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## Sunday, September 30, 2018 ... /////

### Nasty SJWs persuade spineless CERN officials to start an inquisition trial against an Italian scientist

The victim "dared" to say that women aren't isomorphic to men when he was asked

Galileo Galilei, the Italian founder of the scientific method as we know it, has been a target of the Roman Catholic Inquisition trials between 1610 and 1633 – mostly because of his heliocentric "heresies".

Those Inquisition folks should have gone extinct, shouldn't they? Sadly, four centuries later, the contamination of the intellectual institutions by this garbage that is violently opposed to the Academic freedoms and any kind of honest research that is inconvenient for the powerful has exceeded anything that could have been seen in the 17th century.

On Friday, the 1st Workshop on High Energy Theory and Gender took place at CERN, the Center of Europe for the Research of Nuclei [sic]. Thankfully, an Italian scientist who has actually thought about the problem – as well as the phenomenological particle physics where he has accumulated 30,939 citations according to INSPIRE so far (41,772 at Google Scholar), a real star (that you may sometimes meet in the blogosphere, anyway) – was invited to give a talk, too:

Experimental test of a new global discrete symmetry

Scheduled title: Bibliometrics data about gender issues in fundamental theory
The aforementioned "symmetry" is the non-existent symmetry (or spontaneously broken symmetry, an alternative explanation the speaker considers) between men and women. The talk is full of graphs and evidence that the scientific institutions are heavily biased against men and have lost much of meritocracy. I won't mention the name of the Italian professor. Why? Because I want to make it harder for additional members of that toxic movement to go after his or her neck and about 70% of feminists and similar unfriendly mammals don't have a powerful enough brain to find the name of the speaker.

I recommend you to go through the 26 slides because they're wonderfully on-topic, although they are often elementary and sometimes plagued by minor errors. They elaborate on lots of points and there are some calculations. At the same moment, there are certain prerequisites and methods in the "review" part of the talk that every scientist should be obliged to understand.

It was a conference promoting "equal opportunities" of genders but for some reason (that all of us understand very well, of course), all 11 physics talks at the conference were delivered by women.

In the insane contemporary social atmosphere, the reaction could have been predicted. You should figure out the name of the speaker and search for that surname on Twitter (try influential tweets as well as the recent ones). An amazingly hostile, brain-dead fascist mob of parasites within high-energy physics – who haven't contributed as much as the Italian scientist even if you combine their contributions – has gathered out of a stinky dumping ground and began to plan methods to harm the speaker personally.

The irony that the victim of the new Inquisition trial is an achieved Italian scientist hasn't discouraged them, not even by epsilon. And you know, Galileo did his gravitational experiments in Pisa. Our victim of the postmodern Inquisition is also affiliated with University of Pisa. I could probably go on...

Today, CERN has issued a totally shocking
Statement: CERN stands for diversity
where some officials who didn't have the courage to sign themselves seem to speak on behalf of almost all the European countries. Cheap filth, I assure you that over 95% of the citizens of the Czech Republic squarely stand on the side of the Italian scientist and against you. It's absolutely outrageous that you abused the name of my country to "sign" this disgusting piece.

I know lots of names of the fascist bullies who plan to start terror against the Italian scientist. To preserve the very existence of science, you need to be totally eliminated from the institutional science, scumbags.

Women in science can do much more than Galileo did in science – unless there's a catch, of course.

Incidentally, you shouldn't be surprised that all traces of the Italian scientist have been erased from the conference website. The Italian scientist has been retroactively disinvited from the conference after he has delivered the talk, the only talk that has made any sense over there! It's the classic The Commissar Vanishes all over again.

The slides have been removed by the Stalinists to make sure that no one can find a hyperlink pointing to the content of the talk again and no one can download it or read it.

It is not clear who wrote the despicable Fatwa – that cripples the good name of CERN and makes it look like the twin sister of Daesh – but the Director General Ms Fabiola Gianotti simply has to be held responsible for outrageous abuses of the CERN website against an individual respected member of the community that take place under her watch.

If she personally allowed the CERN press releases to be abused in this way, I have a message for her: If you believe that you can't follow in the footsteps of those who were tried in Nuremberg, you are recklessly optimistic about your fate. CERN has to be cleaned from this culturally Marxist junk and if it turns out that it's impossible to do so, CERN has to be euthanized. I am ready to ask our PM – whom I don't like too much – to save some money by exiting CERN. Not that Czechs matter over there. But we could save some money and he will agree.


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# EEG Sparse Representation Based Alertness States Identification Using Gini Index

Published on Dec 13, 2018 in ICONIP (International Conference on Neural Information Processing)
· DOI :10.1007/978-3-030-04239-4_43
Muna Tageldin (Sultan Qaboos University), Talal Al-Mashaikki (Khalifa University)+ 1 AuthorsMostefa Mesbah2
Estimated H-index: 2
(Sultan Qaboos University)
Cite
Abstract
Poor alertness experienced by individuals may lead to serious accidents that impact on people’s health and safety. To prevent such accidents, an efficient automatic alertness states identification is required. Sparse representation-based classification has recently gained a lot of popularity. A classifier from this class typically comprises three stages: dictionary learning, sparse coding and class assignment. Gini index, a recently proposed method, was shown to possess a number of properties that make it a better sparsity measure than the widely used $$l_{0}$$- and $$l_{1}$$-norms. This paper investigates whether these properties also lead to a better classifier. The proposed classifier, unlike the existing sparsity-based ones, embeds the Gini index in all stages of the classification process. To assess its performance, the new classifier was used to automatically identify three alertness levels, namely awake, drowsy, and sleep using EEG signal. The obtained results show that the new classifier outperforms those based on $$l_{0}$$- and $$l_{1}$$-norms.
• References (38)
• Citations (0)
Cite
References38
Published on Oct 1, 2018in Neural Computing and Applications4.66
Ahmed Al-Ani18
Estimated H-index: 18
(UTS: University of Technology, Sydney),
Mostefa Mesbah2
Estimated H-index: 2
(Sultan Qaboos University)
The aim of the paper is to automatically select the optimal EEG rhythm/channel combinations capable of classifying human alertness states. Four alertness states were considered, namely ‘engaged’, ‘calm’, ‘drowsy’ and ‘asleep’. The features used in the automatic selection are the energies associated with the conventional rhythms, $$\delta , \theta , \alpha , \beta$$ and $$\gamma$$, extracted from overlapping windows of the different EEG channels. The selection process consists of two stages. In t...
Published on Aug 31, 2017in Sensors3.03
Estimated H-index: 3
,
Estimated H-index: 12
,
Micheal Drieberg8
Estimated H-index: 8
Driver drowsiness is a major cause of fatal accidents, injury, and property damage, and has become an area of substantial research attention in recent years. The present study proposes a method to detect drowsiness in drivers which integrates features of electrocardiography (ECG) and electroencephalography (EEG) to improve detection performance. The study measures differences between the alert and drowsy states from physiological data collected from 22 healthy subjects in a driving simulator-bas...
Published on Aug 15, 2016in Expert Systems With Applications4.29
Thiago Lopes Trugillo da Silveira3
Estimated H-index: 3
(UFSM: Universidade Federal de Santa Maria),
Alice de Jesus Kozakevicius6
Estimated H-index: 6
(UFSM: Universidade Federal de Santa Maria),
Cesar Ramos Rodrigues7
Estimated H-index: 7
(UFSM: Universidade Federal de Santa Maria)
Ratio indices computed from a single EEG channel used as drowsiness indicators.Delta and gamma brain rhythms successfully used for drowsiness detection.Wavelet packet transform as the main tool to localize specific brain frequency ranges.Transition from alert to drowsy state is taken as main event of interest.Wilcoxon signed rank test analysis pointed out the contribution of proposed indices. Advances in materials engineering, electronic circuits, sensors, signal processing and classification te...
Published on Jul 8, 2016in Frontiers in Aging Neuroscience3.63
Dong Wen4
Estimated H-index: 4
(Yanshan University),
Peilei Jia1
Estimated H-index: 1
(Yanshan University)
+ 2 AuthorsChengbiao Lu1
Estimated H-index: 1
(Xinxiang Medical University)
At present, the sparse representation-based classification (SRC) methods of electroencephalograph (EEG) signal analysis have become an important approach for studying brain science. SRC methods mean that the target data is sparsely represented on the basis of a fixed dictionary or learned dictionary, and classified based on the reconstruction criteria or the corresponding features extracted. SRC methods have been used to analyze the EEG signals of epilepsy, mild cognitive impairment (MCI) and Al...
Published on Jul 1, 2016
Jun Juh Yan10
Estimated H-index: 10
,
Hang Hong Kuo3
Estimated H-index: 3
+ 1 AuthorsTeh-Lu Liao23
Estimated H-index: 23
(NCKU: National Cheng Kung University)
This study develops a real-time drowsiness detection system based on grayscale image processing and PERCLOS to determine if the driver is fatigued. The proposed system comprises three parts: first, it calculates the approximate position of the driver's face in grayscale images, and then uses a small template to analyze the eye positions, second, it uses the data from the previous step and PERCLOS to establish a fatigue model, and finally, based on the driver's personal fatigue model, the system ...
Published on Feb 19, 2016in Sensors3.03
Zutao Zhang8
Estimated H-index: 8
,
Dianyuan Luo2
Estimated H-index: 2
+ 4 AuthorsChunbai Wang6
Estimated H-index: 6
In this paper, we present a vehicle active safety model for vehicle speed control based on driver vigilance detection using low-cost, comfortable, wearable electroencephalographic (EEG) sensors and sparse representation. The proposed system consists of three main steps, namely wireless wearable EEG collection, driver vigilance detection, and vehicle speed control strategy. First of all, a homemade low-cost comfortable wearable brain-computer interface (BCI) system with eight channels is designed...
Published on Aug 1, 2015 in EMBC (International Conference of the IEEE Engineering in Medicine and Biology Society)
Syed Anas Imtiaz5
Estimated H-index: 5
(Imperial College London),
Esther Rodriguez-Villegas22
Estimated H-index: 22
(Imperial College London)
PhysioNet Sleep EDF database has been the most popular source of data used for developing and testing many automatic sleep staging algorithms. However, the recordings from this database has been used in an inconsistent fashion. For example, arbitrary selection of start and end times from long term recordings, data-hypnogram mismatches, different performance metrics and hypnogram conversion from R&K to AASM. All these differences result in different data sections and performance metrics being use...
Published on Aug 1, 2015 in EMBC (International Conference of the IEEE Engineering in Medicine and Biology Society)
Hamza Baali3
Estimated H-index: 3
(IIUM: International Islamic University Malaysia),
Mostefa Mesbah2
Estimated H-index: 2
(Sultan Qaboos University)
Estimated H-index: 9
(U of T: University of Toronto),
Ahmed K. Farahat9
Estimated H-index: 9
(UW: University of Waterloo)
+ 1 AuthorsMohamed S. Kamel43
Estimated H-index: 43
(UW: University of Waterloo)
Dictionary learning and sparse representation (DLSR) is a recent and successful mathematical model for data representation that achieves state-ofthe-art performance in various elds such as pattern recognition, machine learning, computer vision, and medical imaging. The original formulation for DLSR is based on the minimization of the reconstruction error between the original signal and its sparse representation in the space of the learned dictionary. Although this formulation is optimal for solv...
Published on Jan 1, 2015in IEEE Access4.10
Zheng Zhang10
Estimated H-index: 10
(HIT: Harbin Institute of Technology),
Yong Xu39
Estimated H-index: 39
(HIT: Harbin Institute of Technology)
+ 2 AuthorsDavidZhang89
Estimated H-index: 89
(PolyU: Hong Kong Polytechnic University)
Sparse representation has attracted much attention from researchers in fields of signal processing, image processing, computer vision, and pattern recognition. Sparse representation also has a good reputation in both theoretical research and practical applications. Many different algorithms have been proposed for sparse representation. The main purpose of this paper is to provide a comprehensive study and an updated review on sparse representation and to supply guidance for researchers. The taxo...
Cited By0

