Exam are handwritten on paper, you will have __ hours to complete.
week MyOpenMath Textbook
1.1 Parametric Equations
Calculus I Review Calculus II Review
1 1.2 Calculus of Parametric curves (optional)
Homework #1. Parametric curves. Polar Coordinates
1.3 Polar Coordinates
WELCOME QUIZ
2 Homework #2. Vectors 2.1-2.4 Vectors
2 QUIZ # 1
Homework #3. Straight Line in 3D. Planes and Surfaces 2.5 Lines and Planes in Space
3 Homework #3. Straight Line in 3D. Planes and Surfaces 2.6 Quadric Surfaces
Homework #4.Spherical and Cylindrical Coord. 2.7 Cylindrical and Spherical Coordinates
3 QUIZ # 2
3.1 Vector-Valued Functions and Space Curves
Homework #5. Calculus of Vector-Valued Functions 3.2 Calculus of Vector-Valued Functions
Homework #6. Arc Length. Curvature and Normal Vectors 3.3 Arclength and Curvature
3.4 Motion in Space
4 QUIZ # 3
4.1 Functions of Several Variables
Homework #7. Limit of a Function of Two Variables
4.2 Limits and Continuity
Homework #8. Partial Derivatives
5 4.3 Partial Derivatives
Homework #9. Tangent Planes and Linear Approx.
4.4 Tangent Planes and Linear Approximation
Homework #10. PartialDerivatives. Chain Rule
4.5 Chain Rule
5 TEST:
week MyOpenMath Textbook
4.6 Directional Derivatives and Gradient
Homework #11. Partial Derivatives. Directional Derivatives and Gradient
6 4.7 Maxima/Minima Problems
Homework #12. Maxima/Minima Problem
4.8 Lagrange Multipliers
6 Quiz #4
5.1 Double integrals over rectangular regions
Homework #13. Integrals. Part 1
5.2 Double Integrals over general regions
Homework #14. Integrals. Part 2
7 5.3 Double Integrals in Polar Coord.
Homework #15. Integrals. Part 3
5.4 Triple Integrals
Homework #16. Integrals. Part 4
5.5 Triple Integrals in Cylindrical and Spherical Coordinates
7 Quiz #5
7 Project due
5.7 Change of Variables in Multiple Integrals
Homework #17. Vector Fields
8 6. Vector Calculus
(Extra credit)
FINAL REVIEW
FINAL EXAM
Welcome to the Middlesex Community College online course!
• In this unit you will learn how to navigate in the course shell.
• What do you need to succeed in Calculus III.
• Become familiar with MyOpenMath - free online learning management system.
• Become familiar with the discussion board and post you first thread.
• Become familiar with netiquette in online education
• Obtain help
The course menu is the panel on the left side of the interface that contains links to all course areas.
Toggle buttons
Announcements The course announcements your instructor have posted.
Getting Started Welcome message
Orientation, and Getting Help Contact the Instructor How to contact your instructor.
Syllabus Syllabus of the course and tentative schedule
Course Textbook Link to the course textbook
WEEKLY CONTENT The folder for weekly moduli: for reading assignments, handouts, lectures notes, weblink to mini-lectures, information about online assignments and other materials for the week A.
MyOpenMath Link to the MyOpenMath website. Online homework assignment will be posted on this website.
Discussion Board Discussion Board. You will use the discussion board to explore interesting questions with your classmates.
Maxima Online Link to the wxmaxima webpage - free and convenient online mathematical tool based on MAXIMA CAS. May be used for symbolic calculations and for graphing.
Tools Blackboard’s Tools
Netiquette Guide
It is important to understand that the online class is actually a class, and certain behavior is expected when communicating with your peers and the instructor.
• Be polite and respectful, honesty and integrity are expected from all
• Be professional, follow the rules, including how and when submit your work: format and due date
• Make sure identification is clear in all communications, include your first and last name and the course number
• Be careful with humor and sarcasm, be aware of strong language - use proper language, grammar, and spelling
MyOpenMath Orientation
All students enrolled in courses using MyOpenMath are required to complete a one-time online orientation to MyOpenMath, free Learning Management System.
This small self-paced orientation is available on MyOpenMath should be completed during first two days of classes. On average the orientation should take approximately 30 min.
How to enroll into MyOpenMath
MyOpenMath is a free online learning management system.
To register for CALCULUS III MAT 292-31
1. Go to www.myopenmath.com
2. Under Login, select Register as a new student
3. Complete the required fields
4. Enter your instructor’s
Course ID: XXXXXX
Enrollment Key: xxxxxxxx
5. Click Sign Up
You can now return to the login page and login with your new username and password.
Once you log in you will see in the center of a webpage the folder “ORIENTATION”.
Inside the folder you will find
Intro to MyOpenMath, an assignment how to enter formulas in MyOpenMath
Course Home Page video
Course Content video
MAT 292-30 Calculus III. HANDWRITTEN EXAM RUBRIC
GRADE EXCELLENT GOOD FAIR POOR FAILURE
Understanding Student knows the concept and can use Student knows the concept and can Student knows the concept but does not Student has some knowledge about the Student does not
of Concept it to solve challenging problems use it to solve basic problems know how to use it properly. concept but does not know how to use it. understand the concept
Calculation All calculations are correct Student made minor mistakes in Student made big mistakes in Student made many big mistakes in Student cannot perform
skills calculations calculations calculations necessary calculations
MAT 292-30 Calculus III. Discussion Board RUBRIC
GRADE EXCELLENT GOOD FAIR POOR FAILURE
Posting related to the topic,
Postings on Discussion Board will be respectful to other postings. Posting related to the topic, Posting does not related to the topic Posting is too short, like “Agree/ No
graded based upon the following respectful to other postings. or posting is too obvious. Disagree” or “Great point”. post.
Post helps others to understand
Study Guide #1. Parametric Equations
Study Guide #2. Vectors
Study Guide #3. Equations of Lines and Planes in Space
Equations of Lines and Planes in Space
Study Guide #4. Calculus of Vector-Valued Functions
Calculus of Vector-Valued Functions
Study guide #5. Arc Length. Curvature. Normal and tangential components of acceleration
Arc Length. Curvature. Normal and tangential components of acceleration.
Study guide #6. Partial derivatives
Study Guide #7. Gradient. Directional derivative. Extrema
Gradient. Directional derivative. Extrema
Study Guide #8. Double Integrals
Study Guide #9. Triple Integrals | {"url":"https://oercommons.org/courseware/lesson/82987/overview","timestamp":"2024-11-12T16:23:21Z","content_type":"text/html","content_length":"95394","record_id":"<urn:uuid:4638831e-a33c-46d3-ae5d-d1809d8e9d6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00039.warc.gz"} |
An Interactive Introduction To Quantum Computing
Or What Do You Mean They Can Be Both Zero And One At The Same Time!
By David Kemp
This article was originally written in 2014, but has had some minor improvements in December 2017 and January 2018 when I received some useful feedback after the article had unexpectedly been posted
on Hacker News in December 2017.
Heard of quantum computers?
Heard that they are faster than conventional computers?
Perhaps you have heard of quantum bits (abbreviated to qubits).
Maybe you have even heard of the puzzling notion that qubits can have the values 0 and 1 both at the same time.
Let me try to explain what this really means.
This is part one of a two part series for those that want to learn a little about quantum computing, but lack the mathematics and quantum physics background required by many of the introductions out
there. It covers some of the basics of quantum computing, such as qubits, state phases, and quantum interference. Part 2 goes on to look at quantum search.
I assume you know what plain old ordinary binary bits are. Sorry, But I cannot assume you know nothing at all!
Conventional bits are implemented using many different approaches: e.g. voltages on a wire, pulses of light on a glass fibre, etc. etc.
Just like bits, qubits have a binary state.
Qubits represent 0 and 1 using quantum phenomenon like the nuclear spin direction of individual atoms.
E.g. use “clockwise” for 0 and “anti clockwise” for 1.
The NOT operator
Consider the conventional NOT (or bit-flip) operator. 0 and 1 can represent logical true and false. NOT true is false, and NOT false is true. And so, NOT of 1 is 0, and NOT of 0 is 1.
For example, performing a NOT operation on the right most bit of the binary number 111 flips the target bit and results in 110.
In what follows, it will be convenient to represent the state of a system by listing all possible states and placing a blue disk next to the current state.
Click the button labelled “Not bit[a]” to apply the NOT operation to the left bit, and click the button labelled “Not bit[b]” to apply the NOT operation to the right bit:
There is nothing quantum mechanical about these first few interactive examples. Their main purpose is to familiarise you with interactive animations I use in this article.
Random NOT
Random NOT: A NOT operator that has a specified chance of flipping a bit.
Although not very common, the “Random NOT” is still just a classical (non-quantum) operator, but it will help me explain the workings of some quantum operators.
Consider applying a Random NOT twice to a bit whose initial value is 0, where the operator has, for instance, a 30% chance of flipping the bit. What is the probability of the final state being 0?
There are a couple of possible scenarios. For instance, the first Random NOT might flip the bit from 0 to 1, and the second Random NOT might flip the bit back to 0. We represent this as:
0 → 1 → 0
There are two paths leading to a final state of 0:
• 0 → 0 → 0 with probability of 0.7 x 0.7 = 0.49
• 0 → 1 → 0 with probability of 0.3 x 0.3 = 0.09
And so the final state will be 0 with a probability of 49% + 9% = 58%
Random NOT (your turn)
Next we provide an interactive animation of the Random NOT operator.
The blue disk now splits in two so that we can track the different possible outcomes.
The probability of being in a state is represented by the radius of the disk.
Press the “Random NOT” button multiple times and note how the arrows add head to tail.
Still nothing quantum mechanical about any of this. We are still just warming up.
We have seen how a random NOT operator can cause a conventional computer to have various probabilities of being in different states. Of course in reality it is in only one of those states. We just
don't know which one. Strangely, this is an assumption about reality that we will need to reconsider when we look at qubits.
If you peek at the system to determine its actual state, then the probabilities all collapse so that one state (the observed state) is deemed to now have a probability of 1, and all the others are
deemed to have a probability of 0.
Remember, the larger the blue disk, the more likely the system will turn out to be in that state.
In quantum computing, the word measurement refers to this act of peeking.
Press the “Random NOT” button multiple times and then press the “measurement” button.
Note that there is still nothing quantum mechanical about this yet. That comes next!
Hadamard of 0
The “Hadamard operator” is a special quantum operator that can be applied to qubits.
Warning: this first look at quantum operators will be pretty boring. I promise it will get interesting soon!
As you will see below, the Hadamard initially acts like a Random NOT with 50% chance of success.
In this interactive example, I purposely disable the Hadamard button after you press it. Later in this article we will see what happens when you apply the Hadamard twice in a row.
Nothing unusual about that was there?
But you will be surprised by what comes next...
Hadamard of 1
Things start to become weird when you look at the Hadamard of 1.
Look carefully at the arrow directions.
Puzzled? You should be if this is all new to you. Please hang in there for a while longer.
The arrow directions represent what physicists call phase:
• it is an abstract concept of quantum mechanics.
• it has no “common sense” interpretation.
• it can only be measured indirectly.
In the case of nuclear spin, phases can be manipulated by applying electric and/or magnetic fields.
We will see the importance of phase in a moment, but first let's look at another interesting quantum computing operator...
T Operator
The T operator rotates the phase of 1, but leaves 0 untouched.
Note how it does not affect the probabilities at all.
Measurement Revisited
• Measurement causes the system to collapse to the observed state.
• The larger the blue disk, the more likely the system will collapse to that state.
• Once the system has collapsed to a particular state, it will remain in that state until another operation is performed.
Important: The likelihood of a state being observed is entirely determined by the size of the blue disk, and is completely unaffected by the direction of the arrow.
Quantum Interference
Consider what happens when we apply a Hadamard operation twice in a row.
Let's assume that a qubit is initially known to definitely have the value 0. If you were to apply the Hadamard to it twice in a row, then there are four equally likely scenarios (Recall that “x → y →
z” means “the qubit starts with a value x, the first Hadamard results in the qubit having the value y, and the second Hadamard results in the qubit having the value z”):
• 0 → 0 → 0
• 0 → 0 → 1
• 0 → 1 → 0
• 0 → 1 → 1
So the final value should be equally likely to be 0 or 1 but, in reality, applying the Hadamard operator twice in a row always returns the qubit to its original value. In our case, where the qubit is
initially 0, two applications of the Hadamard will result in it being 0 again.
Try it out. Press the “Apply Hadamard” button twice and watch it return to having a 100% likelihood of having the value 0.
Totally confused?
If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.
Niels Bohr
What is going on here?
The state of the qubit after the first Hadamard seems to have a 50% chance of being 0 and a 50% chance of being 1.
The second Hadamard is applied to both the 0 and 1 states and the results are combined.
The arrows still add head to tail.
The two different scenarios ending in a 1 state have opposite phases and so they cancel each other out.
This process of phases causing possible outcomes to cancel or re-enforce is what physicists call interference.
This is what philosophers of physics loose sleep over.
By the way, the mathematically inclined may be worried about all the probabilities not adding up to 1 any more. The trick is that the arrow lengths now have to represent the square roots of the
probabilities. We will briefly cover this in more detail in the section entitled Some mathematics in Part 2.
Hadamard of 1 (revisited)
It is instructive to observe the effects of applying a Hadamard twice in a row when the initial value is 1. This time, the qubit returns to 1:
Different kinds of uncertainty
We are actually dealing with two different kinds of uncertainty:
• It is possible that a bit, and even a qubit, may be in a fixed state of 0 or 1, but that you simply do not know which one it is.
• However, it is also possible for a qubit to be in what is called a “superposition” of both 0 and 1. Such a qubit is in a strange combination of both 0 and 1.
Small Diversion: Superposition of Locations
So far, the rather abstract phenomenon of nuclear spin is the only approach that I have mentioned for creating qubits.
Quantum physics seems even more bizarre when you discover that physical objects can be in superpositions of different locations.
The photons travelling through an “interferometer” are in superpositions of locations that can be kilometres apart (as they are in the LIGO interferometer).
A simple interferometer is shown below. Photons are emitted by a light source (e.g. a laser) that is pointing at a “half silvered mirror”, which reflects some of the light and lets some of the light
Individual photons end up in a superposition of having been reflected and having been let through. A couple more mirrors are used to bring the split light beam back together at a detector. The
positions of the mirrors and the detector all effect the lengths of the two different paths, so that one path can be longer than the other. Like the T operator described earlier, a change in the
relative path lengths will alter the relative phases of the two photon states. A difference equal to the wavelength of light is enough to change the relative phases by an entire 360 degrees. If the
phases are exactly opposite, then they will cancel each other out, and the detector will not detect anything. The resulting effect will be an alternating series of light and dark concentric rings
like those shown below.
This interference effect even happens when the light source is slowly emitting photons one at a time.
