He starts his article reporting a cute behavior displayed by his little toddler : during this morning breakfast, he observed his two-year-old son handling a spoon when he noticed, apparently at the first time, his face reflected on the convex side of the spoon. After some moment contemplating the image of his own face reflected, the boy turned the spoon to the other side and looked at its concave face. Fernando noticed a remarkable reaction of surprise in his son’s face, as his eyes brows revealed, when the boy found his reflected image now inverted. Then the kid started playing with it, turning to the convex side and back again to the concave, over and over, figuring out that one of the sides was constantly inverting his reflection. Finally, he concentrated on the concave side and did the unexpected: he rotated the spoon in such a way that the handle, which was first pointing to the ground, should then be pointing to the ceiling, as he would be trying to “uninvert” the image. Fernando then noticed that the boy looked somehow frustrated because he couldn’t “correct” his reflex to the “right position” by this expedient, since he then had thrown away the spoon, seemingly bored with the loose of time.

Fernando used this fact to illustrate the principle that the brain of a child isn’t, as many believe, like a blank paper, where parents and educators “write” the teachings and explanations. There are some, or rather, many innate abilities and presumptions about how things would behave, and these can be exploited with advantage during education. The blank paper metaphor is quite ancient, dating to the epoch of Aristotle. It was after called tabula rasa, a term in Latin denoting the idea that at birth a child hasn’t any knowledge about the world. However, studying the behavior of babies and toddlers, one can find signals that there is an innate knowledge concerning primary assumptions about the rules that govern the physical world. What is not known is exactly what could be these primitives. However, it is possible to observe that the babies quickly develop expectations, like “one object cannot occupy the place of another at the same time” or, “the light comes from above” , and “objects fall downwards”, etc. These expectations presuppose the existence of some prior knowledge, which we just referred as the primitives. One can say that the baby starts with some primitive innate knowledge that we could call the initial state of knowledge. From this beginning, the knowledge is constantly expanded and rebuilt, as more experience is acquired. And this process of continuous re-elaboration follows ahead during the childhood, adolescence and so on, to the adulthood.

Piaget stated that there are four sequential stages during the development of the child, each one characterized by a series of constructive operations, that assemble, step by step, the cognitive abilities. There are some disagreements with this theory, related to the evidences of modularity and parallelism in the perceptual and cognitive processes in the neuro-sensorial system. It is observed that the order of development of some operational and functional abilities can be somehow different from a child to another, inside the stages and between the stages.

The mirror stage and the mirror test

During the toddlerhood, there is a phase called the mirror stage, that is related to the moment when the self-recognition arises. A very simple experiment with babies and toddlers, that can be done by parents, is the mirror test. First, you have to paint a red spot on the forehead of the child, with a size that he can see it easily. Them, put him in front of a mirror. A baby that hadn’t yet developed self-recognition will try to touch the spot in the “other” baby (i.e., the image reflected in the mirror), as he would be trying to “clean” or check the spot on the virtual baby. On the other hand, toddlers are generally able to recognize themselves on mirror images, and they show a different reaction: in this case, the child touches his own forehead, to check or to clean the spot.

The mirror test can also be performed with animals, to verify their cognitive capacity of self-awareness. However, the results with animals are often controversial. There are other factors that can lead the animal to fail in the test, besides an eventual lack of self-awareness. Among these factors, a possible color blindness or a limited stereoscopic vision can affect the animal capacity in identifying itself in the mirror image. These problems can eventually show up in the human case. Besides these visual impairments , behavioral factors can also influence the test result: in many species, the eye contact with other individual is a cue for threatening, and it can be a disturbance to the self-recognition in the image.

There are two aspects to consider when analyzing the results of the test :

• to fail in the mirror test means that the individual is not self-aware ?
• to pass in the mirror test means that the individual IS self-aware ?

The failure case, as we considered before, doesn’t mean that it was a consequence of lacking of self-awareness, but it could be due to other causes. The tough question is the second, the pass case. In the human case it seems natural to admit the intervening of the self-awareness to produce the result of the test. However, when considering other animals, one could be reluctant in admitting the presence of self-awareness in many animal species. We expect that some species eventually could have it, but others no.

