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# Lorentz covariant derivative of the vielbein determinant

Denote by $\mathcal{D}_m$ the Lorentz covariant derivative, $$\mathcal{D}_m=\partial_m-\frac{1}{2}\omega_m{}^{ab}M_{ab} \tag{1}$$ where indices $m,n,p,\dots$ are world indices, indices $a,b,c,\dots$ are tangent indices, $\omega_{m}{}^{ab}$ is the Lorentz connection and $M_{ab}$ are the Lorentz generators (in an arbitrary representation). I would like to compute the covariant derivative of the determinant of the vielbein, $e:=\text{det}(e_m{}^{a})$, in both the torsion and torsionless cases. Lets consider $D=3+1$ for concreteness.

In the torsionless case, it is known that the Lorentz covariant derivative $\mathcal{D}_m$ coincides with the regular covariant derivative $\nabla_m$ of general relativity (in tosion free case) involving the Christoffel symbols. Therefore, since we know that $\nabla_m\sqrt{-g}=0$ and $\sqrt{-g}=\sqrt{-\text{det}(g_{mn})}=$ $=\sqrt{-\text{det}(e_m{}^{a}e_n{}^{b}\eta_{ab})}=\sqrt{e^2}=|e|$, we can conclude that $\mathcal{D}_me=0$.

How would I go about computing $\mathcal{D}_me$ without using the knowledge $\nabla_m\sqrt{-g}=0$?

In the torsionless case, the spin connection could be expressed in terms of the vielbein via the torsion free condition, which would eliminate $\omega_{m}{}^{ab}$ in $(1)$ from the calculation in favour of the vielbein. Further more the torsion free condition is equivalent to the statement that the vielbein is covariantly constant, $\hat{\nabla}_me_{n}{}^{a}=0$ where $\hat{\nabla}_a=\nabla_m-\frac{1}{2}\omega_{m}{}^{bc}M_{bc}$, that might also come in handy. But these are just ideas of what might be useful in the calculation, i'm not sure how to put them to use. To start with, i'm not even sure how to interpret the expression $M_{bc}e$. Normally we would have $M_{bc}\phi=0$ for any scalar field $\phi$, but I suspect that this is not the case here (given that I anticipate $\mathcal{D}_me=0$ and $\partial_m e\neq 0$). I know that under an infinitesimal local Lorentz transformation parameterized by $K^{ab}=-K^{ba}$, $e$ is invariant, but I don't think I can apply that here.

Once I see how the torsionless case is done, I'll have a crack at the case with torsion myself.

Edit 1

Might someone confirm whether or not the following is valid: \begin{align} \frac{1}{2}\omega_n{}^{bc}M_{bc}\text{det}(e_{m}{}^{a})&=\frac{1}{2}\omega_n{}^{bc}M_{bc}\big(\frac{1}{4!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}e_{m_1}{}^{a_1}e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)\\ &=\frac{4}{4!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}\big(\frac{1}{2}\omega_{nbc}M^{bc}e_{m_1}{}^{a_1}\big)e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\\ &=\frac{1}{3!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}\big(\omega_{nbc}\eta^{a_1b}e_{m_1}{}^{c}\big)e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\\ &=-\frac{1}{3!}\omega_{nc}{}^{a_1}\varepsilon_{a_1a_2a_3a_4}\big(\varepsilon^{m_1m_2m_3m_4}e_{m_1}{}^{c}e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)\\ &=-\frac{1}{3!}\omega_{nc}{}^{a_1}\varepsilon_{a_1a_2a_3a_4}\big(\text{det}(e_{m}^{a})\varepsilon^{ca_2a_3a_4}\big)\\ &=-\frac{1}{3!}\cdot 3!e\omega_{nc}{}^{a_1}\delta^{c}_{a_1}\\ &=0 \end{align} where in the last line I have used the antisymmetry $\omega_{n}{}^{bc}=-\omega_{n}{}^{cb}$. This would imply that $\mathcal{D}_me=\partial_me$ in both the torsion and tosionless case, which I am doubtful of (but coincidentally, it is exactly what I need for a separate calculation I am doing).

Edit 2

Similar to the above I can show that $\nabla_m e=0$, \begin{align} \nabla_m e &=\nabla_m\big(\frac{1}{4!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}e_{m_1}{}^{a_1}e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)\\ &=\partial_m\big(\frac{1}{4!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}e_{m_1}{}^{a_1}e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)~~~~~~~~~~~~~~~~~~~-4\big(\frac{1}{4!} \varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}(\Gamma^p_{mm_1}e_{p}{}^{a_1})e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)\\ &=\partial_me-\frac{1}{3!}\varepsilon^{m_1m_2m_3m_4}\Gamma^p_{mm_1} e\varepsilon_{pm_2m_3m_4}\\ &=\partial_me-e\Gamma^p_{mp} \\ &=\partial_me-e(\frac{1}{2}g^{-1}\partial_mg)\\ &=\partial_me-\partial_me\\ &=0 \end{align}

These two results (in edit 1 and 2) are consistent with the fact that the Vielbein is covariantly constant (true for torsionless case) since on the one hand we have (using $\hat{\nabla}_me_{n}{}^{a}=0$) \begin{align} \hat{\nabla}_me=\hat{\nabla}_m\big(\frac{1}{4!}\varepsilon^{m_1m_2m_3m_4}\varepsilon_{a_1a_2a_3a_4}e_{m_1}{}^{a_1}e_{m_2}{}^{a_2}e_{m_3}{}^{a_3}e_{m_4}{}^{a_4}\big)=0 \end{align} and the the other hand we have \begin{align} \hat{\nabla}_me=(\nabla_m-\frac{1}{2}\omega_m{}^{bc}M_{bc})e=0. \end{align} However there is an inconsistency with the claim that $\mathcal{D}_m$ coincides with $\nabla_m$ in the torsionless case (since according to the above we have $\mathcal{D}_me=\partial_me$ and $\nabla_me=0$). Does anyone see where the problem is?

• For the bounty, please show how to (dis)prove $\mathcal{D}_me=0$, using the definition $\mathcal{D}_m=\partial_m-\frac{1}{2}\omega_m{}^{ab}M_{ab}$ (and if necessary, the torsion free condition). It would be great if you could also provide explanations/resources on the general formulation of covariantly differentiating tensor densities (which the determinant is). – NormalsNotFar Sep 23 '18 at 8:09
• Your question doesn't make much sense. The metric-preserving condition is an external condition you impose on the connection, and it is completely independent of the torsion-free condition. So, you can have torsion-full connections that preserve the metric, or you can have torsion-less connections that fail to preserve the metric. For generic $\omega_m{}^{ab}$ you can't hope to prove anything, because all of these properties must be externally imposed. – Ben Niehoff Sep 26 '18 at 19:26

Notation:

Here in this answer I will break with some conventions present in the OP's question. In particular, greek $$\mu,\nu...$$ indices will denote components taken with respect to a holonomic (coordinate-) frame (also known as "world indices" or "curved indices") and latin $$a,b...$$ indices will denote local Lorentz indices (also known as "flat indices"). I will work in $$n$$ dimensions.

In addition to their question, OP is also confused/interested about the relationship between torsion and the covariant constancy of the volume element. For this reason we will use a connection $$\nabla_\mu$$ which is metric compatible, but not torsionless.

Preliminaries:

On a vector field with world indices, the connection acts as $$\nabla_\mu X^\nu=\partial_\mu X^\nu + \Gamma^\nu_{\mu\rho}X^\rho,$$ where the coefficients $$\Gamma^\nu_{\mu\rho}$$ are not the usual Christoffel symbols, but the Christoffel symbols extended with the contorsion tensor.

On a vector field with local Lorentz indices, the connection acts as $$\nabla_\mu X^a=\partial_\mu X^a+\omega_{\mu}{}^{a}{}_{b}X^b,$$ where the $$\omega_{\mu}{}^{a}{}_{b}$$ coefficients are not the usual Ricci rotation coefficients, but the Ricci rotation coefficients extended with terms involving torsion. Note that I will not use the Lorentz algebra generators directly, the $$\omega_{\mu}{}^{a}{}_{b}$$ coefficients are Lorentz algebra valued, and any other representations will be derived directly from these.

It should also be noted that technically these are the same connection, in two different representations.

The vielbein is $$e^\mu_a$$ and the covielbein is $$\theta^a_{\mu}$$. These are inverses of one another as matrices.

I will also note that the vielbein (and the covielbein) is covariantly constant, even if the connection is torsionful. In fact, they are covariantly constant even if the connection is not even metric-compatible.

Confusion can arise from the fact that the covariant derivative of the vielbein depends on which indices of the vielbein one considers as "tensor indices". If both indices are considered as tensor indices, then $$\theta^a_\mu$$ is just the identity tensor $$\delta^\mu_\nu$$ expressed with mixed indices. It's derivative vanishes as a consistency postulate: $$\nabla_\mu\theta^a_\nu=\partial_\mu\theta^a_\nu+\omega_{\mu}{}^{a}{}_{b}\theta^b_\nu-\Gamma^\rho_{\mu\nu}\theta^a_\rho=0.$$

On the other hand, if a so-called covariant exterior derivative is calculated as $$\nabla_\mu\theta^a_{(\nu)}-\nabla_\nu\theta^a_{(\mu)}=T^a_{\mu\nu},$$ where the indices in parentheses are excempt from the covariant derivative (not considered as tensor indices), then the right hand side of this expression is the torsion 2-form.

