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## Sunday, May 25, 2008

### Wormholes at the LCH?

The LCH is next to open. Hopefully supersymmetry and the Higgs boson will be found. But there are more possible things that we can find there. The recent Planck 2008 congress was devoted to that topic. You can see blog entries covering that in the Dmitry (non equilibrium phenomena) blog, concretely this entry, and previous ones (ok, the entry with the interview to Polyakov is not about Planck 2008, but it is worth reading anyway). Another ongress, PPC2008 was devoted to LHC, and string phenomenology in general. You can read blog entries about it in Tomasso Dorigos blog, particularly in this entry and subsequent ones (advise, there are a lot of them). Another coverage of the congress is here. If some reader has a lot of time and is interested in reading still more about scientific congress and phenomenology he could read the coverage that Marni Dee has made of neutrinos 2008. At the monet of writing this entry her las post is this.

Well, i am going to add some fuel into the phenomenology of LCH possible predictions in this post. I am going to talk about wormholes production in LCH. That possibility heavily relies in the large extra dimensions scenario. You can read a post in the thomaso dorigos series on PPC2008 about the subject, this. Also you could read my own post about large extra dimensions and warped geometries.

As I said in that entry one thing that can possibly found in the LHC, if the LED sceneries are true, are microblack holes. But that is not the whole history. One can find another things such as p-branes or wormholes. You can read about the formers in this paper. Advise, there they talk about a somewhat peculiar type of p-branes, cosmic branes, introduced in this other paper. I am not going to describe those papers, just quote the abstract, which is self explanatory:

We compute the cross section for p-brane creation by super-Planckian scattering processes
in a (n +4)-dimensional space-time with n−m flat large extra dimensions and m
flat small dimensions of size of the fundamental gravitational scale. We find that the
cross section for the formation of a brane with dimension 1 ≤ p ≤ m, completely wrapped
on the small dimensions, is larger than the cross section for the creation of a spherically
symmetric black hole. Therefore, in space times with asymmetric compactifications we
predict that branes are more likely to be created in super-Planckian scattering processes
than black holes. The higher rate of p-brane production significantly enhances possible
detection of non-perturbative gravitational events by future hadron colliders and cosmic
rays detectors.

Now I am going, at last, with the actual topic of this post. I will explain the resuoults of two papers, Time Machine at the LHC and If LHC is a Mini-Time-Machines Factory, Can We Notice?.

The first paper face the question of the actal calculation of the probability of the wormhole beeing formed. The second, the sing that it could leave in the LHC if it actually forms.

The first paper begins with a brief review of how calculation of black hole formation can be obtained. That subject is generically know under the name of "planckian scattering". He beguines with a "quantum gravity" approach, based on the wheeler-de Witt formalism. There the possibility of black hole formation is calculated by considering the kernell of the following transtion amplitude:

$\langle h'', \phi'',\Sigma''|h', \phi',\Sigma'i\rangle= \int exp{i/\hbar S[g,\Psi]}DgDh$

The key in that approach is to study the transition between geometries describing two particles and the geometry describing black holes (or wormholes). For the details the authors refers to the paper of arxiv gr-qc/9404036. which, unfortunately, is unavailable seemingly due to a corruption in the latex that arxiv uses to render the pdf.

Other approach discussed in that brief review of black hole formation is to suppose that that ultra-relativistic particles are represented by plane gravitational waves, which interacting collide and produce a black hole. For 3D geometry the energies required are not available in the LHC, but in the LED sceneries the thing changes. The Schwarzschild radius of a 4+n dimensional black hole of mass M = √s is approximately:

$r_o \approx M^{-1}_{4+n} (s/M^{2}_{4+n})^{1/2n +1)$

( is the square of the center of mass energy of colliding particles, M4+n is the 4+n dimensional Planck mass and the 4 dimensional Planck mass is given by:

$M^2_{pl} \approx V_n M^{2 + n}_{4+n}$

Where Vn is the volume of the extra dimensions. This problem can be achieved using the Aichelburg-Sexl metric, which describe a particle at ultra relativistic speeds and is obtained by doing a Lorentz boost to the Schwarzschild metric and some convenient changes of variables. In the article the details of the calculations are not presented and the author refers to the literature. Maybe some reader is surprised why string theory is not mentioned in this article to analyze the formation of black holes. I guess that it is a good idea to give some things about it. Of course the ultimate reason to even bother about practical formation of black holes depends on warped dimensions. Although warped dimensions can be studied by classical general relativity in 4+1 dimensions (for one warped dimension) the practical motivation to do that depends on string theory. Ok, that was clear, I guess, so I´ll say some quick ideas about string theory and black holes. String theory can describe black holes of an special type, Reissner-Nordstrom black holes, using Dp-branes, and calculate it´s entropy. But the approach used to describe that black holes doesn´t say nothing about the actual formation of them. AS soon as in the eighties that question was studied by Veneziano who calculated the possibility of black hole formation by exchange of gravitons in a sting theoretic description. I don´t really know why exactly that approaches are not followed, but at least I leave Constance of their existence.

The article follows with a very brief introduction to wormholes. That is a topic widely discussed in the net. The reader can find references to it in many places. To avoid to make this entry too long I have made a separate entry in my other blog. I must apologize for English readers because the entry is written in Spanish, anyway, the actual url for it is this.

Anyway, in that entry I only discuss the basic of wormholes. For the article discussed here it is necessary to consider wormholes in a braneworld scenario. There, where the Universe is considered as a 3-brane embedded in a D-dimensional bulk, the four-dimensional Einstein field equations contain the effective four-dimensional stress energy tensor:

$G_{\nu\mu}= M^{-1}_{pl}.T^{eff}_{\nu\mu}$

The effective energy momentum tensor is a sum of the stress energy tensor of a matter confined on the brane, Tμv and correction terms that arise from a projection of the D-dimensional Einstein equation to the four-dimensional space-time. For some particular examples it is possible to show that the four-dimensional effective stress energy tensor violates the NEC meanwhile the total five-dimensional stress energy tensor does respect the NEC.