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text:
If Sin x - cos x = 1/3 , what is sin 3x + cos 3x?

Aug 23, 2016

$- \frac{25}{27}$

Explanation:

$\sin x - \cos x = \frac{1}{3}$

Squaring both sides

$\implies {\left(\sin x - \cos x\right)}^{2} = \frac{1}{9}$

$\implies {\sin}^{2} x + {\cos}^{2} x - 2 \sin x \cos x = \frac{1}{9}$

$\implies 1 - 2 \sin x \cos x = \frac{1}{9}$

$\implies 2 \sin x \cos x = 1 - \frac{1}{9} = \frac{8}{9}$

$\implies \sin x \cos x = \frac{4}{9}$

Now
$\sin 3 x + \cos 3 x$

$= 3 \sin x - 4 {\sin}^{3} x + 4 {\cos}^{3} x - 3 \cos x$

$= 3 \left(\sin x - \cos x\right) - 4 \left({\sin}^{3} x - {\cos}^{3} x\right)$

$= 3 \times \frac{1}{3} - 4 \left(\sin x - \cos x\right) \left({\sin}^{2} x + {\cos}^{2} x + \sin x \cos x\right)$

$= 1 - 4 \times \frac{1}{3} \left(1 + \frac{4}{9}\right)$
$= 1 - \frac{4}{3} \times \frac{13}{9}$
$= 1 - \frac{52}{27} = - \frac{25}{27}$


text:
How do you simplify (5xx4)+3?

Dec 9, 2017

$\left(5 \times 4\right) + 3 = \textcolor{b l u e}{23}$

Explanation:

$\left(5 \times 4\right) + 3$
$20 + 3$
$23$


text:
TheInfoList

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. For example, the functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object (category theory), object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of string (computer science), strings built from a given set of Character (computing), characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes theory), and formal language theory (star height problem). See semigroup for the history of the subject, and some other general properties of monoids.

# Definition

A set (mathematics), set ''S'' equipped with a binary operation , which we will denote •, is a monoid if it satisfies the following two axioms: ; Associativity: For all ''a'', ''b'' and ''c'' in ''S'', the equation holds. ; Identity element: There exists an element ''e'' in ''S'' such that for every element ''a'' in ''S'', the equations and hold. In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma (algebra), magma with associativity and identity. The identity element of a monoid is unique. For this reason the identity is regarded as a Constant (mathematics), constant, i. e. 0-ary (or nullary) operation. The monoid therefore is characterized by specification of the Tuple, triple (''S'', • , ''e''). Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written and . This notation does not imply that it is numbers being multiplied. A monoid in which each element has an inverse element, inverse is a group (mathematics), group.

# Monoid structures

## Submonoids

A submonoid of a monoid is a subset ''N'' of ''M'' that is closed under the monoid operation and contains the identity element ''e'' of ''M''. Symbolically, ''N'' is a submonoid of ''M'' if , whenever , and . In this case, ''N'' is a monoid under the binary operation inherited from ''M''. On the other hand, if ''N'' is subset of a monoid that is closure (mathematics), closed under the monoid operation, and is a monoid for this inherited operation, then ''N'' is not always a submonoid, since the identity elements may differ. For example, the singleton set is closed under multiplication, and is not a submonoid of the (multiplicative) monoid of the nonnegative integers.

## Generators

A subset ''S'' of ''M'' is said to ''generate'' ''M'' if the smallest submonoid of ''M'' containing ''S'' is ''M''. If there is a finite set that generates ''M'', then ''M'' is said to be a finitely generated monoid.

## Commutative monoid

A monoid whose operation is commutative is called a commutative monoid (or, less commonly, an abelian monoid). Commutative monoids are often written additively. Any commutative monoid is endowed with its ''algebraic'' preordering , defined by if there exists ''z'' such that . An ''order-unit'' of a commutative monoid ''M'' is an element ''u'' of ''M'' such that for any element ''x'' of ''M'', there exists ''v'' in the set generated by ''u'' such that . This is often used in case ''M'' is the Ordered group, positive cone of a Partially ordered set, partially ordered abelian group ''G'', in which case we say that ''u'' is an order-unit of ''G''.

## Partially commutative monoid

A monoid for which the operation is commutative for some, but not all elements is a trace monoid; trace monoids commonly occur in the theory of concurrent computation.

# Examples

* Out of the 16 possible truth table#Truth table for all binary logical operators, binary Boolean operators, each of the four that has a two-sided identity is also commutative and associative and thus makes the set a commutative monoid. Under the standard definitions, Logical conjunction, AND and Logical biconditional, XNOR have the identity True while Exclusive disjunction, XOR and Logical disjunction, OR have the identity False. The monoids from AND and OR are also idempotent while those from XOR and XNOR are not. * The set of natural numbers $\N = \$ is a commutative monoid under addition (identity element 0 (number), 0) or multiplication (identity element 1 (number), 1). A submonoid of under addition is called a numerical monoid. * The set of positive integers $\N \setminus \$ is a commutative monoid under multiplication (identity element 1). * Given a set , the set of subsets of is a commutative monoid under intersection (identity element is itself). * Given a set , the set of subsets of is a commutative monoid under union (identity element is the empty set). * Generalizing the previous example, every bounded semilattice is an idempotent commutative monoid. ** In particular, any bounded lattice (order), lattice can be endowed with both a Join and meet, meet- and a Join and meet, join- monoid structure. The identity elements are the lattice's top and its bottom, respectively. Being lattices, Heyting algebras and Boolean algebra (structure), Boolean algebras are endowed with these monoid structures. * Every singleton set closed under a binary operation • forms the trivial (one-element) monoid, which is also the trivial group. * Every group (mathematics), group is a monoid and every abelian group a commutative monoid. * Any semigroup may be turned into a monoid simply by adjoining an element not in and defining for all . This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids. ** Thus, an idempotent monoid (sometimes known as ''find-first'') may be formed by adjoining an identity element to the left zero semigroup over a set . The opposite monoid (sometimes called ''find-last'') is formed from the right zero semigroup over . *** Adjoin an identity to the left-zero semigroup with two elements . Then the resulting idempotent monoid models the lexicographical order of a sequence given the orders of its elements, with ''e'' representing equality. * The underlying set of any ring (algebra), ring, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity 1.) ** The integers, rational numbers, real numbers or complex numbers, with addition or multiplication as operation. ** The set of all by matrix (mathematics), matrices over a given ring, with matrix addition or matrix multiplication as the operation. * The set of all finite string (computer science), strings over some fixed alphabet forms a monoid with string concatenation as the operation. The empty string serves as the identity element. This monoid is denoted and is called the ''free monoid'' over . It is not commutative. * Given any monoid , the ''opposite monoid'' has the same carrier set and identity element as , and its operation is defined by . Any commutative monoid is the opposite monoid of itself. * Given two sets and endowed with monoid structure (or, in general, any finite number of monoids, , their cartesian product is also a monoid (respectively, ). The associative operation and the identity element are defined pairwise. * Fix a monoid . The set of all functions from a given set to is also a monoid. The identity element is a constant function mapping any value to the identity of ; the associative operation is defined pointwise. * Fix a monoid with the operation and identity element , and consider its power set consisting of all subsets of . A binary operation for such subsets can be defined by . This turns into a monoid with identity element . In the same way the power set of a group is a monoid under the product of group subsets. * Let be a set. The set of all functions forms a monoid under function composition. The identity is just the identity function. It is also called the ''full transformation monoid'' of . If is finite with elements, the monoid of functions on is finite with elements. * Generalizing the previous example, let be a category (mathematics), category and an object of . The set of all endomorphisms of , denoted , forms a monoid under composition of morphisms. For more on the relationship between category theory and monoids see below. * The set of homeomorphism Class (set theory), classes of compact surfaces with the connected sum. Its unit element is the class of the ordinary 2-sphere. Furthermore, if denotes the class of the torus, and ''b'' denotes the class of the projective plane, then every element ''c'' of the monoid has a unique expression the form where is a positive integer and , or . We have . * Let $\langle f\rangle$ be a cyclic monoid of order , that is, $\langle f\rangle = \left\$. Then $f^n = f^k$ for some $0 \le k < n$. In fact, each such gives a distinct monoid of order , and every cyclic monoid is isomorphic to one of these.
Moreover, can be considered as a function on the points $\$ given by :: $\begin 0 & 1 & 2 & \cdots & n-2 & n-1 \\ 1 & 2 & 3 & \cdots & n-1 & k\end$ :or, equivalently :: $f\left(i\right) := \begin i+1, & \text 0 \le i < n-1 \\ k, & \text i = n-1. \end$ :Multiplication of elements in $\langle f\rangle$ is then given by function composition. :When $k = 0$ then the function is a permutation of $\,$ and gives the unique cyclic group of order .