The connection is assumed to be metric compatible, however, so we have $$0=\nabla_\mu g_{ab}=\partial_\mu g_{ab}-\omega_{\mu}{}^{c}{}_{a}g_{cb}-\omega_{\mu}{}^{c}{}_{b}g_{ac}=-(\omega_{\mu ba}+\omega_{\mu ab}),$$ where we have used that in a local Lorentz frame, $$g_{ab}$$ has constant components. Ergo, the connection is skew-symmetric in $$ab$$ even if it is torsionful, only metric compatibility is required.

Densities in vielbein formalism - covariant derivative of densities:

For the sake of simplicity, I will assume that the manifold we are working on is orientable, and a positive orientation has been chosen. I will express everything in either positive coordinates, or positive frames. This way, we can get rid of the absolute value in the transformation rule for densities.

Let $$\pi_{\mu_1...\mu_n}$$ denote the Levi-Civita symbol. It is not assumed to be a geometric object at all. A totally antisymmetric tensor $$\rho_{\mu_1...\mu_n}$$ has only one independent component, and it can be written as $$\rho_{\mu_1...\mu_n}=\rho\cdot\pi_{\mu_1...\mu_n},$$ where $$\rho=\rho_{12...n}$$.

It is not difficult to check that $$\rho$$ transforms as a scalar density of weight 1 (without the absolute value sign, but we got rid of it by choice of orientation).

Hence, we obtain a duality between scalar densities of weight 1, and totally antisymmetric covariant tensors of order $$n$$ ($$n$$-forms).

The canonical volume element on a pseudo-Riemannian manifold is defined as $$\mu_{\mu_1...\mu_n}=\sqrt{|\det g|}\pi_{\mu_1...\mu_n}=|\det\theta|\pi_{\mu_1...\mu_n}.$$

Note the greek indices - this form of the volume tensor is taken with respect to a holonomic frame.

The form of the volume tensor in a local Lorentz frame can be calculated as $$\mu_{a_1...a_n}=\mu_{\mu_1...\mu_n}e^{\mu_1}_{a_1}...e^{\mu_n}_{a_n}=|\det\theta|\pi_{\mu_1...\mu_n}e^{\mu_1}_{a_1}...e^{\mu_n}_{a_n}=\det\theta\det e \pi_{a_1...a_n}=\pi_{a_1...a_n}.$$ Therefore, if we define the single, independent component of the volume tensor as the function $$f$$ multiplying the Levi-Civita symbol $$\pi_{a...}$$, then we arrive at the important result that in a local Lorentz frame, the component of the volume density/tensor is 1.

The covariant derivative preserves symmetries, so if $$\rho_{a_1...a_n}$$ is a completely antisymmetric tensor field expressed in a local Lorentz frame, we can calculate its covariant derivative as $$\nabla_\mu\rho_{a_1...a_n}=(\nabla_\mu\rho_{1...n})\pi_{a_1...a_n},$$ so let us calculate $$\nabla_\mu\rho_{1...n}$$ for a generic $$\rho$$.

We have $$\nabla_\mu\rho_{1...n}=\partial_\mu\rho_{1...n}-\omega_\mu{}^b{}_{1}\rho_{b2...n}-...-\omega_\mu{}^b{}_{n}\rho_{12...b} \\ =\partial_\mu\rho-\rho(\omega_\mu{}^b{}_{1}\pi_{b2...n}+\omega_\mu{}^b{}_{n}\pi_{12...b}) \\ =\partial_\mu\rho-\rho\omega_{\mu}{}^b{}_b=\partial_\mu\rho.$$

Due to the antisymmetry of $$\pi$$, in each contraction in the parentheses, only one term survives - the one where $$b$$ has the same value as the second lower index of $$\omega$$, so the entire parenthetical expression reduces to a trace, and we have utilized the fact that the connection coefficients have vanishing trace due to antisymmetry (which itself stems from metric-compatibility, as stated in the beginning).

We have obtained that when expressed in local Lorentz frames, densities can be partially differentiated.

However we have also obtained that $$\mu_{a_1...a_n}=1\cdot\pi_{a_1...a_n},$$ and so $$\nabla_\mu \mu=\partial_\mu 1=0,$$ and we obtained the desired result.

Also note that we have only assumed compatibility, but not torsionlessness on the connection $$\nabla$$.

• Rereading this, the answer is completely unintelligible. I will rework it in a couple of hours – Bence Racskó Sep 23 '18 at 13:21
• Answer has been reworked. – Bence Racskó Sep 23 '18 at 16:00


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User:Modest Genius

Hello and welcome to Modest Genius's user page. Please use my talk page for any comments you wish to leave me.

Biographical details

I used to have my full name on here, but people started getting this as the top hit when they googled my name. You can easily find it in the article history if you're interested. I currently live in the UK and am 37 years old (using the {{age}} template).

I have a PhD in astrochemistry and work as part of the editorial staff of Science, one of the world's top scientific journals. I completed my PhD at the University of Nottingham, where I was subsequently a postdoctoral researcher. After that I worked at the Royal Astronomical Society, on their journal Monthly Notices of the Royal Astronomical Society (MNRAS). I have also held visiting positions at the Southern African Large Telescope and European Southern Observatory. As a professional astronomer, I conducted research into the interstellar medium, primarily using observational optical and infrared astronomy; I continue to conduct astronomical research in my spare time. These days I work in scientific publishing as a journal editor. I therefore declare a potential conflict of interest on all those topics.

Hobbies and interests

I have a professional interest in physics and astronomy, and have always been interested in history, particularly English history from the Heptarchy onwards, and European history between classical antiquity and the Napoleonic wars. I'm also interested in politics, and am a keen reader of newspapers and other news sources, particularly The Guardian.

Wikipedia interests

I've been editing Wikipedia since summer 2005, when I got annoyed by finding minor mistakes in my favourite online reference and just had to fix them. I signed up for an account in November 2005. To begin with, I mostly just fixed mistakes in pages I was looking at anyway, and didn't go out of my way to add anything. Over the years I've become more involved in the running of the place, particularly the Main Page, and done some content creation.

I can mostly be found hanging around Talk:Main Page, WP:ERRORS, WP:In the news/Candidates and their associated pages. Related to this, I sometimes contribute to writing or improving articles which appear on the In the News section. I'm also a member of WikiProject Astronomy, often contributing to areas raised at WT:AST and WT:ASTRO. I tend to have a slow burning 'project' article, usually on a topic in astronomy, though I rarely have enough real-life time to move these along very quickly. I have rollback and reviewer user rights, though I don't tend to use either of them very often.

Licensing

In general, I am averse to anyone using my work for commercial gain without permission. I'm not a fan of Wikipedia's licenses allowing this, although if I want to contribute here I don't have much choice. However, I am a great fan of releasing content to others to use, so long as they don't make money from my time and effort. Similarly, I don't like anyone claiming someone else's work as their own or without proper attribution, and try not to do so myself.

In order to allow others to make use of content they created, and to reduce the problems created with multiple licences, I hereby release any edit I mark as 'minor' into the public domain.

Barnstars

• The "Barnstar of Fighting USA Electioncruft" 65.190.89.154 (talk) 22:50, 4 January 2008 (UTC)
 The Tireless Contributor Barnstar For being diligent in spotting our poor spelling and grammar on the main page I award you this barnstar. Keep up the awesome work! :) Master of Puppets Call me MoP! :D 09:42, 21 November 2008 (UTC)
 The In The News Barnstar For your seemingly never-ending and always useful efforts at WP:ITN/C and WP:ERRORS. HJ Mitchell | Penny for your thoughts? 02:10, 20 August 2010 (UTC)
 The Original Barnstar This barnstar is awarded to everyone who - whatever their opinion - contributed to the discussion about Wikipedia and SOPA. Thank you for being a part of the discussion. Presented by the Wikimedia Foundation. — Preceding unsigned comment added by Philippe (WMF) (talk • contribs) 20:26, 21 January 2012
 The Minor barnstar I made this change with haste unbecoming of the edit's importance. Thank you for fixing it. This barnstar recognises the importance of wiki-gnoming and of users, like you, who have at one point courteously and quietly fixed the edits of people in far too much of a rush. Happy editing! AGK [•] 22:26, 27 February 2013 (UTC)
 The Main Page Barnstar Awarded to Modest Genius for becoming only the fourth editor to make 1,000 edits to WP:ERRORS. With many thanks for all the hard work you do in trying to keep the main page error-free. BencherliteTalk 10:13, 17 July 2013 (UTC)
 The Editor's Barnstar Re. your assistance, thanks very much for digging out that page for me, it's much appreciated. I'd been searching for ages and had nearly given up. Cheers! ——SN54129 16:22, 6 November 2019 (UTC)

Articles

This is a list of some of the articles I've worked on which have been been promoted to GA or FA (when/if that ever happens), recognised with a slot in ITN or DYK, or that I originally created. On top of these, there are a whole load of ITN blurbs, OTD items etc. that I've either proposed or improved; I've limited this list to articles I've contributed substantial content to.

Wow that's a much shorter list than I thought it would be. Either I've forgotten some of them, or I don't write much content...