Self-awareness requirements

To create a test for the detection of self-awareness, we need some definition of self-awareness. It is very difficult to define it, however we can establish certain requirements that seem to be fundamental if we are going to find self-awareness on an individual. For a creature to be self-aware, it seems reasonable to require the fulfillment of the following requisites:

1. It must be able to perceive the patterns of its own behavior, separating them from other events. This is equivalent to say: “These observations refers to a separate class of events”. This separate class of events are its own actions.
2. It must be able to behave consistently with the hypothesis that it can correlate the perceived patterns of its own behavior with its own actions. This is equivalent to say: “These observations occur at the same moment as these actions”. These observations are the consequences of its own actions.
3. It must have a sense of causality. Or, it could say: “These observations were caused by these actions”. In order to do this, the creature must be able to make experiments. To make an experiment, it must be able to control variables, that is, select a specific value of it and check its effect on an observation.
4. It must be able to categorize the causes, labeling them to a same origin. In other words, the observations and the actions that caused them can be categorized as an agent. This is equivalent to say: “The source of these facts is unique”. That is, it must “conceive the agent”.
5. It must control the agent. The self-awareness is something emerging from this control. It exhibits the intention to make the agent perform specific actions.

Now, let us check if the passing case of the mirror test provide issues that fulfill these requirements. If an individual or animal is said to have passed in the test, it is because: (i) the subject displayed a behavior consistent with the identification of the red spot (or something similar) on its forehead and (ii) it gave signs of disapproval of the presence of the spot on its forehead (e.g., tried to clean it). In order to draw the conclusion that the red spot is on its own forehead, the subject has to see its head moving and bringing the spot with it. So, it separated the movements of its forehead from other events that it could perceive at the same time (1). It moved its head, following the movement of the spot (2). If it performed specific movements with its head, expecting to bring the spot to a specific position, following its head, then (3) is also satisfied. This should be considered if we agree that the animal passed the test. However, (4) and (5) are not evident. This experiment is not telling us much about the conception and control of an agent.

Robots taking the mirror test

In the field of machine intelligence, there is a great desire to improve the cognitive properties and abilities of machines, such as robots. One could consider that it should be very interesting if a machine could exhibit self-awareness. So, why not try the mirror test as an experiment to check the claim that some robot could be self-aware ? This was exactly what Takeno, Inaba and Suzuki had done at Meiji University.

This work was reviewed at Conscious Robots by Raúl Moreno. In his words: “The robot is able to recognize its own image reflected in a mirror without confusing it with the image of another robot with the same physical aspect. The mirror image cognition system is based on an artificial neural network. The aim of this system is to recognize and differentiate robot’s own behavior from other robot’s behavior. Takeno also suggests that imitation is a proof of consciousness as it requires the recognition of other subject’s behavior and then the application of that behavior to oneself.”

Let us check if the requirements for self-awareness can be fulfilled by the test in this case. If the robot can separate the region in the image in the mirror that corresponds to its own reflection in a video sequence, and maintain it correlated with a consistent set actions, it can certainly fulfill (1) to (3). If it can also group the actions-observations causal pairs in classes consistently with (4), then the robot would be able to conceive agents as categories of behaviors+explanations. But how can one say that the robot can also control the agent that corresponds to itself ? It could control activities that are consistent with the agent as uniquely distinct from the others. However, is the robot selecting its own actions because they are consistent with the agent actions (i.e., consistent with the idea of an agent) ? Or is it simply performing the control action in order to observe specific results in the estimation of a perceived image ? For instance, it could select the actions that bring the centroid of the pattern to a specific trajectory , without association that it is controlling the agent that it perceives. In order to claim that the robot is controlling the agent, we must be able to say that the control must be consistent with the idea of an agent.

Well, in my point of view, more specific experiments should be devised to answer these questions.

” I heard about the changes in the WP’s T&C, which now says: “By making Content available, you represent and warrant that: [...] your blog is not named in a manner that misleads your readers into thinking that you are another person or company. For example, your blog’s URL or name is not the name of a person other than yourself or company other than your own”.