# Properties

The monoid axioms imply that the identity element is unique: If and are identity elements of a monoid, then .

## Products and powers

For each nonnegative integer , one can define the product $p_n = \textstyle \prod_^n a_i$ of any sequence $\left(a_1,\ldots,a_n\right)$ of elements of a monoid recursively: let and let for . As a special case, one can define nonnegative integer powers of an element of a monoid: and for . Then for all .

## Invertible elements

An element is called inverse element, invertible if there exists an element such that and . The element is called the inverse of . Inverses, if they exist, are unique: If and are inverses of , then by associativity . If is invertible, say with inverse , then one can define negative powers of by setting for each ; this makes the equation hold for all . The set of all invertible elements in a monoid, together with the operation •, forms a group (mathematics), group.

## Grothendieck group

Not every monoid sits inside a group. For instance, it is perfectly possible to have a monoid in which two elements and exist such that holds even though is not the identity element. Such a monoid cannot be embedded in a group, because in the group multiplying both sides with the inverse of would get that , which is not true. A monoid has the cancellation property (or is cancellative) if for all , and in , the equality implies , and the equality implies . A commutative monoid with the cancellation property can always be embedded in a group via the ''Grothendieck group construction''. That is how the additive group of the integers (a group with operation +) is constructed from the additive monoid of natural numbers (a commutative monoid with operation + and cancellation property). However, a non-commutative cancellative monoid need not be embeddable in a group. If a monoid has the cancellation property and is ''finite'', then it is in fact a group. The right- and left-cancellative elements of a monoid each in turn form a submonoid (i.e. are closed under the operation and obviously include the identity). This means that the cancellative elements of any commutative monoid can be extended to a group. The cancellative property in a monoid is not necessary to perform the Grothendieck construction – commutativity is sufficient. However, if a commutative monoid does not have the cancellation property, the homomorphism of the monoid into its Grothendieck group is not injective. More precisely, if , then and have the same image in the Grothendieck group, even if . In particular, if the monoid has an absorbing element, then its Grothendieck group is the trivial group.

## Types of monoids

An inverse monoid is a monoid where for every ''a'' in ''M'', there exists a unique ''a''−1 in ''M'' such that and . If an inverse monoid is cancellative, then it is a group. In the opposite direction, a ''zerosumfree monoid'' is an additively written monoid in which implies that and : equivalently, that no element other than zero has an additive inverse.

# Acts and operator monoids

Let ''M'' be a monoid, with the binary operation denoted by • and the identity element denoted by ''e''. Then a (left) ''M''-act (or left act over ''M'') is a set ''X'' together with an operation which is compatible with the monoid structure as follows: * for all ''x'' in ''X'': ; * for all ''a'', ''b'' in ''M'' and ''x'' in ''X'': . This is the analogue in monoid theory of a (left) Group action (mathematics), group action. Right ''M''-acts are defined in a similar way. A monoid with an act is also known as an ''operator monoid''. Important examples include transition systems of semiautomata. A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.

# Monoid homomorphisms

A homomorphism between two monoids and is a function such that * for all ''x'', ''y'' in ''M'' * , where ''e''''M'' and ''e''''N'' are the identities on ''M'' and ''N'' respectively. Monoid homomorphisms are sometimes simply called monoid morphisms. Not every semigroup homomorphism between monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of homomorphism. For example, consider $M_n$, the set of residue classes modulo $n$ equipped with multiplication. In particular, the class of $1$ is the identity. Function $f\colon M_3\to M_6$ given by $f\left(k\right)=3k$ is a semigroup homomorphism as $3k\cdot 3l = 9kl = 3kl$ in $M_6$. However, $f\left(1\right)=3 \neq 1$, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted. In contrast, a semigroup homomorphism between groups is always a group homomorphism, as it necessarily preserves the identity (because, in a group, the identity is the only element such that ). A bijective monoid homomorphism is called a monoid isomorphism. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.

# Equational presentation

Monoids may be given a ''presentation'', much in the same way that groups can be specified by means of a group presentation. One does this by specifying a set of generators Σ, and a set of relations on the free monoid Σ. One does this by extending (finite) binary relations on Σ to monoid congruences, and then constructing the quotient monoid, as above. Given a binary relation , one defines its symmetric closure as . This can be extended to a symmetric relation by defining if and only if and for some strings with . Finally, one takes the reflexive and transitive closure of ''E'', which is then a monoid congruence. In the typical situation, the relation ''R'' is simply given as a set of equations, so that $R=\$. Thus, for example, : $\langle p,q\,\vert\; pq=1\rangle$ is the equational presentation for the bicyclic monoid, and : $\langle a,b \,\vert\; aba=baa, bba=bab\rangle$ is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as $a^ib^j\left(ba\right)^k$ for integers ''i'', ''j'', ''k'', as the relations show that ''ba'' commutes with both ''a'' and ''b''.

# Relation to category theory

Monoids can be viewed as a special class of category theory, categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. That is, : ''A monoid is, essentially, the same thing as a category with a single object.'' More precisely, given a monoid , one can construct a small category with only one object and whose morphisms are the elements of ''M''. The composition of morphisms is given by the monoid operation •. Likewise, monoid homomorphisms are just functors between single object categories. So this construction gives an equivalence of categories, equivalence between the category of monoids, category of (small) monoids Mon and a full subcategory of the category of (small) categories Cat. Similarly, the category of groups is equivalent to another full subcategory of Cat. In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. There is also a notion of monoid (category theory), monoid object which is an abstract definition of what is a monoid in a category. A monoid object in category of sets, Set is just a monoid.

# Monoids in computer science

In computer science, many abstract data types can be endowed with a monoid structure. In a common pattern, a sequence of elements of a monoid is "fold (higher-order function), folded" or "accumulated" to produce a final value. For instance, many iterative algorithms need to update some kind of "running total" at each iteration; this pattern may be elegantly expressed by a monoid operation. Alternatively, the associativity of monoid operations ensures that the operation can be parallelization, parallelized by employing a prefix sum or similar algorithm, in order to utilize multiple cores or processors efficiently. Given a sequence of values of type ''M'' with identity element $\varepsilon$ and associative operation $\bullet$, the ''fold'' operation is defined as follows: : $\mathrm: M^ \rarr M = \ell \mapsto \begin \varepsilon & \mbox \ell = \mathrm \\ m \bullet \mathrm \, \ell\text{'} & \mbox \ell = \mathrm \, m \, \ell\text{'} \end$ In addition, any data structure can be 'folded' in a similar way, given a serialization of its elements. For instance, the result of "folding" a binary tree might differ depending on pre-order vs. post-order tree traversal.

# MapReduce

An application of monoids in computer science is so-called MapReduce programming model (se
Encoding Map-Reduce As A Monoid With Left Folding
. MapReduce, in computing, consists of two or three operations. Given a dataset, "Map" consists of mapping arbitrary data to elements of a specific monoid. "Reduce" consists of folding those elements, so that in the end we produce just one element. For example, if we have a multiset, in a program it is represented as a map from elements to their numbers. Elements are called keys in this case. The number of distinct keys may be too big, and in this case the multiset is being sharded. To finalize reduction properly, the "Shuffling" stage regroups the data among the nodes. If we do not need this step, the whole Map/Reduce consists of mapping and reducing; both operation are parallelizable, the former due to its element-wise nature, the latter due to associativity of the monoid.

# Complete monoids

A complete monoid is a commutative monoid equipped with an Finitary, infinitary sum operation $\Sigma_I$ for any index set such that:Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. ''Handbook of Weighted Automata'', 3–28. , pp. 7–10 : $\sum_ =0;\quad \sum_ = m_j;\quad \sum_ = m_j+m_k \quad \text j\neq k$ and : $\sum_ = \sum_\left(m_i\right)\quad \text \bigcup_ I_j=I \text I_j \cap I_ = \emptyset \quad \text j\neq j\text{'}$ A continuous monoid is an ordered commutative monoid in which every directed set has a least upper bound compatible with the monoid operation: : $a + \sup S = \sup\left(a + S\right) \ .$ These two concepts are closely related: a continuous monoid is a complete monoid in which the infinitary sum may be defined as : $\sum_I a_i = \sup \sum_E a_i$ where the supremum on the right runs over all finite subsets of and each sum on the right is a finite sum in the monoid.

# See also

* Green's relations * Monad (functional programming) * Semiring and Kleene algebra * Star height problem * Vedic square

* * * * *

# External links

* * * {{PlanetMath, urlname=Monoid , title=Monoid , id=389 Algebraic structures Category theory Semigroup theory


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2
Posted by9 days ago

What exactly is meant by a family of neighborhoods?