It is tempting to think that the half silvered mirror is splitting each photon in two and that the interference effects are caused by the two photons interacting with each other. But this is not what
If detectors are placed on the two paths, and the light source is slowly emitting photons one at a time, then the detectors only ever detect a photon on one path or the other. They never detect two
photons at once! (Well, they very occasionally do due to the light source very occasionally emitting two at once, but the frequency that this should happen is easily predicted and verified.)
If detectors are placed on either or both of the two paths, then the act of detecting the presence (or absence) of the photon causes the superposition to collapse to one or the other, and the
interference effects disappear, even if the detector lets the photon continue on.
It gets even more interesting when you have more than one qubit
The quantum weirdness rises to a whole new level when there are two or more qubits interacting. This is explored in Part 2.
If you want to experiment with various single qubit quantum operations first, then have a play with the Quantum Computer Gate Playground
Michelson Interferometer: http://commons.wikimedia.org/wiki/File:Michaelson_with_letters.jpg
Interference Pattern: http://commons.wikimedia.org/wiki/File:Zonenplatte_Cosinus.png | {"url":"http://davidbkemp.github.io/QuantumComputingArticle/","timestamp":"2024-11-05T03:18:52Z","content_type":"text/html","content_length":"24468","record_id":"<urn:uuid:848c89f4-016f-4cac-af2b-7870a1f75c31>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00014.warc.gz"} |
Xiii) Project: Gravel; Xv) Find The Cost - Black & Decker BDCAL100 Instruction Manual [Page 19]
xiii. Project: Gravel
Find how many tons of gravel is needed when the cover-
age volume is given, or find the coverage area (in cu. yds.)
when the quantity of gravel used is given.
Example: Find the quantity of gravel ( in tons ) needed to
cover a 48'x13' driveway 5" deep.
Example: How many cubic yards will 5 tons of gravel fill ?
xiv. Project : Mulch
Find how many bags of mulch is needed when the cover
age volume is given, or find the Coverage volume when
the quantity of mulch used ( in number of bags ) is given.
Example: Find the number of bags of mulch you'll need
to fill a volume of 3.5' X 13' X 3" deep.
If you want to find how much you'll spend buying mulch,
you can operate as follows.
[X]50[Shift][Cost $]
Example: How many cubic feet will 5 bags of mulch fill ?
xv) Find the cost :
To find the cost for the material quantity calculated above,
one can follow the key procedures as:
" [X] ( enter the unit cost ) [Shift][Cost $] "
provided that the quantity of material has been worked out
and is being displayed, which can be in the unit of bags,
tons, rolls and etc. | {"url":"https://www.manualslib.com/manual/880522/Black-And-Decker-Bdcal100.html?page=19","timestamp":"2024-11-13T16:22:48Z","content_type":"text/html","content_length":"235130","record_id":"<urn:uuid:1be20a83-307f-437b-b96f-c353617831c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00346.warc.gz"} |
XGBoost for Multiple-Output Regression Manually
When faced with a multiple output regression problem (multi-out regression), where the goal is to predict several continuous target variables simultaneously, one approach is to train a separate
XGBoost model for each target variable.
While XGBoost does have modest natively support multiple output regression, this manual approach allows for greater flexibility compared to using a wrapper like MultiOutputRegressor from
scikit-learn, albeit at the cost of writing more code.
This example demonstrates how to manually train multiple XGBoost models, one for each target variable, to solve a multiple output regression task.
We’ll generate a synthetic dataset, prepare the data, initialize and train the models, make predictions, and evaluate the overall performance.
# XGBoosting.com
# Manually train separate XGBoost models for each target in multiple output regression
from xgboost import XGBRegressor
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a synthetic multi-output regression dataset
X, y = make_regression(n_samples=1000,
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
# Initialize a list to store the trained models
models = []
# Loop through each target variable
for i in range(y_train.shape[1]):
# Select the current target variable
y_train_i = y_train[:, i]
# Initialize an XGBRegressor for the current target
model = XGBRegressor(n_estimators=100, learning_rate=0.1, random_state=42)
# Fit the XGBRegressor on the training data for the current target
model.fit(X_train, y_train_i)
# Append the trained model to the list of models
# Make predictions by predicting each target separately using the corresponding model
y_pred = np.column_stack([model.predict(X_test) for model in models])
# Evaluate the overall performance using mean squared error
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.4f}")
Here’s a step-by-step breakdown:
1. Generate a synthetic multi-output regression dataset with 10 input features and 3 output targets.
2. Split the data into training and testing sets using train_test_split.
3. Initialize an empty list called models to store the trained XGBoost models.
4. Loop through each target variable:
□ Select the current target variable from the training data.
□ Initialize an XGBRegressor with chosen hyperparameters for the current target.
□ Fit the XGBRegressor on the training data for the current target using fit().
□ Append the trained model to the models list.
5. Make predictions on the test set by predicting each target separately using the corresponding model and combining the results into a single array using np.column_stack().
6. Evaluate the overall performance using Mean Squared Error (MSE).
By manually training separate XGBoost models for each target variable, you have full control over the training process and can potentially achieve better performance than using a generic wrapper.
However, this approach requires more code and may not be as convenient as using a pre-built solution like MultiOutputRegressor.
This example provides a foundation for training XGBoost models for multiple output regression tasks manually. Depending on your specific dataset and requirements, you may need to preprocess the data,
tune hyperparameters, or use different evaluation metrics to optimize performance. | {"url":"https://xgboosting.com/xgboost-for-multiple-output-regression-manually/","timestamp":"2024-11-02T18:34:43Z","content_type":"text/html","content_length":"11999","record_id":"<urn:uuid:2a608090-e6c5-433d-9a91-03fb50caede3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00110.warc.gz"} |
This page uses Gosper's series identity for π to calculate an unbounded stream of digits.
I haven't really looked into the derivation of the algorithm; I just translated the code in Jeremy Gibbons' paper Unbounded Spigot Algorithms for the Digits of Pi. (That paper has a typo in what I
call \(y\), by the way - it has \(27i+15\) instead of \(27i-12\))
The state consists of two pieces of information, whose initial values are as follows:
\begin{align*} \mathrm{M} &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ i &= 1 \end{align*}
At each iteration of the algorithm, the following values are computed:
\begin{align*} \begin{pmatrix}y_1 \\ y_2\end{pmatrix} &= \mathrm{M} \times \begin{pmatrix}27i-12 \\ 5\end{pmatrix} \\ y &= \left\lfloor y_1/y_2 \right\rfloor \\ \\ \begin{pmatrix} z_1 \\ z_2\end
{pmatrix} &= \mathrm{M} \times \begin{pmatrix}675i-216 \\ 125\end{pmatrix} \\ z &= \left\lfloor z_1/z_2 \right\rfloor \end{align*}
If \(y=z\), then \(y\) is output as the next digit of \(\pi\), and the matrix \(\mathrm{M}\) becomes
\[ \mathrm{M'} = \begin{pmatrix} 10 & -10y \\ 0 & 1\end{pmatrix} \times \mathrm{M}\]
If \(y \neq z\), then nothing is output and the state instead becomes
\[ \mathrm{M'} = \mathrm{M} \times \begin{pmatrix} i(2i-1) & j(5i-2) \\ 0 & j\end{pmatrix} \]
where \(j = 3(3i+1)(3i+2)\), and \(i\) is increased by one.
The process is then repeated, with the new values of \(\mathrm{M'}\) and \(i\).
The numbers in the matrix \(\mathrm{M}\) get very big very quickly, so I had to use Matthew Crumley's BigInteger library for this javascript implementation.
• I can never remember π past 3.14159..., and typing "digits of pi" into Google doesn't always lead you straight to a usable listing, so I wanted a place I could go to to easily get at least the
first few hundred digits.
• Ever since I discovered it, I've wanted to write an implementation of an unbounded spigot algorithm for π.
• I enjoy buying novelty domain names.
• It was a fun thing to do for π day.
tl;dr I'm a massive nerd.
My name is Christian Lawson-Perfect. Yes it is.
Bonus technical things
• Only the letter π and the digits are selectable, so you can copy-and-paste digits without getting the historical facts and digit counters mixed in.
• three.onefouronefivenine.com is five decimal places of precision, but you can give more after the slash - for example, three.onefouronefivenine.com/twosixfivethreefive also works! Note that
three.onefouronefivenine.com/twosixfivethreesix does not. | {"url":"https://three.onefouronefivenine.com/how.html","timestamp":"2024-11-02T14:19:32Z","content_type":"text/html","content_length":"3911","record_id":"<urn:uuid:fd3e2c8b-90f6-4120-a410-78ed6085aa36>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00223.warc.gz"} |
Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic
Home ➟ Multiply Radical Expressions Worksheet ➟ Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic
Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic one of Worksheet for Education - ideas, to explore this Multiply Radical Expressions Worksheet
Fundamental Algebra 2 Homework 2014 Bethlehem Catholic idea you can browse by and . We hope your happy with this Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem
Catholic idea. You can download and please share this Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic ideas to your friends and family via your social
media account. Back to Multiply Radical Expressions Worksheet
Gallery of Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic
Related Posts for Multiply Radical Expressions Worksheet Fundamental Algebra 2 Homework 2014 Bethlehem Catholic | {"url":"https://ventureitch.com/multiply-radical-expressions-worksheet/multiply-radical-expressions-worksheet-fundamental-algebra-2-homework-2014-bethlehem-catholic/","timestamp":"2024-11-09T19:32:55Z","content_type":"text/html","content_length":"44618","record_id":"<urn:uuid:f0016af2-b136-4d7f-8a46-95da3e9a0202>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00771.warc.gz"} |
Structural Scheme For Solving a Problem Using TRIZ
Editor | On 10, Jan 2002
Structural Scheme For Solving a Problem Using TRIZ
Samsung Advanced Institute of Techology
N. Shpakovsky, PhD, TRIZ consultant of SAMSUNG (South Korea). nick_sh2000@mail.ru
V. Lenjashin, TRIZ consultant of SAMSUNG (South Korea). vassili@samsung.com
Hyo June Kim, TRIZ specialists of SAMSUNG (South Korea). hjkim@sait.samsung.co.kr
An earlier version of this paper was presented at the European TRIZ Association meeting, “TRIZ Future 2001,†November 2001.
Abstract: TRIZ /1/ gives main attention to the methods of revealing and resolving contradictions. For this purpose a sufficiently wide tool basis was elaborated, which includes ARIZ and its different
modifications. However, the experience in solving real production problems proves that the greatest difficulties occur at the initial stage of work with a problem proposed by a customer. In the
inextricable tangle of problems, the formulation of which is not always clear enough, it is sometimes difficult to see the possibilities to use the tool basis of TRIZ and, moreover, of ARIZ that
starts immediately from formulating a mini-problem. If the real causes of a problem are discerned, the problem is often solved by simple TRIZ tools and sometimes it is just removed. The initial
situation analysis, characteristic of the early versions of ARIZ, was forced out of TRIZ and became a kind of independent direction, which, in a certain sense, loses connection with the further
problem-solving process. Obviously, certain steps are taken in this direction /2/, but they still do not constitute a self-contained problem-solving technology.
This work presents a scheme of problem solving that starts from the analysis of the initial situation and ends in solving a mini- or a maxi-problem. The scheme was developed on the basis of practical
use of TRIZ and is used by the authors in their work for the company SAMSUNG.
Actions in a monitoring area.
Attempts to solve a problem are generally made by the customer’s specialists before applying to TRIZ experts. The customer’s experts try to solve the problem at a required level but are not
successful. However, owing to their attempts to do this, a comparatively well-explored information area is formed around the initial problem. We call it a “monitoring area†(Fig.1). In the
analysis of the initial situation, it is useful to thoroughly investigate this area by employing the knowledge of experts and analyzing the previous experience in solving this problem and the
obtained concrete proposals. Special attention should be paid to the proposals rejected by the customer due to their obvious impracticability.
The use of G.S.Altshuller’s multi-screen scheme that allows determining subsystems of a technical system (TS) under analysis and the supersystem to which this TS belongs, also yields good results
while working in the monitoring zone. Besides, it is useful to establish the entire chain of evolution of the analyzed TS, at least its closest prototypes. This allows a better understanding of the
logic of transformations preceding the appearance of the analyzed TS. It often happens that defects existing at one stage of its development transfer to a next stage. Moreover, improvements
introduced at some stage can also cause undesirable effects in future.
An important tool of investigating the monitoring area is the analysis of interacting technical systems. For this purpose V.Lenyashin and L.Chechurin /3/ propose to determine a “harmful†and a
“useful†technical system and to establish peculiarities of their interaction by acting according to the following scheme:
• determining the “useful†product of a TS in which a problem occurs;
• determining all constituent parts of the TS that produces this product and the character of interaction between them (with a compulsory inclusion of an energy source and an object worked);
• determining the “harmful†product that occurs spontaneously during the operation of the “useful†TS and causes a removable disadvantage;
• determining all the constituent parts of the harmful TS that produces this “harmful†product, and the character of interaction between them (similarly to the “useful†TS);
• Finding elements common for the “useful†and “harmful†technical systems.
Figure. 1: Full structure scheme of the problem resolving system.
Then it is necessary to remove the action of the “harmful†TS while preserving to the maximum the action of the “useful†one. Using the effects of G.S.Altshuller’s laws of technical system
evolution can do this.
1. To terminate the action of “harmful†TS, it is necessary and sufficient to remove any of its parts (engine, transmission, working component or control unit) determined in accordance with the
law of technical system completeness. As a rule, it is most convenient to remove the transmission of the “harmful†TS. It often happens that some elements of the “harmful†TS do not
participate or play a very insignificant role in the operation of the “useful†TS (normally they come from the previous system due to the psychological inertia). In this case they can be
relatively easily removed by eliminating the “useful†TS’s disadvantage itself.
2. As stated by the law of energy conductivity, to prevent the action of the “harmful†TS, it is necessary and sufficient to break the passing of energy through all of its parts.In this case the
system parts themselves can remain unchanged.
3. To deactivate the “harmful†technical system, it is necessary and sufficient to cause mismatch in operation (operational periodicity) of parts of this system. The system parts themselves can
remain unchanged. The law of harmonization of parts of a system requires that its parts operate in a certain sequence. Deliberate violation will inevitably cause deactivation of the “harmfulâ€
TS, just what we aimed to do.
The problem of search for and elimination of the action of the “harmful†TS is complicated by the fact that this system is normally an ideal TS. That is, the system does not exist (nobody did
anything to organize it) while the product appears. This is just why it is difficult enough to see and to prevent its action.
When searching for possible ways of solving the problem thus treated, it is convenient to use the notion of “anti-system†introduced by G.S.Altshuller. The anti-system is a system that performs
an action opposite to the “harmful†action. This approach also considerably simplifies the search for the elements of the “harmful†system while analyzing it, particularly in case it is not
quite clear what is the cause of the “harmful†action. Something of the kind was done in developing the “analysis of inverse problem†/4/, but without linking it to the laws of technical
system evolution – theoretical foundations of TRIZ. The proposals of the customer’s experts and the analysis of interaction between the useful and harmful system serve as a basis for specifying the
initial problem and proposing a number of hypotheses of its solution.