I've also reviewed a few articles for Good Article status:

Images

Images I have uploaded (either locally or to Commons):

format: plain


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Cannon-Thurston fibers for iwip automorphisms of $F_N$

Abstract : For any atoroidal iwip $\phi \in Out(F_N)$ the mapping torus group $G_\phi=F_N\rtimes_\phi e$ is hyperbolic, and the embedding $\iota: F_N \overset{\lhd}{\longrightarrow} G_\phi$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat \iota: \partial F_N \to \partial G_\phi$. We prove that for any $\phi$ as above, the map $\hat \iota$ is finite-to-one and that the preimage of every point of $\partial G_\phi$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_\phi$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there are at most $4N-5$ distinct $F_N$-orbits of such {\em singular} points in $\partial G_\phi$ (for the translation action of $F_N$ on $\partial G_\phi$). By contrast, we show that for $k=1,2$ there are uncountably many points $S\in \partial G_\phi$ (and thus uncountably many $F_N$-orbits of such $S$) with exactly $k$ preimages in $\partial F_N$.
Type de document :
Article dans une revue
Journal of the London Mathematical Society, London Mathematical Society, 2015, 91 (1), pp.203--224. 〈10.1112/jlms/jdu069〉

https://hal.archives-ouvertes.fr/hal-01318434
Contributeur : Aigle I2m <>
Soumis le : jeudi 19 mai 2016 - 15:44:36
Dernière modification le : lundi 4 mars 2019 - 14:04:19

Citation

Ilya Kapovich, Martin Lustig. Cannon-Thurston fibers for iwip automorphisms of $F_N$. Journal of the London Mathematical Society, London Mathematical Society, 2015, 91 (1), pp.203--224. 〈10.1112/jlms/jdu069〉. 〈hal-01318434〉

Métriques

Consultations de la notice


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# WiQEE : A Wiki for mathematical proofs

The Quod Erat Edificandum Wiki – WiQEE for short – is a mix of two ideas. First of all, it is a documentation platform, whose main content is the articles written by its contributors. Secondly, it aims to become a formally-verified library of mathematical proofs and programs, that can be extracted to executable form, or composed together to prove more complicated theories.

Every page on this wiki is generated from a source file, written in a language called CaPriCon. The source of a page can be downloaded from its menu. If you want to know how the sausage gets made, I invite you to take a look at this index’s source for an example of such source files.

If you do so, you may notice that the source doesn’t look that different from the article itself. That is because CaPriCon is an exclusively literate programming language, in which most text is simply text, and the code is specially identified through dedicated syntaxes. The comment formatting language itself is Markdown in this case, although it would be trivial to adapt it to other textual markup languages such as TeX.

## Literate Programming in Wiki Form

Everything you’ve read until now is a comment of the “index.md” program. If you want to write programs, though, you will probably need to write some code, which in this case consists in a sequence of steps that are executed in the order in which they appear. The first sort of code that you can execute is a code paragraph, and is indicated in the source by starting a line with >.

Such code is executed, and its output is added to the document.

You can also write some inline code between “mustaches” (a mustache is a pair of nested {} inside the source), which will be executed as part of a paragraph, and whose output . If all works well, you should see a small image of steps next to the generated output. Clicking on those steps will, as in the previous case and without surprises, show you the steps that were taken to produce this output.

## A Concatenative Scripting Language

If you peeked at the examples given before, you may have noticed something strange. In order to produce some output, you’d expect a sentence to look like print "...", but instead the program was written in the reverse order, as "..." print.

Indeed, in the tradition of PostScript, CaPriCon is a concatenative, stack-based language. In such a language, programs are represented as sequences of symbols, that are executed from first to last, and that may keep intermediate results on a shared stack. All the power of the language comes from the words that are made available, rather than from the particular syntax of those words.

Concatenative languages have a few interesting properties that make them perfect candidates for integration into a literate environment like the one CaPriCon provides :

• simplicity : every program is only a sequence of words that are evaluated in order, nothing more. You don’t have to learn any fancy syntax, only the words and what they mean.

The CaPriCon language only has ~50 basic words in its vocabulary, which are referenced in this lexicon. Beyond that, you can define your own words by composing the existing ones into so-called “quotes”, written { word1..wordn }.

• exploration : concatenative languages are very well-suited to implement incremental development environments, in which you can devise programs (and proofs) by walking through their execution step by step, with complete control and visibility over the state of the environment at every stage.

To experience this process of exploration, you can install the interactive CaPriCon interpreter (for Linux or OSX), that is used to power this site. This interpreter can be run as-is, but it is usually preferrable to load a prelude before any useful interaction can occur.

A prelude is simply a file containing the basic definitions needed to write simple programs. Without one, you’re left with the basic functions, which require you to be much more verbose. The default CaPriCon prelude can be found in the above packages, under the name “prelude”.

If you still aren’t quite sure how to go about writing your own scripts, this tutorial may help walk you through the first hurdles.

## Proofs With Prismatic Constructions

While CaPriCon can be a decent addition to a documentation format, it was first designed as an interactive interface for the construction of proof terms, described in the Calculus of Prismatic Constructions (hence the name CaPriCon), which is an extension of the basic CoC with a lifting operator for naturally inductive values.

That operator is called , and it works by projecting inductive values onto a larger version of themselves, universe-wise. For example, given the “inductive” type , and a hypothesis , represents a computationally sound proof of . Each branch of the base inductive type is “projected” through to a parallel branch in a larger universe, with additional “rays” to provide reflection into that universe. thus acts like a sort of prism (in fact, an endoprism) that continuously projects inductive values onto their induction principle, giving rise to the eponymous prismatic constructions.

This generalization can be applied to model many common cases which previously required the introduction of abstract-only inductive types, including but not limited to : general dependent inductive types, complete with recursive and inductive-inductive types; co-inductive types and their associated semantics; and even some kinds of quotient types, though the full scope of those quotient types is still to be discovered.

#### Talk is cheap, show me the code

From the previous overview, may seem quite complex, but it is actually implemented as an almost linear operation on De Bruijn indices. You can look at CaPriCon’s Haskell implementation of this model to convince yourself of that. All in all, the whole interpreter/typechecker/pattern-matcher boils down to ~300 lines of index-shifting code.

If you feel hesitant about trusting a bunch of Haskell functions written by a stranger on the Internet, don’t hesitate to start hijacking bits of it to write CaPriCon interepreters of your own, in your language of choice. Mine just happens to be Haskell, but maybe yours will bring something new and interesting to the table as well.

## Contributing to WiQEE, or creating your own

It goes without saying that I encourage you to contribute to WiQEE in any way you’d like. You may find the project template for this site in this Git repository, and the source for the actual pages in this other Git repository.

If you want to contribute to this specific site, you can use the power of Pull Requests on the WiQEE-pages repository (the second one). I can’t promise an immediate reply, but I’m working on automating the builds to allow every contributor to participate, without needing my validation at every turn.

If like me you lack that sort of patience, you can also test your pages by running your own version of this site (how exciting !). To do so, you have to clone the first repository, apply your modifications in the src/pages subdirectory, and run make over the whole thing. You’ll need Pandoc and Sass to be installed, for Markdown and SCSS processing, as well as CaPriCon for obvious reasons. Once make has run its course, you should find a workable HTML version of your WiQEE in the public subdirectory.

The basics :


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# How to set -dAutoRotatePages#/None in MikTeX

I'm using MikTeX 2.9 on Windows 7.

I understand one needs to tell GS explicitly not to auto rotate pages by supplying:

-dAutoRotatePages#/None

as parameter to ps2pdf (or gswin32c, as indicated here).

Is there a way to also supply this parameter to \MiKTeX-2.9\miktex\bin\latex.exe? That is, can I run something like:

latex.exe [parameter] main.tex

and the produced DVI will not exhibit undesired autorotated pictures?

The reason for asking this is that I use Texniccenter, and the build setup (to DVI) only allows to pass options to latex.exe.

-
Welcome to TeX.sx! Usually, we don't put a greeting or a "thank you" in our posts. While this might seem strange at first, it is not a sign of lack of politeness, but rather part of our trying to keep everything very concise. Upvoting is the preferred way here to say "thank you" to users who helped you. –  ienissei Jun 3 '12 at 7:32
A tip: If you indent lines by 4 spaces or enclose words in backticks , they'll be marked as code, as can be seen in my edit. You can also highlight the code and click the "code" button (with "{}" on it). –  ienissei Jun 3 '12 at 7:32
Running latex only generates a .dvi file, it does not run GhostScript. That may be done by your editor in a 'build' system, which is where you would need to add the parameter. Exactly how do you run ps2pdf (the final stage of PDF creation using the dvips route)? –  Joseph Wright Jun 3 '12 at 8:11
You have to set the parameter in your editor and not in MiKTeX. What Editor are you using? You'll find answers for TeXnicCenter and TeXWorks here: tug.org/PSTricks/main.cgi?file=pdf/pdfoutput The option for ps2pdfis set in the same way. –  Herbert Jun 3 '12 at 8:15
@ Joseph, Herbert: I don't run ps2pdf at all. If I run pdflatex, the unwanted autorotation does not occur. –  Johannes Jun 3 '12 at 13:29

## Install GhostScript

Make sure you have GhostScript installed and you have set PATH to its location. gswin64c.exe is for 64 bit version and gswin32c for 32 bit version. I use 64 bit version in this answer.

## Create a batch file

The following batch will compile your TeX input file 4 times, convert the resulting DVI to PS and finally convert the PS to PDF.