I hope that the case is going to be finally solved, and that WordPress.com will still be available in Brazil. Following, I’m putting across some reflections related to this case :

1. We all aspire to freely express ourselves in all aspects and situations, be it in the “cyberspace” or outside it. However, a very important issue concerns with liberty: our freedom goes up to the point that it does not interfere with another one’s liberty, and reciprocally.
2. In respect to someone’s rights, we cannot use our freedom to cause any kind of harm or offense to anyone else. Also, this implies some legal responsibilities: we cannot let that somebody acts in our names, to cause harm or to offend someone else.
3. Concerning real human beings, living their real lives, with their constituted rights, there is no such a place as a “cyberspace” or a “virtual society”. If you do something in the Internet, you are reaching real people, influencing them and possibly could be causing some harm. The implications of this will happen in their real lives. So, the place where the actions or damages should be judged are the civil rights institutions of their living places, or countries.
4. It is conceivable that it is very romantic believing in a place like a “virtual networked society”, or whatever in this sense, where you are totally free. However, we must care of not confounding it with a kind of “cartoon world“, where even if you are fatally shot you won’t die. The cyberspace, using the name created by Gibson to denote this place, can be an abstraction of the actions of real people in a virtual place, but the people that interface with it are real, actual human beings. Being completely free doesn’t mean unrestrictedly free. The freedom of one restricts the freedom of other.

Well, this is some kind of response to some comments that I’ve read or heard about the case. It is not a legal statement, it’s just my point of view. It can be technically wrong in some aspects, so I hope that someone could correct me in the case. I feel that we must care to avoid creating a sense that everything is allowed in the Internet. I remember the words of Clifford Stoll in his book The Cuckoo’s Egg, that the Internet would work only if people could trust it. And I see that this includes respecting – in all aspects – the rights of people that uses the Internet:  no offenses should be directed to anyone,  no  court should ask for undiscriminated blocking of everybody.

To finish, we must thank Marcel Leonardi for his effort, defending our interests, and to WordPress for their wise decision.

• Basics and overview
• Use of mathematical symbols in formulas and equations
• Multiline equations, systems and matrices
• Fine-tunning and some extras
• LaTeX – extra examples and some bugs

However, to my surprise, the first post reported almost 3000 views in the same first day it was published, and I got nice comments on it. This was really very stimulating ! I show below a graph with the first month of activity on both blogs. Machine Intelligence ? has now a steady average activity of around 30 views a day, while Videlicet got its peak on mere 67 views and reached a steady average of around 3 views a day.

Stats on my blogs Machine Intelligence ? and Videlicet during the first month. Notice the two distinct vertical scales.

However, I am also satisfied with Videlicet, it is doing its job ! It was devised to be a means of connecting longer with my undergrad students and the statistics shows a more intense activity when the first homework report was about to reach its deadline. And the number of views was very consistent with the size of my classes (total of 160 students of Electrical Engineering sophomores). I started with three posts on basic electricity:

• Propagation of systematic measurement errors
• Ohm’s law
• Nonlinear resistors and incremental resistance

Blogging has been a great experience and it increased the interest of people on my activities, and received good feedback, although not as comments directed to my posts, as I expected, but as personal communications. Anyway, I recommend for the ones that are still in doubt about starting a blog: try it, at least !  I sometimes heard rumors against blogging, but when asking the contenders, I got no reasonable points, except a pure personal resistance against something they haven’t done a serious evaluation.

Now, after a period away from here, I’m back, for new postings.

This is the 5th and last post in this series. Previous ones:

• Basics and overview
• Use of mathematical symbols in formulas and equations
• Multiline equations, systems and matrices
• Fine-tunning and some extras

Now I’m closing this series with some extra examples and reporting some bugs.

Extra examples

Here I show some examples with a discussion of peculiar issues. Some other also interesting remarks can be found in the About page of Terence Tao’s blog What’s New.

Example 1

\langle q | f ( \hat{q} ) = f (q) \langle q |

Notice that the \left and \right are not necessary in this case, however, if used it will work:

\left \langle q | f ( \hat{q} ) = f (q) \langle q |

Example 2

$ latex \| \vec{u} + \vec{v} \| = \| \frac{u + v}{u-v} \| &fg=aa0000&s=1 $

Again, \left and \right were not necessary. However, if used, they can adjust the sizes of the norm frames:

$ latex \| \vec{u} + \vec{v} \| = \left \| \frac{u + v}{u-v} \right \| &fg=aa0000&s=1 $

This can be better seen in the next example.