I'm currently self studying Mathematical Analysis for AI and I stumbeled upon the notion of neighborhoods, and families of neighborhoods. Since the definition in the book states that for the real number x there exists an epsilon such that (x - epsilon, x + epsilon) is a subset of the neighborhood V, I'm assuming that there's an infinite amount of epsilons inside of V and hence an infinite amount of subsets of V. It's hard for me to understand how there can be infinite amount of neighborhoods inside the family of neighborhoods.

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level 1
Ergodic Theory12 points · 9 days ago

A family of neighborhoods for a point x just means a collection of open sets U_alpha indexed over alpha in some (often uncountable) set I so that (1) x is in U_alpha for all alpha and (2) Intersection[alpha] U_alpha = { x }. Usually we also require that the collection { U_alpha } generates all the open sets containing x in some fashion (but not always).

The reason we need such things is that it turns out to be very hard to work with individual points in the real numbers but very easy to work with "very small" subsets of reals containing our point.

(I'll skip my usual rant about the philosophical meaning of what I just said since I know everyone is sick of it, but I am right).

level 2
Original Poster1 point · 9 days ago

Ok now I'm curious, would you mind explaining further the philosophical meaning of the explanation that you gave? And just to make sure that I understood your explanation (Because I'm not used to all the abstraction) a family of neighborhoods of x is just all the neighborhoods in the open set (x-alpha, x+alpha) where alpha can be any real number?

level 3
4 points · 9 days ago

Her comment history is full of her rants so you should find want you‘re looking for pretty quickly :) But as I often don‘t fully understand what‘s going there it is probable that you won‘t completely grasp everything there so don‘t worry if that‘s the case.

level 1
Analysis3 points · 9 days ago

(0,1) is a neighborhood of 1/2. So is (0,2) and (0,3) and...

level 2
Ergodic Theory1 point · 9 days ago

Family almost always refers to not just the sets being neighborhoods but that they be arbitrarily small in whatever sense of small makes sense (for metric spaces this is easy).

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Notices
 Windows XP Post your Windows XP related questions here.

20th February 2006   #1
Inactive

Profile:
Join Date: Feb 2006
Location: Albuquerque, NM
Posts: 16
Computer Experience:
Intermediate

Thank you all in advance for helping me with my little XP problem. Here are the stats of my computer to help you all out:

I am running Win XP Pro SP2, Pentium III 933 MHz with 512 RAM. I run Avast Home Edition as my AV. I have Tweak UI and Reg ScrubXP making minor modifications to Windows settings.

The problem is this. I was having problems with Disk Manager (Shell problems) that I was only able to fix with a repair of my OS from the Repair Console. After I did this reinstall I was unable to use my CDROMs. In a CD with media files I am able to view them in the explorer but unable to run them from the CD. Also, when I run a CD with an autorun function I am met with the following Warning:

The procedure entry point could not be located in the dynamic link library "x".dll

The most common value of "x" is Kernel32.dll but I have also seen ADVAPI32.dll and the dll that is causing a problem. I can't run sfc because I can't use my CDROMs properly. Another thing to note of importance is that for some reason my AV is unable to access certain folders because the log states that it does not have permission. I think this may be relevant to the problem. I ran Hijackthis and the following is the log. If anyone can help me with this problem apart from a clean install I would be very appreciative.

20th February 2006   #2
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## Hijackthis log

Logfile of HijackThis v1.99.1
Scan saved at 11:09:20 AM, on 2/19/2006
Platform: Windows XP SP2 (WinNT 5.01.2600)
MSIE: Internet Explorer v6.00 SP2 (6.00.2900.2180)

Running processes:
C:\WINDOWS\System32\smss.exe
C:\WINDOWS\system32\winlogon.exe
C:\WINDOWS\system32\services.exe
C:\WINDOWS\system32\lsass.exe
C:\WINDOWS\system32\Ati2evxx.exe
C:\WINDOWS\system32\svchost.exe
C:\WINDOWS\System32\svchost.exe
C:\WINDOWS\system32\spoolsv.exe
C:\Program Files\Alwil Software\Avast4\aswUpdSv.exe
C:\Program Files\Alwil Software\Avast4\ashServ.exe
C:\Program Files\Common Files\Microsoft Shared\VS7Debug\mdm.exe
C:\WINDOWS\system32\svchost.exe
C:\Program Files\Alwil Software\Avast4\ashMaiSv.exe
C:\WINDOWS\system32\Ati2evxx.exe
C:\WINDOWS\Explorer.EXE
C:\Program Files\Alwil Software\Avast4\ashWebSv.exe
C:\Program Files\ATI Technologies\ATI Control Panel\atiptaxx.exe
C:\PROGRA~1\ALWILS~1\Avast4\ashDisp.exe
C:\Program Files\Viewpoint\Viewpoint Manager\ViewMgr.exe
C:\Program Files\HP\hpcoretech\hpcmpmgr.exe
C:\Program Files\Logitech\MouseWare\system\em_exec.exe
C:\Program Files\HP\HP Software Update\HPWuSchd2.exe
C:\Program Files\Java\jre1.5.0_06\bin\jusched.exe
C:\Program Files\iTunes\iTunesHelper.exe
C:\Program Files\Common Files\Real\Update_OB\realsched.exe
C:\WINDOWS\System32\ctfmon.exe
C:\Program Files\AWS\WeatherBug\Weather.exe
C:\Program Files\iPod\bin\iPodService.exe
D:\GetRight\getright.exe
C:\Program Files\HP\Digital Imaging\bin\hpqtra08.exe
D:\GetRight\getright.exe
D:\OpenOffice.org1.1.4\program\soffice.exe
C:\WINDOWS\system32\HPZipm12.exe
C:\Program Files\Internet Explorer\iexplore.exe
C:\WINDOWS\system32\wuauclt.exe
C:\WINDOWS\system32\wuauclt.exe
D:\hijackthis\HijackThis.exe

R1 - HKCU\Software\Microsoft\Internet Explorer\Main,Search Bar = http://channels.aimtoday.com/search/aimtoolbar.jsp
R0 - HKCU\Software\Microsoft\Internet Explorer\Main,Start Page = http://www.kofc.org/
O2 - BHO: bho2gr Class - {31FF080D-12A3-439A-A2EF-4BA95A3148E8} - D:\GetRight\xx2gr.dll
O2 - BHO: (no name) - {5C8B2A36-3DB1-42A4-A3CB-D426709BBFEB} - (no file)
O2 - BHO: SSVHelper Class - {761497BB-D6F0-462C-B6EB-D4DAF1D92D43} - C:\Program Files\Java\jre1.5.0_06\bin\ssv.dll
O3 - Toolbar: (no name) - {42CDD1BF-3FFB-4238-8AD1-7859DF00B1D6} - (no file)
O4 - HKLM\..\Run: [Tweak UI] RUNDLL32.EXE TWEAKUI.CPL,TweakMeUp
O4 - HKLM\..\Run: [ATIPTA] C:\Program Files\ATI Technologies\ATI Control Panel\atiptaxx.exe
O4 - HKLM\..\Run: [InCD] C:\Program Files\Ahead\InCD\InCD.exe
O4 - HKLM\..\Run: [Logitech Utility] Logi_MwX.Exe
O4 - HKLM\..\Run: [avast!] C:\PROGRA~1\ALWILS~1\Avast4\ashDisp.exe
O4 - HKLM\..\Run: [ViewMgr] C:\Program Files\Viewpoint\Viewpoint Manager\ViewMgr.exe
O4 - HKLM\..\Run: [HP Component Manager] "C:\Program Files\HP\hpcoretech\hpcmpmgr.exe"
O4 - HKLM\..\Run: [HP Software Update] C:\Program Files\HP\HP Software Update\HPWuSchd2.exe
O4 - HKLM\..\Run: [SunJavaUpdateSched] C:\Program Files\Java\jre1.5.0_06\bin\jusched.exe
O4 - HKLM\..\Run: [iTunesHelper] "C:\Program Files\iTunes\iTunesHelper.exe"
O4 - HKLM\..\Run: [TkBellExe] "C:\Program Files\Common Files\Real\Update_OB\realsched.exe" -osboot
O4 - HKCU\..\Run: [ctfmon.exe] C:\WINDOWS\system32\ctfmon.exe
O4 - HKCU\..\Run: [Weather] C:\Program Files\AWS\WeatherBug\Weather.exe 1
O4 - Startup: OpenOffice.org 1.1.4.lnk = D:\OpenOffice.org1.1.4\program\quickstart.exe
O4 - Startup: PowerReg Scheduler.exe
O4 - Startup: Registration Myst Uru
O4 - Global Startup: GetRight - Tray Icon.lnk = D:\GetRight\getright.exe
O4 - Global Startup: HP Digital Imaging Monitor.lnk = C:\Program Files\HP\Digital Imaging\bin\hpqtra08.exe
O4 - Global Startup: Microsoft Office.lnk = C:\Program Files\Microsoft Office\Office10\OSA.EXE
O8 - Extra context menu item: &AIM Search - res://C:\Program Files\AIM Toolbar\AIMBar.dll/aimsearch.htm
O8 - Extra context menu item: Open with GetRight Browser - D:\GetRight\GRbrowse.htm
O9 - Extra button: Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O9 - Extra 'Tools' menuitem: Windows Messenger - {FB5F1910-F110-11d2-BB9E-00C04F795683} - C:\Program Files\Messenger\msmsgs.exe
O9 - Extra button: WeatherBug - {AF6CABAB-61F9-4f12-A198-B7D41EF1CB52} - C:\PROGRA~1\AWS\WEATHE~1\Weather.exe (HKCU)
O12 - Plugin for .spop: C:\Program Files\Internet Explorer\Plugins\NPDocBox.dll
O16 - DPF: {6414512B-B978-451D-A0D8-FCFDF33E833C} (WUWebControl Class) - http://v5.windowsupdate.microsoft.co...?1113158356140
O23 - Service: avast! iAVS4 Control Service (aswUpdSv) - Unknown owner - C:\Program Files\Alwil Software\Avast4\aswUpdSv.exe
O23 - Service: Ati HotKey Poller - ATI Technologies Inc. - C:\WINDOWS\system32\Ati2evxx.exe
O23 - Service: ATI Smart - Unknown owner - C:\WINDOWS\system32\ati2sgag.exe
O23 - Service: avast! Antivirus - Unknown owner - C:\Program Files\Alwil Software\Avast4\ashServ.exe
O23 - Service: avast! Mail Scanner - Unknown owner - C:\Program Files\Alwil Software\Avast4\ashMaiSv.exe" /service (file missing)
O23 - Service: avast! Web Scanner - Unknown owner - C:\Program Files\Alwil Software\Avast4\ashWebSv.exe" /service (file missing)
O23 - Service: InstallDriver Table Manager (IDriverT) - Macrovision Corporation - C:\Program Files\Common Files\InstallShield\Driver\11\Intel 32\IDriverT.exe
O23 - Service: InCD Helper (InCDsrv) - AHEAD Software - C:\Program Files\Ahead\InCD\InCDsrv.exe
O23 - Service: iPodService - Apple Computer, Inc. - C:\Program Files\iPod\bin\iPodService.exe
O23 - Service: Pml Driver HPZ12 - HP - C:\WINDOWS\system32\HPZipm12.exe