Save it as combo.bat, for example, in any directory. Set the PATH system variable to this batch location such that your operating system can find the batch when you call it later.

rem This is combo.bat. It takes a single argument which is a file name without extension

echo off

rem remove the previous PDF if any
del %1.pdf

rem compile 4 times
latex -draftmode -interaction=batchmode %1
latex -draftmode -interaction=batchmode %1
latex -draftmode -interaction=batchmode %1
latex %1

dvips -t unknown %1

gswin64c -r10000 -dCompatibilityLevel=1.5 -dAutoRotatePages=/None -dPDFSETTINGS=/prepress -dNOPAUSE -dBATCH -sDEVICE=pdfwrite -sOutputFile=%1.pdf %1.ps

rem remove unnecessary files
del %1.aux
del %1.dvi
del %1.log
del %1.out
del %1.ps
del %1.toc
del %1.nav
del %1.snm

## Configure TeXnicCenter

Configure TeXnicCenter as follows:

-
This will produce a PDF file (and probably one where the autorotation issue is fixed). However, will it also produce a "correct" DVI file? –  Johannes Jun 3 '12 at 13:42
@Johannes: Are you sure that latex will automatically rotate its content in DVI? I don't think latex` does rotation automatically. –  kiss my armpit Jun 3 '12 at 13:45
See the example files I posted in response to Ulrike's comment. Not sure which binary/program is automatically rotating content (latex/dvips/ps2pdf?) -- all I can tell is that in YAP, the final output appears rotated. –  Johannes Jun 4 '12 at 9:45


text:
# Buttons labelled with the number of times they have been clicked

I am new to functional programming and I decided to learn it thru Elm instead of haskell. I wrote a practice code where there are 2 buttons and their names correspond to the number of times they are clicked. The following is my implementation:

import Html exposing (..)
import Html.Attributes exposing (..)
import Html.Events exposing (onClick)

main = Html.beginnerProgram {model=model, view=view, update=update}

-- model
type alias Model =
{ a : Int
, b : Int
, total : Int}

model : Model
model = Model 0 0 0

-- view
type Msg = Add Foo
type Foo = A | B

view : Model -> Html Msg
view model =
div []
[ button [onClick (Add A)] [text (toString model.a)]
, button [onClick (Add B)] [text (toString model.b)]
, div [] [ text (toString model.total)]
]

-- update
update : Msg -> Model -> Model
update msg model =
case msg of
case foo of
A ->
let
a = model.a + 1
total = a + model.b
in
{model | a = a, total = total}
B ->
let
b = model.b + 1
total = b + model.a
in
{model | b = b, total = total}

I feel like that the above code is a dumb implementation of what I wanted to accomplish. Can somebody suggest a better implementation?

## migrated from stackoverflow.comNov 21 '17 at 18:26

This question came from our site for professional and enthusiast programmers.

• I recommend you configure your editor to format-on-save using elm-format – user633183 Nov 21 '17 at 16:22
• Thank you. I just learned how to do that. BTW, any idea why elm-format uses 4-spaces indentation? It's a bit too much for me. I think 2 spaces are enough. – jocab Nov 22 '17 at 10:12

Here are a few things I'd recommend changing:

1. There's no need to store the sum of a and b. You can always compute it in the view with a + b. This simplifies the update function a lot.

2. You can match Foo A and Foo B in update instead of a nested case.

3. Optionally, you can destructure model into {a, b} in view. I don't have any opinion on this. Either way is fine.

Here's the final elm-format'd code:

module Main exposing (..)

import Html exposing (..)
import Html.Attributes exposing (..)
import Html.Events exposing (onClick)

main =
Html.beginnerProgram { model = model, view = view, update = update }

type alias Model =
{ a : Int
, b : Int
}

model : Model
model =
Model 0 0

type Msg

type Foo
= A
| B

view : Model -> Html Msg
view { a, b } =
div []
[ button [ onClick (Add A) ] [ text (toString a) ]
, button [ onClick (Add B) ] [ text (toString b) ]
, div [] [ text (toString (a + b)) ]
]

update : Msg -> Model -> Model
update msg model =
case msg of

• There's one more change I'd personally make: use <| to reduce parentheses: text (toString a) -> text <| toString a and similar for the two lines below. – Dogbert Nov 22 '17 at 4:17


text:
# Canonical transformation generated by hamiltonian?

Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any sense to me. Typically, the generating function is a function of some of the "old" coordinates as well as the new (transformed) ones, but the hamiltonian is a function of q and p.

Please tell me whether the above statement is rubbish, or whether it can indeed be interpreted in a meaningful way.

-

The statement isn't that $H$ is a generating function of the canonical transformation, in the sense used in canonical transformations, which must indeed be a function of one new and one old set of phase space coordinates. The statement is that $\hat H$ is the infinitesimal generator of the Lie group of translations in the sense of the theory of Lie algebras, namely a basis vector of the Lie algebra. These are different words – generating function and generator. Both of them "generate" the transformations of the phase space variables in some way but the two ways are technically different. In classical physics, the infinitesimal generator generates infinitesimal canonical transformations via the Poisson bracket (the classical limit of the commutator of linear operators), $$q_i \to q_i+\delta q_i,\quad \delta q_i = \epsilon\{ H,q_i \}$$ where $q_i$ may be both coordinates or momenta.


text:
Question 330958

Slope of *[tex \Large \frac{3}{4}] means that for every positive difference of 4 in the horizontal direction, there is a positive difference of 3 in the vertical direction.  So pick any point you like in *[tex \Large \mathbb{R}^2].  Then move 4 units to the right, stop and move 3 units up, stop and mark your second point.  Two points is sufficient to uniquely determine a line in *[tex \Large \mathbb{R}^2].

The slopes of perpendicular lines are negative reciprocals of each other.

*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1\ \perp\ L_2 \ \ \Leftrightarrow\ \ m_1\ =\ -\frac{1}{m_2}\ \text{ and } m_1,\, m_2\, \neq\, 0]

John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it


text:
The sum of two consecutive integers is 17, what are the integers?

1 Answer
Jul 3, 2015

The numbers are $8 \mathmr{and} 9$

Explanation:

Let the two consecutive numbers be :

color(blue)((x) and (x+1)

As per the condition given:
$\textcolor{b l u e}{\left(x\right) + \left(x + 1\right)} = 17$

$2 x + 1 = 17$

$2 x = 16$

$x = 8$

color(blue)(x=8
So the numbers are as follows $8 \mathmr{and} 9$


text:
## Predicting Entropy

Volume: $\Delta S = nR\ln \frac{V_{2}}{V_{1}}$
Temperature: $\Delta S = nC\ln \frac{T_{2}}{T_{1}}$

ng1D
Posts: 43
Joined: Wed Sep 18, 2019 12:17 am

### Predicting Entropy

What compositions and conditions of a system would help predict which of two systems has the greater entropy?

805291863
Posts: 100
Joined: Sat Sep 14, 2019 12:16 am

### Re: Predicting Entropy

Entropy depends on the number of molecules and the number of states the molecules can be in. Therefore, the greater the number of molecules, the greater the amount of entropy. Also, the more complex the molecule, the greater the amount of entropy

305385703
Posts: 102
Joined: Thu Jul 11, 2019 12:15 am

### Re: Predicting Entropy

Entropy is best determined by the amount of substance, the phase of the substance, and the complexity of the molecule.


text:
Open access peer-reviewed chapter

# Spirulina Phycobiliproteins as Food Components and Complements

By Dragana Stanic-Vucinic, Simeon Minic, Milan R. Nikolic and Tanja Cirkovic Velickovic

Submitted: October 9th 2017Reviewed: January 11th 2018Published: June 27th 2018

DOI: 10.5772/intechopen.73791

## Abstract

Spirulina has a documented history of use as a food for more than 1000 years, and has been in production as a dietary supplement for 40 years. Among many of Spirulina bioactive components, blue protein C-phycocyanin and its linear tetrapyrrole chromophore phycocyanobilin occupy a special place due to broad possibilities for application in various areas of food technology. The subject of this chapter is up-to-date food applications of these Spirulina components, with a focus on their use as food colorants, additives, nutriceuticals, and dietary supplements. Their other actual and future food application possibilities will also be briefly presented and discussed.

### Keywords

• Spirulina
• C-phycocyanin
• phycocyanobilin
• food components
• food complements

## 1. Introduction: Spirulina as a superfood

The blue-green microalgae Spirulina has been used in human nutrition for centuries. It is still used as food in some places, such as the Lake Chad area, where it is sold as dried bread called “dihe” [1]. For human consumption, the commercial production of Spirulina dates back to the 1970s. Spirulina used in human nutrition is the dried commercial biomass of two cyanobacteria species traditionally called Spirulina platensis and Spirulina maxima, which belong to the genus Arthrospira spp. Taxonomically, these organisms are classified in kingdom Bacteria; phylum Cyanobacteria; order Oscillatoriales; family Phormidiaceae [2].

Spirulina is filamentous, helical, photosynthetic cyanobacteria naturally inhabiting alkaline brackish and saline waters in tropical and subtropical regions. Biochemical analysis has revealed its exceptional nutritive properties, so it is referred in the literature as “super food” or “food of the future” [2]. Spirulina is one of the richest natural sources of proteins and essential amino acids, as well as an excellent source of vitamins (primarily A, K, and vitamin B complex), macro- and micro-elements (calcium, potassium, magnesium, iron, iodine, selenium, chromium, zinc, and manganese), essential fatty acids, including γ-linoleic acid (GLA), glycolipids, lipopolysaccharides, and sulfolipids [3]. Spirulina is especially rich in a variety of pigments, such as chlorophylls, β-carotene, xanthophylls, and phycobilins (phycobiliproteins) (Table 1).