After discussing how wormholes are described in branworld sceneries the actual question of their formation is discussed. The approach followed mimetizes the approach followed for black holes. It is not very detailed and it is based in the following cross section:

$\sigma pp\rightarrow wh(s) \approx \sum_{i,j}\int_{\tau_m}^1 d\tau \int_{\tau}^1 dx/x fi(x)fj(\tau/x)\sigma ij\rightarrow wh(\tau s)$

Here √s is the center of mass energy, x and τ/x are the parton momentum fractions, and fi are the parton distribution functions. The parameter τm = M2min/s where Mmin corresponds to the minimum mass for a valid wormhole description.

σij→wh(s) is the geometrical cross section of the wormhole production and depends in a form factor F. The form factor F(√s/MD) incorporates the theoretical uncertainties in description of the process, such as the amount of the initial center mass energy that goes into the wormhole, the distribution of wormhole masses as function of energy. These corrections are similar to corrections in the formula for black hole production.

The second paper centers on the description of the observable traces of the existence of a wormhole. Much emphasizes is made in the possibility of the wormhole behaving as a time machine. For this it´s two motous must be shifting away at a considerable speed for some lapse of time. Anyway, the authors prefer to use MTM (minitime machine) instead of wormhole. Anyway the actual possibilities of detection discussed in the paper can be summarized as follows:

(i) change of the energy spectrum due to the frequency-filtration property of MTM,
(ii) possible production of anomalously energetic particles, accelerated by passing many times
through gravitational field inside the MTM,
(iii) acceleration of particle decays, since the proper time of a particle moving inside MTM can
strongly exceed the laboratory time,
(iv) CPT and naive unitarity violation (thermalization) due to effective non-local interactions
caused by MTM and to possible ambiguity in the population of closed world-lines inside MTM,
(v) collective effects due to conversion of a single particle into a bunch of its co-existing copies
within the MTM

## Sunday, May 18, 2008

### Black holes information distortion paradox

A few days ago a friend of mine, graduate theoretical physician, but not an active physician nowadays, and an ocasional reader of this blog,let me know of a new in the media versing about a resolution of the "black hole information paradox". The new was published in many webs, for example here. By the same time a thread was opened in physics phorums about the topic, concretelly Physicists Demonstrate How Information Can Escape From Black Holes based in LQG.

Ok, I suposse that I would have to somehow give an opinion about the new. I have waited a bit to see if, apart of physics phorums people, some of the big (or even not so bigs) ones on the blogosphere said something about the particular. Afther all Astekhar, the mainauthor of the paper behind the new, is one of the greatests personalities in LQG (the whole field begins with a work of him about new variables in canonical gravity) and the theme is very catchy, to say the less. For some reason there has not been such an entry in referential blogs, so I´ll try to give my humble opinion abot the particular.

First of all to say that I find that the claim of the new somewhat distort the actual nature of the achivement. Suposedly the paper solves the questions in the framework of LQG. Afther all in the news release you can read:

"Once we realized that the notion of space-time as a continuum is only an approximation of reality, it became clear to us that singularities are merely artifacts of our insistence that space-time should be described as a continuum."

The idea of discrete space-time strongly suggest that they are talking about LQG. Well, in the physics porum post somone pointed to the arxiv paper behind the news release, concretelly this, Information is Not Lost in the Evaporation of 2-dimensional Black Holes. The first bad thing comes in the title, "2-dimensional black holes". That is they solve the problem in a simplified modell, that always opens the possibilitie that the problem could not be solved in the full environment, afther all 3-d quantum gravity is very diferent from 4-d one.

Anyway, let´s see what is going on. In the last post I talked about LQG and I did a brief description of how LQG treats black holes (or at least one of the ways they do it when they face the singularity problem). As not every reader of this blog is assumed to speak spanish I´ll re-explain it. They don´t work in the full LQG framework but in modell with reduced simmetry. They get the Scharschild solution (a solution for vacuum Einstei equations, statif and with radial symmetry) and write the hamiltonial constraint equation fo it. They treat the radious as a discrete time coordinate. That results in a diference equation that can be solved and they show that they can evolve the solution for negative values of the radious, so they, seemengly, advoid the central singularity. Before reading the paper of Astekharet all I tried to figure how they could have procceded. To begin with the information paradox problem is related to matter in the vecinity of the horizont. So they would need to introduce in the description mttr in some wayor another. The original work of Hawkings that raised the whole problem used a fiexed Schwarschild background and an scalar field propagating in it. By the properties of quantization of fields in a courved backgrounds it was known that a vacuum state contianing no particles for aan observer is transformed in a state containg particles by a bogoliougov transformation for another non inertial observer. Playing with that, and with the conformal diagrams of black holes, Hawking derived that black holes actually emit radiation, in thermal quilibrium. That raises the problem that the black hole aan evaaporate because that procces. But the b-h was formed by matter in a pure state, and the thermally described matter is in a mixed stte, so the evolution would be not unitary (that is a bried description of the problem we are trating, for the ssafe of someone wouldn´t know it). Canonical LQG, the one in which Astekhar usually works, normally trates pure gravity, althought it can describe non fermionic matter also. Knowing that I thoguht that they would use some variant of the singularity removal approach including a klein-gordon field. Well, I was too naive.

They work in something called "Callen-Giddings-Harvey-Strominger (CGHS) black holes". I had not previous knowledge of that model, but the names behind it sound me "stringy", in particular Strominger is mainly an string theorist. Well, I was not wrong this time. The paper makes begins with a brief description of the hawking problems, some previous aproachs to the solution (açone by Hawking himself aaproach based in the maldacnena AdS/CFT corresondence) and just afther that talks about some workd in the early ninties triying to solve it in a toy two dimensional modell, the CGHS.