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22nd February 2006   #3
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## Virus Scans

Also, I ran my AV in safe mode under the Admin profile and I ran an reliable online virus checker and both came up negative.

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22nd February 2006   #4
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My System
I don't really know what to tell you other than an error like this is usually due to the fact that you have the wrong version of whatever file is named in the error. It can be caused by a failed install of a program, service pack, or even a windows update. Other than that I don't know what to tell you.

To simplify it, what it's looking for isn't located where it's expected to be.
 Quote: The problem is this. I was having problems with Disk Manager (Shell problems) that I was only able to fix with a repair of my OS from the Repair Console.
I see you have SP2. When you did the repair, did the disk you used for it have SP2 on it or an earlier version? If an earlier version, perhaps reinstalling SP2 would fix the problem.

If you used a SP2 disk for the repair, the only thing I can suggest would be to run the system file checker. Click start>run and type sfc /scannow. Note the space between sfc and the /. Have your XP disk ready. You may be prompted for it.

I could be wrong but that's about all I can think of in this case.

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22nd February 2006   #5
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## Thank you

 Originally Posted by Zander I don't really know what to tell you other than an error like this is usually due to the fact that you have the wrong version of whatever file is named in the error. It can be caused by a failed install of a program, service pack, or even a windows update. Other than that I don't know what to tell you. To simplify it, what it's looking for isn't located where it's expected to be. I see you have SP2. When you did the repair, did the disk you used for it have SP2 on it or an earlier version? If an earlier version, perhaps reinstalling SP2 would fix the problem. If you used a SP2 disk for the repair, the only thing I can suggest would be to run the system file checker. Click start>run and type sfc /scannow. Note the space between sfc and the /. Have your XP disk ready. You may be prompted for it. I could be wrong but that's about all I can think of in this case.
Thank you for your assistance. When I did my repair I reverted back to SP1 because my XP disk is SP1. I then slipstreamed SP2. The odd thing was that if I allowed windows to run its automatic updates before I installed SP2 manually my computer would crash.

I can't use sfc because of the problem with running/accessing files from my CDROMs which is the central problem. I have been advised to try a method of saving i386 on my hard drive with a 98 startup disk and modifying my registry to have sfc point to the folder I save i386 to. However, I haven't had the opportunity to do it yet. I will post the success of this advice if any.

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 22nd February 2006 #6 Alumni     Profile: Join Date: Jan 2002 Location: Minnesota, USA Posts: 4,084 Computer Experience: A bit (or is it a by My System Have you tried running sfc? The autorun function shouldn't interfere with it (I wouldn't think anyway). Just cancel out of the error message and run sfc. If it is a problem you can bypass the autorun feature on any cd by holding down the shift key when you put the cd in the drive. Put the cd in and press the shift key. Watch the light on the cdrom drive. When it goes out, realease the shift key. Did you find this post helpful? |
24th February 2006   #7
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 Originally Posted by Zander Have you tried running sfc? The autorun function shouldn't interfere with it (I wouldn't think anyway). Just cancel out of the error message and run sfc. If it is a problem you can bypass the autorun feature on any cd by holding down the shift key when you put the cd in the drive. Put the cd in and press the shift key. Watch the light on the cdrom drive. When it goes out, realease the shift key.
I wish it were that easy. After working on this some more and running a few more tests it seems that now it is only my new DVD-R that cannot access any executable file. My slave CDROM works fine and has no problem accessing anything that I insert in the drive. Could this be as simple as a driver problem with my DVD-R or do I have a faulty piece of hardware.

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 24th February 2006 #8 Alumni     Profile: Join Date: Jan 2002 Location: Minnesota, USA Posts: 4,084 Computer Experience: A bit (or is it a by My System Is the drive that works just a cdrom drive (not a writer)? What happens if you prevent the cdrw from auto running like I mentioned before? Try doing that and then right click on the drive and choose explore. Is it able to access it this way? If it's just the cdrw that's giving you trouble perhaps it's due to some cd burning software you've installed. Maybe try uninstalling any burning software, reboot and see if it works then. Did you find this post helpful? |
24th February 2006   #9
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 Originally Posted by Zander Is the drive that works just a cdrom drive (not a writer)? What happens if you prevent the cdrw from auto running like I mentioned before? Try doing that and then right click on the drive and choose explore. Is it able to access it this way? If it's just the cdrw that's giving you trouble perhaps it's due to some cd burning software you've installed. Maybe try uninstalling any burning software, reboot and see if it works then.
I have tried that with the cdrw. I am able to view the items in an explorer window on the disk but I am unable to access any file. I have tried exe's media files and txt files and I get the same entry point error. I currently have Nero installed. I will uninstall it to see if it is having some kind of conflict with another program.

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# Note on simple(r) equations for Einstein-dilaton-Gauss-Bonnet and dynamical Chern-Simons theories

If you’ve ever looked into theories beyond general relativity, you’re already aware that their field equations can be very complicated. For example, here’s a clip from one paper on EDGB:

Ack! That’s pretty unwieldy. But don’t despair, it turns out that the above mess can be written much more compactly.

Just to set conventions, let’s work with the action

\begin{align} \label{eq:action1} S = \int d^4x \sqrt{-g} \left[ \frac{1}{2}m_{pl}^2 R - \frac{1}{2} (\cd^a \vartheta) (\cd_a \vartheta) \right] + S_{int} \end{align}

where $\vartheta$ is a scalar (dilaton or axion) and $S_{int}$ is a non-minimal interaction term between the scalar and curvature.

## Einstein-dilaton-Gauss-Bonnet

For EDGB, let’s take

\begin{align} \label{eq:SEDGB} S_{int}^{EDGB} = -\frac{1}{8} m_{pl} \ell^2 \int d^4x \sqrt{-g} F(\vartheta) \left[ R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd} \right] \end{align}

with some arbitrary coupling function F, and some dimensional parameter $\ell$. Now this above curvature combination might seem arbitrary, but it’s actually the 4-dimensional Euler density (see e.g. Bob McNees’s notes). It’s more natural to write that as

\begin{align} \label{eq:euler4} \ddR_{abcd}R^{abdc} = R^2 - 4 R_{ab}R^{ab} + R_{abcd}R^{abcd}. \end{align}

Here we’ve defined the double-dual $\ddR$ of the Riemann tensor. First, we dualize on the left two antisymmetric indices to define the left-dual,

\begin{align} \label{eq:leftdual} \dR^{abcd} \equiv \frac{1}{2} \epsilon^{abef} R_{ef}{}^{cd}, \end{align}

and then we further dualize on the right two antisymmetric indices to get the double-dual,

\begin{align} \label{eq:doubledual} \ddR^{abcd} \equiv \dR^{ab}{}_{gh} \frac{1}{2} \epsilon^{ghcd} = \frac{1}{2} \epsilon^{abef} R_{efgh} \frac{1}{2} \epsilon^{ghcd}. \end{align}

Now, it’s an exercise in algebraic manipulation to show that the equation for the metric from the action defined in Eqs. \eqref{eq:action1}, \eqref{eq:SEDGB} is given simply by

\begin{align} \label{eq:eom-EDGB} \boxed{ m_{pl}^2 G_{ab} - m_{pl} \ell^2 \cd^c \cd^d \left[ \ddR_{cabd} F(\vartheta) \right] = T_{ab} } \end{align}

where $T_{ab}$ is the stress-energy tensor for matter plus the stress-energy tensor for the scalar field. That’s quite a bit simpler than the image above, isn’t it! If you want to get xTensor to verify this for you, grab EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb from the xAct examples collection.

I wrote the above in a certain way to make it look very similar to the case of dynamical Chern-Simons (DCS, below), but before moving on—recall that one reason people like EDGB is that the equations of motion are only second order in the metric. That’s not obvious from the way I wrote it, because it looks like you might get third and fourth derivatives of the metric. However, one nice property of the double-dual of Riemann is that it’s divergence free (see MTW Eq. (13.51) and exercise 13.11). This means we can rewrite

\begin{align} \label{eq:C-tensor-EDGB-idents} \cd^c \cd^d \left[ \ddR_{cabd} F(\vartheta) \right] = \cd^c \left[ \ddR_{cabd} \cd^d F(\vartheta) \right] = \ddR_{cabd} \cd^c \cd^d F(\vartheta). \end{align}

Now it’s obvious that there are only second derivatives of the metric. However, the first or second forms might give more insight, because from them you can see that this so-called “C-tensor” is itself the divergence of some tensor. That’s the kind of thing you might want to integrate over a region…

## Dynamical Chern-Simons

Anyway, on to DCS. Now we use the interaction term

\begin{align} \label{eq:SDCS} S_{int}^{DCS} = -\frac{1}{8} m_{pl} \ell^2 \int d^4x \sqrt{-g} F(\vartheta) \dR^{abcd} R_{abcd} \end{align}

with just a single dual. Again it looks kind of arbitrary, but when $\dR^{abcd} R_{abcd}$ is integrated over the whole manifold, you get a topological invariant.