SubstanceQuantity/activity per serving (3 g*)% DV**SubstanceQuantity/activity per serving (3 g*)% DV**
Total carbohydrates<1 g<1Chromium50 μg41
Proteins2 g4Sodium35 mg<2
Vitamin A (as β-carotene)11,250 IU230Potassium60 mg2
Vitamin K75 μg94C-phycocyanin240 mg
Vitamin B129 μg150GLA32 mg
Iron7 mg39Chlorophyla a30 mg
Magnesium15 mg4Total carotenoids15 mg
Manganese0.4 mg20Superoxid dismutase2500 U

### Table 1.

Nutritional profile of commercial Spirulina powder (Nutrex, Hawaii, USA).

Recommended daily value of Spirulina powder.

Percent daily values (DV) are based on a 2000 calories diet.

A huge number of in vitro and in vivo studies, published in the last few decades, have revealed potentially beneficial effects of Spirulina on human health. Health benefits mainly arise from the antioxidant effect of algae as a whole, or from its individual ingredients, such as phycobiliproteins (Section 2). Moreover, the presence of significant amounts of GLA, sulfated polysaccharide (calcium spirulin), and sulfolipids additionally contribute to health-promoting activities of Spirulina [3].

Several dried biomass products of Spirulina have categorized as “generally recognized as safe” (GRAS) by the Food and Drug Administration (FDA) of USA. A recommended dosage for adults is usually in the range of 3–10 g of Spirulina per day, while maximum daily intake should not exceed 30 g [3]. Extensive safety studies of Spirulina did not show the presence of cyanobacterial toxins [1]. Spirulina production requires the use of high quality nutrients and accurate determination of heavy metals in the culture medium, as well as in the biomass. Heavy metal analysis of commercial Spirulina products did not found to exceed the regulatory levels [2]. Nevertheless, it should be paid much attention during Spirulina cultivation to prevent contamination with heavy metals or the other cyanobacteria, capable to produce toxins.

## 2. Phycobiliproteins

Phycobiliproteins are photosynthetic antenna pigments in the cyanobacteria, red and cryptophyte algae, that efficiently harvest light energy, which is subsequently transferred to chlorophylls during photosynthesis. Therefore, phycobiliproteins significantly contribute to the global photosynthesis. Phycobiliproteins are deeply colored, highly fluorescent, and water-soluble proteins with high propensity to form oligomers (hexamers) that constitute the building blocks of the extra-membranous antenna complex, phycobilisomes. Its intensive color arises from covalently attached linear tetrapyrrole chromophores (phycobilins) via thioether bonds to the cysteine residues [4].

Phycobilins are produced by heme metabolism. Heme is synthesized from protoheme IX by ferrochelatase. Then, heme oxygenase cleaves heme and biliverdin IXα is obtained. Biliverdin IXα is reduced by ferredoxin-dependent bilin reductases to obtain phycobilins. Final step in phycobiliproteins biosynthesis is the covalent attachment of bilin chromophores to the apoproteins, catalyzed by phycobiliprotein lyases. Slow spontaneous in vitro attachment of tetrapyrrole chromophores to the apoproteins has low fidelity and mixture of oxidation products is obtained [5].

Spirulina produces two phycobiliproteins: C-phycocyanin (C-PC) as the major pigment and allophycocyanin (APC), which is present in much smaller quantities, approximately at an 10:1 ratio [3]. C-phycocyanin level varies based on growing conditions, and may constitute up to 20% of the dry weight of Spirulina [6]. C-phycocyanin and APC are homologous proteins and both bind phycocyanobilin (PCB) chromophore [7, 8]. The presence of the third phycobiliprotein, red phycoerythrin, in Arthrospira platensis is the subject of debate. While some studies have found that Spirulina produces small amounts of phycoerythrin, the other ones did not detect phycoerythrin in Spirulina [9].

### 2.1. Structure and physicochemical properties of C-phycocyanin and phycocyanobilin

C-phycocyanin (CAS registry number 11016-15-2) is water-soluble, intensive blue protein with strong fluorescence. It is the heterodimer consisting of α- (~18 kDa) and β-subunits (~19 kDa), which form αβ monomers, further aggregating to trimmers (αβ)3 and hexamers (αβ)6. Hexamer form represents a functional unit of phycobilisomes. C-phycocyanin is α-helicoidal protein, with one well-defined domain (similar to the globins) observed within the 3D structure of both chains. Color and intensive fluorescence of C-PC arises from PCB, covalently attached to Cys-84 of α-subunits, while β-subunit binds two PCB molecules via Cys-82 and Cys-153 residues [8]. Allophycocyanin has similar structure and physicochemical properties as C-PC. Unlike C-PC, β-subunit of APC binds only one PCB molecule (at Cys-84) [7, 8]. Amino acid variation of phycocyanins between cyanobacteria and red algae species are very minor [10].

The VIS absorption spectrum of the native C-PC has pronounced specific peak at 620 nm, arising from bound PCB. Phycocyanobilin has a molecular weight of 586.7 g/mol and characteristic fluorescence spectrum with an emission peak at 640 nm. Spectra of free PCB differ from spectra of native protein, in sense of intensity and shape of absorption and emission bands [11]. Bilin chromophore is a very sensitive indicator of the conformational state of the protein, enabling monitoring of C-PC denaturation/renaturation by standard spectroscopic methods. Thermal denaturation of C-PC induces shift of absorption maximum from 620 to 600 nm with significant decrease in protein absorbance (color intensity) and fluorescence [12]. Changes of PCB conformation upon denaturation induce these phenomena: chromophore in native protein has stretched conformation, while denaturation changes PCB conformation to the cyclic, similar to the free chromophore [13].

#### 2.1.1. Production, isolation, and purification of C-phycocyanin and phycocyanobilin

Thanks to the high protein (C-PC) content, as well as large availability, Arthrospira platensis is culture of choice for C-PC production. Spirulina growth requires dry, hot, and sunny climatic conditions [14]. Photoautotrophic Spirulina production is outdoor method, used for commercial production of C-PC at tropical and subtropicals regions, in open ponds and raceways. In the mixotrophic production, Spirulina cultivation is performed in an enclosed reactor with the addition of glucose, yielding a higher amount of C-PC. Spirulina can grow even heterotrophically, but in this case small yield of pigments is obtained [10]. Presence of covalently attached chromophore makes recombinant production of C-PC more complicated in comparison to other proteins. Complete synthesis of C-PC depends not only on co-expression of α- and β-chains, but also on parallel synthesis of PCB and its covalent attachment to protein [15].

Crucial parameters for C-PC production are lighting conditions (light spectrum, quality, intensity, and cycle), climatic conditions (pH and temperature), and media type. Their optimization strategies are reviewed in [16], with higher productivity in closed bioreactor systems than open ponds. Utilization of agricultural waste to replace the synthetic chemicals in algae cultivation media could also have enviro-economical impact.

Isolation of C-PC in high yield requires efficient extraction process. There are several effective approaches used for C-PC extraction: freezing and thawing, homogenization with mortar and pestle, sonication, high pressure homogenization, osmotic shock (using distilled water), acid treatment, enzymatic treatment (by lysozyme), organic solvent extraction, etc. [17]. Potential applications of C-PC in medicine or for research purposes (as fluorescent tag) require its high purity. The purity of C-PC is evaluated using ratio between absorbance at 620 and 280 nm (A620/A280). C-PC preparations with A620/A280 greater than 0.7 is considered as food grade, while preparations with A620/A280 more than 3.9 and 4 have reactive and analytical grade of purity, respectively [14]. C-phycocyanin price strongly depends on its purity, ranging from $200 to$2.2 million per kilogram. Numerous different procedures for C-PC purification (usually after protein precipitation with ammonium sulfate) use one or more chromatographic steps (ion-exchange chromatography, hydrophobic chromatography, gel filtration, hydroxyapatite chromatography, and expanded bed adsorption chromatography) or two-phase aqueous extraction [10]. Changing light conditions during cultivation of Spirulina (blue and red light vs. normal) could increase yield and purity of C-PC [18].

Phycocyanobilin (CAS 20298-86-6) isolation requires cleavage of thioether bond between apoprotein and bilin chromophore, by acid hydrolysis, enzymatic cleavage, or alcohol reflux. The most common procedure for the cleavage of PCB from C-PC is still conventional reflux in methanol, lasting up to 16 hours [10]. Performing ethanolysis in the sealed vessel at 120°C decreases reaction time to 30 minutes, and obtained PCB has higher purity in comparison to conventional reflux method [17]. Phycocyanobilin can be produced in mammalian cells by metabolic engineering, introducing genes for heme oxygenase-1 and PCB:ferrodoxin oxidoreductase, with simultaneous knock-down of biliverdin reductase A to prevent PCB reduction to phycocyanorubin [19].