Just before writing the actual equation of the model the aouthor make a very courious advise:

"Although our
considerations are motivated by loop quantum gravity, in
this Letter we will use the more familiar Fock quantization
since the main argument is rather general"

So they say that we are not going to see a formalism related to LQG, alathought LQG is behing the scene. Well, that means taht we must belief in LQG, but we are not ging to see it. Ok, lets belive, at least for a while. Let´s see (part of)the lagrangian describing the model:

$S(g, \Phi, f) =1/G\int d^2 V e^{-2\Phi}(R + 4g^{ab}$...

Now it is when one can beguin to be really surprised. We have that Phi is said to be a dilaton. But a dilaton is a field related to string theory. All of the strings theories have a dilaton. So we are in a modell inspired by string theory (an aspect that it is not mentionesd anywhere in the paper). R is the curvature and f is an scalar field. Well, ok, no problem, someone would expect their appearence.

Afther that they introduce the equatios associated to the lagrangian and begins the task of finding solutons resembling a black hole suited for their purposes. First they affront the classical case. They do it in a perturbative, recoursive, way. That is, they choose a candidate metric, calculate the stress tensor for the fields and reintroduce it in the Einstein equation. By dong that they find that the metric an develope a singularity that they can identify as a proper black hole.

Afther that they consider a quantized version, they add hats xD). Not, serously, they use a fock space tratement (in teh spirit of the Wald aproach to quantization in courved backgrounds, but this time quantizing the metric also). They afrront the uestion of quantization (solving the conmutator eqations to say that) by a bootstrapping procedure, a recoursive way similar to the classical one. They do the suual stuff of identifiying the average values with the classical solutions,but they face a problem when the metric becomes singular, and they cann not continuate the bootstraping. Afther that they use another procedure, a mean field approach MFA. They argue that the e relevant part to solve the information paradox depends on the behaviour of the MFA in the near future ifinity and with some 3 extra sumptions ( they explain that two of them aare commonly accepted and that the other is very natural) they can calculate the S-Matrix and se that it is unitary.

Well, the detaills of how valids are the asumptions (2-D space time, MFA, asymtotic regions, etc) is something that, fourtunately, I don´t need to judge. The key point that I want to raise is that what we see in the paper is very, very, far from any formalisms related to LQG. So to claim that this can be seem as a trioumph of LQG, if they don´t bring a future a paper (or smewhat point me that I am missing something important) where they addapt the calculations to something more LQG like, is to somewhat distort the truth. Or, at least, a too propagandistic deformation of facts ;-).

P.S. Seriously, I would like to leave this tasks to the famous physicists bloggers. For example, I am still wating Sean Carrol to post about the 't Hoof paper I writted about two posts above.

## Tuesday, May 13, 2008

### The trouble with LQG

A través del excelente blog física en la ciencia ficción llegué a otro blog, la bella teoria. Supongo, no lo sé con certea, que eso de "la bella teoria" debería ir por la teoria de cuerdas, que en su momento alguna gente consideró matemáticamente bella y elegante. En todo caso en el blog ví que se trataban con cierta frecuencia temas sobre teoria de cuerdas (aunque no exclusivamente). También observé que algunas de esas entradas contenían algunos errores elementles. De hecho le señalé al autor uno en una de las entradas, a lo que me respondio dándome las gracias por el aviso. Posteriormente he visto algún que otro error, pero no me he molestado en írselos indicando uno a uno. Yo no me considero una autoridad en teoria de cuerdas ¿alguien aparte de Lubos Motl se atreve a asumir ese rol xD? y soy consiciente de que es probable que en este blog haya errores en algún que otro aspecto. Además el tono de ese blog es claramente divulgativo, para un público general y las entradas estan escritas de manera amena y cuidda (con fotos y cosas así). Dado que no hay en español muchos blogs (si es que hay algún otro) que cumpla similar función (yo, por ejemplo, oriento este blog a gente de un nivel de conocimientos bastante mas elevado) no tiene sentido ir señalando fallitos, que es algo que supongo podriá resultar irritante.

El caso es que el autor ahora ha ido a leer el polémico libro de Lee Smollin "the trouble with physics". Tras hacer una presentación del mismo en una entrada publico una segunda titulada Gravedad cuántica, continuando la revolución de Einstein Cómo quiera que es muy poco probable que la gente que lea su blog (y aparentemetne el mismo autor) este al tanto de los detalles de las "string wars" en los blogs Motl ,Distler, String Coffe y not even wrong, y de las críticas que los teoricos de cuerdas han elaborado sobre el estado actual de la LQG dejé una respuesta, que no ha recibido ulterior réplica dónde resumía algunas de ellas. Por su interés para los lectores de este blog dejo aquí esa respuesta:

"Salvador, lo que estas comentando es lo que se conoce cómo LQG. loop quantum gravity, o gravedad cuántica de lazos en español.

Dado que en tu anterior post hablabas del libro "the trouble with physics" asummo que has sacado de allí la información sobre este post. Aprovecho pués para comentarte ambos temas en esta respuesta.

Llevo siguiendo la LQG, a nivel técnico, no de divulgacion, desde el 2003, cuando se hizo populara en una hábil maniobra publicitria de algunos de su máximos represetantes. Fué un producto muy bien vendido, sobre todo mediante un paper de review de Robert Thieman. Te presentan la idea general, explicando cómo el programa de cuantización canónica lleva a que el operador área esta cuantizado. Luego hay argumentaciones de que, mediante algo conocido cómo "double special relativity", que extiende el formalismo de la relatividad especial al caso de una longitud privilegiada (la de Planck) se llega a que la velocidad de la luz en el vacio depende de la frecuencia, para luego señalar que los satelites GLAST podrian llegar a comprobar eso en un lapso de tiempo breve (su lanzamiento estaba previsto para el 2005).