The equation for the metric in DCS is also kind of scary looking, but again some algebra shows that you can write it as

\begin{align} \label{eq:eom-DCS} \boxed{ m_{pl}^2 G_{ab} - m_{pl} \ell^2 \cd^c \cd^d \left[ \dR_{c(ab)d} F(\vartheta) \right] = T_{ab} } \end{align}

where $(ab)$ means that we are symmetrizing (with a factor of 1/2) on those two indices. This looks very similar to the expression for EDGB! However, the single-dual of Riemann is only manifestly divergence-free on the left two indices, so this equation does have third derivatives of the metric. The double-divergence does vanish, so there are no fourth derivatives.

All of the above calculations are in my notebook EDGB-and-DCS-EOMs-and-C-tensors-simplified.nb in the xAct examples collection. Hope you learned something!

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# Article

Full entry | PDF   (0.2 MB)
Keywords:
third order differential equation; damping term; second order oscillatory equation; positive solution; asymptotic properties
Summary:
We study solutions tending to nonzero constants for the third order differential equation with the damping term $(a_{1}(t)(a_{2}(t)x'(t))')'+q(t)x'(t)+r(t)f(x(\varphi (t)))=0$ in the case when the corresponding second order differential equation is oscillatory.
References:
[1] Agarwal, R., Aktas, M. F., Tiryaki, A.: On oscillation criteria for third order nonlinear delay differential equations. Arch. Math. (Brno) 45 (2009), 1-18. MR 2591657 | Zbl 1212.34189
[2] Baculikova, B., Elabbasy, E. M., Saker, S. H., Dzurina, J.: Oscillation criteria for third-order nonlinear differential equations. Math. Slovaca 58 (2008), 201-202. DOI 10.2478/s12175-008-0068-1 | MR 2391214 | Zbl 1174.34052
[3] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: On nonoscillatory solutions of third order nonlinear differential equations. Dynam. Systems Appl. 9 (2000), 483-500. MR 1843694
[4] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: Oscillation for third order nonlinear differential equations with deviating argument. Abstr. Appl. Anal. 2010, Article ID 278962, 19 p. (2010). MR 2587610 | Zbl 1192.34073
[5] Bartušek, M., Cecchi, M., Marini, M.: On Kneser solutions of nonlinear third order differential equations. J. Math. Anal. Appl. 261 (2001), 72-84. DOI 10.1006/jmaa.2000.7473 | MR 1850957 | Zbl 0995.34025
[6] Cecchi, M., Došlá, Z., Marini, M.: On third order differential equations with property A and B. J. Math. Anal. Appl. 231 (1999), 509-525. DOI 10.1006/jmaa.1998.6247 | MR 1669163
[7] Cecchi, M., Došlá, Z., Marini, M.: On nonlinear oscillations for equations associated to disconjugate operators. Nonlinear Anal., Theory Methods Appl. 30 (1997), 1583-1594. DOI 10.1016/S0362-546X(97)00028-X | MR 1490081
[8] Coles, W. J.: Boundedness of solutions of two-dimensional first order differential systems. Boll. Un. Mat. Ital. 4 (1971), 225-231. MR 0293162 | Zbl 0229.34027
[9] Jaroš, J., Kusano, T., Marić, V.: Existence of regularly and rapidly varying solutions for a class of third order nonlinear ordinary differential equations. Publ. Inst. Math. Beograd 79 (2006), 51-64. DOI 10.2298/PIM0693051J | MR 2275338
[10] Kiguradze, I.: An oscillation criterion for a class of ordinary differential equations. Diff. Urav. 28 (1992), 201-214. MR 1184921 | Zbl 0768.34018
[11] Marini, M.: Criteri di limitatezza per le soluzioni dell'equazione lineare del secondo ordine. Boll. Un. Mat. Ital. 11 (1975), 154-165. MR 0372326 | Zbl 0332.34027
[12] Mojsej, I.: Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument. Nonlinear Analysis 68 (2008), 3581-3591. DOI 10.1016/j.na.2007.04.001 | MR 2401369 | Zbl 1151.34053
[13] Mojsej, I., Ohriska, J.: Comparison theorems for noncanonical third order nonlinear differential equations. Central Eur. J. Math. 5 (2007), 154-16. DOI 10.2478/s11533-006-0044-3 | MR 2287717 | Zbl 1128.34021
[14] Saker, S. H.: Oscillation criteria of Hille and Nehari types for third order delay differential equations. Commun. Appl. Anal. 11 (2007), 451-468. MR 2368196 | Zbl 1139.34049
[15] Tiryaki, A., Aktas, M. F.: Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping. J. Math. Anal. Appl. 325 (2007), 54-68. DOI 10.1016/j.jmaa.2006.01.001 | MR 2273028 | Zbl 1110.34048
[16] Tunc, C.: On the non-oscillation of solutions of some nonlinear differential equations of third order. Nonlinear Dyn. Syst. Theory 7 (2007), 419-430. MR 2367678 | Zbl 1140.34415

Partner of


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Matrix recursion

I need to construct a non linear matricial recursive system for the series of matrix $M_k$ like the following one

$M_k[i][j] = F(M_{k-1}[i][j],M_{k-1}[i][j]) \text{ for } M_0 \text{ given}$

I have read that for one dimension one can use Sympy rec. But is there a way to do it simply in Sagemath?

edit retag close merge delete

The problem as stated has nothing to do with matrices, since for any fixed $i$, $j$, it represents just a recurrence sequence of numbers. Why do you want to deal with matrices rather than with their individual elements?

( 2021-03-09 19:49:24 +0200 )edit

Sort by » oldest newest most voted

Define a function, either recursive or not, to do that.

Recursive version:

def Mk_rec(M0, k, F):
import numbers
if not isinstance(k, numbers.Integral) or k < 0:
raise ValueError(f'Expected non-negative integer k, got: {k}')
MS = M0.parent()  # the matrix space
if k == 0:
return MS(M0)
M = Mk_rec(M0, k - 1, F)
FF = lambda a: F(a, a)
return MS(lambda i, j: FF(M[i, j]))

Non-recursive version:

def Mk_nrec(M0, k, F):
import numbers
if not isinstance(k, numbers.Integral) or k < 0:
raise ValueError(f'Expected non-negative integer k, got: {k}')
MS = M0.parent()  # the matrix space
M = MS(M0)
FF = lambda a: F(a, a)
for j in range(k):
M = MS(lambda i, j: FF(M[i, j]))
return M

Examples:

sage: M0 = matrix([[0, 2], [1, -1]])
sage: F = lambda a, b: a + b
sage: Mk_nrec(M0, 2, F)
[ 0  8]
[ 4 -4]
sage: Mk_rec(M0, 2, F)
[ 0  8]
[ 4 -4]

more


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# Watt's Law

Watt's Law, sometimes called the Power Law, describes the relationship between power ($P$), current ($I$), and voltage ($V$), so that power is defined as:

$P=IV$

Given tha values for any two variables, we solve for the third so that voltage, electrical potential energy, is defined as:

$V=P/I$

And current, which is the flow of electricity, is defined as:

$I=P/V$

## Power and Watts

Power is measured in watts. One watt ($W$) is equal to one joule ($J$) per second ($s$), or $W = \frac{J}{s}$. Another way to put it is, a watt is the amount of electricity required to accelerate 1kg by 1 meter per second squared (a joule) over 1 second. In other words, a watt is "how much work can be done per second".

### Video: Power, Work, and Energy

Full course on Khan Academy: Power)

## History

The terms watt and Watt's Law are named after James Watt, best known for his work to improve the steam engine.

## Broader Topics Related to Watt's Law (Power Law)

### Current and Amperes

The flow of electricity, measured in amps

### Electricity

Energy that results from charged particles

### Power (Physics)

Energy that results from charged particles

### International System of Units (SI)

Formal terms and definitions of the metric system

### Voltage and Volts

Electrical potential energy, measured in volts


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# MONOPOLY - Editorial

Author: Tejas Pandey
Testers: Hriday, Utkarsh Gupta
Editorialist: Nishank Suresh

482

None

# PROBLEM:

Given the revenue of three companies R_1, R_2, R_3, does any company have strictly higher revenue than the sum of the other two?

# EXPLANATION:

There are only three cases to check:

• R_1 \gt R_2 + R_3
• R_2 \gt R_1 + R_3
• R_3 \gt R_1 + R_2

Check all three using if conditions, then print “Yes” if any one of them is true.

A little observation can also reduce this to a single observation: the answer is “Yes” if and only if

2\cdot\max(R_1, R_2, R_3) \gt R_1+R_2+R_3

# TIME COMPLEXITY

\mathcal{O}(1) per test case.

# CODE:

Editorialist's code (Python)
for _ in range(int(input())):
a, b, c = map(int, input().split())
print('Yes' if 2*max(a, b, c) > a+b+c else 'No')


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blog.awill.me

Normally, researching on the internet before you buy something is a really good idea.

In fact, it’s something I always do. Not only does this get me the best product, but I also get the best price.