## 3. Food applications of C-phycocyanin and phycocyanobilin

### 3.1. Stability and technologies to improve stability

Natural food colorants are often sensitive to heat, light, oxygen, acidic conditions, and exposure to oxidants, such as ascorbic acid and trace metal ions. Generally speaking, natural C-PC is not a particularly stable protein. It was found to be unstable to heat and light in aqueous solution. The presence of photosensitive PCB makes C-PC sensitive to light and prone to free-radical oxidation [20]. The optimum pH range for C-PC was found to be 5.0–6.0 [21] and it is insoluble in acidic solution (pH 3) [22]. The critical temperature for C-PC stability is 47°C, with a sharp drop in the protein half-life values above this temperature. At 50°C, the C-PC solution showed maximum stability at pH 6.0, while at 60°C the maximum protein stability was at pH 5.5 [23]. Exposure to light of 3 × 105 lux for 24 hours in aqueous solution at pH 5 and 7 caused ~80% of its degradation [22]. Therefore, although C-PC has high potential for applications in food industry, biotechnology, and medicine, stability issue is one of the limiting factors for its successful application.

### 3.2. Safety and bioavailability

Numerous toxicological studies, such as acute, sub-chronic, chronic, mutagenic, teratogenic/developmental toxicity, carcinogenic, and multiple generational/reproduction tests, have confirmed excellent safety profile of Arthrospira platensis and Arthrospira maxima (Class A rating by the dietary supplements information expert committee of the US pharmacopeial convention). They were of paramount importance in the determination that water extracts of Spirulina or C-PC are safe as well. Only very rare, single-case events of adverse incidences associated with consumption of Spirulina have been reported [30].

Desert Lake Technologies, LLC got GRAS notification in 2012 for its CyaninPlus™ product, consistent with section 201(s) of the Federal Food, Drug and Cosmetic Act. It is a water extract of the Spirulina platensis or Spirulina maxima intended for use as an ingredient in food at levels of up to 250 mg per serving, equivalent to approximately 125 mg of C-PC.

In animal models, C-PC possesses low toxicity and lack of adverse effects. For example, in acute oral toxicity study, the measured LD50 values were estimated to be greater than 3 g/kg for rats and mice, without mortality even at the highest dose of C-PC from Arthrospira maxima tested (3 g/kg o.p.). No changes in behavior or histopathology, or effect on body weight were observed [31]. Furthermore, acute and sub-chronic oral toxicity study revealed that C-PC (isolated from Spirulina platensis) at high concentrations [0.25–5.0 g/kg body weight (w/w)] did not induce any symptoms of toxicity nor mortality of the albino rats [32]. In human randomized, double-blind, placebo-controlled study, high dose of C-PC-enriched aqueous extract from Spirulina platensis, equivalent to ~1 g phycocyanin per day (the highest dose generally recognized as safe by the US FDA), after 2 weeks showed safety regarding anticoagulant activity and platelet activation status markers, but reduced levels of aspartate transaminase and alanine transaminase in conjunction with rapid and robust relief of chronic pain [33]. Unlike cancer cells, C-PC is non-toxic to normal cells, for example, platelets and erythrocytes [34]. Although Spirulina is not regarded as source of allergens, there is one case report describing anaphylaxis caused by C-PC [35]. To conclude, animal and clinical scientific studies support that Spirulina and C-PC, its most abundant organic component, are safe for human consumption, in agreement by their more than 1000 years use in diet.

Bioavailability is the term used to describe how much of the nutrient are easily absorbed into the body and so is able to have an active effect. Spirulina is extremely digestible, high energy but low calorie and low fat natural food. Many studies demonstrated in vivo effects of orally administered Spirulina or C-PC [1, 2, 3]. Our research group has shown that C-PC is rapidly digested by pepsin in simulated gastric fluid, releasing chromopeptides varying in size 2–13 amino acid residues. Released chromopeptides had significant antioxidant activity and metal-chelating property, with cytotoxic effect on cancer cell lines positively correlating their antioxidative capacity, with chromophore portion being most responsible for these effects [36]. There is no literature data related to transport of PC-derived peptides or PCB from gastrointestinal tract to circulation. Our previous studies demonstrated that PCB binds to human serum albumin (HSA) with high affinity (2.2 × 106 M−1) [37], and stabilizes protein structure [38], suggesting that in circulation HSA most likely transports PCB to tissues, similar to other bioactive food-derived substances. Many studies observed in vitro effects of C-PC in cell culture, but the location of protein inside cells is controversial and it is still unknown whether C-PC requires a transport protein carrier to enter cells. For skin delivery and protection from oxidative stress damage, C-PC is characterized by a reduced bioavailability, due to its high molecular weight, and therefore it is encapsulated in hyalurosomes as carrier [39].

### 3.3. Interactions with food matrix components

In addition to their sensitivity to light, heat, and oxidants, natural food colors are prone to interact with other food ingredients, especially if they are carrying proteinous component, such as C-PC. Several studies have found that both C-PC and PCB binds to food matrix components, such as proteins, lectins, saccharides, lipids, and polyphenols [40, 41, 42, 43, 44, 45, 46, 47].

C-phycocyanin non-covalently binds to bovine serum albumin (BSA), with binding constant 6.8 × 105 M−1 and n = 1.2, as determined by fluorescence quenching of BSA. FT-IR, and synchronous fluorescence spectroscopy confirmed the conformation of BSA has been affected the interaction with C-PC [40]. In the recent study, we found that PCB also interact with BSA, showing high affinity binding (Ka = 2 × 106 M−1), determined by protein fluorescence quenching and microscale thermophoresis. Two binding sites were detected on BSA, at the inter-domain cleft and at subdomain IB, with stereo-selective binding of the P pigment conformer to the protein. Although complex formation partly masked the antioxidant properties of PCB and BSA, a mutually protective effect against free radical-induced oxidation was found [41]. Additionally, PCB binding to major whey protein β-lactoglobulin changes its secondary and tertiary structure, causing higher resistance to digestion by pepsin and pancreatin [42].

C-phycocyanin also interacts with food-derived lectins. Jacalin, tumor-specific lectin from cempedak, binds C-PC specifically in a carbohydrate-independent manner, and with affinities better than that for porphyrins. The binding pattern involves both ionic and hydrophobic interactions and more than one contact site [43]. Concanavalin A and peanut agglutinin can also interact with C-PC, although the nature of the interaction is distinctly different from that for jacalin. The legume lectins bind C-PC via two distinct sites, and the binding is weaker in the presence of their specific carbohydrate ligands. Therefore, lectins are proposed as useful carrier for targeted delivery of C-PC in photodynamic therapy [44].

Well-known cryoprotecting disaccharide trehalose interacts with C-PC and decreases the internal protein dynamics, slowing down molecular motions responsible for its unfolding and denaturation [45]. Although it was found that C-PC interacts with lipids at the air-water interface, the oxidation of monogalactosyldiacylglycerol could not be prevented by the introduction of C-PC molecules at the lipid-water interface [46]. Formation of complex between C-PC fragments and polyphenols, in order to obtain more stable blue color for application in food, feed, cosmetic, and pharmaceutical products, was recently patented [47].

### 3.4. Health-promoting effects

A good part of the bioactivity properties of Spirulina are assigned to the pronounced antioxidant capacity of C-PC, mainly attributed to its chromophore (PCB) moiety (see below). This phycobiliprotein has proven (in)dependent therapeutic effects, such as anticancer, antiinflammatory, and antimicrobial effects, immune enhancement function, liver, and kidney protection, among others. These benefits were subject of many recent excellent review articles (e.g. [48]). As potential safe and non-toxic compounds, C-PC and PCB become a new hot spot in the medicine. The complex mechanisms of its pharmacological actions begin to be fully understood at the molecular level. C-phycocyanin is currently not in clinical use, because positive health-related reports are not integrated deeply and accurately enough, putting limitations to its application as a drug. Otherwise, C-PC-encapsulated chitosomes, capable of preserving the protein stability in the gastrointestinal tract and with enhancing efficacy are in development [49].

Phycocyanobilin is potent inhibitor of certain NADPH oxidase isoforms, likely because in mammalian cells it is rapidly reduced to phycocyanorubin, a close homolog of bilirubin. Over-activity of NADPH oxidase causes oxidative stress, and is known to mediate and/or exacerbate numerous pathological conditions [50]. In vitro, PCB is capable to modulate other important markers of oxidative stress and endothelial dysfunction, such as eNOS and/or VCAM-1, and to markedly up-regulate heme oxygenase-1, a key enzyme responsible for generation of a potent antioxidant bilirubin [51].

The suitable clinical dose of PCB remains to be defined. Without mass-produced pigment derived from commercially available PCB-enriched Spirulina extracts, bioengineered organisms, or chemically synthesized pigment, ingestion of whole Spirulina is still the least expensive way to benefit from this phytonutrient. A tablespoon of Spirulina powder (about 15 g) contains approximately 100 mg of PCB, daily dose that might be effective [50]. Interestingly, no relevant data about relative absorption and bioefficacy of free PCB or Spirulina-bound pigment exist for either rodents or humans.

#### 3.4.1. Antioxidant properties

Proteins bearing colored prosthetic groups, such as a highly conjugated linear tetrapyrrole chromophore in C-PC, can be both the source and target of reactive species in biological systems. An extremely high antioxidant capacity of C-PC was unambiguously established, based on experiments carried out both in vivo and in vitro. It not only scavenges, for instance, peroxyl, hydroxyl, and superoxide radicals, but also inhibits the lipid peroxidation mediated by reactive oxygen species. The bilin group seems to be the main target, since the in vitro radical assisted bleaching of PCB color in protein clearly indicates its involvement in the scavenging of reactive species [52]. A key contribution of the structural components and various modulating factors on the antioxidant activity of C-PC will be briefly mentioned here, as they can influence protein utility as a food supplement and therapeutic agent.