De ahí pasan a argumentar que deducen la entropia de un agujero negro, reproduciendo la fórmula conocida mediante la aproximación semiclásica de que depende de una potencia del área. No entran en los detalles, que dependen de una construccion de relatividad general poco conocida llamada "isolated horizonts" (y también de el concepto relacionado de "dynamic horizonts". Cómo casi nadie esta familiarizado con eso conceptos consiguen que en una primera lectura uno no caiga en que en el cálculo se esta asumiendo, como punto de partida, que la entroia depende del área. Además la teoria de la LQG tiene un parámetro libre, concido cómo parámetro de inmirizzi. Ajustando ese valor se consigue que la fórmula obtenida se ajuste. Total, que el único punto medianamente serio del cálculo es que el área esta cuantizada qu ehay que dividir el área total entre la unidad mínima de área para encontrar el número posible de microestados.

Luego pasan a cosmologia. Anuncian qu econsiguen demostrar el "rebote", es decir que no puede haber un big crunch pués usando factores cuánticos se comprueba que el colapso se detiene en un radio mínimo y luego hay un nuevo big bang. Claaro, eso no se hace dentro de la teoria completa sino restringiéndose a "minisuperespacios", es decir, subgrupos de las métricas posibles. Vale, se admite que es una aproximación, el problema es que no dan ningún criterio que permita hacerse una idea de cuan buena o mala es esa aproximación.

Por cierto, la LQG canónica, basada directamente en la gravedad clásica de Einstein y luego cuantizada de un modo raro, presuntamente no perturbativo, es estática. Para tener un espaciotiempo dinámico tiene que irse a un formalismo lagrangiano (las spinfoams). El caso es que las spin-foams no parten de la relatividad de Einstein sino de unas teorias clásicas que con ciertas restricciones, son más o menos clasicamente equivalentes a la relatividad de Einstein. Además hay varias de ess teorias clásicas y cada una lleva a una teoría cuántica esencialmente diferente.

Y bueo esas dos son las líneas fundamentales, pero aparte hay otras que asumen de partida que el espaciotiempo es discreto, lo cuál es mucho suponer. Y no quda muy claro la relación entre ellas.

Mas importante es que esas teorias cuánticas deberían tener un limite clásico que reprodujera la gravedad clásica de Einstein. Pué sbien, no han conseguido eso de una manera medianamente clara (lo mejor que tiene es obtener, mediante una serie de aproximaciones de dudosa justificación) reproducir, en el marco de algunas spinfoams algo que debería ser similar al potencial de Newton.

Vamos, que no digo yo que la LQG sea un sinsentido, pero desde luego tiene muchos problemas bastante serios.

Respecto a su mejor punto, la posibilidad de confirmación de la dispersión de la luz en el vacio, al final los satélites GLAST han tardado muchísimo en lanzarse, (deben estar a punto de salir ahí fuera por estas fechas). Un experimento en tierra hace unos meses dió un resultado preliminar en esa línea, pero no es conluyente (y hay ciertas teorias de cuerdas, bastante raras todo sea dicho, las cuerdas de liouville, que predicen algo similaar. De hehco fué un fisico de cuerdas quien hizo populñaar ese experimento).

Y, por si no esta claro, la teoria de cuerdas también es una gravedad cuántica, así que no se puede presentar esto cómo si fuera le más de lo más en gravedad cuántica y dejar de lado la teoría de cuerdas.

En fín, no me parece mal que comentes sobre la LQG, pero cuidado, que no es oro todo lo que reluce con es gente."

Bien, tras esa entrada Salva ha publicado otra . Más allá de los agujeros negros. No he leido "the trouble with hysics, sólo los comentarios, unos favorables (en la comunidad LQG) otros adversos, así que no tengo la certeza, pero si la firme sospecha, de que los datos de esa entrada salen de ese libro. En cualquier caso lo que si conozco es el tema específico tratado (agujeros negros en LQG) pués lei en su momento unos cuantos artículos originales sobre el particular. La entrada en sí no contiene ningún error, pero creo que es interesante que complete la respuesta que reproduje antes para explicar aalgo respecto a la singularidad de los agujeros negros en LQG.


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How to do integrals involving Gauss function and cosine function?

How can I calculate the integral: $$\int_{-\infty}^{+\infty}e^{-\frac{(x-a)^2}{0.01}}\cos(bx)dx$$ I can not find in the references. Excuse my bad English.

-
You could expand the cosine in a Taylor series around $x=a$... –  Fabian Oct 19 '12 at 16:21

$$\cos(bx)=\frac12\left(\mathrm e^{\mathrm ibx}+\mathrm e^{-\mathrm ibx}\right)\;,$$
A reference is Rudin, Principles of mathematical analysis, Example 9.43. Here the integral $$\int_{-\infty}^{\infty}e^{-x^2}\cos(xt)\,dx$$ is calculated using the theory of ordinary differential equations. The integral is $$\sqrt{\pi}\exp\left(-\frac{t^2}{4} \right).$$ (Hint) In your integral after introducing new variable you should calculate $$\int_{-\infty}^{\infty}e^{-z^2}\cos(cz)\,dz$$ and $$\int_{-\infty}^{\infty}e^{-z^2}\sin(cz)\,dz.$$

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# Sum of (i^2)/(4^i) where i is from 0 to infinity.

1. Jun 21, 2004

### Johnny Leong

Sum of (i^2)/(4^i) where i is from 0 to infinity.

2. Jun 21, 2004

### NateTG

You mean $$\sum_{i=1}^\infty \frac{i^2}{4^i}$$? (I don't like $$\frac{0^2}{4^0}$$ - it's not necessarily clear what it is.)

Well, for $$i \geq 4$$ ,
$$2^i \geq i^2$$,
so
$$\frac{i^2}{4^i} < \frac{2^i}{4^i} = \frac{1}{2^i}$$
so
$$\sum_{i=1}^\infty \frac{i^2}{4^i} < \frac{1}{4}+\frac{4}{16}+\frac{9}{64} + \sum_{i=4}^{\infty} \frac{1}{2^i}$$

Perhaps you can find some better bounds?

3. Jun 21, 2004

### Muzza

I'm sorry, but what's unclear about 0^2/4^0?