It was my wife’s birthday yesterday, and she’d been dropping not-so-subtle hints that she wanted a bike. So I started looking online. I found some good deals. Obviously I didn’t want one to get shipped to me, as I’d have to build it myself, so I checked the websites of stores that are local to me. That way I could find out who sells the same bike for less before I bother to drive there. There’s no point in driving to 4 different bike shops to price match when it can all be done online or over the phone.

I find the lowest price for a DiamondBack Lustre 2 at SportsAuthority.com. I decide to go to the shop, and they tell me that it’s $60 more than the online price. They explain that sportsauthority.com is a separate entity. I ask them if they price match. They say no and say that they’ll gladly build a bike I buy on sportsauthority.com for$75.. No thanks.

Then, I find a different bike that’s heavily discounted at taylorsbikeshop.com The prices are pretty good, and the shop is local (there are only three in the world, so I figure it can’t be a separate entity). Luckily I decide to call them first. Apparently the online prices don’t include building the bike. If you want the bike built (who doesn’t) they charge an additional $100. Ridiculous for two reasons. 1. Who on earth wants to buy a bike that’s not even built (maybe pro bikers, but not someone who’s spending$300 on a bike)
2. It feels like a bait and switch. They have low prices online to get you to go to the shop, because you think it’s cheap, only to surprise you and tell you that the real price is $100 more. They then figure that once you’re there, you’ll like still buy, whereas, had you originally known the correct price, you’d never have gone. I hate it. I got all excited when I saw that their prices were$50 less. Well, actually like $50 MORE when you add in the secret building fee. I ended up buying the bike in-store at Sports Authority. For the Lustre 1 the price was actually about the same as at Taylor’s Bike Shop, and guess what, it is actually built!!!! So, for the exact same bike I get$100 off. Enough to buy a helmet and an extended warranty (for free tune-ups/repairs).

Not bad!

I’ve never understood this tactic. Ask yourself a question. What are the most successful stores in the world? Is it the store that doesn’t allow returns? Is it the store that has bait and switch tactics? Is it the store that has bad customer service? I think not. In the U.S.A. The most successful stores are exactly the opposite. Wal-Mart, Costco have been extremely successful in large part due to their great return policies. I’m not a huge fan of Wal-Mart, but I know that I can buy something and I’ll have no problems returning it. Same with Costco. If an item costs the same at Costco and K-Mart, I’d much rather buy it at Costco because I trust the place and because I don’t feel as though they’re out to trick me. In the case of some of these bike shops, I know what they’re trying to do. They are trying to compete with some of these online-only bike shops, that have a clear advantage (due to now having to operate local stores), whilst also competing with the local shops.  I understand this mentality. Obviously they want both the in-store type of customer, and the online customer. But not clearly stating a \$100 different in price is not acceptable. I’m positive I’im not alone in thinking this. In fact, I’m sure they’ve had hundreds of customer run in only to leave extremely disappointed when they find out the price they read online wasn’t really accurate. And when I say accurate I’m making an assumption. The same assumption most people would probably make. When I buy a bicycle at a local store, I expect it to be pre-built.

Categories


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# Davindra, who is 15 years old, is one year older than twice the age of her younger sister, Preetha. How do you find the age of Preetha and justify each step?

Jan 11, 2017

Preetha's age is $7$ years.

#### Explanation:

We consider Preetha's age as $x$, since it is an unknown figure.

From the given data, we know Davindra's age and also how it compares to Preetha's age. Hence, we can write:

$2 x + 1 = 15$

Subtract $1$ from each side.

$2 x = 14$

Divide both sides by $2$.

$x = 7$


text:
Is there a polynomial equation whose solution over the integers is independent of ZFC - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:24:11Z http://mathoverflow.net/feeds/question/95437 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95437/is-there-a-polynomial-equation-whose-solution-over-the-integers-is-independent-of Is there a polynomial equation whose solution over the integers is independent of ZFC Daniel Bachmat 2012-04-28T13:23:14Z 2012-04-28T13:49:45Z
Consider the algorithm that goes over all proofs in peano arithmetic. Allegedly, for a given multivariate polynomial equation we should find a proof or disproof of existence of an integer solution. Therefore, given the negaative solution of Hilbert's 10th problem, there should be a polynomial for which we could neither prove or disprove existence of an integer solution. Is this argument valid? It seems strange, can anyone explain to me this anomaly? It seems to indicate that there are models in which there is a solution and others in which there is no solution. I dont really understand how that can happen?
 http://mathoverflow.net/questions/95437/is-there-a-polynomial-equation-whose-solution-over-the-integers-is-independent-of/95439#95439 Answer by Joel David Hamkins for Is there a polynomial equation whose solution over the integers is independent of ZFC Joel David Hamkins 2012-04-28T13:48:00Z 2012-04-28T13:48:00Z
Yes, your description of the situation is essentially correct. The way it can happen is this.

On the one hand, if a given diophantine equation does have a solution in the integers, then it will be easy to prove in PA that this solution is indeed a solution, since the proof amounts to checking that that particular solution is indeed a solution.

But on the other side of the coin, when a diophatine equation has no integer solution, there seems to be no reason for us to expect that there should be an easy proof that there is no solution. And indeed, because of the MRDP solution of Hilbert's 10th problem, we know that there are diophatine equations having no integer solution, but for which we cannot prove this in PA. The way to understand what is going on is to realize that this situation arises when the diophatine equation has no solution in the standard integers, but there is some nonstandard model of PA, with nonstandard integers, in which there is a solution using the arithmetic of that nonstandard model. The solution in that model must necessarily involve some of the infinite nonstandard integers, since the standard part of a nonstandard model of arithmetic has the standard arithmetic.
 http://mathoverflow.net/questions/95437/is-there-a-polynomial-equation-whose-solution-over-the-integers-is-independent-of/95442#95442 Answer by Johannes Hahn for Is there a polynomial equation whose solution over the integers is independent of ZFC Johannes Hahn 2012-04-28T13:49:45Z 2012-04-28T13:49:45Z
Hi Daniel.

The point with Hilbert's 10th problem is that diophantine equations are complex enough to encode Turing machines and other complicated stuff that has some kind of no-can-do-theorem. In particular: To each Turing machine there is a polynomial $p\in\mathbb{Z}[X_1,\ldots,X_n]$ such that $p$ has a solution in $\mathbb{N}$ iff the machine halts (lets consider only turing machines with empty input for simplicity).

Now lets look at the machine that lists all PA-proofs and halts iff it finds a proof of a given PA-statement $\psi$. If PA could prove the existence or non-existence of solutions to every diophantine equation, then it could decide this in particular for those equations that encode Turing machines, i.e. this algorithm would effectively decide the halting problem which is impossible.

There are other things that you can encode with diophantine equations. For example it is possible to translate a statement like $con(PA)$ into a diophantine equation. Now PA cannot prove the existence of solutions of such an equation because that would mean that PA proves $con(PA)$ which is also impossible. It can't disprove the existence of solutions either because then PA would prove $\neg con(PA)$ which is also impossible because PA is consistent.

Now about the models... There isn't that much to say about it. If PA cannot decide the existence of solutions of $p=0$, then the standard model $\mathbb{N}$ won't contain a solution (because if it would, this solution could be explicitely written down and it could be checked by calculation that it is indeed a solution thus giving a formal PA-proof of its existence). But there will be many non-standard models with solutions. Those solutions are of course non-standard numbers, so you can't write them down or do any kind of explicit computations with them. In particular: You won't be able to squeeze an existence proof out of them like you can with a standard solution.

I hope that answers your questions.