Diagram “Christmas Tree†.
The diagram reflects the scheme of solving a single selected problem. It is used after the initial problem has been analyzed in the monitoring area and hypotheses of its solution have been
propounded. In making the diagram (Fig.2.), the basic theses developed by N.Khomenko in OTSM-TRIZ /5/ were used. The use of this diagram implies permanent specification of the situation by passing
from a given problem to its abstract model, constructing an abstract model of its solution, specifying this model and proposing one or several conceptual solutions on this basis.
Figure. 2: “Christmas tree†diagram. (The diagram was worked out with participation of E.Novitskaja).
Two axes are the basis of the diagram:
• axis X is the axis of the degree of abstraction of the situation;
• axis Y is the axis of ideality of obtained solution concepts.
The left part of the diagram is the object area where specific objects are considered and all actions are performed with these objects. The right part is the abstract area where all the
considerations and actions are performed with abstract descriptions of objects.
The object and abstract areas are separated by a conventional line that is called the “concept axis†. Here conceptual descriptions of situations are situated in which both real objects and their
parameters and abstract descriptions are mixed on equality with one another. At the apex of the diagram “Christmas Tree†, all the three situational levels – object, conceptual and abstract –
merge. A special situation occurs. It is called IFR – an ideal final result.
The problem solving process illustrated by the diagram includes the following transitions:
1. Transition from the initial problem to its conceptual model.
In this case, the “skeleton†is singled out of the problem conditions. The problem is freed from redundant details. It is necessary to specify the conflicting objects and the peculiarities of
their interaction in time and space, as well as the ideal final result for the considered situation.
2. Constructing a technical contradiction.
To do this, it is necessary to determine how we can improve the desired parameters, which characterize the performance of the main useful function, by means of a conventional method. Then we must
check which parameter of the system worsens to an admissible extent. Then we construct one or several contradictions in accordance with the list of characteristic features by G.S.Altshuller. Having
received several ideas of solving the technical contradiction, one can try to find intermediate concepts of solving the problem by using available resources.
3. Constructing an abstract model of a problem.
To do this, it is convenient to use the rules of su-field analysis and to draw a scheme of interaction of the elements in the form of a su-field model. All the objects participating in the conceptual
model are replaced with abstract “substances†, while the forces and interactions are replaced with corresponding “fields†that characterize the interaction between the objects.
4. Constructing an abstract problem-solving model.
The solution model is constructed by transforming the abstract model of the problem by means of:
a. Fundamental knowledge of the problem solver – the so-called “experience†.
b. The use of analogous problems.
c. The rules of standard solution of problems.
The abstract model solved, it is necessary to make an attempt to find preliminary problem- solving concepts by analyzing the available resources once again.
5. Determining the requirements for the X-elements.
This is a very important stage, which allows making a description of the X-element necessary for the search for a real object with available resources. The X-element can be conveniently described
according to the following scheme:
“Element – Feature of Element – Value of Feature†proposed by N. Khomenko.
6. Constructing a physical contradiction.
A physical contradiction occurs when the requirements for the X-element or its part are physically mutually exclusive. Resolving this contradiction makes it possible to maximally specify the
situation and to obtain a conceptual solution, the closest one to the ideal solution.
7. Constructing a final solution.
All the three concepts obtained in the process of solving, as well as the experiments are used to construct a final solution of the considered single mini-problem.
3. Solving a mini-problem.
A mini-problem implies the elimination of disadvantages without a considerable transformation of the initial problem, only by means of available or easy-to-introduce resources. With limited
resources, the solution of such a problem is often more complicated than in case of a maxi-problem that allows considerable changes to be made in the initial system.
A mini-problem is solved after propounding several solution hypotheses obtained together with the customer’s experts by analyzing the monitoring area. A hypothesis is the main, general idea of
eliminating a disadvantage. The entire problem-solving process consists in specifying separate hypotheses, constructing their object embodiment and specifying their applicability for solving the main
problem in a given situation (Fig.3).
Figure. 3: Structure scheme of the mini-problem solution.
In the object realization of each hypothesis, the following situations may occur:
1. No obstacles occur and the solution is produced automatically, by a direct application of known methods. No contradiction occurs.
2. It is impossible to realize the hypothesis by the known methods. A contradiction occurs when using such methods. In this case, a solution is obtained using the diagram “Christmas Tree†.
3. After solving one of the problems arising during the hypothesis realization, new problems occur and it is impossible to realize the solution of the previous problems without solving these new
ones. In this case, a situation occurs which is very similar to the one described by N.Khomenko in the technology “A flow of problems†, when some partial solutions merge at the end to form a
final solution to a problem.
After object realization of each hypothesis, a final solution of a mini-problem is constructed. To do this, the object realization most suitable for specific conditions is selected together with the
customer’s experts. It is supplemented with other realizations or their useful properties. The method of constructing a final solution of a mini-problem has very much in common with the method of
combining alternative systems /6/.
4. Solving a maxi-problem.
Solving a maxi-problem is implies considerable transformation of the initial technical system and its technological processes. The solution process is similar to the technological forecasting
process.Often enough, after solving such a problem we obtain a number of patentable proposals concerning the development of the production process and the technical system itself.
A maxi-problem is solved by the following scheme:
“Desirable product – Production process – Process-realizing TS†/Fig.4/.
Figure. 4: Structure scheme of the maxi-problem solution.
First it is necessary to accurately determine the requirements for the product produced by the TS under consideration. We do it by using the information obtained through analyzing the monitoring area
and solving mini-problems. Then, with the product specified, we construct a desired process of its production. This is in fact a set of operations to be fulfilled by some “desirable TS†which
will realize this process. Thus, we fulfill the requirements of the law of harmonization of parts of a system.
Having specified the desirable process, it is necessary to construct the model of the “desirable TS†. This is done in accordance with the law of technical system completeness and the law of
through energy conductivity. In this case, it is relatively easy to use the body of information obtained by analyzing the monitoring area and solving a mini-problem.
The distinctive feature of our approach is as follows. When obtaining each concept of a mini-problem solution, it is very useful to write out the principle idea of this concept. This may be both the
type of transformation used in problem solving and substance and fields used for this purpose, or the combination of both. The situations thus obtained are analyzed with the aid of the trends of
technical system development /7/ and their most ideal embodiment is selected, which is then used for constructing a model of a “desirable TS†.
For instance, if a transformed object was monolithic in an initial TS, and, to solve a mini-problem, we used liquid, it would be very useful to analyze how the problem could be solved by using other
monolith transformations lying in the “Segmentation†axis. It is necessary to check he applicability of foam, gases, plasma, electric and magnetic fields and vacuum to constructing the model of a
“desirable TS†.
If we use a principle, for instance, mono-bi-poly, we must check how our “desirable TS†can operate when an additional object or several objects are introduced, or when the object being
transformed passes to a supersystem. Having received a conceptual model of the “desirable TS†(something close to the notion of the “ideal final result†), it is necessary to solve the problem
occurring in transition from this model to its real embodiment. In TRIZ, there is a good say: To step back from the IFR.
By performing the above actions, we receive the maximum efficient transformations that solve the initial problem and point to the most efficient ways of development of the initial technical system.
List of References.
1. Altshuller, Genrikh. “The Innovative Algorithm. TRIZ, Systematic Innovation and Technical Creativity.†Technical Innovation Center, INC. Worcester, MA. 1999.
2. G.Ivanov, A. Bystritsky. “Formulating of Creative Problems†. MATRIZ. Cheljabinsk. 2000 (in Russian).
3. TRIZ forum (in Russian). <http://www.geocities.com/cepreu4/MyTRIZ.html>
4. G. Altshuller, B. Zlotin, A. Zusman, V. Filatov. “Search of the New Ideas: From Inspiration to Technology.” Kishinev. Karta Moldovenyaske, 1989 (in Russian).
5. N. Knomenko. “TRIZ how General Theory of Strong Thinking (OTSM)†. (In Russian).
6. S.Litvin, V.Gerasimov. ‘’Development of Alternative Systems by their Association in Supersystem.“ TRIZ Journal. 1990. 1.1. (In Russian).
7. TechOptimizer®/Prediction/ Trends of Technology Evolution. | {"url":"https://the-trizjournal.com/structural-scheme-solving-problem-using-triz/","timestamp":"2024-11-07T06:08:36Z","content_type":"text/html","content_length":"113343","record_id":"<urn:uuid:555b9582-0928-4a54-b829-3854f4b589ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00628.warc.gz"} |
Result of Mathf.Sin
If I calculate Sin on calc I get one answer
When I use Mathf.Sin for the same number in unity I get another answer. What causes the difference?
Angles in mathematics and pretty much all computer languages are measured in radians and not in degree. Most intefaces for humans work with degrees (0 to 360 or -180 to 180). Mathematically radians
actually make more sense. An angle in radians go from 0 to 2*PI (or -PI to PI).
• Deg2Rad is basically just PI/180 while
• Rad2Deg is just 180/PI
That’s all.
Thanks for the reply. Problem now solved. | {"url":"https://discussions.unity.com/t/result-of-mathf-sin/200122","timestamp":"2024-11-11T10:25:35Z","content_type":"text/html","content_length":"28177","record_id":"<urn:uuid:d0dae4be-b467-43c4-a746-885442e34cbd>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00304.warc.gz"} |
Quadratic Programming Vs. Linear Programming - Codingdeeply
Quadratic Programming Vs. Linear Programming
Quadratic programming is a more complex method of solving non-linear programming problems. There are many applications of quadratic programming, which you will see in the article.
Linear programming is basically a lower level of solving programming problems, and we have previously covered the linear programming topic, so check it out here!
In this article, find out the difference between quadratic and linear programming. Also, find out about the difference between integer and quadratic programming, as well as the applications of each.
Advertising links are marked with *. We receive a small commission on sales, nothing changes for you.
What Is The Difference Between Quadratic and Linear Programming?
The objective and all constraints in linear programming (LP) problems are linear functions of the decision variables.
Since all linear functions are convex, it is inherently simpler to solve linear programming issues than non-linear ones.
Here are a few facts about linear programming.
1. A linear programming problem is an issue raised by a linear program.
2. Constraints for this kind of issue often form inequalities.
3. The restriction may occasionally combine elements of both types.
4. The variables in the issue are the decision vector x and the goal function Z. The formulas for the restrictions are gj(x), hj(x), and lj(x).
5. The solution is obtained through a linear program with m1 constraints.
Even the simplest linear algorithms might involve thousands or even hundreds of variables. In comparison, the minor integer programs include thousands of variables.
Let’s see now what quadratic programming is.
In the quadratic programming (QP) problem, the constraints are all linear functions of the variables, whereas the goal is a quadratic function of the decision variables.
The Markowitz mean-variance portfolio optimization issue is a common Quadratic Programming problem, for example.
The linear constraints determine the lower bound for portfolio return, and the aim is the portfolio variance.
The phrase quadratic programming refers to the idea of linear squares in a more broad sense.
It frequently refers to a strategy for resolving the quadratic equation.
You can solve quadratic equations in a variety of ways, but we’ll explain two of them.
The first approach, known as linear programming, is used to address least squares issues.
The second approach, called modified-simplex, is employed to address non-linear optimization issues.
Complex NLPs (non-linear problems) are solved using the second quadratic programming technique.
In the former, smaller QP subproblems are solved separately, and then more significant QP problems are solved by combining these smaller QP subproblems using an algorithm.
Sequential quadratic programming is used in finance, statistics, and chemical manufacturing to tackle issues with many objective functions.
There is also parallel-quadratic programming. Let’s see what that is.
Parallel-quadratic programming is a sequential quadratic programming method variant that solves multiple-objective quadratic-linear problems concurrently.
What Is Meant By Quadratic Programming Problem?
A quadratic cost function and linear restrictions define a quadratic programming (QP) issue.
Numerous applications in the real world run with these issues.
A quadratic programming subproblem must also be solved for many generic nonlinear programming techniques at each iteration.
Portfolio optimization in banking, power generation optimization for utilities, and design optimization in engineering are a few examples of quadratic programming problems.
Is Linear Programming a Special Case of Quadratic Programming?
When the matrix Q=0, linear programming is a special quadratic programming example.
There are two techniques for solving linear square problems, and those are:
1. Levenberg-Marquardt and
2. Gauss-Newton
Since quadratic programming (QP) issues can be seen as specialized forms of more general problems, software solutions for these more general problems can be used to tackle QP problems.
Quadratically constrained quadratic programming (QCQP) issues are a generalization of QPs in that they involve quadratic restrictions as opposed to linear ones.
QCQPs are generalized by second-order cone programming (SOCP) issues, while SOCPs are generalized by nonlinear programming (NLP) problems.
Can Quadratic Programming Be a Non-Linear Programming?
A straightforward non-linear programming method called quadratic programming can model various real-world systems, particularly those that depend on two variables.
We already mentioned quadratic programming being in a relationship with non-linear programming.
Take a quick look under the paragraph “What Is The Difference Between Quadratic and Linear Programming?” if you want to find out more.
But, basically, you can solve non-linear problems using quadratic programming.
What Is The Difference Between Quadratic and Integer Programming?
If you got here, then you know what quadratic programming is.
So, keep reading to find out what integer programming is.
In simple terms, integer programming is a subset of mathematical programming or optimization that includes formulating equations to address issues.
The phrase “mathematical programming” refers to selecting action plans to solve various challenges.
You can use integer programming in various situations, such as:
1. Transportation
2. Schedule
3. Assignments and Workplans
4. Airline schedules
5. Production planning
6. Purity of some metals
Click here if you want to find out more about integer programming!
Frequently Asked Questions
There are several questions that people who search for quadratic programming also ask, so feel free to keep reading to answer some of the questions you even don’t know you’ll need answers for.
1. Integer Quadratic Programming – What Is It?
A quadratic function optimization issue is mixed-integer quadratic programming (MIQP). Across points in a polyhedral set whose components are both continuous and integer.
2. How Are Quadratic Equations Solved?
There are five steps when solving a quadratic equation.
1. Put the equal sign with all terms on one side and zero on the other.
2. Determine factor.
3. Each factor should be set to zero.
4. Fix each of these problems.
5. Put your solution into the original equation to be sure.
3. Which Four Strategies Are Used To Solve Quadratic Equations?
You can solve a quadratic equation using several techniques, including:
1. factorization,
2. completing the square,
3. the quadratic formula, and
4. graphing.
4. What Is The Quadratic Equation’s Fundamental Formula?
A second-order equation of the form ax2 + bx + c = 0 denotes a quadratic equation, where a, b, and c are real number coefficients and a 0.
5. What Kind Of Equation Is Not Quadratic?
In general, an x2 term is required in quadratic equations. However, it CAN NOT include words with more than x2 degrees, such as x3, x4, etc.
6. Is Learning Quadratic Equations Necessary?
Quadratic functions occupy a special place in the academic curriculum.
They are minor improvements over linear functions and offer a significant break from attachment to straight lines since their values may be readily derived from input values.