4. Jun 21, 2004

### Gokul43201

Staff Emeritus
$$\sum_{i=1}^\infty \frac{i^2}{4^i} > \frac{1}{4}+\frac{4}{16}+\frac{9}{64} + \sum_{i=4}^{\infty} \frac{1}{4^i}$$

That gives a range of 0.7 to 0.9. Wonder if it isn't just 3/4 or 4/5 ?

5. Jun 21, 2004

Mathematica's saying 20/27 (~.740741). But, like most things Mathematica, I have no idea how it produced that number.

6. Jun 21, 2004

### Johnny Leong

Actually, this question could not give an accurate answer. The answer should just be an approximation, right? Because the terms in the summation are not having some sequence properties.

7. Jun 22, 2004

### Johnny Leong

I think Gokul43201's answer would not be a good one because after I have read a book, I found that when we do some approximation, the approximation's value had better dominate the value of the original question.

8. Jun 22, 2004

### HallsofIvy

Staff Emeritus
What in the world does that mean? What do you mean by "dominate the value of the original question? An approximation is an approximation. There exist good approximations and bad approximations. The ideal is to get as close as possible to the true value for the work done.

9. Jun 22, 2004

### arildno

$$F(x)=\sum_{i=0}^{\infty}i^{2}x^{i}$$
$$\sum_{i=0}^{\infty}i^{2}x^{i}=\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}$$
$$\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}=\frac{d}{dx}\sum_{i=0}^{\infty}ix^{i+1}-\sum_{i=0}^{\infty}ix^{i}$$
$$\sum_{i=0}^{\infty}ix^{i}=x\frac{d}{dx}\sum_{i=0}^{\infty}x^{i}$$
Hence, we have:
$$F(x)=\frac{d}{dx}(x^{2}\frac{d}{dx}\frac{1}{1-x})-x\frac{d}{dx}\frac{1}{1-x}$$
or:
$$F(x)=\frac{x^{2}+x}{(1-x)^{3}}$$
The sum of the original series is found by evaluating $$F(\frac{1}{4})$$

Last edited: Jun 22, 2004
10. Jun 22, 2004

### Gokul43201

Staff Emeritus
Which is 20/27, as Mathematica magically guessed.

Nice, arildno.

11. Jun 22, 2004

A very clever little trick, indeed.

12. Jun 23, 2004

### Johnny Leong

I mean the approximation should be an upper bound to the accurate answer to the original question.

13. Jun 23, 2004

### Johnny Leong

What are you doing, arildno? You are professional but I do not understand. Why you do like this?

14. Jun 23, 2004

### arildno

OK, I'll break up this in tiny pieces; then pin-point what you don't understand.

1. Changing perspective from number to function:
We start out with the series:
$$S=\sum_{i=0}^{\infty}i^{2}(\frac{1}{4})^{i}$$
Our aim is to find the number S!
However, this is difficult to do as it stands; what I want to do, is to change the problem slightly, so that:
a) I gain access to powerful solving techniques in the new problem to be solved
(which is not accessible in the original problem)
b) I can easily find the answer to the original problem once I have found the answer to the new problem

That is why I change perspectives to try to simplify the expression for the following function:
$$F(x)=\sum_{i=0}^{\infty}i^{2}x^{i}$$

a): Since I now have a power series function , I can use the extraordinarily powerful teqnique of differentiation to help me solve the new problem (simplifying the expression for F(x)!)
b) If I can do this, then I can find the answer to my original problem simply by
computing $$F(\frac{1}{4})$$

I'll let you ponder on this for a while, if you have some specific questions, pleasy notify..


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# blog migration to another site

I grew frustrated with data entry on this site, and switched to a github site that allows me to use a native editor to write and update entries.  The new site contains the items posted here, and a few more, and it’s where I’ll be blogging from now on.

The new site is at http://dankelley.github.io/blog.

# one-dimensional optimization

Abstract. The method of one-dimensional optimization is explained in the context of capillary-gravity waves.

1. Introduction.

R provides functions for both one-dimensional and multi-dimensional optimization. The second topic is much more complicated than the former (see e.g. Nocedal 1999) and will be left for another day.

A convenient function for 1D optimization is optimize(), also known as optimise(). Its first argument is a function whose minimum (or maximum) is sought, and the second is a two-element vector giving the range of values of the independent variable to be searched. (See ?optimize for more.)

2. Application.

As an example, consider the phase speed of deep gravity-capillary waves, which is given by $\omega/k$ where $\omega$ is the frequency and $k$ is the wavenumber, and the two are bound together with the dispersion relationship $\omega^2=gk+\sigma k^3/\rho$, where $g$ is the acceleration due to gravity, $\sigma$ is the surface tension parameter (0.074 N/m for an air-water interface) and $\rho$ is the water density (1000 kg/m^3 for fresh water). This yields wave speed given by the following R function.

phaseSpeed <- function(k) {
g <- 9.8
sigma <- 0.074  # water-air
rho <- 1000  # fresh water
omega2 <- g * k + sigma * k^3/rho
sqrt(omega2)/k
}

Readers with a background in the topic of waves may know that there is a minimum phase speed at wavelengths $\lambda$ of about 0.02m, or a wavenumber $k=2\pi/\lambda$ of roughly 300. It always makes sense to plot a function to be optimized, if only to check that it has been coded correctly, so that's the next step. We’ll use a range of half an order of magnitude (factor of 3) smaller and larger $k$.

k <- seq(100, 1000, length.out = 100)
par(mar = c(3, 3, 1, 1), mgp = c(2, 0.7, 0))
plot(k, phaseSpeed(k), type = "l",
xlab = "Wavenumber", ylab = "Phase speed")

Wave speed for capillary-gravity waves in infinitely deep water

The results suggest that the range of $k$ illustrated in the diagram contains the minimum, so we provide that to optimize().