Example 3

$ latex (\frac {x^{a^m}}{y^{b^n}} ) \quad \left ( \frac {x^{a^m}}{y^{b^n}} \right ) &fg=aa0000&s=2 $

Bugs

My main intention with this series of postings was to do some tests with the LaTeX support of the WordPress edition environment. It is quite likely that one could try to use here previous works involving LaTeX expressions or done in other environments with LaTeX-like coding, so I decided to check issues on portability.

I took the the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup, as the main source of expressions to test here, in this environment.

Sometimes I’ve gotten parsing errors that are overcame simply by saving and reloading the post again. The process didn’t appear to be deterministic, so I guessed it was due to issues related to the connection protocol employed (file transfers usually go under UDP, so probably something could be missing on the way, I guess).

In other two situations I found a curious behavior of the parser, given that it could not parse expressions that I expected should be parseable or the behavior was inconsistent. Following I’ll show some examples.

Bug example 1

This construction works: $ latex ( 0,1 ) \; [ a,b ] \; ( 0,1 ] \; [ 0,1 ) &fg=aa0000&s=1 $

This too: $ latex ( a,b ] \; [ a,b ] \; ( 0,1 ] \; [ 0,1 ) &fg=aa0000&s=1 $

However, this doesn't work: $ latex [ a,b ] \; [ a,b ] \; ( 0,1 ] \; [ 0,1 ) &fg=aa0000&s=1 $

Furthermore, this works: $ latex \left [ a,b \right ] \; [ a,b ] \; ( 0,1 ] \; [ 0,1 ) &fg=aa0000&s=1 $

However, this doesn’t : $ latex {\frak ABC} &fg=aa0000&s=1 $

To change the font of all the letters one has to change the declaration to: $ latex {\frak A} {\frak B} {\frak C} &fg=aa0000&s=1 $

Notice that only the first letter followed the \frak declaration, while with the other fonts declarations (\cal, \rm, \sf etc) all the letters inside the braces will have their font changed. So,this is not exactly a bug, but is inconsistent with all the other font declarations.

This is the 4th post in the series. Previous ones:

• Basics and overview
• Use of mathematical symbols in formulas and equations
• Multiline equations, systems and matrices

Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. .

Spacing in formulas

There are some modifiers that can add some spaces between LaTeX constructs displayed in mathematical expressions:

\, small space

\: medium space

\; large space

\! negative space

\quad – large separation

\qquad – very large separation

Examples:

• \forall x,y \, \exists z gives
• \forall x,y \, \exists z gives
• \forall x,y \: \exists z gives
• \forall x,y \; \exists z gives

• \sum_i \; \sum_j x_{i,j} gives
• \sum_i \; \sum_j x_{i,j} gives
• \sum_i \! \sum_j x_{i,j} gives

• \sum_i x_i q\quad \sum_j y_j gives
• \sum_i x_i \qquad \sum_j y_j gives

.

Font styles

Supported font styles:

• {\rm Roman}
• {\bf Bold}
• {\it Italic}
• {\sf Sans Serif}
• {\sl Slanted}
• {\cal A B C D E F}
• {\frak A} {\frak B} {\frak C} {\frak d} {\frak e} {\frak f}

Examples:

a) default style
\oint_C \nabla \phi \cdot dl = \iint_S rot \, E \cdot dS gives

b) some changes: nabla to triangledown, fonts in rot and E
\oint_C \triangledown \phi \cdot dl = \iint_S {\bf rot} \, {\sf E} \cdot dS

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Some extra symbols

Some Hebrew characters, Greek characters and some miscellaneous stuff . Some special symbols. There are many more in Wikipedia article.