C-phycocyanin is a more efficient peroxynitrite scavenger than free PCB due to (additional) interactions with tyrosine and tryptophan residues of the apoprotein [53]. Differences in the amino acid composition affect C-PC antioxidant capacity. Selenium-C-phycocyanin purified from Se-enriched Spirulina platensis exhibits stronger antioxidant free radicals scavenging activity than standard protein, attributed to the incorporation of selenoamino acids into the polypeptide chains of protein [54]. C-phycocyanin (from Spirulina fusiformis) exposed to blue light shows better in vitro antioxidant property than protein exposed to normal light, due to marginal changes in the apoprotein cysteine content [55]. Interestingly, bilin group is not the main target of C-PC reaction with hypochlorous acid and singlet oxygen [56].

C-phycocyanin generates hydroxyl radicals in the light, while scavenging them in the dark. Radical generation ability disappears, but scavenging greatly increases in denaturated protein, confirming the role of phycobilin moiety in scavenging. Trypsin hydrolysis of C-PC demonstrated the apoprotein portion also made a significant contribution to the antioxidant activity [57]. The heat denatured (spray-dried) C-PC shows the same level of activity as the intact protein, finding important for preparation and utilization of C-PC [58]. C-phycocyanin can be cloned and expressed in Escherichia coli, to reduce the cost and time for protein production. Recombinant holoprotein (α-subunit) not only retained the spectroscopic characteristics of the native protein, but also its bioactive properties, including powerful radical scavenging activity [59]. Although less potent, recombinant apo-C-PC β-subunit acts as an antioxidant on human erythrocytes as well [60]. Other bioactivities of recombinant biliproteins should be further studied to provide additional health benefits.

### 3.5. Food colorants

In the last decades, consumers are becoming more educated and aware of what they eat, demanding for clean labeling of the food/beverage products and making the pressure to food industry to switch from artificial to natural ingredients and additives. The main consumers of vividly colored food products are children. Due to their low weight, they are at constant risk to exceed recommended daily intake (mg/kg weight) of artificial colorants. Nowadays, the leading confectioners switched to natural colors to avoid obligatory label warning for acceptable daily intake levels of the colorings. Consequently, market of natural food colors is in prominent expansion, expecting to reach $2.5 billion by 2025. Compared with other natural pigments, natural blue pigments are rare, because a complex combination of molecular features (such as π-bond conjugation, aromatic ring systems, heteroatoms, and ionic charges) is required to absorb red light (~600 nm region) [61]. Anthocyanins are the primary source of blue color in plants, but their color is pH dependent. On the other hand, fungi and microorganisms produce many blue compounds in response to stress or predators and therefore their unpredictable biological activities make their safety for food use questionable. None of discovered natural blue pigments cannot reach shade, brilliance, vividness, molar absorptivity, and stability of Brilliant Blue FCF (Blue 1 or E133), the most used of approved synthetic blue food colorants, and concomitantly to be safe and cost-effective [61]. In this moment, the only permitted natural blue food colorants are gardenia blue (in Japan), blue anthocyanins and Spirulina color (composed mainly of C-PC). Although gardenia blue and blue anthocyanins have better stability to heat and light than C-PC [22], only C-PC can offer brightness, brilliance, and shade most similar to Brilliant Blue FCF, making this protein much more acceptable and ensuring seamless switches from artificial to natural food colors for existing food products. Trichotomine, indole alkaloid from kusagi berries (native in China and Japan), is the most promising natural bright blue colorant due to molar absorptivity (70,000 M−1 cm−1) similar to that Brilliant Blue FCF (134,000 M−1 cm−1). Limited supply of kusagi berries and the low concentration of pigment make this option economically unjustified [61]. In contrast, although PCB have relatively low molar absorptivity (37,900 M−1 cm−1) [37], Spirulina can be sustainably produced in huge, almost unlimited amounts, and the high pigment concentration provide its extraction in cost-effective way. Demand for C-PC as a natural blue food colorant has experienced exponential growth in the past 5 years, especially after FDA approval of Spirulina extract as a food colorant for gum and candy in 2013, with market estimated at more than$50 million. In 2014, its application was expanded to frosting, ice cream and frozen desserts, dessert coatings and toppings, dry beverage mixes and powders, yogurts, custards, puddings, cottage cheese, gelatin, breadcrumb, and ready-to-eat cereals. In 2015, coatings in dietary supplements and pharmaceuticals were also approved and, in this moment, C-PC is the only approved natural blue colorant in the US, Europe, and Asia. In US FDA Code of Federal Regulations, Spirulina extract is approved as color additive exempt from certification, prepared by the filtered aqueous extraction of the dried biomass of Spirulina platensis and containing phycocyanins as the principal coloring components [62].

Commercial powder formulations of C-PC, such as Linablue® (DIC Corporation, Japan), are declared as completely soluble in cold and warm water and <20% ethanol, making a homogeneous transparent solution, with stable color shade in the pH range 4.5–8.0 (except at C-PC pI value around pH 4.2), which can be improved by the presence of protein-containing ingredients; low thermal stability, which can be improved in high density sucrose solutions; with low light stability, which can be improved in the presence of antioxidant like ascorbate; and with no tongue dyeing effect. In combination of C-PC with red, yellow, and other natural colorants, it is possible to obtain vibrant green, purple, and other natural colors. As FDA is still considering the petition for copper chlorophyllin, natural green food color often involves C-PC or Spirulina extract mixed with safflower or turmeric extract (curcumin).

Due to its refreshing ice cool color, C-PC is also increasingly promoted as natural color for alcoholic beverages, such as FIRKIN Blue gin.

In comparison to artificial colors, natural colorings are less vivid, and interactions with food matrix components can result in further decrease in their vibrancy, or unwanted change in color and flavor. For example, our research group observed an instant clear color change from blue to green when PCB interacts with BSA [41]. Therefore, switching from artificial to natural colorings in existing food products can be challenging and complex.

### 3.6. Functional food additives, nutraceuticals, and dietary supplements

In last decade, there is an increase in chronic diseases and increasing costs of health care due to busy lifestyles and unhealthy nutrition. On the other hand, people are more health conscious and more interested in health-promoting products to improve their health quality. This imposed a demand for functional food ingredients and additives, nutraceuticals and dietary supplements of natural origin.

There are several studies dealing with incorporation of Spirulina or its proteins into different food, such as biscuits, pasta, milk-based products, various breads, and crisps, in order to create protein-enriched functional food products. In all these studies the food was fortified by whole Spirulina powder or biomass, except in the study where isolated C-PC was used [63]. Spirulina was incorporated into pasta (e.g. [64, 65, 66]), biscuits and cookies (e.g. [67, 68, 69]), extruded products (e.g. [70, 71]), ice cream [72], yoghurt and acidophilus milk (e.g. [73, 74], baby food formulas [75], and bread [76]. However, all these food products were thermally processed leading to destruction of C-PC and resulting in green-yellowish color of product due to partially retained carotenoids and chlorophyll. This fact was ignored in almost all studies, except in the study monitoring C-PC degradation at 615 nm [71].

Although protein component of C-PC added nutritive value to these products, bioactivity of sensitive PCB component cannot be exploited. The only way to take full advantage of health-promoting effects of bilin component, is addition of Spirulina biomass/powder, PCB-enriched Spirulina or C-PC alone, after all thermal food-processing steps. Similarly, although Spirulina fortification of milk have positive influence on viability of milk fermenting microbiota [74], their activity also decrease content of precious bilins. In the most of the studies dealing with Spirulina biomass-food enrichment, the limiting factor for quantity of added Spirulina was consumer acceptance due to sensory characteristics related to flavor and taste. The use of C-PC or Spirulina protein isolate/concentrate fraction would reduce undesirable fishy off-flavor of whole algae biomass and in that way notably improve consumer acceptance.

Possible advantages of joint administration of flavanol-rich cocoa powder and Spirulina, or PCB-enriched Spirulina extracts was proposed [77]. As inhibitor of NADPH oxidase, PCB would minimize NADPH oxidase-derived oxidative stress, while flavanols would promote vasodilation by up-regulation of NO production. Cocoa-Spirulina powder blended with milk (cow’s, soy, and rice) can yield a drink with a tasty rich chocolate flavor, as cocoa can mask the unpleasant flavor and odor of Spirulina/C-PC. These two nutraceuticals could complement each other actions in prevention of senile dementia by optimizing cerebrovascular perfusion, and by suppressing cerebral oxidant stress. Combined supplementation with PCB, citrulline, taurine, and supranutritional doses of folic acid and biotin could help in slowing the progression of diabetic complications, based on their complementary action on the oxidative stress and the associated loss of NO bioactivity [78].

Besides being component of many dietary supplements, C-PC also becomes popular component of different wellness bioactive drinks, providing attractive blue color and nutraceutical properties [e.g. Ocean Mist by Allgalio Biotech, B Blue bioactive drink by B blue, Bloo tonic by Cidererie Nicol, Holy water by Juice Generation, Natura blue by Natura4Ever, Smart chimp by Smart chimp and many other drinks based on Blue Majik (C-PC-enriched organic extract of Spirulina platensis by E3Live) made by other producers].

Purified PCB is still not available as a nutraceutical supplement, but new research turned toward methods for efficient cleavage of PCB from the C-PC [17]. Further stabilization will enable commercially available PCB as food colorant and dietary supplement.