o <- optimize(phaseSpeed, range(k))
phaseSpeed(o$minimum)  ## [1] 0.2321  This speed is not especially fast; it would take about a heartbeat to move past your hand. (Or, about as fast as the waves that pulse your heart … presumably with different physics, though!) 3. Exercises 1. Use str(o) to learn about the contents of the optimized solution. 2. Use abline() to indicate the wavenumber at the speed minimum. 3. Try other functions that are of interest to you. 4. Use the multi-dimensional optimizer named optim() on this problem. References Jorge Nocedal and Stephen J. Wright, 1999. Numerical optimization. Springer series in operations research. New York, NY, USA. # using the plyr package in R Abstract. The merits of the plyr package are illustrated. 1. Introduction. The base R system provides lapply() and related functions, and the package plyr provides alternatives that are worth considering. It will be assumed that readers are familiar with lapply() and are willing to spend a few moments reading the plyr documentation, to see why the illustration here will use the ldply() function. The test task will be extraction of latitude (and then both latitude and longitude) from the section dataset in the oce package. (Users of that package may be aware that there is a built-in accessor for doing this, so results can easily be checked.) 2. Methods First, load the data library(oce) data(section)  Next, find latitudes using lapply lat <- unlist(lapply(section[["station"]], function(x) x[["latitude"]]))  Next, find latitudes with ldply library(plyr) lat <- ldply(section[["station"]], function(x) x[["latitude"]])  3. Results The reader can check that the results match, although ldply() returns a data frame, not a vector as in the first method. Tests of speed indicate such a difference too small to be of much interest, at least in this case library(microbenchmark) microbenchmark(ldply(section[["station"]], function(x) x[["latitude"]])$V1,
unlist(lapply(section[["station"]], function(x) x[["latitude"]])))

## Unit: milliseconds
##                                                               expr   min
##        ldply(section[["station"]], function(x) x[["latitude"]])$V1 18.99 ## unlist(lapply(section[["station"]], function(x) x[["latitude"]])) 18.36 ## lq median uq max neval ## 20.26 20.56 21.02 36.05 100 ## 19.71 19.93 20.64 63.18 100  . 4. Discussion Since ldply() returns a data frame, it is more flexible than unlist(), which returns a vector. For example, the following creates a data frame with columns for lat and lon: latlon <- ldply(section[["station"]], function(x) c(x[["latitude"]], x[["longitude"]]))  A station plot is produced as follows. if (!interactive()) png("plyr.png", unit = "in", res = 150, width = 7, height = 7, pointsize = 10) mapPlot(coastlineWorld, projection = "orthographic", orientation = c(20, -40, 0)) mapPoints(latlon$V2, latlon$V1, pch = "+", cex = 1/2, col = "red") if (!interactive()) dev.off()  Map of station locations, produced by extracting locations using a function from the plyr package 5. Conclusions The effort of learning how to use the plyr package is likely to pay off in more flexible code, particularly because of the use of data frames in that package. On this theme, note that the author of plyr is developing a similar package called dplry, which centres more closely on data frames and offers many new features; see http://blog.rstudio.org/2014/01/17/introducing-dplyr/ for a blog item introducing dplyr. # using R markdown on wordpress Abstract. An explanation is given for using the vim editor to compose wordpress blog items in normal-markdown or R-markdown format. 1. The gist of R markdown format. R markdown is an extension of normal-markdown that permits the embedding of R code and output (including figures). WordPress seems not to handle the figure-inclusion mechanism, but there is a workaround: create the figures locally then use the ‘insert media’ button on wordpress to upload and include them in the text. For more details of R-markdown format, see e.g. the rstudio documentation. 2. Setting up wordpress to accept markdown format. As noted in a comment on this posting, go to the “Dashboard” and then select “Settings”, then select “Writing” and then click the box next to “Use markdown for posts and pages”. 