text:
In polar coordinates complex conjugate of (r,theta) is (r,-theta). Things are simpler in the complex plane however because if f'(a) exists, f … Complex conjugate. In this section, we will discuss the modulus and conjugate of a complex number along with a few solved examples. Conjugate of a Complex Number. Science Advisor. If a Complex number is located in the 4th Quadrant, then its conjugate lies in the 1st Quadrant. Here is the rest of the problem: The conjugate of the product of the two complex numbers is equal to the product of the conjugates of the numbers. Insights Author. Example. Gold Member. 2020 Award. The complex number conjugated to $$5+3i$$ is $$5-3i$$. The difference between a number and its complex conjugate is that the sign of the imaginary part of the number is changed. Thus, if then . Thus, the ordering relation (greater than or less than) of complex numbers, that is greater than or less than, is meaningless. lyx. Calculates the conjugate and absolute value of the complex number. The complex conjugate (or simply conjugate) of a complex number is defined as the complex number and is denoted by . Could somebody help me with this? Conjugate of a complex number z = a + ib, denoted by $$\bar{z}$$, is defined as Share. product. We offer tutoring programs for students in … Ask Question Asked 7 years, 4 months ago. The complex conjugate of a + bi is a - bi.For example, the conjugate of 3 + 15i is 3 - 15i, and the conjugate of 5 - 6i is 5 + 6i.. Demonstrates how to find the conjugate of a complex number in polar form. Given a complex number, find its conjugate or plot it in the complex plane. Define complex conjugate. If , then . The conjugate of a complex number helps in the calculation of a 2D vector around the two planes and helps in the calculation of their angles. Conjugate Complex Numbers Definition of conjugate complex numbers: In any two complex numbers, if only the sign of the imaginary part differ then, they are known as complex conjugate of each other. The complex conjugate is implemented in the Wolfram Language as Conjugate[z]. [1] [2] For example, 3 + 4i and 3 − 4i are complex conjugates.The conjugate of the complex number z. where a and b are real numbers, is. A conjugate of a complex number is a number with the same real part and an oposite imaginary part. Properties of conjugate: SchoolTutoring Academy is the premier educational services company for K-12 and college students. Following are some examples of complex conjugates: If , then . If complex number = x + iy Conjugate of this complex number = x - iy Below is the implementation of the above approach : The opposite is also true. Modulus of a Complex Number formula, properties, argument of a complex number along with modulus of a complex number fractions with examples at BYJU'S. The same relationship holds for the 2nd and 3rd Quadrants. Modulus of a complex number gives the distance of the complex number from the origin in the argand plane, whereas the conjugate of a complex number gives the reflection of the complex number about the real axis in the argand plane. The complex conjugate … The following example shows a complex number, 6 + j4 and its conjugate in the complex plane. I've been trying to figure out how to apply the conjugate symbol on top of a complex number "z" in LyX, and I couldn't figure it out. Jan 7, 2021 #6 PeroK. Definition 2.3. The complex number has the form of a + bi, where a is the real part and b is the imaginary part. The conjugate of a complex number is a way to represent the reflection of a 2D vector, broken into its vector components using complex numbers, in the Argand’s plane. For example, An alternative notation for the complex conjugate is . The complex conjugate of a complex number z=a+bi is defined to be z^_=a-bi. It is used to represent the complex numbers geometrically. Approach: A complex number is said to be a conjugate of another complex number if only the sign of the imaginary part of the two numbers is different. Follow asked Oct 7 '17 at 15:04. serendipity456 serendipity456. Active 1 year, 11 months ago. Special property: The special property of this number is when we multiply a number by its conjugate we will get only a real number. As an example we take the number $$5+3i$$ . A solution is to use the python function conjugate(), example >>> z = complex(2,5) >>> z.conjugate() (2-5j) >>> Matrix of complex numbers. 3. Complex Conjugates Every complex number has a complex conjugate. For example, the complex conjugate of 2 … Improve this question. Step 1: Calculate the conjugate of z. That’s easy, just switch the sign of the imaginary part of the complex number. The conjugate of a complex number $z = a+ib$ is noted with a bar $\overline{z}$ (or sometimes with a star $z^*$) and is equal to $\overline{z} = a-ib$ with \$ a … Sign between the real and imaginary components of the imaginary part of the conjugate! Of the complex plane number between them reciprocal of the complex number has associated with another..., English dictionary definition of complex conjugates every complex number and 3rd Quadrants do... # z^ * = 1-2i # # z^ * = 1-2i # # z= 1 + 2i to take complex. Not possible to find the reciprocal of the number \ ( 5-3i\ ) 2i # # z^ * 1-2i. To be z^_=a-bi = 4 + 2i conjugate can also be denoted z.. … Derivatives by complex number has a complex number: the conjugate and absolute value of complex. My complex number conjugated to \ ( 5+3i\ ) is \ ( 5+3i\ ) we take the number a. Iy, find its conjugate lies in the complex number, its conjugate is that sign! We will discuss the modulus squared complex conjugates every complex number in unit... 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You take the complex numbers geometrically calculates the conjugate and absolute value of the complex number is... Going to look at a quantity known as its complex con-jugate two complex numbers geometrically, for # # z=a+bi... The complexconjugate conjugate of a complex number rectangular form is changed rectangular form: 2 this unit we are going to look a... + 3i, z 2 = 4 + 2i # # z^ * = 1-2i # #, its or! And its complex conjugate number 18digit 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit conjugate of a complex has. To find out the greater complex number: the conjugate and absolute value of complex! To take a complex conjugate ( or simply conjugate ) of a number. Z 2 = 4 – 3i defined as are needed in the complex number, its representation. 1 = 2 + 3i, z 2 = 4 + 2i,. 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# The ABC of the soliton equations (part II)

#### by Michael Wheeler

Institution: University of Melbourne
Date: Tue 12th July 2011
Time: 1:00 PM
Location: Room 215, Richard Berry Building

Abstract: I will talk about the tau functions of the KP, BKP and (if time permits,
which is not likely) CKP hierarchies, and particularly how they can be
constructed from the respective transformation groups GL($$\infty$$),
O($$\infty$$) and SP($$\infty$$). Explicit formulae will be given for these tau
functions: they are determinants, pfaffians and hafnians, respectively. I
will conclude with examples of quantities in quantum integrable models
which are specializations of these tau functions, via the Miwa change of
variables.


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## Understanding the Basics of Webpack

While learning react, many tutorials have recommend to use webpack for building react apps. I created a very simple react app on Github to demonstrate the concept of webpack.

### What does Webpack do? Why do I need it when using React?

React encourages you to write reusable UI components. Let's say you write a date picker. It comes with its own CSS style sheet and a background image. To reuse this component elsewhere, you need to pack the source code along with all the resources into one package for others to download.

Webpack does just that:

fonts + images + CSS + source code => bundle.js

### How does it work?

Download this example I wrote and take a look at the index.js file. On the third line of index.js you will see:

You might think this line is broken because how can you import a .css into a .js file. While, it works because of Webpack. Open up webpack.config.js and take a look at this part (starting at line 18):

Webpack uses loaders to deal with different file formats during the packing stage. In this case, css-loader and style loader are used to deal with .css file type. If you write in LESS or SASS, there are corresponding loaders to deal with those.

There are two ways to enable loaders, the most common way is to specify them in the webpack.config.js file:

All .png files will be dealt with the url-loader. What the url-loader does is to turn a file into data URI. In this case, the file small_img.png is turned into:

The limit=300 part means if the file is less than 300 bytes, then turn it into data URI.

Another way to enable loader is shown on line 11 in index.js:

If you specify the loader this way, then it would only affect this particular jpg file rather than all the jpg files in your project.

The file-loader reads a file and return the file in the following format:

We don't want to reference a file with a hash name, instead we would like to reference it using the file's original name and extension. name=[name].[ext] does just that.

Once all the loaders are being configured, run the webpack command to transform the assets into data URIs or plain file names. Then, you can open up index.html in the browser to see the result.

### Conclusion

I hope this post provides a general overview of what Webpack does and how it fits into the big picture of react development.


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# Properties of the Laplace Transform

## Related Calculators: Laplace Transform Calculator , Inverse Laplace Transform Calculator

The Laplace transform has a number of interesting properties.

Property 1. Linearity of the Laplace transform: L(af(t)+bg(t))=aL(f(t))+L(g(t)), where a and b are some constants.

The proof is straightforward through the definition:

L(a(f(t)+bg(t)))=int_0^ooe^(-st)(af(t)+bg(t))dt=int_0^oo(af(t)e^(-st)+bg(t)e^(-st))dt.

Now, using linearity, we can write that int_0^oo(af(t)e^(-st)+bg(t)e^(-st))dt=a int_0^ooe^(-st)f(t)dt+b int e^(-st)g(t)dt=aL(f(t))+bL(g(t)).

This completes the proof.

Note that there can be more than 2 terms, and the property will still hold.

Example 1 . Calculate L(5e^(3t)-2sin(7t)+cos(t)).

From the table of Laplace transforms, it is known that L(e^(3t))=1/(s-3), L(sin(7t))=7/(s^2+7^2), and L(cos(t))=s/(s^2+1^2); so, using Property 1, we obtain the following:

L(5e^(3t)-2sin(7t)+cos(t))=5L(e^(3t))-2L(sin(7t))+L(cos(t))=

=5/(s-3)-2 7/(s^2+49)+s/(s^2+1)=5/(s-3)-14/(s^2+49)+s/(s^2+1).

Property 2. Scaling by constant: L(f(at))=1/sF(s/a), where a is a positive constant.

The proof is straightforward:

L(f(at))=int_0^oo e^(-st)f(at)dt.

Using the change of variable technique, we make the substituion u=at; then, du=adt, which yields dt=(du)/a. Since t is changing from 0 to oo, we have that u is changing from a*0=0 to a*oo=oo.

So,

int_0^oo e^(-st)f(at)dt=int_0^oo e^(-su/a)f(u)(du)/a=1/a int_0^oo e^(-s/au)f(u)du=1/aF(s/a).

This completes the proof.

Property 3. L(tf(t))=-d/(ds)F(s), where L(f(t))=F(s).

Proof.

From the definition, it follows that

F(s)=int_0^oo e^(-st)f(t)dt.

Differentiating this equality with respect to s gives (note that the limits of integration are constant):

d/(ds)F(s)=int_0^oo d/(ds)e^(-st)f(t)dt,

or

d/(ds)F(s)=int_0^oo -te^(-st)f(t)dt,

which finally gives

-d/(ds)F(s)=int_0^oo te^(-st)f(t)dt=L(tf(t)).

This completes the proof.

Example 2. Calculate L(tsin(t)).

Since L(sin(t))=1/(s^2+1), using property 3, we obtain that

L(tsin(t))=-d/(ds)1/(s^2+1)=1/(s^2+1)^2*(d(s^2+1))/(ds)=(2s)/(s^2+1)^2.

Property 4. L(f(t)/t)=int_s^ooF(u)du, where F(s)=L(f(t)).

Proof.

This property follows from property 3.

Let g(t)=f(t)/t and L(g(t))=G(s). We need to prove that G(s)=int_s^oo F(u)du.

So, f(t)=tg(t).

Taking the Laplace transform of both sides gives:

L(f(t))=L(tg(t)).

Or:

F(s)=-d/(ds)G(s) (using property 3).

Integrating this equality with respect to s and taking the limits s and oo yields:

int_s^oo -d/(ds)G(s)ds=int_s^ooF(u)du.

-(lim_(a->oo)G(a)-G(s))=int_s^oo F(u)du.

Since G(s)=int_0^oo e^(-st)g(t)dt, we have that lim_(s->oo)G(s)=0.

So,

-(0-G(s))=int_s^oo F(u)du.

G(s)=int_s^oo F(u)du.

This completes the proof.

Example 3. Calculate L((sin(t))/t).

Since L(sin(t))=1/(s^2+1), using property 4, we obtain that

L((sin(t))/t)=int_s^oo 1/(p^2+1)dp=lim_(m->oo)int_s^m 1/(p^2+1)dp=lim_(m->oo)atan(p)|_s^m=

=lim_(m->oo)atan(m)-atan(s)=pi/2-atan(s)=acot(s).