10th International Brazilian Meeting on Cognitive Science
Main theme: Situated and Embodied Cognition
When: December 7-11, 2015
Venue: University of São Paulo, São Paulo city, Brazil
Download the final version of the conference complete program with keynote talks, round tables and papers & posters presentations scheduling and abstracts:
For more informations, check the website, Facebook page and twitter:
website: http://www.lsi.usp.br/ebicc-2015/
Facebook page: https://www.facebook.com/ebicc2015
Twitter: @ebicc2015
10th International Brazilian Meeting on Cognitive Science
Main theme: Situated and Embodied Cognition
When: December 7-11, 2015
Venue: University of São Paulo, São Paulo city, Brazil
website: http://www.lsi.usp.br/ebicc-2015/
Facebook page: https://www.facebook.com/ebicc2015
SCOPE
The central theme of EBICC-2015 is Situated Embodied Cognition – Information and Autonomous Action, and focuses on questions concerning the role of the informational coupling between the cognitive agent, and the environment and its epistemological implications.
This theme provides the main guideline for the debate, and alternative views and proposals are expected and welcomed for work submission and presentation.
The key questions under the main theme are:
SUBJECTS
In addition to the central theme, we expect and encourage the submission of works in all subjects relevant to cognitive science research and thinking, and here we indicate some examples of topics for submission:
SUB-THEMES
EBICC-2015 features several sub-themes that can be considered as suggestions for submission topics:
• Information, mind, and cognition.
• Information, structure, and cognition.
• Logic, information, and intelligence.
• Semiotics and cognition.
• Language, communication, and cognition.
• Perception, action, and attention.
• Art, emotion, neuromodulation, and cognition.
• History of the concepts of cognition, mind, and brain.
• Computational modeling of cognition.
• Machine learning, neuromathematics, and cognition.
• Cognitive computational architectures.
• Cognitive robotics.
• Cognition, technology, and action.
• Cognitive appliances and technological applications of cognitive systems.
• Agents and cognitive games and their educational and socio-cultural roles.
• Ubiquitous Computing and its relations with cognition and society.
• Computational ubiquity, telepresence and personal identity.
• Ethical consequences of information technology and cognitive issues.
• Information and autonomous action: ethical issues and epistemological questions.
CATEGORIES OF SUBMISSION
EBICC-2015 will accept the following categories for submissions:
So, thanks to everybody for the interest in my blog, and see you soon !
Find below:
1) PDF File with the accesses per country since the beginning of this statistics
Machine Intelligence at WordPress – Oct 9 2012 stats.com
2) Picture with the more recent accesses stats
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.
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 :
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.
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:
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.
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 :
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.
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.
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:
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:
Now I’m closing this series with some extra examples and reporting some bugs.
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.
\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 |
$ 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.
$ latex (\frac {x^{a^m}}{y^{b^n}} ) \quad \left ( \frac {x^{a^m}}{y^{b^n}} \right ) &fg=aa0000&s=2 $
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.
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 $
This expression parses : $ latex {\cal ABC} &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:
Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. .
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:
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Supported font styles:
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 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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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
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
\left \langle \psi \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:
Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup.
.
You can present equations with several lines, using the array statement. Inside its declaration you must :
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}
.
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}
.
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 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:
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.
.
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”
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How to display subscripts and superscripts, indexes and exponents:
Subscripts:
Superscripts:
Composition with preceding indexes: {}_1^2 \Psi_3^4 gives
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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}
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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
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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:
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. .
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:
.
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 symbols
\circ \bigcirc \Diamond \Box \triangle
\vartriangle \triangledown \triangleleft \triangleright \vartriangleright \vartriangleleft
\angle \sphericalangle \measuredangle 45^\circ
\perp \mid \nmid \| \asymp \parallel
.
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
.
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
.
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 only. For matrices, see next post.
\binom{n}{p} = \frac{n!}{p!(n-p)!}