3. Guidelines for vim editor. WordPress will insert line breaks whenever they are found in the text, so it is important not to use auto-fill in Vim. This can be set up by inserting the line autocmd! BufNewFile,BufReadPre,FileReadPre *.Rmd so ~/.vim/rmd.vim  in the ~/.vimrc file, and then creating a new file in ~/.vim named rmd.vim that contains " wrap long lines in the editor window ... set wrap " ... and prevent automatic end-of-line insertion set textwidth=0 wrapmargin=0  4. Guidelines for emacs editor. (Something may be added here if someone tells me what to add…) 5. Converting R markdown to normal markdown R-markdown may contain R code chunks (with the token {r} after the triple-backtick). When these are run through the knit() function provided in the knitr package, the braces are removed, which is the standard for syntax-enabled markdown as used on wordpress. Conversion from R markdown to normal markdown can be done easily inside R, with the following (in this case for the file used for the present posting). library(knitr) knit("editing_vim.Rmd")  # colourizing along a trajectory Abstract. Demonstrate how to colourize an (x,y) curve according to z. 1. Introduction. In Oceanography it can be useful to use colour to display z values along an (x,y) trajectory. For example, CTD data might be displayed in this way, with x being distance along track, y being depth, and z being temperature. This post shows how one might do this generally with fake data. 2. Methods. The R code given below demonstrates this with fake data. The core idea is to use segments(), here with head() and tail() to chop up the trajectory. library(oce) x <- 1:50 y <- x * 2/10 + x^2/10 z <- seq(0, 5, length.out = length(x)) zlim <- range(z) npalette <- 10 palette <- oceColorsJet(npalette) drawPalette(zlim = zlim, col = palette) plot(x, y, type = "l") segments(head(x, -1), head(y, -1), tail(x, -1), tail(y, -1), col = palette[findInterval(z, seq(zlim[1], zlim[2], length.out = npalette))], lwd = 10) points(x, y, pch = 20)  3. Results. The graph shows that this works reasonably well. The dots will probably not be useful in an actual application, and are used here just to indicate colourization in groups. Demonstration of colourizing an (x,y) trajectory according to the value of a third variable, z. 4. Conclusions. The method works well, and is flexible in terms of colour schemes. Appendix 1: exercises find a way to blend colour between the points, perhaps by defining a euclidian distance in (x,y) space (which will of course require scaling if x and y have different units or ranges) and then using approx(). # overshoot in butterworth low-pass filters Abstract. Demonstrate overshoot for butterworth filters in R. 1. Introduction. Butterworth filters with order other than 1 have an overshoot phenomenon that can be problematic in some cases. For example, if smoothing is used on an estimate of kinetic energy, overshoots might yield negative values that are nonphysical. This post simply illustrates this with made-up data that the reader can experiment with. 2. Methods. The code given below creates the graph shown in the next section library(signal) n <- 100 x <- 1:n y <- ifelse(0.3*n < x & x < 0.7*n, 1, 0) if (!interactive()) png("butter.png", width=7, height=5, unit="in", res=150, pointsize=9) plot(x, y, type='o', pch=20, ylim=c(-0.1, 1.1)) b1 <- butter(1, 0.1) y1 <- filtfilt(b1, y) points(x, y1, type='o', pch=20, col='red') b2 <- butter(2, 0.1) y2 <- filtfilt(b2, y) points(x, y2, type='o', pch=20, col='blue') b3 <- butter(3, 0.1) y3 <- filtfilt(b3, y) points(x, y3, type='o', pch=20, col='forestgreen') if (!interactive()) dev.off()  3. Results The graph illustrates; for the colour coding, see the code above. 4. Conclusions Caution is recommended in using butterworth filters of order 2 and higher. The advantage of the rapid cutoff in frequency space may be outweighed by the disadvantage of overshooting in time space. # cabelling Abstract R code is provided in aide of laboratory demonstration of cabelling. 1. Introduction Setting up a cabelling experiment requires creating two watermasses of equal density, and if only S and T can be measured, that means calculating densities. Using a TS diagram and graphical interpolation is one approach to that task, but another is to use R to do the calculation. 2. Methods The code given below will do the calculation for specified Sa, Ta and Sb, where the second letter indicates the watermass. The code uses uniroot() to find the temperature Tb that yields equal densities for watermasses “a” and “b”. # Alter next three lines as desired; a and b are watermasses. Sa <- 30 Ta <- 10 Sb <- 40 library(oce) # Should not need to edit below this line rho0 <- swRho(Sa, Ta, 0) Tb <- uniroot(function(T) rho0-swRho(Sb,T,0), lower=0, upper=100)$root
Sc <- (Sa + Sb) /2
Tc <- (Ta + Tb) /2
## density change, and equiv temp change
drho <- swRho(Sc, Tc, 0) - rho0
dT <- drho / rho0 / swAlpha(Sc, Tc, 0)
if (!interactive()) png("cabelling.png", width=7, height=7,
unit="in", res=200, pointsize=12)
plotTS(as.ctd(c(Sa, Sb, Sc), c(Ta, Tb, Tc), 0), pch=20, cex=2)
drawIsopycnals(levels=rho0, col="red", cex=0)
segments(Sa, Ta, Sb, Tb, col="blue")
text(Sb, Tb, "b", pos=4)
text(Sa, Ta, "a", pos=4)
text(Sc, Tc, "c", pos=4)
legend("topleft",
legend=sprintf("Sa=%.1f, Ta=%.1f, Sb=%.1f  ->  Tb=%.1f, drho=%.2f, dT=%.2f",
Sa, Ta, Sb, Tb, drho, dT),
bg="white")
if (!interactive()) dev.off()