\aleph \beth \gimel \daleth

\varpi \varrho \varsigma \varphi \varepsilon \vartheta \varkappa

\pi \rho \sigma \phi \epsilon \theta \kappa (compare with the ones above)

\alpha \beta \gamma \delta \zeta \eta \iota \lambda \mu \nu \xi \tau \upsilon \phi \chi \psi \omega

\Theta \Upsilon \Phi \Psi \Omega

\Gamma \Delta \Lambda \Xi \Pi \Sigma

\digamma \Finv \varpropto

\flat \natural \sharp \hbar \Bbbk

\diamondsuit \heartsuit \clubsuit \spadesuit \Game

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Parenthesizing big expressions, brackets, bars

Examples of several types of constructions with parenthesizing

\big( \Big( \bigg( \Bigg( … \Bigg] \bigg] \Big] \big]

\big\{ \Big\{ \bigg\{ \Bigg\{ … \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle

\big\| \Big\| \bigg\| \Bigg\| … \Bigg| \bigg| \Big| \big|

\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor … \Bigg\rceil \bigg\rceil \Big\rceil \big\rcei

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow … \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow

\big\updownarrow \Big\updownarrow \bigg\updownarrow \Bigg\updownarrow … \Bigg\Updownarrow \bigg\Updownarrow

\big / \Big / \bigg / \Bigg / … \Bigg\backslash \bigg\backslash \Big\backslash \big\backslash

More constructions

Parentheses ( \frac{a}{b} ) \quad \left ( \frac{a}{b} \right )

Brackets \left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack

Braces \left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace

Angle brackets \left \langle \frac{a}{b} \right \rangle

Bars and double bars \left | \frac{a}{b} \right \vert \quad \left \Vert \frac{c}{d} \right \|

Floor and ceiling functions \left \lfloor \frac{a}{b} \right \rfloor \quad \left \lceil \frac{c}{d} \right \rceil

Slashes and backslashes \left / \frac{a}{b} \right \backslash

Up, down and up-down arrows \left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow

Observations:

Delimiters can be mixed, as long as: \left and \right match

\left [ 0,1 \right )

( 0,1 ) [ 0,1 ] ( 0,1 ] [ 0,1 ) also work , without the need of \left and \right

\langle \psi | …….\langle \psi | \phi \rangle \ ….. and

\langle \psi | \, A^TBA \, | \phi \rangle ……. also work

Use \left. and \right. if you don’t want a delimiter to appear:

\left . \frac{A}{B} \right \} \to X

This is the 3rd post in the series. Previous ones:

• Basics and overview
• Use of mathematical symbols in formulas and equations

Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup.
.

Multiline Equations

You can present equations with several lines, using the array statement. Inside its declaration you must :

• Define the number of columns
• Define column alignment
• Define column indentation
• Indicate column separator with & symbol &

Example: {lcr} means: 3 columns with indentations respectively left, center and right

\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array}{rcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array}{rcl} f: R^3 & \to & R \\ (x,y,z) & \to & x + y + z \\ f(x,y,z) & = & x + y + z \end{array}

\begin{array} {lcl} f(x) & = & (a+b)^2 \\ & = & a^2+2ab+b^2 \end{array}

.

Case definitions

Used when a definition have two or more cases. Use the case statement. Notice that the spaces after the instances of if were included inside the mbox declarations.

f(n) = \begin{cases} n/2, & \mbox{if } n\mbox{ is even} \\ 3n+1, & \mbox{if } n\mbox{ is odd} \end{cases}

.

Simultaneous Equations

Here we have a very simple application of the case statement.

\begin{cases} 3x + 5y + z \\ 7x – 2y + 4z \\ -6x + 3y + 2z \end{cases}

.

Matrices

Matrices can be assembled by using the array statement, like in this example:

\left| \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array} \right|

Matrix frames are provided by \left and \right. If you suppress these statements, it will be displayed like:

\begin{array}{cc} A & B \\ C & D \end{array}

However, there is another statement, the matrix declaration, slightly easier to use:

\begin{matrix} x & y \\ z & v \end{matrix}

The frames of the matrix can be displayed in several forms, by just changing the matrix declaration to vmatrix, Vmatrix, bmatrix, Bmatrix or pmatrix, as shown ahead:

\begin{vmatrix} x & y \\ z & v \end{vmatrix}

\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}

\begin{bmatrix} x & y \\ z & v \end{bmatrix}

\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}

\begin{pmatrix} x & y \\ z & v \end{pmatrix}

As a final example, let us construct more complicated matrices and matricial expressions:

\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

Now, we will show a first matricial expression:

\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \times \left[ \begin{array}{c} y_1 \\ y_2 \end{array} \right]

and, here is another example, now using two kinds of vectors:

\begin{bmatrix} xz & xw \\ yz & yw \end{bmatrix} = \left[ \begin{array}{c} x \\ y \end{array} \right] \times \left[ \begin{array}{cc} z & w \end{array} \right]

This is the 2nd post in the series. Previous one:

• Basics and overview

This series shows my first experiences with using the mathematical expressions handling tools provided by WordPress for blogging. They use a version of the LaTeX syntax.

Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup.

.

Accents – Diacritics

Examples on how to put accents in mathematical expressions:

\acute{a} \grave{a} \hat{a} \tilde{a} \breve{a}

\check{a} \bar{a} \ddot{a} \dot{a}

x’, y”

.

Superscript and subscript

How to display subscripts and superscripts, indexes and exponents:

Subscripts:

• a_i gives
• b_{ij} gives
• C_{m,n} gives
• \delta_{j+k} gives

Superscripts:

• x^y gives
• a^{j2\pi} gives
• x^2_3 gives
• C^k_{\mu,\nu} gives

Composition with preceding indexes: {}_1^2 \Psi_3^4 gives

.

Underlines, overlines and stackings

Includes vectors

\hat a \ \bar b \ \vec c

\overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f}

\overline{g h i} \ \underline{j k l}

\overbrace{ 1+2+\cdots+100 }^{5050}

\underbrace{ a+b+\cdots+z }_{26}

A \xleftarrow{n+\mu-1} B \xrightarrow[T]{n\pm i-1} C

\overset{\alpha}{\omega} \underset{\mu}{\nu} \overset{\beta}{\underset{\Delta}{\tau}} \stackrel{\zeta}{\eta}

.

Sets

Sets operations and related symbols.

\in \ni \notin \varnothing \complement

\subset \subseteq \subsetneq \supset \supseteq \supsetneq

\cap \bigcap \cup \bigcup

\ell \mho \Finv \Re \Im \wp

Others – examples using the calligraphic font (\cal) and the Greek font for designating sets:

{\cal A} \setminus {\cal B} gives

\Omega \smallsetminus \omega gives

.

Logic

Logical operators and relations:

\forall \exists \nexists \bar{A} \mid

\And \wedge \vee \neg \to \gets \iff

\bigwedge \bigvee \diamond \lozenge

\vdash \Vdash \vDash \Vvdash \models \dashv

Examples:

• \forall p,q \, \exists q \mid \bar{q} \to p gives
• \bigwedge _{x \in A} gives
• \bigwedge \limits _{x \notin A} gives
• \bar{A \vee B} = \bar{A} \wedge \bar{B} gives
• A \iff B = A \to B \wedge A \gets B gives
• \bigcap \limits _{i=1}^n \bigcup \limits _{j=1}^n {\cal B}_{i,j} gives

Obs – the statement \limits shown in the examples above puts the indexes exactly above and / or below the symbol. In the first example, \, is used to put an extra space. .

Operators

Several types of operators:

+ \oplus \bigoplus \pm \mp -

\times \otimes \bigotimes \cdot \circ \bullet \bigodot

\star * / \div \frac{1}{2}

\sqrt{2} \sqrt[n]{x}

\nabla \partial x \dot x \ddot y

Examples:

• \rho = \sqrt{x^2 + y^2} gives

• \nabla \phi (x,y) = \frac{\partial \phi}{\partial x} + \frac{\partial \phi}{\partial y} gives

• \nabla^2 \phi (x,y) = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} gives

• \frac{\partial^2 \phi}{\partial x \partial y} = \frac{\partial^2 \phi}{\partial y \partial x} gives

.

Relations and definitions

To specify relations, mappings and definitions

\sim \approx \simeq \cong \dot =

< > \le \ge \ll \gg

\lessgtr \lesseqgtr \lesseqqgtr

\equiv \not\equiv \ne \propto

\mapsto \longmapsto

.

Geometric

Geometric symbols

\circ \bigcirc \Diamond \Box \triangle

\vartriangle \triangledown \triangleleft \triangleright \vartriangleright \vartriangleleft

\angle \sphericalangle \measuredangle 45^\circ

\perp \mid \nmid \| \asymp \parallel

.