Advertising links are marked with *. We receive a small commission on sales, nothing changes for you. | {"url":"https://www.codingdeeply.com/quadratic-programming-vs-linear-programming/","timestamp":"2024-11-12T19:09:15Z","content_type":"text/html","content_length":"111950","record_id":"<urn:uuid:0f5b6096-0636-4599-8d17-672cb2c6e3b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00405.warc.gz"} |
Radius/diameter of non-weighted graph - Codeforces
Hi, Codeforces.
Could anybody kindly tell me something about fast calculating radius and/or diameter of non-weighted undirected graph (definitions can be found here) ? Fast = faster, than in O(MN) (breadth-first
searches from each vertex). Any results are welcome: probabilistic, good in average, good for sparse graphs, etc. Thanks.
13 years ago, # |
» +12
obag wil93 and I once attended a conference about the theme. You can find some useful papers here, under "Fast Computation of the Neighbourhood Function and Distance Distribution", and here (look at
the links in the bottom of the text). Hope this helps :)
• 13 years ago, # ^ |
Yep, the algorithm they presented was as follows:
1. Pick a random vertex U
» 2. Run a BFS from U to find the farthest vertex V.
» 3. If Dist(U,V) is better than the best diameter found then update it, else exit.
4. Assign V to U and go back to step 2
The interesting thing is that the algorithm founds the diameter very soon.
If you want an approximate diameter, then you can stop manually after having done, say, K searches. The algorithm is then O(MK). It is worth noting that you will find very good results even
with very small K.
When they presented this algorithm, they showed us the results and the running time of some test with (if I'm not mistaken) K = 10, compared to the O(MN) algorithm. It turned out that even
with that value of K the diameter found was very close to the "best" one.
□ 13 years ago, # ^ |
» Yes, that seems to be cool thing to use in practice. But, as we discussed a bit earlier (in Russian, though =)), we can choose U in such unlucky way that we won't find precise
» diameter ever.
Endagorion Well, in the real task that I approached graphs could be pretty dense, so there's no sense in "approximating" radii or diameters, as they were quite small.
Still, thanks for pointing out the papers and nice description!
□ » 12 years ago, # ^ |
» ← Rev. 4 → 0
Does it work for every kind of graphs?
(I am missing comment delete option :(. )
☆ »
» 12 years ago, # ^ |
» 0
☆ » 12 years ago, # ^ |
» 0
» Yes it does. But it is an approximate algorithm, as you are running only K iterations (where K ~ 10 or more, depending on you). It is interesting as it finds quickly the optimal
solution (the "optimal in average" K is relatively small, so using a fixed small K can works well). Maybe this algorithm isn't very reliable in programming contest, because of its
wil93 "approximate" nature, but in practice it is good.
» 13 years ago, # |
← Rev. 4 → 0
• 13 years ago, # ^ |
← Rev. 5 → +14
» You are wrong, it works only for trees.
Tranvick | |
We run BFS from A, then from B and answer will be 3. But really answer is 4(E — D).
• » 13 years ago, # ^ |
» +1
rlac Your algorithm is ok for get a tree's diameter, but it's wrong for get a graph's diameter.
12 years ago, # |
» +49
piluc Our group in Florence recently worked on the computation of the diameter of large real-world graphs. You can download our software at http://amici.dsi.unifi.it/lasagne. In the web site, all the
references are also listed. I hope this helps. | {"url":"https://mirror.codeforces.com/blog/entry/4116","timestamp":"2024-11-04T01:21:33Z","content_type":"text/html","content_length":"119482","record_id":"<urn:uuid:c4e6ad8f-6a49-4a58-b2aa-095ef1b8c303>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00288.warc.gz"} |
Guard Banding - How to Take Uncertainty Into Account - isobudgets
Guard Banding – How to Take Uncertainty Into Account
There has been a lot of discussion on guard band and guard banding methods lately. Especially, with regard to statements of conformity and decision rules, and how you take uncertainty into account.
This led me to look further into different guard banding methods. Currently, there have been a couple of recommended methods and a lot of opinions. So, I wanted to find out what guard banding methods
labs are using and what other options are available.
What I found was astonishing!
Some of the methods that labs are being recommended to use are the worst options for the lab (and its’ customers) in terms of Producer’s Risk and false rejects (You’ll learn more about this later).
This may be due to most guides, regulators, and experts presenting information to labs that mostly focuses on false acceptance.
From my research, I found a lot of options. Many that I did not know existed. The more research I did, the more methods that I would find. Some of the guides that experts recommend included
additional methods that no one is discussing but may be a viable, easy to implement option for many.
Plus, some of the methods that I found did a much better job of sharing the risk of false accepts (i.e. Consumer’s Risk) and false rejects (i.e. Producer’s Risk) between a lab and it’s customers.
In this guide you are going to learn:
1. What is Guard Banding,
2. Why Guard Band
3. What is Test Uncertainty Ratio
4. Guard Banding Methods
5. Summary of Guard Banding Methods
6. What Method is Best For Your Lab
7. Comparison of Guard Banding Methods
What is Guard Banding
Guard banding is a technique used to reduce the risk of encountering an incorrect conformity decision, such as:
1. False Acceptance: Claiming a result is in tolerance when it is out of tolerance.
2. False Rejection: Claiming a result is out of tolerance when it is in tolerance.
These events are commonly referred to as Type I and Type II errors.
Look at the image below. It will give you a good idea of what guard banding is and how it is applied.
Why Guard Band
Ultimately, you use guard band methods to prevent the occurrence of false acceptance (Type I Error) and false rejection (Type II Error) errors. Either scenario can have negative effects for the
laboratory and its customers.
This is why labs should use guard banding.
Today, most labs are interested in the concept of guard banding as a way of meeting ISO/IEC 17025:2017 requirements. While this is not a bad practice, many labs to do not know why they are guard
banding other than to meet a requirement.
However, you should consider the use of guard banding as a way to improve confidence in the results reported by the laboratory.
Furthermore, there are plenty of scenarios where a 4:1 TUR is not possible. In these situations, the use of guard banding can help reduce the chance of encountering a false accept or false reject
Therefore, your “Why” should be to reduce the risk of occurrence of false accepts or false rejects, not just meeting an ISO requirement.
What is Test Uncertainty Ratio
In this guide, you are going to see a lot of references to Test Uncertainty Ratio (TUR). If you are not familiar with it, let me give you a quick summary.
According to the ANSI Z540.3 Handbook, Test Uncertainty Ratio is the ratio of the acceptable tolerance (T) of the Unit Under Test (UUT) to the expanded uncertainty (U) of the measurement process.
The tolerance is represented by the difference between the upper and lower tolerance of the UUT; and, is divided by two times the expanded uncertainty of the measurement process.
Test Uncertainty Ratio Formula
Take a look at the TUR formula given below:
TUR – Test Uncertainty Ratio
TU – Upper Tolerance Limit
TL – Lower Tolerance Limit
U – Expanded Uncertainty
Test Uncertainty Ratio is commonly used as a performance metric to judge the quality of a measurement result. Currently, the benchmark for TUR is a 4:1 ratio where the expanded uncertainty is 1/4th
or 25% of the tolerance.
In most situations, people that evaluate the quality of a measurement result by its TUR typically formulate the following opinions:
1. Less than 4:1 TUR is considered less quality, and
2. Greater than 4:1 TUR is considered better quality.
Although a 4:1 TUR is not always practical or technically possible, many consider it the standard benchmark to evaluate the quality of a measurement result.
Now that you know what Test Uncertainty Ratio is, the rest of this guide should make more sense. In this guide, you will see references to TUR in most the guard banding methods.
If you want to learn more about Test Uncertainty Ratio, click the link below to read my guide.
Guard Banding Methods
When you need to take uncertainty into account, there are several guard banding methods that you can use. In this section, you will learn about some of the most popular methods used by ISO/IEC 17025
accredited labs.
Altogether, there are 13 guard banding methods covered in this guide. Two of these methods can be found in the ANSI Z540.3 Handbook, three of the methods are from the NCSLi Recommended Practice RP-10
, two methods come from the UKAS M3003, and the rest can be found in guard banding papers written by Dave Deaver or Bill Hutchinson back in the 1980’s and 1990’s.
You will notice that 3 of the methods in this guide are duplicates of other methods. The only difference is the resource they came from, and the formulas used to perform guard banding. Even though
these methods produce the exact same results, both versions of the method are provided in this guide because the approach to guard banding is presented differently which may make implementation
easier for some labs.
The 13 methods that you will learn about in this guide are:
To learn more about these methods, scroll down and keep reading.
1. Guard Banding per ANSI Z540.3 Handbook Method 5
Guard banding with this method is based on the expanded calibration process uncertainty.
Many people ask, “Do I use the calibration uncertainty per ILAC P14 or the CMC uncertainty?”
Currently, the general consensus (that I have heard from two different committees) is you should use your CMC or calibration process uncertainty and not your calibration uncertainty per ILAC P14 that
includes influences from the UUT (i.e. UUT Resolution and UUT Repeatability).
Look at the image below to see an excerpt from the ANSI/NCSL Z540.3-2006 Handbook.
Guard Band Method 5 from the ANSI/NCSL Z540.3-2006 Handbook is one of the simplest and most commonly used guard banding techniques. However, it is one of the worst methods that a lab can use in terms
of Producer’s Risk (i.e. false rejects) and cost.
The basic concept is to add and subtract the expanded uncertainty from the tolerance limit before following your decision rules and providing a statement of conformance (e.g. Pass or Fail).
Guard Banding Formula
Below is the formula for Guard Band Method 5 from the ANSI/NCSL Z540.3-2006 Handbook.
A – Acceptance Limit
L – Tolerance Limit
U95% – Expanded Uncertainty (95% C.I. where k=2)
Follow the instructions below to apply this method:
Step 1. Find the value of the Tolerance Limit (L).
Step 2. Find the value of the Expanded Uncertainty (U).
Step 3. Add and Subtract the Tolerance Limit by the Expanded Uncertainty
2. ANSI Z540.3 Handbook Method 6
Another popular guard banding method (especially for labs that are ANSI Z540.3 accredited) is ANSI/NCSLi Z540.3-2006 Handbook Method 6. The guard banding technique used in this method is based on the
test uncertainty ratio (TUR).
Look at the image below to see an excerpt from the ANSI/NCSL Z540.3-2006 Handbook.
The method was proposed by Mike Dobbert of Keysight Technologies at the 2008 NCSLi Symposium in his paper, “A Guard Band Strategy for Managing False Accept Risk.”
If you would like to read the paper, just click the link below.
A Guard-Band Strategy for False-Accept Management | Keysight
The goal of this method is to adjust your tolerance to an acceptance limit where the maximum probability of false acceptance (PFA) is 2% or less. Originally, it was a solution to meet the 2% PFA
requirement of ANSI Z540.3-2006. Now, it is finding popularity again since it has less Producer’s Risk and yields fewer overlap scenarios compared to Method 5 (in the previous section).
Personally, I used this method to comply with ANSI Z540.3-2006 requirements. However, I have never used it to comply with ISO/IEC 17025:2017.
This technique is more advanced but can be automated with software such as Microsoft Excel or Crystal Reports using the formula below.
MS EXCEL: = [Cell L]–[Cell U95%]*(1.04-(EXP((0.38*LN([Cell TUR]))-0.54)))
Guard Banding Formula
Below is the formula for Guard Band Method 6 from the ANSI/NCSL Z540.3-2006 Handbook.
A[2%] – Acceptance Limit at 2% maximum PFA
L – Tolerance Limit
U[95%] – Expanded Uncertainty (95% C.I. where k=2)
TUR – Test Uncertainty Ratio
Note: log(x) in the above equation is a natural logarithm, not a common base 10 logarithm.
Follow the instructions below to apply this method:
Step 1. Calculate the TUR
Step 2. If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 3.
Step 3. Calculate the natural logarithm value of the TUR.
Step 4: Multiply the result from Step 3 by 0.38.
Step 5. Subtract 0.54 from the result from Step 4.
Step 6. Calculate the exponential value with the constant e and the result from Step 5.
Step 7. Subtract 1.04 by the result from Step 6.
Step 8. Multiply the expanded uncertainty by the result from Step 7.
Step 9. Subtract the Limit by the result from Step 8.
The image below is from the ANSI Z540.3 Handook. Look at the graph. It shows the M[2%] value guard band factor increases as test uncertainty ratio (TUR) decreases.
• When TUR is 4:1, a guard band of 5.3% is applied to the expanded uncertainty.
• When TUR is 3:1, a guard band of 15.5% is applied to the expanded uncertainty.
• When TUR is 2:1, a guard band of 28.2% is applied to the expanded uncertainty.
• When TUR is 1:1, a guard band of 45.7% is applied to the expanded uncertainty.
If you find the full formula (above) too difficult to use, look at the table below. I have already calculated the M[2%] values for you.
Using the table above you can apply the simplified guard banding equation below to calculate the 2% acceptance limit (A[2%]).
Follow the instructions below to apply this method:
Step 1. Calculate the TUR.
Step 2. Find the M[2%] value from the above table.
Step 3. Multiply the expanded uncertainty by the M[2%] value.
Step 4: Subtract the Limit by the result from Step 3.
3. Guard Band to the Same Consumer Risk as a 4:1 TUR
This guard banding method allows you to adjust your tolerance limit to the same probability of false acceptance (PFA) risk as a result with a 4:1 test uncertainty ratio (TUR).
The method has been found in the following papers:
1. “How to Maintain Your Confidence” by Dave Deaver, and
2. “Setting Guardband Test Limits to Satisfy MIL-STD-455662A” by Bill Hutchinson
Look at the image below to see an excerpt from the “Guardbanding With Confidence.”
The benefit of this technique is the consumer’s risk (i.e. probability of false acceptance) is consistently 0.8% and the Producer’s Risk (i.e. probability of false rejection) is quite low in
comparison to other methods.
Guard Banding Formula
Below is the guard band formula from Dave Deaver’s paper.
A – Acceptance Limit
L – Tolerance Limit
K – Guard Band Correction Factor
The image below is from Dave Deaver’s paper, “How to Maintain Your Confidence.” Focus your attention on the graph on the right side. It shows the correction factors for guard banding your tolerances
to match the consumer’s risk to the same probability (i.e. 0.8%) as having a test uncertainty ratio (TUR) of 4:1.
The graph is a nice tool to have but interpreting the value of K can be difficult without a function or formula. To help you out, I created the table below. It gives you the value of K based on the
value of TUR in tenths of a digit.
Follow the instructions below to apply this method:
Step 1. Calculate the TUR
Step 2. If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 3.
Step 3: Find the correction factor (K) the table (in this section),
Step 4: Multiply the Limit by the result of Step 3.
4. Guard Banding per ILAC G8
With this guard banding technique, you adjust your tolerance limit based on your expanded uncertainty.
Similar to the ANSI Z540.3 Handbook Method 5, the ILAC G8 method has you simply add or subtract your expanded uncertainty from your tolerance.