### 3.7. Future carriers of bioactive substances and food additives with promising techno-functional and food-preserving properties

In recent years, the food industry is in increasing search for new sources of inexpensive food protein having nutritional and techno-functional characteristics similar to high-cost animal proteins. In addition to plant one, proteins extracted from microalgae are becoming favorable alternative due to availability and sustainability of their production on one hand, and due to their extraordinary nutritional and bioactive properties, as well as suitable functional properties on the other hand.

In this moment, there are only few studies investigating the functional properties of Spirulina protein concentrate/isolate, with phycobilins being the main functional protein component. Proteins isolated from Spirulina are quite capable of reducing the interfacial tension at the aqueous-air interface at relatively lower bulk concentrations than common food proteins [79]. In comparison with soy protein isolates, Spirulina protein isolate (SPI) demonstrates lower water, but higher oil absorption capacity. SPI showed good emulsifying and foaming capacity, and ability to form protein films and gels [80]. This study demonstrated that emulsifying capacity, the emulsion aging stability, the emulsion microstructure and opacity as well as the foaming capacity and the foam stability were pH dependent. Also, emulsifying and foaming capacities have shown to be positively correlated to the protein solubility [80]. Spirulina protein isolate forms gels after heating (90°C) and cooling, showing fairly low minimum critical gelling concentrations (1.5 wt% in aqueous solution) compared to other food (soy) proteins [81]. Spirulina protein concentrate (SPC) have shown higher emulsifying and similar foaming capacity, when compared with soybean meal [82]. The rheological and textural parameters increased linearly with increased C-PC addition (0.25–1.25% w/w) in oil-in-water emulsions, suggesting C-PC emulsion stabilizing role [83]. From food technology point of view, these studies imply that C-PC or Spirulina proteins are promising food ingredients and additives, and that further studies are needed to fully exploit their most likely excellent functional properties.

In contrast to other natural food colors, but similar to other food-derived proteins, C-PC can be used to modify techno-functional properties of food matrices or as carrier of bioactive substances. As a biodegradable, biocompatible, and poor immunogenic protein molecule, C-PC is suitable as carrier for preparation of protein-based nanoparticles. Drug delivery via their loading into C-PC nanoparticles have shown to be more effective and safer [84, 85]. By analogy, C-PC-based nanoparticles can be used for food applications in the future as carrier for other active substances, acting together in synergistic manner and complementing mutual benefits. In order to fully utilize all benefits of valued bilin component, C-PC should be added only after all thermal pretreatments.

The natural food colorings are often associated with functional properties. C-Phycocyanin/PCB with their extraordinary antioxidative activities could have role in maintaining of the lipid oxidative stability, especially in food products with high lipid contents. Addition of C-PC was found to inhibit linoleic acid peroxidation and decrease TBARS value in liposome-meat system [86]. Some studies demonstrated that C-PC exhibited antibacterial and antifungal potential [87, 88], suggesting that C-PC/PCB can also can serve as antimicrobial agent. Silver nanoparticles-based antimicrobial packaging is a promising form of active food packaging, and C-PC was used for synthesis of bio-silver nanoparticles [89]. Incorporation of Spirulina powder in strudels can significantly retard lipid oxidation and reduced the number of yeast and mold resulting in prolonged shelf life [68]. Therefore, C-PC/PCB in addition to their role as food colors can contribute to food preservation and improvement of food shelf life and/or to the reduce addition of non-natural food preservatives.

## 4. Other applications of C-phycocyanin and phycocyanobilin

As we have seen, C-PC and PCB have excellent antioxidant properties. Irradiation of these molecules with visible light produces reactive oxygen species, making them good candidates for application in photodynamic therapy (PTD). Indeed, it was shown that anticancer activities of C-PC against breast cancer MCF-7 cells increases upon exposure to He-Ne laser (632.8 nm wavelength) [90]. C-phycocyanin has specific affinity for tumor-associated macrophages (TAM), which have been proposed to be a “target for cancer therapy”. Formation of non-covalent conjugate between Zn-phthalocyanine and C-PC resulted in an enhanced photodynamic effect with selective accumulation in the tumor site, probably through the specific binding of C-PC to TAMs [91].

In comparison to other fluorophores, phycobiliproteins have a high molar extinction coefficient and fluorescence quantum yield, as well as a large Stokes shift. Therefore, C-PC could be a good candidate for applications as fluorescent marker. When C-PC is extracted by low ionic strength buffers monomers will be the dominant form, inducing decrease of protein fluorescence. Therefore, in order to obtain stabilized highly fluorescent oligomers cross-linking of C-PC is needed. Chemically stabilized C-PC, fused to the biospecific domains such as streptavidin, is used as a biospecific fluorescent assay. Further, C-PC fluorescence can be also used for in vivo monitoring of cyanobacterial growth and detection of toxic cyanobacteria in drinking water [15]. Furthermore, strong quenching of C-PC fluorescence Hg2+ ions implies its potential application as biosensor for heavy metals in aquatic systems [92]. Interestingly, PCB synthesized in mammalian cells through metabolic engineering could be useful optogenetic tool for regulation of cell processes by light [19].

Artificial photosynthesis is currently a hot topic in science and technology. Consequently, there are growing demands for designing photoelectron-chemical (PEC) cells, capable to perform artificial photosynthesis. In PEC devices, light-harvesting proteins (such as C-PC) are used to “sensitize” metal and semiconductor surfaces. BioPEC solar hydrogen generator with a hematite-phycocyanin hybrid photo anode was designed. In order to obtain PEC cells with higher performances, the stability of immobilized C-PC needs to be improved [93].

Beside the application for food and drink coloring, C-PC is also used as a cosmetics colorant in lipsticks, eyeliners, and eye shadows preparations [94].

## 5. Conclusion

The vast majority of studies regarding Spirulina bioactive components used whole algal (dried) mass or its aqueous extracts. Major Spirulina deep blue color protein C-phycocyanin and its bilin chromophore have remarkable potential for use in food technology, as safe food colorant, functional food additive, nutraceutical and/or dietary supplement, given their excellent health-related properties, and opportunity for sustainable and relatively inexpensive mass production.

## Acknowledgments

This study was supported by the grant No. 172024 obtained from the Ministry of Education and Science of the Republic of Serbia.

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© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Dragana Stanic-Vucinic, Simeon Minic, Milan R. Nikolic and Tanja Cirkovic Velickovic (June 27th 2018). Spirulina Phycobiliproteins as Food Components and Complements, Microalgal Biotechnology, Eduardo Jacob-Lopes, Leila Queiroz Zepka and Maria Isabel Queiroz, IntechOpen, DOI: 10.5772/intechopen.73791. Available from:

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# What is the continuity of f(t) = 3 - sqrt(9-t^2)

What is the continuity of $f\left(t\right)=3-\sqrt{9-{t}^{2}}$
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$f\left(t\right)=3-\sqrt{9-{t}^{2}}$ has domain $\left[-3,3\right]$
For a in $\left(-3,3\right)$, $\underset{t\to a}{lim}f\left(t\right)=f\left(a\right)$ because
$\underset{t\to a}{lim}\left(3-\sqrt{9-{t}^{2}}\right)=3-\underset{t\to a}{lim}\sqrt{9-{t}^{2}}$
$=3-\sqrt{\underset{t\to a}{lim}\left(9-{t}^{2}\right)}=3-\sqrt{9-\underset{t\to a}{lim}{t}^{2}}\right)$
$=3-\sqrt{9-{a}^{2}}=f\left(a\right)$
So f is continuous on (−3,3).
Similar reasoning will show that
$\underset{t\to -{3}^{+}}{lim}f\left(t\right)=f\left(-3\right)$ and
$\underset{t\to {3}^{-}}{lim}f\left(t\right)=f\left(3\right)$
So f is continuous on [−3,3].


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How do you display text in the center of the screen using printf?

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you could:
use strlen() to obtain the length of the string. taken you know the width of the screen (in characters. usually 85 I think), you can calculate the number of whitespaces before the text to center it on screen.

you could then, either add whitespaces to your string (or a new string) and print them, or
depending on what platform you're developing on, set the position of the text manually.
in good old c/c++ you had gotoXY(), but in for example Visual C++ this is removed..
In windows, everything is a window .. so you'll have to use some windows function to set the position.
Here are some wrapper functions I wrote way back:
void setXY( COORD coords) {	SetConsoleCursorPosition(GetStdHandle(STD_OUTPUT_HANDLE), coords);}COORD getXY() {	CONSOLE_SCREEN_BUFFER_INFO tempInfo;	GetConsoleScreenBufferInfo(GetStdHandle(STD_OUTPUT_HANDLE), &tempInfo);	return tempInfo.dwCursorPosition;}

[edited by - BiTwhise on January 13, 2004 2:20:02 PM]

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assuming that the string you are printing is less than the width of the screen:

#define SCREEN_WIDTH 80 /* change if not correct */

f(char *p)
{
char buf[SCREEN_WIDTH+1] = "\0";
int i, n, half_n, width_half;
n = strlen(p);
half_n = n / 2;
width_half = SCREEN_WIDTH / 2;
for (i = 0; i < width_half - half_n; i++)
buf = '' '';
buf[i] = ''\0'';
strcat(buf, p);
printf("%s\n", p);
}

I didn''t test this code, but something like this should work

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