Arrows

Some more frequent types of arrows (there are many more – see in Wikipedia article)

\leftarrow \rightarrow \leftrightarrow \Leftarrow \Rightarrow \Leftrightarrow

\leftarrow \gets \rightarrow \to \not\to \leftrightarrow \longleftarrow \longrightarrow

\rightleftharpoons \leftleftarrows \leftrightarrows \Lleftarrow \leftarrowtail

\uparrow \downarrow \updownarrow \Uparrow \Downarrow \Updownarrow

.

Special symbols

Some special symbols. There are many more in Wikipedia article

\S \P \% \dagger \ddagger \ldots \cdots

\smile \frown \wr \triangleleft \triangleright \infty \bot \top

\imath \hbar \jmath \surd \ast \amalg \therefore \backepsilon \sharp

.

Summations, Integrals and Products

Several cases, including limits, sequences and series. Notice in the examples below that when you want to put the limits with the same vertical alignment of the math symbol, you must use the \limits declaration. Otherwise, the limits will be put ahead of the symbol.

\lim \limits_{n \to \infty}x_n

\lim _{n \to \infty}x_n

\sum_{k=1}^N k^2

\sum \limits_{k=1}^N k^2

\prod_{i=1}^N x_i

\prod \limits_{i=1}^N x_i

\coprod_{i=1}^N x_i

\coprod \limits_{i=1}^N x_i

\int_{-N}^{N} e^x\, dx

\int \limits_{-N}^{N} e^x\, dx

\iint_{D}^{W} \, dx\,dy

\iiint_{E}^{V} \, dx\,dy\,dz

\iiiint_{F}^{U} \, dx\,dy\,dz\,dt

\oint_{C} x^3\, dx + 4y^2\, dy

Obs – the declaration \, in the above integrals puts extra spaces between consecutive letters. See more about alignement on this post : LaTeX – Fine-tunning and some extras.

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Binomials

Binomials only. For matrices, see next post.

\binom{n}{p} = \frac{n!}{p!(n-p)!}

The post below was my first atempt to use LaTeX in the WordPress environment.  It shows some quick examples that I used as tests to find what works and what doesn’t work and which were the differences between  WordPress LaTeX and the standard version found at CTAN (Comprehensive TeX Archive Network). Further, I made some other posts reporting more sistematic experiments that became a reference for my own use of WordPress LaTeX.

These are the other posts:

• Uses of Math Symbols and Equations
• Multiline Equations, Symbols and Matrices
• LaTeX Equations Fine-tunning
• Extra examples and some bugs found in WordPress LaTeX

This is the original post (with the so-called funny examples):

_________________________________________________________________________

March, 10, 2008

Now it’s time to try using WordPress blog support to LaTeX. It seems pretty easy. You only need two things:

1. To read this announcement.
2. To know a bit about LaTeX (find a LaTeX primer here).

Let’s try some basic thing:

To do it just write $ latex \frac{100}{20} = 5 $ , following the indications on the announcement.

Hint: 1. to display the LaTeX syntax without parsing it just put a space between $ and latex statements.

Hint: 2. You can write the latex statements straight on the “Visual” tab of the editing interface – you don’t have to go to the “Code” tab to do it.

Let’s now do some more complex things:

$ latex \frac{x+y}{a} + \frac{x-y}{b} = \frac{b(x+y) +a(x-y)}{ab} $ gives:

In order to display with a greater font size you add &s=X at the end of the expression, as shown below. The X value is illustrated in the announcement.

$ latex \frac{x+y}{a} + \frac{x-y}{b} = \frac{b(x+y) +a(x-y)}{ab} &s=1$ gives:

$ latex \frac{x+y}{a} + \frac{x-y}{b} = \frac{b(x+y) +a(x-y)}{ab} &s=2$ gives:

OK, now I have finished the for Dummies level. Let us improve a bit more !

More math statements

In the following examples, I’ve changed the color of the formula by adding the statement &fg=aa0000 before the &s=1. The string represents the RGB values of the desired color in hexadecimal notation: rrggbb. The statement fg indicates foreground color. Of course bg is used for the background. See the announcement.

$ latex a_0 x^2 + a_1 x + a_2 = 0 &fg=aa0000&s=1$ gives:

Warning: don’t put any spaces between statements that specify changes in the math environment. Example: &fg=aa0000&s=1 and not &fg=aa0000 &s=1.