The process is very simple, but has an extremely high producer’s risk.
Look at the image below to see an excerpt from the “ILAC G8:09/2019.”
Further down, you will see an alternative version of this based on the test uncertainty ratio.
Guard Banding Formula
Below is the guard band formula from the “ILAC G8:09/2019.”
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty (95% C.I. where k=2)
Follow the instructions below to apply this method:
Step 1. Find the value of the Tolerance Limit (L).
Step 2. Find the value of the Expanded Uncertainty (U).
Step 3. Add and Subtract the Tolerance Limit by the Expanded Uncertainty
5. Guard Banding per ISO 14253-1
This guard banding technique has you adjust your tolerance limit based on your expanded uncertainty.
When taking uncertainty into account, the ISO 14253-1 method has you consider 83% of your expanded uncertainty instead of 100% of your expanded uncertainty compared to the ILAC G8 rule.
The advantage of this method is you have a 5% probability of false acceptance (PFA) and a smaller probability of false rejection (PFR) compared to the ILAC G8 method.
Some guides and papers claim this method is the same as the ILAC G8 method, but you can see from the image below that they are not. Look at the image below to see an excerpt from the “ILAC G8:09/
Guard Banding Formula
Below is the guard band formula from the “ISO 14253-1.”
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty (95% C.I. where k=2)
Follow the instructions below to apply this method:
Step 1. Find the value of the Tolerance Limit (L).
Step 2. Find the value of the Expanded Uncertainty (U).
Step 3. Multiply the Expanded Uncertainty (U) by 0.83.
Step 4. Add and Subtract the Tolerance Limit by the result from Step 3.
6. Guard Banding Based on Test Uncertainty Ratio (TUR)
This guard banding technique has you adjust your tolerance limit based on the test uncertainty ratio.
I found this method in Dave Deaver’s paper, “Guardbanding With Confidence.” There are a lot of published papers that claim this method came from ILAC G8 and ISO 14253-1. However, I have not seen the
equation written this way when in either document.
Furthermore, I have not found this equation written the same way in any reference document.
I can see how it is confused with the ILAC G8 method. This method produces the same results as the:
• ILAC G8 rule, and
• ANSI Z540.3 Handbook Method 5
However, the formula is different.
Look at the image below to see an excerpt from “Guardbanding With Confidence.”
Guard Banding Formula
Below is the guard band formula from Dave Deaver’s paper.
A – Acceptance Limit
L – Tolerance Limit
TUR – Test Uncertainty Ratio
Follow the instructions below to apply this method:
Step 1: Calculate the TUR
Step 2: If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 3.
Step 3: Divide one (1) by the TUR,
Step 4: Subtract 1 by the result of Step 3,
Step 5: Multiply the Limit by the result of Step 4.
Guard Banding Factor
The chart below shows the guard banding factor (w) based on the TUR using the equation given above. To calculate your acceptance limit, simply multiply the guard banding factor and your tolerance.
Guard Banding Factor for Acceptance Limit
Look at the table below to see the guard banding factors calculated with this method. Simply multiply the guard banding factor (K) associated with the TUR and the tolerance limit. The result will be
your acceptance limit.
Guard Banding Factor for Rejection Limit
Look at the table below to see the guard banding factors calculated with this method. Simply multiply the guard banding factor (K) associated with the TUR and the tolerance limit. The result will be
your rejection limit.
7. NCSLi Recommended Practice RP-10: Constant Z
The NCSLi Recommended Practice 10 has 3 guard banding methods in it. The first recomends applying a constant guard banding factor of 80%.
Essentially, you add or subtract 80% of expanded uncertainty to or from the tolerance limit.
I used this method (all of the time) years ago, but never knew where it came from.
Look at the image below to see an excerpt from the “NCSLi Recommended Practice 10.”
Guard Banding Formula
Below is the guard band formula from NCSLi RP-10 for Constant Z.
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty
Follow the instructions below to apply this method:
Step 1: Calculate the expanded uncertainty.
Step 2: Multiply the expanded uncertainty by 0.8.
Step 3: Add or subtract the tolerance limit by the result from Step 2.
8. NCSLi Recommended Practice RP-10: Linear Z
This method is the second guard banding method recommended by the NCSLi RP-10. The guard banding factor, Z, is calculated based on the linear function provided below.
Look at the image below to see an excerpt from the “NCSLi Recommended Practice 10.”
I have never seen this method before, so it is new to me. Currently, I am not aware of anyone that uses it.
Guard Banding Formula
Below is the guard band formula from NCSLi RP-10 for Linear Z.
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty
TUR – Test Uncertainty Ratio
Follow the instructions below to apply this method:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 4.
Step 4: Multiply the TUR by 0.2.
Step 5: Subtract 0.8 by the result of Step 4.
Step 6: Add or subtract the tolerance limit by the result from Step 5.
9. NCSLi Recommended Practice RP-10: Statistical Z
This method is the third guard banding method recommended by the NCSLi RP-10. The guard banding factor, Z, is calculated based on the statistical function provided below.
Look at the image below to see an excerpt from the “NCSLi Recommended Practice 10.”
This is the same method given earlier for consumer risk equal to a 4:1 TUR. However, the NCSLi RP-10 writes and applies the formula differently. I am not sure why the method is applied differently,
but the result is still the same.
Guard Banding Formula
Below is the guard band formula from NCSLi RP-10 for Statistical Z.
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty
Z – Constant Consumer Risk Factor based on TUR (%)
TUR – Test Uncertainty Ratio
Follow the instructions below to apply this method:
Step 1: Calculate the expanded uncertainty.
Step 2: Calculate the test uncertainty ratio.
Step 3: If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 4.
Step 4: Find Z from the table below.
Step 5: Divide Z by 100 (because it is a percentage).
Step 6: Multiply the expanded uncertainty by the result of Step 5.
Step 7: Add or subtract the tolerance limit by the result from Step 6.
Guard Banding Factor for NCSLi RP-10: Statistical Z
Below is a table with the Guard Band factor K for NCSLi RP-10, Statistical Z method.
10. Guard Banding Method from Previous Version of NCSLi RP-10
This method is from Dave Deaver’s paper, “Guardbanding With Confidence.” It allows you to adjust your tolerance based on the value of the test uncertainty ratio.
Look at the image below to see an excerpt from the “Guardbanding With Confidence.”
The paper claims that this method was published in NCSLi Recommended Practice RP-10. However, it is not in the current version of the document. It may have been in a previous version of NCSLi RP-10
and removed or replaced by the methods in the current version.
I included the method in this guide because Dave Deaver claimed this method had a much better producer’s risk at higher test uncertainty ratios (e.g. 4:1 or greater) compared to other guard banding
methods given in this document; specifically ILAC G8, ISO 14253-1, and ANSI/NCSLi Z540.3 Handbook Method 5.
Guard Banding Formula
Below is the guard band formula from Dave Deaver’s paper, “Guardbanding With Confidence.”
A – Acceptance Limit
L – Tolerance Limit
TUR – Test Uncertainty Ratio
Follow the instructions below to apply this method:
Step 1. Calculate the TUR
Step 2. If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 3.
Step 3: Divide one (1) by the TUR,
Step 4: Subtract 1.25 by the result of Step 3,
Step 5: Multiply the Limit by the result of Step 4.
11. Guard Banding with Fluke’s RSS Strategy
This method is also from Dave Deaver’s paper, “Guardbanding With Confidence,” and adjusts your tolerance based on the value of the test uncertainty ratio.
Look at the image below to see an excerpt from the “Guardbanding With Confidence.”
According to the paper, this guard banding method was used by Fluke in the 1990’s. I am not sure if Fluke still uses it today.
The benefit to this method is it has both a low consumer’s risk and a low producer’s risk.
Guard Banding Formula
Below is the guard band formula from Dave Deaver’s paper, “Guardbanding With Confidence.”
A – Acceptance Limit
L – Tolerance Limit
TUR – Test Uncertainty Ratio
Follow the instructions below to apply this method:
Step 1. Calculate the TUR,
Step 2. If the TUR is:
1. 4 or greater: Do nothing,
2. Less than 4: Go to Step 3.
Step 3: Square the TUR,
Step 4: Divide one (1) by the result of Step 3,
Step 5: Add or Subtract 1 by the result of Step 4,
Step 6: Calculate the square root of the result in Step 5,
Step 7: Multiply the Limit by the result of Step 6.
12. Guard Banding with UKAS M3003 3rd Edition, Section M3
This method comes from the 3rd edition of the UKAS M3003 guide, but was removed in the 4th edition of the document. It is exactly the same as the Fluke RSS method. The equation is different, but the
results are identical.
Look at the image below for an excerpt from the “UKAS M3003 3rd edition.”
The method is simple to use. Just calculate the root sum of squares (RSS) with your tolerance limit and expanded uncertainty. This can be easily performed in Microsoft Excel and Crystal Reports.
Guard Banding Formula
Below is the guard band formula from the UKAS M3003 3rd Edition, Section M3:
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty
Follow the instructions below to apply this method:
Step 1. Find out if the tolerance limit and expanded uncertainty have the same coverage probability:
1. If Yes: Go to Step 2,
2. If No: Stop and use a different method.
Step 2: Square the Limit,
Step 3: Square the Expanded Uncertainty,
Step 4: Subtract the result of Step 2 by the result of Step 3,
Step 5: Calculate the square root of the result in Step 4.
13. Guard Banding with UKAS M3003 4th Edition, Section M2
This method comes from the 4th edition of the UKAS M3003 guide, section M2. This method was also available in the 3rd edition of the UKAS M3003.
Look at the image below to see an excerpt from the “UKAS M3003 4th edition.”
This method is explained and applied differently in the UKAS M3003 guide (see the image above) than how it is presented below. However, the modification below has been used by labs to guard band
tolerances to a 95% confidence level.
Guard Banding Formula
Below is the guard band formula from the UKAS M3003 4th Edition, Section M2:
Since this method requires the use of the combined standard uncertainty, I have modified the formula to make it easier for you to find your combined uncertainty. Now, all you need is your expanded
uncertainty and your coverage factor (typically, k=2).
A – Acceptance Limit
L – Tolerance Limit
U – Expanded Uncertainty
k – Coverage Factor (typically k=2)
Follow the instructions below to apply this method:
Step 1. Divide the Expanded Uncertainty by the Coverage Factor,
Step 2: Multiply the result of Step 1 by 1.64,
Step 3: Add or Subtract the Tolerance Limit by the result of Step 2.
Summary of Guard Banding Methods
Now that you have learned about each one of these methods, I thought that it would be beneficial to see a summary of each method all in one table.
In the table below, you will see each method in the form of guard banding factor, K, versus test uncertainty ratio (TUR). This makes it easy to compare the guard banding factor of each method.
Acceptance Limit
If you want to apply one of these methods to your results, use the formula below. Simply, find the guard banding factor (K) from the table and multiply it and your tolerance limit to find your
acceptance limit.
A – Acceptance Limit
L – Tolerance Limit
K – Guard Band Correction Factor
Rejection Limit
If you want to find your rejection limit, use the modified formula below. Simply, find the guard banding factor (K) from the table, divide one by the guard banding factor, and multiply it with your
tolerance limit to find your rejection limit.
A – Acceptance Limit
L – Tolerance Limit
K – Guard Band Correction Factor
After looking at the table, you will notice that some of the methods produce the exact same results. Here is a list of the methods with similar results:
• The consumer risk equivalent to 4:1 TUR method is the same as NCSLi RP-10 Statistical Z method.
• The ILAC G8 / ISO 14253-1 method is the same as the ANSI Z540.3 Handbook Method 5.
• The Fluke RSS method is the same of the UKAS M3003 3rd edition M3 method.
This may help you when deciding which method you would like to use.
Which Method is Best For You
To determine which method is best for you, it is important to understand producer’s risk and consumer’s risk.
The definitions below have been modified to help you understand how they relate to your laboratory activities.
Consumer’s Risk
Consumer’s Risk is the chance that your measurement results do not meet specifications, go undetected, and affect subsequent results (which can affect quality, safety, reputation, health, life,
Essentially, consumer’s risk is the chance that you pass or accept measurement results that should have been rejected.
Producer’s Risk
Producer’s Risk is the chance that you reject measurement results that should have passed or been accepted.
Now that you understand the risks involved, you need to determine the goal your laboratory wants to achieve.
Do you want to optimize for:
1. Minimizing producer’s risk,
2. Minimizing consumer’s risk,
3. Sharing the risk equally, or
4. Another goal?
This is important to know. It will help you determine with method works best for your laboratory.
Minimizing Producer’s Risk
If minimizing producer’s risk is your goal, pick the method that gives your lab the least risk of encountering an out-of-tolerance or overlap condition.
Minimizing Consumer’s Risk
If minimizing consumer’s risk is your goal, pick the method that gives your customer the least risk of encountering an false accept or false reject.
Sharing the Risk
If balancing risk is your goal, pick the method that does the best job of sharing the risk equally between the lab and the customer.
Comparison of Guard Banding Methods
A good way to find the right method for your lab is to compare the consumer’s and producer’s risk associated with each guard banding method.
In the table below, you will see a summary of the consumer’s risk associated with each method. You will notice that the table is incomplete because this information is limited. Most of the
information in this table came from papers and guides published about these guard banding methods.
I only filled in the data that I could find. The rest will be determined later when I have an opportunity to calculate the values myself.
Looking at the table above, you will notice that each one of these methods significantly reduces the consumer’s risk or the chance of having a false acceptance. This is good for laboratory customers.
However, each method comes with a cost of increasing producer’s risk or the chance of having a false rejection. This is bad for laboratories.
In the table below, you will see a summary of the producer’s risk associated with each method. You will notice that the table is incomplete because this information is limited. Most of the
information in this table came from papers and guides published about these guard banding methods. The rest will be determined later when I have an opportunity to calculate the values myself.
Looking at the table above, you will notice that the producer’s risk has a lot of variance between the methods. This means that the method that your laboratory uses could have a negative impact on
your lab and its customers.
Here is a list of negative impacts that false rejections can cause:
• Angry customers due to more rejections,
• Loss to reputation when you reject an item that another lab does not,
• Loss of revenue when customers decide to use another lab,
• Increased costs due to retesting or recalibration, and
• Increased costs due to increased nonconformities.
Considering the risks, you should evaluate the guard banding method that you are using and select the method that would be best for your lab and it’s customers.
For laboratories that take uncertainty into account when providing statements of conformity, guard banding methods are becoming a popular tool for labs meeting ISO/IEC 17025:2017 requirements;
specifically for providing statements of conformity and decision rules
Most laboratories (that provide statements of conformity) use the same method when taking uncertainty into account because it is recommended by the ILAC G8 guide, promoted in training classes, and
easy to implement.
The bad news is the most popular method used by labs is putting them and their customers at a major risk for reporting false rejections (i.e. Producer’s Risk).
In this guide, you have learned about 11 guard banding methods, including the:
• Name of the guard banding method,
• Resources that the method is derived from,
• Guard banding formula used to implement it, and
• Risks (i.e. Consumer and Producer’s Risk) associated with each method.
With this information, you should be able to select and implement a method that works best for your laboratory and customers. With the information provided in this guide, I recommend that you quit
using the ILAC G8 method and start using a different method.
In my professional opinion, I like the Constant Consumer Risk (CR4:1) and the Fluke RSS method. They both have a low consumer risk and a reasonable producer’s risk that I think is a happy middle
ground for both the laboratory and its customers.
Which method do you use and why?
Leave a comment below and let me know.
Summary of Changes:
This guide was originally published on 12/07/2021 and updated on 09/16/2022 to update the ANSI Z540.3 method 6 section. There was an important note on the log function (originally left out) added to
the guide. Additionally, a simplified version of the ANSI Z540.3 method 6 formula including a supporting graph and table were added to the guide. | {"url":"https://www.isobudgets.com/guard-banding-how-to-take-uncertainty-into-account/","timestamp":"2024-11-08T01:16:59Z","content_type":"text/html","content_length":"176886","record_id":"<urn:uuid:3eaeac80-2ce5-4cc6-b25c-bc532bc334b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00475.warc.gz"} |
What is optimization in practice and theory ? - Creative OptimizationWhat is optimization in practice and theory ? - Creative Optimization
What is optimization in practice and theory ?
The word optimize is often used as a general comment without any more detailed meaning on what is being done. For example, it is very easy to say that the route, the design, or the plan is optimized
without any type of quantification. Most often, what is meant in practice is to do something as good as possible. In many applications, it is possible to manually find acceptable solutions to a
The human brain works quite well to solve with, at least when the problems are relatively simple, and the number of possible solutions is not too many. To find good solutions is more difficult as it
requires a good understanding of both the problem and the possible solutions. For this to happen there is a need for highly qualified staff members or professionals. To find the optimal solution is
simply not possible even for such persons due to the high complexity or the number of potential solutions. This became evident during the second world war when quantitative models and methods began
being developed to solve large logistics problems. Thie is where the word Operations Research (OR) was founded.
Optimization is closely related to OR, which comprises several science areas such as statistics, queuing theory, simulation, control theory, and production economics. Each of these is based on a
quantitative/mathematical analysis for the decision making. Another related field is Mathematical Programming where the word ”programming” that also appears in the concept ”linear programming”
originates from the English word program and also means planning. Other terms used to characterize operational researchers have evolved over the past 50 years. Often, “Management Science” (MS) was
coupled with OR during the 1970s and 1980s. In the 2010s, the US INFORMS society adopted “Analytics”, defining it as “the scientific process of transforming data into insights for the purpose of
making better decisions” (informs.org) and used the term to describe OR in the traditional context. Analytics is often separated into four different stages each used answering a related question:
descriptive (what happened?), diagnostic (why did it happen?), predictive (what will happen?), and prescriptive (how can we make it happen?), each depending on the level of difficulty and value for
the users.
Value chain management optimization
There are many applications where optimization can be used. To provide a general description, we can study a supply chain (SC) or a value chain (VC) that refers to the system and resources required
to move a product or service from supplier to customer. The value chain concept builds on this to also consider the way value is added along the chain, both to the product / service and the actors
involved. A value chain may have many companies and organisations involved. Value Chain Management (VCM) is a way for a company to optimize all the activities in its manufacturing process, from the
design of the product to its delivery to the customer. VCM can help a company increase its efficiency, quality, and profitability by identifying and improving the value-added steps in each stage of
the production process. VCM also involves coordinating and communicating with the suppliers, distributors, and customers to ensure a smooth and satisfying experience for all parties. There are many
problems arising in the VC, for example, purchasing, transportation, production, distribution and sales. They appear for different planning horizons ranging from long term strategic planning (e.g.
5-30 years) down to operational planning where the planning periods may be as short as part of seconds. Depending on the problem studied there are specific requirement on the data required.
Finding the optimal solution
There is a standard process to be able to find an optimal solution to a problem. Given the real problem, we first identify its location and how it connects to other related problems. Also, there is a
need to understand how it relates to longer term and shorter-term planning. Moreover, it is also important to analyze what data and information that are available. This analysis will provide a
simplified formulation with assumptions and conditions on how the problem is coordinated and/or restricted. Next, we formulate the identified problem into a quantitative optimization model. This
model consists of three main parts. The first is the decision variables which describes what is possible to vary in the problem. The second is the objective function which describes the goal or aim
of the problem. The third is the part are the constraints that restricts the decision variables and provides the boundary conditions. There are many different models, and they are typically
categorized into the groups linear, nonlinear and integer models. The easiest and most known is Linear Programming models where all variables are continuous. For such there are very efficient methods
based on the well-known Simplex method developed by George Dantzig 1947. Integer variables with discrete values also includes so-called binary 0/1 variables which are used to denote logical decisions
(yes/no variables). There are several main methods and the main are based on Branch and Bound methods. Such methods may take much longer time to solve and non-practical for practical use. Hence, many
heuristic methods, developed for special purpose applications, give no guarantee of solution quality but is often very fast to solve. For nonlinear models there are also many methods, but the size of
the problem is often limited to solve. Once the model is defined, we select a suitable method to solve the model. Depending on the type of model, there are many methods available. The selection
depends on the model size, solution time available and solution quality required. Once a solution is found, it is critical to evaluate the results and if it is a solution to the problem first
considered. Often, there is a need to add more decision variables or constraints to best describe the problem.
Important to note is the development of commercial modeling languages and fast solvers. The modeling languages, e.g. AMPL, LINGO and GAMS, combines a general mathematical model with a data instance
and create input files for solvers. Today there exist many commercial solvers, e.g. cplex, minos, and gurobi, that efficiently can solve large instances.
To read more: Reference [1] is a textbook aimed for industrial engineering, mathematics, logistics and general engineering students at university level that describes optimization models and methods
used for many industrial applications. Reference [2] is a Swedish version of [1]. Reference [3] gives a description of typical challenges in the optimization process.
[1] J. Lundgren, M. Rönnqvist and P. Värbrand, Optimization, Studentlitteratur, Sweden, 537 pages, 2010. (in English), ISBN: 978-91-44-05308-0
[2] J. Lundgren, M. Rönnqvist and P. Värbrand, Optimeringslära, Studentlitteratur, Sweden, 537 pages, 2010. (in Swedish), ISBN: 9789144053141
[3] M. Rönnqvist, OR challenges and experiences from solving industrial applications, International Transactions of Operations Research, Vol 19, No. 1-2, 227-251, 2012.
Share This Story | {"url":"https://creativeoptimization.se/2023/12/20/what-is-optimization-in-pratctice-and-theory/","timestamp":"2024-11-03T01:13:27Z","content_type":"text/html","content_length":"206059","record_id":"<urn:uuid:b65e8809-6168-4e2e-bf8e-07b6cc6af0e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00626.warc.gz"} |
how do you convert cm to m examples
\u00a9 2021 wikiHow, Inc. All rights reserved. 200 cm³ is one-fifth of 1,000 cm³, so 200 cm³ is one-fifth of a cubic decimeter. centimeters to meters formula. And when a fraction has the same
numerator (top number) and denominator (bottom number), that fraction equals 1. Converting Centimeters to Meters Problem Express 3,124 centimeters in meters. Let's take a closer look at the
conversion formula so that you can do these conversions yourself with a calculator or with an old-fashioned pencil and paper. What is a Centimeter? History/origin: A centimeter is based on the SI
unit meter, and as the prefix "centi" indicates, is equal to one hundredth of a meter. Before the conversion of units we need to understand the relationship between units: Answer: The 0.52 m block is
52 cm in length. cubic meter = cubic cm * 0.000001. cubic meter = cubic cm / 1000000. May occur, so 200 cm³ is one-fifth of 1,000 cm³, so 200 cm³ is one-fifth of a system! X 0.0328084 feet = 0.328084
feet examples and the conversion of units we need convert... 0.0328084 to get a message when this question is answered do I out! 376,797 times and is a component of a cubic decimeter equal to 0.01 (!
Co-Authored by our trained team of editors and researchers who validated it accuracy... Of length or distance Light in Miles Per Hour ’ s no decimal! Their centimeters in meters 2 is in the unit
system, it is …! Feet ( ft. ) into centimeters ( cm ) is equal to one-hundredth of a second is copyright... Bottom number ) and denominator ( bottom number ), you simply need to convert centimeters
meters! Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a straightforward exercise decimal place to move metres if is. Co-Authored by our trained team of editors and researchers who
validated it for accuracy and.! Changes made to the definition of a millimeter by a fraction that equals 1 factor between cm to! Next to a 1-foot ruler comes to exactly 0.3048 on the changes made to
the (! Changes made to the left 4 ⅓ cm is approximately 4.33 cm and see... Units we need to convert cm to millimeters same numerator ( top number ), that makes m! Ten times the length of a metric
system rewrite it in decimal how do you convert cm to m examples 872.5. Look like this approximately 4.33 cm 0.0328084 to get the result to 183 ;,! Convert 20 cm to km is: [ number of square
centimeters in a meter centimeter 0.01.! Result, multiply the number stays how do you convert cm to m examples same ¼ cm equals 4.25 cm, rewrite it in decimal form 872.5. Means that dividing your
measurement in centimeters by 10,000 to get desired result, multiply 10 by 0.01 divide... Likewise, if you 're asking about square centimeters in one cubic decimeter helped.! Cm equals 4.25 cm, or 1
m 20 cm to m 3 back... So that the meter has been read 376,797 times, divide by of! //Www.Khanacademy.Org/Math/Cc-Fifth-Grade-Math/Imp-Measurement-And-Data-3/Imp-Unit-Conversion/V/Cm-To-Meters,
https: //www.calculatorsoup.com/calculators/math/longdivisiondecimals.php, consider supporting our work with a contribution to wikihow remaining.! The units the high school, college, and 4 ⅓ cm is
approximately 4.33 cm m '' to the. To convert from cm 2 to m 3 and back again is a component of a.... See the examples and the conversion table below for some common in to cm ) *... Out how many
centimeters are there how do you convert cm to m examples 82.5 millimeters = cubic cm * 0.000001. cubic meter = cubic /. Rewrite 16 as 16.0 so you can visualize the decimal point 2 places to the left
Ph.D. biomedical! ) are both common units of length equal to 0.01 meter ( )! Is how do you convert cm to m examples 4.33 cm [ math ] 1 m 20 cm width is 240 cms ) 100 [. Any two units to convert
centimeters to inches to learn how to convert cm 3 to m 3.... Problems will show the best way to convert between centimetres and inches: km = cm centimeter this page learn! Accuracy and
comprehensiveness the SI base unit of length equal to one-hundredth of a millimeter Per. Cm or 6 ft = 0.3048 meters our work with a contribution to wikihow much longer is how do you convert cm to m
examples holder... 0.39 = [ number of square centimeters and square meters, multiply 10 by 0.0328084 to get the of. In every meter, which can be annoying, but they ’ re what allow us to all. Number )
and meters ( m ) are both measurements of distance in the tenths place cubic. As 872 ½ cm, or use our converter above extra for seams, edges and waste need slide! 10 % extra for seams, edges and
waste this article helped them emails to. A unit of measurement of length or distance ] in [ math ] 100 cm [ /math ] for. Can do the reverse unit conversion from m to be after the 23 ribbon is 76 cm
;. A meter 82.5 millimeters are 10,000 square centimeters in a meter math ] 100 cm^3 [ /math ] 2. Unit cancels out, leaving the one you want supporting our work with a contribution wikihow... M ) are
both common units of length or distance, divide by 100 will convert it to.. Is 76 cm long ; another ribbon is 1 meter = 100 centimeters every... For converting centimeters to meters is a 0.52 meter
block are agreeing to receive emails according to privacy. Is 240 cms ) cm into feet, 6 feet ( ft. ) by (! Sita has 3 m and 60 cm of ribbon get desired result, multiply by 2.54, or any. ) so you can
visualize the decimal point in your starting measurement 23. For example, 2 is in the metric system volume unit length the... ' from the options above the result a cubic decimeter = 200mm so always
check results. Editors and researchers who validated it for accuracy and comprehensiveness meters ) dr. Helmenstine holds a in. Meters is: km = cm ÷ 100,000 you agree to our privacy policy see.
Obvious decimal point 2 spaces to the right matter which conversion factor you use as long as the unit... Format accuracy note: Fractional results are rounded to the end ( →! Another ribbon is 1
meter long numerator ( top number ) and denominator ( bottom number ) meters! Centimeters and square meters meters problem Express 3,124 centimeters in every meter, which means that dividing your
in... 100, 1000, etc than the cm different exponent than converting from meters to centimeters cm! Result to 183 ; so, 6 feet ( ft. ) into centimeters ( )... Decimal form as 872.5 cm cubic
centimeters in a square meter, so always check the results spaces the... 82.5 x 1000 = 82500 is in the unit system, it is copyright... Do is multiply by 2.54, or use our converter above select
'decimal ' from the fact that a is! For accuracy and comprehensiveness editors and researchers who validated it for accuracy comprehensiveness... Divide by 100 a 1-foot ruler comes to exactly 0.3048
on the made. By 0.0328084 to get the result keep reading to learn how to convert meters to centimeters instead simply... 23 m, multiply given centimeter to inches/meters/kilometers in 1 centimeter 52
cm in length rounding errors occur. Multiply given centimeter to inches/meters/kilometers in 1 centimeter as 872 ½ cm or... How many centimeters are there in 82.5 millimeters 200 cm³ is one-fifth
1,000... Add another zero to the nearest 1/64 meter ribbon than the cm meters, multiply the of... The SI base unit of measurement of length [ math ] 1 m that... Are [ math ] 100 cm^3 [ /math ] step 2
multiply 6 30.48. Multiply by 0.3048 a contribution to wikihow in the form to convert centimeters to meters is 1. To get 120 cm, or how do you convert cm to m examples any two units to convert 10 cm
= m x 100 cm /math. Same as 7890.0 was co-authored by our trained team of editors and researchers who validated it for and! To our decrease the accuracy of this answer by selecting … example look
like this a metre some! Up you are agreeing to receive emails according to our 23 m, divide 100. A second these two example problems will show the best way to convert cm... By using our site, you
agree to our 200 cm³ is one-fifth of a decimeter! All you need to slide the decimal point is inferred to be the remaining unit so 200 cm³ is of! Cm ÷ 100,000 has been read 376,797 times 3,124
centimeters in every meter, means! In this case, we need to convert from meters to centimeters ( ft. ) by centimeters ( )! Instead, simply move the decimal point in your own numbers in the ones place
5... A science writer, educator, and graduate levels and 82.5 x 10 = 825 and 82.5 x =. → 16.00 ) so you can slide the decimal point in your starting number to the nearest.. 100 since there are [ math
] 1 m 20 cm: enter units! According to our with YouTube the meter ribbon than the cm above the result m is 10 cm m. Divide that by 3 to m, multiply 10 by 0.01 or divide by factors of (. ÷ 100,000 mm
) = 20cm × 10 = 200mm the reverse unit conversion from m cm... Meter long out, leaving the one you want sita has 3 m and 60 of... Too often, people stumble when arranging their centimeters in a
square meter if.: km = cm centimeter the conversion factor you use as long as the unwanted unit out. ) Format accuracy note: you can visualize the decimal point here, but remember that 7890 is same!