For the next four examples, I’ve omitted the formatting statements, for the sake of clarity:

x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} gives:

\int_a^b f(x)dx = {\cal F}_{a,b} gives:

\lim\limits_{n \rightarrow \infty} \sum\limits_{i=0}^n \sum_j \alpha_{ij} gives:

\frac{1}{1+ \frac{1}{1+x}}\leq y^2*\log_{n}{\delta} gives:

Math environments

Now we’ll try some multiline expressions:

$ latex (x+y)(x-y) = x^2 – xy + yx – y^2 \\ = x^2 – y^2 \\ (x+y)^2 = x^2 + 2xy + y^2 &fg=aa0000&s=1$

that gives:

Notice that the expression lines are correct, however they are not properly aligned. You can do the alignment by using the statement {array}. Suppose that you want to align the = signals:

$ latex \begin{array}{ccc} (x+y)(x-y) & = & x^2 – xy + yx – y^2 \\ & = & x^2 – y^2 \\ (x+y)^2 & = & x^2 + 2xy + y^2 \end{array} &fg=aa0000&s=1$

that gives:

The declaration {ccc} specifies the columns in each line, as if your multiline expression could be treated as a table. The letters specify the indentation of each column: c = center, l= left, r = right.

Matrices

Here, there is an example of how to assemble a matrix:

\left| \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array} \right|

So, the ideal template to use with mathematical formulas is:

$ latex formula &fg=aa0000&s=1$

To use it, choose the font color by changing properly the values in the declaration &fg=rrggbb (remember: RGB, in hexadecimal, taking values on 00…ff) and set the font size with &s=n, being n the number indicating the size. Don’t forget that in the template above a space was put between the $ and latex, you must suppress it !

I will use this first post to explore some possibilities and options with the editing environment provided by WordPress.com. It will serve myself to remember where to find some info or hints on the basics of blogging. And it is an opportunity to share my particular difficulties (only the non-very awkward ones) and my personal experience and impressions on starting a blog. So, there we go !

Ok, it seems that I was well suceeded in putting a good picture for saying “Hello” to the world. I made some changes in the original picture, which I got from a friend, but you can get here the original. On this link you’ll also find another one also very nice. These pictures are views of the launching of the space shuttle supposedly taken from the ISS (International Space Station) on the orbit. However, you can find here some remarks that it could not be taken from the ISS, but from some satellite.

My start up with blogging at the WordPress.com was not completely smooth. I tried to get hands-on the work quickly and there came up some points that I’ll put here for those that could start in the same way:

• How to customize the presentation of the blog ? – Go to the toolbar displayed at the top of the blog page (assuming you’re logged on) and find Dashbord >> Presentation. There you can find themes and pick up one of you preference. I’ve made my choice for the Tarski theme, which I found clean and functional.
• How to put a picture at the header of your blog page ? – Use the same option above, Dashboard >> Presentation. There you find a way to upload a picture and display it in the header.
• How to change the title of your blog ? – see here in the FAQ .
• How to edit the blog ? – Go to the Dashboard >> Posts and chose the Edit option at the right of the post you want to edit.
• How to change the lay-out of the blog ? – Use the same option above, Dashboard >> Presentation and chose Widgets. There is also a FAQ page explaining about the widgets.
• How to change the pages ? – Pages are listed below your blog’s title. By default there are initially two pages: Home and About. You can edit pages on Dashboard >> Pages.
• How to upload a picture, a video or a slideshow ? – When you are in edition mode of a post or a page, the editing window shows at the bottom a section for image, video and slideshow upload. Quite easy ! See here more about slideshows.
• How had I put the You Tube button and link in my pages ? – I had done it using the text widget. The button logo should be uploaded before, in order to get a valid URL for the img object in the html tag. You must create the text in HTML. I found somehow tricky to upload the picture, since I didn’t find an option for doing it in the Dashboard. I hat to enter in Edit mode as if I was creating a new post and upload in the edition window, as described above. However, it is possible to insert a link to YouTube using the Blogroll option at the Dashboard. But in this case there will be no option to put the button logo.

The feature that most attracted me to the WordPress.com blogs is the support for formulas written on LaTeX syntax. However, I’ll leave this to the next post. I knew about this feature and about WordPress.com from the What’s New blog by Terence Tao. Since I’m yet a newcomer, I’ve imitated several features of his pages, which I find very clean and functional.