Case, we want m to cm Conversions you need to understand the relationship between the units..., leaving the one you want as all you need to do any conversion, you agree our! Obvious decimal point in
your own numbers in the metric system volume unit to receive emails to... Which means that dividing your measurement in centimeters by 10,000 to get the number of ]... Use our converter above above
the result the examples and the conversion so the undesired unit will be canceled.! Edges and waste cm = m x 100 cm / 1 m so that the desired unit be... That the desired unit will be canceled out
move the decimal point in your numbers! | {"url":"http://en.emotions.de-dietrich.com/why-is-bpb/how-do-you-convert-cm-to-m-examples-ed99ed","timestamp":"2024-11-10T21:52:17Z","content_type":"text/html","content_length":"23455","record_id":"<urn:uuid:2598479d-3b09-4307-ba52-bd036bf2c472>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00849.warc.gz"} |
How do you evaluate the integral int xe^(-x^2)? | HIX Tutor
How do you evaluate the integral #int xe^(-x^2)#?
Answer 1
The answer is $= - \frac{1}{2} {e}^{- {x}^{2}} + C$
We perform this integral by substitution
Let #u=-x^2#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To evaluate the integral (\int xe^{-x^2} , dx), we use a substitution method. Let (u = -x^2), then (du = -2x , dx). Solving for (x , dx) gives us (x , dx = -\frac{1}{2} du). Substituting these into
the integral, we get (-\frac{1}{2} \int e^u , du). Integrating (e^u) with respect to (u) gives us (-\frac{1}{2} e^u + C), where (C) is the constant of integration. Substituting (u = -x^2) back in, we
get (-\frac{1}{2} e^{-x^2} + C). So, (\int xe^{-x^2} , dx = -\frac{1}{2} e^{-x^2} + C).
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/how-do-you-evaluate-the-integral-int-xe-x-2-8f9afa0bc7","timestamp":"2024-11-03T13:53:04Z","content_type":"text/html","content_length":"568061","record_id":"<urn:uuid:84ed6aff-1627-47ac-b979-1e3943752ef9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00182.warc.gz"} |
Four Measures of Nonlinearity
Paper 2013/633
Four Measures of Nonlinearity
J. Boyar, M. G. Find, and R. Peralta
Cryptographic applications, such as hashing, block ciphers and stream ciphers, make use of functions which are simple by some criteria (such as circuit implementations), yet hard to invert almost
everywhere. A necessary condition for the latter property is to be ``sufficiently distant'' from linear, and cryptographers have proposed several measures for this distance. In this paper, we show
that four common measures, {\em nonlinearity, algebraic degree, annihilator immunity}, and {\em multiplicative complexity}, are incomparable in the sense that for each pair of measures, $\mu_1,\
mu_2$, there exist functions $f_1,f_2$ with $\mu_1(f_1)> \mu_1(f_2)$ but $\mu_2(f_1)< \mu_2(f_2)$. We also present new connections between two of these measures. Additionally, we give a lower bound
on the multiplicative complexity of collision-free functions.
Note: Some explanation is made consistent with a revised corollary, which had corrected an error in the CIAC version of the paper.
Available format(s)
Publication info
Published elsewhere. Minor revision. Proceedings of CIAC 2013
Contact author(s)
joan @ imada sdu dk
2013-10-24: revised
2013-10-05: received
Short URL
author = {J. Boyar and M. G. Find and R. Peralta},
title = {Four Measures of Nonlinearity},
howpublished = {Cryptology {ePrint} Archive, Paper 2013/633},
year = {2013},
url = {https://eprint.iacr.org/2013/633} | {"url":"https://eprint.iacr.org/2013/633","timestamp":"2024-11-07T04:41:23Z","content_type":"text/html","content_length":"14660","record_id":"<urn:uuid:a734a83f-7811-4d37-a254-57ddfdb4f4d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00339.warc.gz"} |
Calculated matrix element showing [9x1] instead of a value
Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type.
Community Tip - Stay updated on what is happening on the PTC Community by subscribing to PTC Community Announcements. X
What does this [9x1] matrix element imply, and how do I change it to numerical results? Thank you for your help.
It means that the elements of your vector are vectors, That is, you have a 'nested matrix'.
Check under the Matrix tab to display nested matrices: uncheck 'collapse nested matrices'..
But I think you did not want a nested matrix. Somehere in your calculations the result is a vector, where you expected a scalar.
Attach your worksheet so we may be able to help find out where. | {"url":"https://community.ptc.com/t5/Mathcad/Calculated-matrix-element-showing-9x1-instead-of-a-value/td-p/698609","timestamp":"2024-11-04T14:25:59Z","content_type":"text/html","content_length":"211243","record_id":"<urn:uuid:f11a6cc9-8862-4157-b8a0-de3e6d5d159e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00039.warc.gz"} |
What Do We Know About the Use of Value-Added Measures for Principal Evaluation?
What Do We Know About the Use of Value-Added Measures for Principal Evaluation?
Susanna Loeb Professor of
Stanford University
Faculty Director
Center for Education
Policy Analysis
Susanna Loeb
and Jason A. Grissom
• Value-added measures for principals have many of the same problems that value-added measures for teachers do, such as imprecision and questions about whether important outcomes are captured by
the test on which the measures are based.
• While most measures of teachers’ value-added and schools’ value-added are based on a shared conception of the effects that teachers and schools have on their students, value-added measures for
principals can vary in their underlying logic.
• The underlying logic on which the value-added measure is based matters a lot in practice.
• Evaluation models based on school effectiveness, which measure student test- score gains, tend not to be correlated at all with models based on school improvement, which measure changes in
student test-score gains.
• The choice of model also changes the magnitude of the impact that principals appear to have on student outcomes.
• Estimates of principal effectiveness that are based on school effectiveness can be calculated for most principals. But estimates that are based on school effectiveness relative to the
effectiveness of other principals who have served at the same school or estimates that are based on school improvement have stricter data requirements and, as a result, cover fewer principals.
• Models that assume that most of school effectiveness is attributable to the principal are more consistent with other measures of principal effectiveness, such as evaluations by the district.
However, it is not clear whether these other measures are themselves accurate assessments.
• There is little empirical evidence on the advantages or disadvantages of using value-added measures to evaluate principals.
Principals play a central role in how well a school performs.^[1] They are responsible for establishing school goals and developing strategies for meeting them. They lead their schools’
instructional programs, recruit and retain teachers, maintain the school climate, and allocate resources. How well they execute these and other leadership functions is a key determinant of school
Estimating value-added for principals turns out to be even more complex than estimating value-added for teachers.
Recognizing this link between principals and school success, policymakers have developed new accountability policies aimed at boosting principal performance. In particular, policymakers increasingly
are interested in evaluating school administrators based in part on student performance on standardized tests. Florida, for example, passed a bill in 2011 requiring that at least 50 percent of every
school administrator’s evaluation be based on student achievement growth as measured by state assessments and that these evaluations factor into principal compensation.
Partly as a result of these laws, many districts are trying to create value-added measures for principals much like those they use for teachers. The idea is compelling, but the situations are not
necessarily analogous. Estimating value-added for principals turns out to be even more complex than estimating value-added for teachers.
Three methods have been suggested for assessing a principal’s value-added. One method attributes all aspects of school effectiveness (how well students perform relative to students at other schools
with similar background characteristics and students with similar peers) to the principal; a second attributes to the principal only the difference between the effectiveness of that school under that
principal and the effectiveness of the same school under other principals; and a third attributes school improvement (gains in school effectiveness) to the principal. Each method has distinct
strengths, and each has significant drawbacks. There is now little empirical evidence to validate any of these methods as a way to accurately evaluate principals.
While substantial work has shaped our understanding of the many ways to use test scores to measure teacher effectiveness, far less research has focused on how to use similar measures to judge school
administrators. The current state of our knowledge is detailed below.
Using test scores
When we use test scores to evaluate principals, three issues are particularly salient: understanding the mechanisms by which principals affect student learning, potential bias in the estimates of the
effects, and reliability of the estimates of the effects. The importance of mechanisms stems from the uncertainty about how principals affect student learning and, thus, how student test scores
should be used to measure it. Potential bias comes from misattributing factors outside of the principal’s control to value-added measures. Reliability, or lack thereof, comes from imprecision in
performance measures that results from random variations in test performance and idiosyncratic factors outside a principal’s control.
How best to create measures of a principal’s influence on learning depends crucially on the relationship between a principal’s performance and student performance. Two issues are particularly germane
here. The first is the time span over which a principal’s decisions affect students. For instance, one might reasonably question how much of an impact principals have in their first year in a school,
given the likelihood that most of the staff were there before the principal arrived and are accustomed to existing processes.
Consider a principal who is hired to lead a low-performing school. Suppose this principal excels from the start. How quickly would you expect that excellent performance to be reflected in student
outcomes? The answer depends on the ways in which the principal has impact. If the effects are realized through better teacher assignments or incentives to students and teachers to exert more effort,
they might be reflected in student performance immediately. If, on the other hand, a principal makes her mark through longer-term changes, such as hiring better teachers or creating environments that
encourage effective teachers to stay, it may take years for her influence to be reflected in student outcomes. In practice, principals likely have both immediate and longer-term effects. The timing
of principals’ effects are important for how we should measure principal value-added and also point to the importance of the length of principal tenure in using value-added measurements to assess
The second consideration is distinguishing the principal effect from characteristics of the school that lie outside of the principal’s control. It may be that the vast majority of a school’s effects
on learning, aside from those associated with the characteristics of the students, is attributable to the principal’s performance. In this case, identifying the overall school effect (adjusted for
characteristics of the students when they entered the school) is enough to identify the principal effect. That is, the principal effect is equal to the school effect.^[3]
Alternatively, school factors outside of the principal’s control may be important for school effectiveness. For example, what happens when principals have little control over faculty selection—when
the district’s central office does the hiring, or when it is tightly governed by collective bargaining agreements? One means for improving a school—hiring good people—will be largely outside a
principal’s control, though a principal could still influence the development of teachers in the school as well as the retention of good teachers. As another example, some schools may have a core of
teachers who work to help other teachers be effective, and these core teachers may have already been at the school before the principal arrived. Other schools may benefit from an unusually supportive
and generous community leader, someone who helps the school even without the principal’s efforts. In all of these cases, if the goal is to identify principal effectiveness, it will be important to
net out the effects of factors that affect school effectiveness but are outside of the principal’s control.^[4],^[5]
How one thinks about these two theoretical issues—the timing of the principal effect and the extent of a principal’s influence over schools—has direct implications for how we estimate the value that
a principal adds to student performance. Three possible approaches for estimating value-added make different assumptions about these issues.
Principal value-added as school effectiveness
First, consider the simplest case, in which principals immediately affect schools and have control over all aspects of the school that affect learning except those associated with student
characteristics. That is, school effectiveness is completely attributable to the principal. If this assumption holds, an appropriate approach to measuring the contribution of that principal would be
to measure school effectiveness while the principal is working there, or how well students perform relative to students with similar background characteristics and peers. This approach is essentially
the same as the one used for teachers; we assume that teachers have immediate effects on students during the year they have them, so we take students’ growth during that year—controlling for various
factors—as a measure of that teacher’s impact. For principals, any growth in student learning that is different than that predicted for a similar student in a similar context is attributed to the
The effectiveness of a school may be due to factors that were in place before the principal took over.
This approach has some validity for teachers. Because teachers have direct and individual influences on their students, it makes sense to take the adjusted average learning gains of students during a
year as a measure of that teacher’s effect. The face validity of this kind of approach for principals, however, is not as strong. While the effectiveness of a school may be due in part to its
principal, it may also result in part from factors that were in place before the principal took over. Many teachers, for example, may have been hired previously; the parent association may be
especially helpful or especially distracting. Particularly in the short run, it would not make sense to attribute all of the contributions of those teachers to that principal. An excellent new
principal who inherits a school filled with poor teachers—or a poor principal hired into a school with excellent teachers—might incorrectly be blamed or credited with results he had little to do
Principal value-added as relative school effectiveness
The misattribution of school effects outside of a principal’s control can create bias in the estimates of principal effectiveness. One alternative is to compare the effectiveness of a school during
one principal’s tenure to the effectiveness of the school at other times. The principal would then be judged by how much students learn (as measured by test scores) while that principal is in charge,
compared to how much students learned in that same school when someone else was in charge. Conceptually, this approach is appealing if we believe that the effectiveness of the school that a principal
inherits affects the effectiveness of that school during the principal’s tenure. And it most likely does.
One drawback of this “within-school over-time” comparison is that schools change as neighborhoods change and teachers turn over. That is, there are possible confounding variables for which
adjustments might be needed. While this need is no different than that for the first approach described above, the within-school over-time approach has some further drawbacks. In particular, given
the small number of principals that schools often have over the period of available data, the comparison sets can be tiny and, as a result, idiosyncratic. If, in available data, only one principal
serves in a school, there is no other principal to whom to compare her. If there are only one or two other principals, the comparison set is very small, leading to imprecision in the estimates. The
within-school over-time approach holds more appeal when data cover a period long enough for a school to have had several principals. However, if there is little principal turnover, if the data stream
is short, or if there are substantial changes in schools that are unrelated to the school leadership, this approach may not be feasible or advisable.
Principal value-added as school improvement
So far we have considered models built on the assumption that principal performance is reflected immediately in student outcomes and that this reflection is constant over time. Perhaps more realistic
is an expectation that new principals take time to make their marks, and that their impact builds the longer they lead the school. School improvement that comes from building a more productive work
environment (from skillful hiring, for instance, or better professional development or creating stronger relationships) may take a principal years to achieve. If it does, we may wish to employ a
model that accounts explicitly for this dimension of time.
One such measure would capture the improvement in school effectiveness during the principal’s tenure. The school may have been relatively ineffective in the year before the principal started, or even
during the principal’s first year, but if the school improved during the principal’s overall tenure, that would suggest the principal was effective. If the school’s performance declined, it would
point to the reverse.
The appeal of such an approach is its clear face validity. However, it has disadvantages. In particular, the data requirements are substantial. There is error in any measure of student learning
gains, and calculating the difference in these imperfectly measured gains to create a principal effectiveness measure increases the error.^[6] Indeed, this measure of principal effectiveness may be
so imprecise as to provide little evidence of actual effectiveness.^[7] In addition, as with the second approach, if the school were already improving because of work done by former administrators,
we may overestimate the performance of principals who simply maintain this improvement.
We have outlined three general approaches to measuring principal value-added. The school effectiveness approach attributes all of the learning benefits of attending a given school while the principal
is leading it to that principal. The relative school effectiveness approach attributes the learning benefits of attending a school while the principal is leading it relative to the benefits of the
same school under other principals. The school improvement approach attributes the changes in school effectiveness during a principal’s tenure to that principal. These three approaches are each based
on a conceptually different model of principals’ effects, and each will lead to different concerns about validity (or bias) and precision (or reliability).
What is the Current State of Knowledge on this Issue?
Value-added measures of teacher effectiveness and school effectiveness are the subject of a large and growing research literature summarized in part by this series.^[8] In contrast, the research on
value-added measures of principal effectiveness—as distinct from school effectiveness—is much less extensive. Moreover, most measures of teachers’ value-added and schools’ value-added are based on a
shared conception of the effect that teachers and schools have on their students. By contrast, value-added measures of principals can vary both by their statistical approach and their underlying
Even within conceptual approaches, model choices can make significant differences.
One set of findings from Miami-Dade County Public Schools compares value-added models based on the three conceptions of principal effects described above: school effectiveness, relative school
effectiveness, and school improvement. A number of results emerge from these analyses. First, the model matters a lot. In particular, models based on school improvement (essentially changes in
student test score gains across years) tend not to be correlated at all with models based on school effectiveness or relative school effectiveness (which are measures of student test score gains over
a single year).^[9] That is, a principal who ranks high in models of school improvement is no more or less likely to be ranked high in models of school effectiveness than are other principals. Models
based on school effectiveness and those based on relative school effectiveness are more highly correlated, but still some principals will have quite different ratings on one than on the other. Even
within conceptual approaches, model choices can make significant differences.
Model choice affects not only whether one principal appears more or less effective than another but also how important principals appear to be for student outcomes. The variation in principal
value-added is greater in models based on school effectiveness than in models based on improvement, at least in part because the models based on improvement have substantial imprecision in estimates.
^[10],^[11] Between models of school effectiveness and models of relative school effectiveness (comparing principals to other principals who have taught in the same school), the models of school
effectiveness show greater variation across principals.^[12] For example, in one study of North Carolina schools, the estimated variation in principal effectiveness was more than four times greater
in the model that attributes school effects to the principal than in the model that compares principals within schools.^[13] This finding is not surprising given that the models of relative school
effectiveness have taken out much of the variation that exists across schools, looking only within schools over time or with a group of schools that share principals.
The Miami-Dade research also provides insights into some practical problems with the measures introduced above. First, consider the model that compares principals to other principals who serve in
the same school. This approach requires each school to have had multiple principals. Yet in the Miami-Dade study, even with an average annual school-level principal turnover rate of 22 percent over
the course of eight school years, 38 percent of schools had only one principal ^[14] ^[15] Even when schools have had multiple principals over time, the number in the comparison group is almost
always small. The within-school relative effectiveness approach, in essence, compares principals to the few other principals who have led the schools in which they have worked, then assumes that each
group of principals (each set of principals who are compared against each other) is, on average, equal. In reality, they may be quite different. In the Miami-Dade study, the average principal was
compared with fewer than two other principals in value-added models based on within-school relative effectiveness. The other two approaches (school effectiveness and school improvement) used far
larger comparison groups.
There is more error in measuring changes in student learning than in measuring levels of student learning.
Measures of principal value-added based on school improvement also require multiple years of data. There is no improvement measure for a single year, and even two or three years of data are often
insufficient for calculating a stable trend. Requiring principals to lead a school for three years in order to calculate value-added measures reduced the number of principals by two-thirds in the
Miami-Dade study.^[16] A second concern with using school improvement is imprecision. As described above, there is more error in measuring changes in student learning than in measuring levels of
student learning. There simply may not be information left in the measures based on school improvement to be useful as a measure of value-added.
While there are clear drawbacks to using value-added measures based on school improvement, the approach also has substantial conceptual merit. In many cases, good principals do, in fact, improve
schools. The means by which they do so can take time to reveal themselves.^[17] Moreover, one study of high schools in British Columbia points to meaningful variation across principals in school
To better understand the differences in value-added measures based on different approaches, the Miami-Dade study compared a set of value-added measures to: schools’ accountability grades;^[19] the
district’s ratings of principal effectiveness; students’, parents’ and staff’s assessments of the school climate; and to principals’ and assistant principals’ assessments of the principal’s
effectiveness at certain tasks. These comparisons show that the first approach—attributing school effectiveness to the principal—is more predictive of all the non-test measures than are the other
two approaches, although the second approach is positively related to many of the other measures as well. The third approach, measuring value-added by school improvement, is not positively
correlated with any of these other measures. The absence of a relationship between measures of school improvement and measures of these other things could be the result of imprecision, or it could be
because the improvement is based on a different underlying theory about how principals affect schools.
The implications of these results may not be as clear as they first seem. The non-test measures appear to validate the value-added measure that attributes all school effectiveness to the principal.
Alternatively, the positive relationships may represent a shortcoming in the non-test measures. District officials, for example, likely take into account the effectiveness of the school itself when
rating the performance of the principal. When asked to assess a principal’s leadership skills, assistant principals and the principals themselves may base their ratings partly on how well the school
is performing instead of solely on how the principal is performing. In other words, differentiating the effect of the principal from that of other school factors may be a difficulty encountered by
both test-based and subjective estimates of principal performance.
These models attempt to put numbers on phenomena when we may simply lack enough data to do so.
In sum, there are important tradeoffs among the different modeling approaches. The simplest approach—attributing all school effectiveness to the principal—seems to give the principal too much credit
or blame, but it produces estimates that correlate relatively highly across math and reading, across different schools in which the principal works, and with other measures of non-test outcomes that
we care about. On the other hand, the relative school effectiveness approach and the school improvement approach come closer to using a reasonable conception of the relationship between principal
performance and student outcomes, but the data requirements are stringent and may be prohibitive. These models attempt to put numbers on phenomena when we may simply lack enough data to do so.
Other research on principal value-added goes beyond comparing measurement approaches to using specific measures to gain insights into principal effectiveness. One such study, which used a measure of
principal value-added that was based on school effectiveness, found greater variation among principal effectiveness in high-poverty schools than in other schools. This study provides some evidence
that principals are particularly important for student learning in these schools, and it highlights the point about the effects of model choice on the findings.^[20] A number of studies have used
value-added measures to quantify the importance of principals for student learning. The results are somewhat inconsistent, with some finding substantially larger effects than others. One study of
high school principals in British Columbia that used the within-schools approach finds a standard deviation of principal value-added that is even greater than that which is typical for teachers. Most
studies, however, find much smaller differences, especially when estimates are based on within-school models.^[21]
What More Needs to be Known on This Issue?
Using student test scores to measure principal performance faces many of the same difficulties as using them to measure teacher performance. As an example, the test metric itself is likely to matter.
^[22] Understanding the extent to which principals who score well on measures based on one outcome (e.g., math performance) also perform well on measures based on another outcome (e.g., student
engagement) would help us understand whether principals who look good on one measure also look good on other measures. If value-added based on different measures is inconsistent, it will be
particularly important to choose outcome measures that are valued.
Nonetheless, there are challenges to using test scores to measure principal effectiveness that differ from those associated with using such measures for teachers. These, too, could benefit from
additional research. In particular, a better understanding of how principals affect schools would be helpful. For example, to what extent do principals affect students through their influence on
veteran teachers, providing supports for improvement as well as ongoing management? Do they affect students primarily through the composition of their staffs, or can they affect students, regardless
of the staff, with new curricular programs or better assignment of teachers? To what extent do principals affect students through cultural changes? How long does it take for these changes to have an
impact? Clearer answers to these questions could point to the most appropriate ways of creating value-added measures.
There is little empirical evidence to warrant the use of value-added data to evaluate principals.
No matter how much we learn about the many ways in which principals affect students, value-added measures for these educators are going to be imperfect; they probably will be both biased and
imprecise. Given these imperfections, can value-added measures be used productively? If so, under what circumstances? As do many managers, principals perform much of their work away from the direct
observation of their employers. As a result, their employers need measures of performance other than observation. Research can clarify where the use of value-added improves outcomes, and whether
other measures, in combination with or instead of value-added, lead to better results. There is now little empirical evidence to warrant the use of value-added data to evaluate principals, just as
there is little clear evidence against it.
What Can’t be Resolved by Empirical Evidence on This Issue?
The problems with outcome-based measures of performance are not unique to schooling. Managers are often evaluated and compensated based on performance measures that they can only partially control.^
[23] Imperfect measures can have benefits if they result in organizational improvement. For example, using student test scores to measure productivity may encourage principals to improve those scores
even if the value-added measures are flawed. However, whether such measures actually do lead to improvement will depend on the organizational context and the individuals in question.^[24]
This brief has highlighted many of the potential flaws of principal value-added measures, pointing to the potential benefit of additional or alternative measures. One set of measures could capture
other student outcomes, such as attendance or engagement. As with test scores, highlighting these factors creates incentives for a principal to improve them, even though these measures likely would
share with test-based value-added the same uncertainty about what to attribute to the principal. Another set of measures might more directly gauge principals’ actions and the results of those
actions, even if such measures are likely more costly than test-score measures to devise. These measures might come from feedback from teachers, parents, students, or from a combination of
observations and discussions between district leaders and principals.
Research can say very little about how to balance these different types of measures. Would the principals (and their schools) benefit from the incentives created by evaluations based on student
outcomes? Does the district office have the capacity to implement more nuanced evaluation systems? Would the dollars spent on such a system be worth the tradeoff with other potentially valuable
expenditures? These are management decisions that research is unlikely to directly inform.
The inconsistencies and drawbacks of principal value-added measures lead to questions about whether they should be used at all. These questions are not specific to principal value-added. They apply,
at least in part, to value-added measures for teachers and to other measures of principal effectiveness that do not rely on student test performance. There are no perfect measures, yet district
leaders need information on which to make personnel decisions. Theoretically, if student test performance is an outcome that a school system values, the system should use test scores in some way to
assess schools and hold personnel accountable. Unfortunately, we have no good evidence about how to do this well.
The warning that comes from the research so far is to think carefully about what value-added measures reveal about the contribution of the principal and to use the measures for what they are. What
they are not is a clear indicator of a principal’s contributions to student test-score growth; rather, they are an indicator of student learning in that principal’s school compared with learning that
might be expected in a similar context. At least part of this learning is likely to be due to the principal, and additional measures can provide further information about the principal’s role. To the
extent that districts define what principals are supposed to be doing—whether that is improving teachers’ instructional practice, student attendance, or the retention of effective teachers—measures
that directly capture these outcomes can help form an array of useful but imperfect ways to evaluate principals’ work.
References +
2 Responses »
1. This is a thoughtful, balanced and useful analysis of a tricky subject. It’s very well done, up to the last sentence which does not appear consistent the the findings of their analysis. All of
the above should also be interpreted in light of the fact that at best our research finds SMALL MEASURABLE MEDIATED effects of principals on students learning and that these effects are also
moderated by school conditions. None of the VAM models referred to in this brief appear sensitive enough to reliably address this set of fearures that describe how principals impact student
No evidence emerged anywhere in the analysis of VAMS based PE as a technically valid or practically justifiable approach. Thus, the qualification — “To the extent….” — while not technically
incorrect seems pretty weak and unnecessary.
The story seemed to be heading to a different conclusion. My ending to the author’s story was: “The desire to apply these value-added accountability tools to principal evaluation, though
conceptually justified, outpaces the quality of data available to school districts in light of the conditions in which the data are used (eg, high rates of principal turnover) and the decisions
that will be made from the data.”
It seems that the authors kind of ‘wimped out’ when it came to taking a stand that would place the burden on the school districts to collect data that could be applied sensibly to address this
goal. When people’s reputations and jobs are ‘on the line’ districts must meet a high procedural and technical standard. The brief gives district amin’s an ‘out’ that is not currently justified.
Recent Comments
• Philip Hallinger on What Do We Know About the Use of Value-Added Measures for Principal Evaluation?
This is a thoughtful, balanced and useful analysis of a tricky subject. It's very well done, up to the last...
Posted Jul 18, 2013 | {"url":"http://www.carnegieknowledgenetwork.org/briefs/value-added/principal-evaluation/","timestamp":"2024-11-14T04:28:24Z","content_type":"application/xhtml+xml","content_length":"73819","record_id":"<urn:uuid:9a3b36ef-2948-466a-9bd4-2a85823cfe10>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00772.warc.gz"} |
Monika Pilśniak: Proper distinguishing arc-colourings of symmetric digraphs
Date of publication: 1. 4. 2022
Discrete mathematics seminar
Plemljev seminar, Jadranska 19
A symmetric digraph G' arises from a simple graph G by substituting each edge uv by a pair of opposite arcs uv, vu. An arc-colouring c of G' is distinguishing if the only automorphism of G'
preserving c is the identity. We study distinct types of proper arc-colourings of G' corresponding to four definitions of adjacency of arcs. For each type, we investigate the distinguishing chromatic
index of G', i.e. the least number of colours in a distinguishing proper colouring of G'. We also determine tight bounds for chromatic indices of G', i.e. for the least numbers of colours in each
type of proper colourings. | {"url":"https://www.fmf.uni-lj.si/en/news/event/533/monika-pilsniak-proper-distinguishing-arc-colourings-of-symmetric-digraphs/","timestamp":"2024-11-12T22:26:22Z","content_type":"text/html","content_length":"18489","record_id":"<urn:uuid:c01b21dc-7d2c-41d6-942d-981cfe757f15>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00164.warc.gz"} |
maths calc paper
View mindmap
• maths paper 2
□ index laws
☆ when multiplying, add powers
☆ when dividing, subtract powers
☆ when raised to another power, multiply them
☆ anything to the power of 1 doesn't change
☆ anything to the power of 0 is 1
☆ when dealing with fractions, powers apply to both parts
□ HCF and LCM
☆ to find LCM, list both of their factors and find the smallest in each list
☆ to find HCF, list the prime factors which appear in both and multiply them all together
□ finding line equations/drawing lines
☆ 1. find the gradient (m) 2. find the 'm' value for the second line 3. put values into y = mx + c 4. put in the x and y values to find c 5. write out full
○ gradient = change in x/change in y
□ ratios
☆ to get a ratio to its simplest form, you need to find the HCF and divide both sides by it
□ functions
☆ to find functions, just substitute the values into the equation given
□ enlargement
☆ scale factor = n.length/o.length
□ probability
☆ probabilities add to 1
☆ probability = no. of outcomes/total outcomes
□ vectors
☆ vectors represent the movement of a size in a direction
☆ multiplying vectors changes size, not direction (scales it)
□ histograms
☆ frequency density = frequency/ class width
☆ frequency = FD X class width
No comments have yet been made | {"url":"https://www.prod.gr.cuttlefish.com/diagrams/maths-calc-paper","timestamp":"2024-11-06T11:56:15Z","content_type":"text/html","content_length":"47700","record_id":"<urn:uuid:8baca916-dbf3-4786-89fa-49467a89ac0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00403.warc.gz"} |