If run non-interactively, the code will produce a PNG file like that given below.

3. Results

The legend summarizes the results, indicating also the density change and the temperature change that would be equivalent to that density change (at the midpoint, “c”).

TS diagram for the setup of a cabelling experiment.

4. Conclusions

If the design goal is that the density mismatch between watermasses “a” and “b” should be, say, 10 percent of the density difference of the mixture watermass (“c”), then in the illustrated case the temperature would have to be controlled to within a quarter of a degree Celcius — a task that is challenging enough to argue against this as an informal classroom demonstration.

Exercises.

1. Alter the R code to calculate Sb in terms of Tb.
2. Consider (calculate or measure) the convection associated with sidewall heat conduction.


text:
# Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

```@article{Ruiz2020BesovCV,
title={Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities},
author={Patricia Alonso Ruiz and Fabrice Baudoin and Li Chen and Luke G. Rogers and Nageswari Shanmugalingam and Alexander Teplyaev},
journal={Journal of Functional Analysis},
year={2020}
}```
• Published 10 November 2018
• Mathematics
• Journal of Functional Analysis
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2020
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery
Besov spaces associated with non-negative operators on Banach spaces
• Mathematics
• 2020
Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces \$B^{ s, A }_{ q, X }\$ for non-negative operators \$A\$ on Banach spaces \$X\$
BV functions and fractional Laplacians on Dirichlet spaces
• Mathematics
• 2019
We study \$L^p\$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based
Heat Kernels and Besov Spaces on Metric Measure Spaces
• Mathematics
• 2019
Let (M, ρ, μ) be a metric measure space satisfying the volume doubling condition. Assume also that ( M, ρ, μ) supports a heat kernel satisfying the upper and lower Gaussian bounds. We study the
Sobolev spaces and Poincar\'e inequalities on the Vicsek fractal
• Mathematics
• 2022
In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of
Equivalence of Besov spaces on p.c.f. self-similar sets
• Mathematics
• 2020
On p.c.f. self-similar sets, of which the walk dimensions of heat kernels are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces
Yet another heat semigroup characterization of BV functions on Riemannian manifolds
• Mathematics
• 2020
This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main
\$L^p\$-Poincar\'e inequalities on nested fractals.
• Mathematics
• 2020
We prove on some nested fractals scale invariant \$L^p\$-Poincar\'e inequalities on metric balls in the range \$1 \le p \le 2\$. Our proof is based on the development of the local \$L^p\$-theory of

## References

SHOWING 1-10 OF 81 REFERENCES
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2020
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery
BV functions and fractional Laplacians on Dirichlet spaces
• Mathematics
• 2019
We study \$L^p\$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based
Heat kernel and Lipschitz-Besov spaces
• Mathematics
• 2015
On a metric measure space (M, ρ, μ) we consider a family of the Lipschitz-Besov spaces Λ s,q that is defined only using the metric ρ and measure μ, and a family of Besov spaces B s p,q that is
Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces
• Mathematics
• 2012
Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant
Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces
We give a heat-kernel characterisation of the Besov-Lipschitz spaces Lip (α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of
Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients
• Mathematics
• 2015
Preface 1. Introduction 2. Review of basic functional analysis 3. Lebesgue theory of Banach space-valued functions 4. Lipschitz functions and embeddings 5. Path integrals and modulus 6. Upper
Sobolev Spaces and Calculus of Variations on Fractals
• Mathematics
• 2018
The present note contains a review of \$p\$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize
Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces
• Mathematics
• 2016
The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N1,1-spaces) and the theory of heat semigroups (a concept related to N1,2-spaces) in


text:
# What is int(lnx^9)/x^4 dx?

Nov 8, 2015

It is $9 \left[- \frac{1}{3} {x}^{-} 3 \ln x - \frac{1}{9} {x}^{-} 3 + C\right]$ Which we may prefer to write:
$- \frac{3 \ln x + 1}{x} ^ 3 + C$

#### Explanation:

$\ln {x}^{9} = 9 \ln x$, so

$\int \ln {x}^{9} / {x}^{2} \mathrm{dx} = 9 \int {x}^{- 4} \ln x \mathrm{dx}$ Now use integration by parts.

Let $u = \ln x$ and $\mathrm{dv} = {x}^{-} 4 \mathrm{dx}$,

so $\mathrm{du} = {x}^{- 1} \mathrm{dx}$ and $v = - \frac{1}{3} {x}^{-} 3$

And the integral becomes:

$9 \left[- \frac{1}{3} {x}^{-} 3 \ln x + \frac{1}{3} \int {x}^{-} 3 {x}^{-} 1 \mathrm{dx}\right]$

$= 9 \left[- \frac{1}{3} {x}^{-} 3 \ln x + \frac{1}{3} \int {x}^{-} 4 \mathrm{dx}\right]$

$= 9 \left[- \frac{1}{3} {x}^{-} 3 \ln x - \frac{1}{9} {x}^{-} 3 + C\right]$

Which we may prefer to write:

$= - \frac{3 \ln x + 1}{x} ^ 3 + C$


text:
# Metric Spaces: Contraction Mappings

Definition Let $$A$$ be a mapping of a metric space $$R$$ onto itself. Then $$x$$ is a fixed point of $$A$$ if $$Ax = x$$. Suppose there exists $$\alpha < 1$$ such that $$\rho(Ax, Ay)$$ for all $$x, y \in R$$, then $$A$$ is a contraction mapping. Every contraction mapping is automatically continuous, since $$Ax_n \rightarrow AX$$ whenever $$x_n \rightarrow x$$.

Theorem 1 (Fixed point theorem)

Every contraction mapping $$A$$ defined on a complete metric space $$R$$ has a unique fixed point.

Proof

Keep contracting, i.e. applying $$A$$.

Theorem 1’

Given a continuous mapping of a complete metric space $$R$$ onto itself, suppose $$A^n$$ is a contraction mapping ($$n$$ an integer $$>1$$). Then $$A$$ has a unique fixed point.

Let $$x = \lim_{x \rightarrow \infty} A^{kn} x_0$$ and $$Ax = \lim_{k \rightarrow \infty} A A^{kn} x_0$$.

Theorem 2 (Picard)

Given $$n$$ functions $$f_i (x, y_1, \ldots, y_n)$$ defined and continuous on an $$(n+1)$$-dimensional domain $$G$$ containing the point $(x_n, y_{01}, \ldots, y_{0n})$ suppose each $$f_i$$ satisfies a Lipschitz condition of the form $|f_i(x, y_1, \ldots, y_n) - f_i(x, \tilde{y}_1, \ldots, \tilde{y}_n)| \leq M \max_{1 \leq i \leq n} |y_i - \tilde{y}_i|$ in the variables $$y_1, \ldots, y_n$$. Then there is an interval $$|x - x_i| \leq \delta$$ in which the system of differential equations $\frac{dy_i}{dx} = f_i (x, y_1, \ldots, y_n) \quad (i = 1, \ldots, n)$ has a unique solution $y_1 = \phi_1(x), \ldots, y_n = \phi_n(x)$ satisfying the initial conditions $\phi_1(x_0) = y_{01}, \ldots, \phi_n(x_0) = y_{0n}$

The Fredholm equation (of the second kind) and Volterra equation are defined in the book.

## Problems

1. Let $$A$$ be a mapping of a metric space $$R$$ into itself. Prove the condition $\rho(Ax, Ay) < \rho(x, y) \quad (x \neq y)$ is insufficient for the existence of a fixed point of f$$A$$.

Let $$R$$ be $$(0, 1)$$ and $$A = \frac{1}{2}$$, then fixed point would be $$0$$, but $$0 \notin R$$. Need $$R$$ to be complete.

2. Let $$F(x)$$ be a continuously differentiable function on the interval $$[a, b]$$ such that $$F(a) < 0$$, $$F(b)> 0$$ and $0 < K_1 \leq F^\prime(x) \leq K_2 (a \leq x \leq b)$ Use Theorem 1 to find the unique root of the equation $$F(x) = 0$$

Hint. Introduce the auxiliary function $$f(x) = x - \lambda F(x)$$, and choose $$\lambda$$ such that the theorem works for the equivalent equation $$f(x) = x$$.

$f(x) = x - \lambda F(x) \quad f^\prime(x) = 1 - \lambda F^\prime(x)$ Select λ so that $$|f^\prime(x)| = |1 - \lambda F^\prime(x)| < 1$$, whenever $$\lambda F^\prime(x) < 1$$ since $$F^\prime(x)$$ is strictly positive. Can do this by setting \$λ = \frac{1}{K_1 + K_2} since $\lambda F^\prime(x) \leq K_2 = \frac{K_2}{K_1 + K_2} < 1$ By Theorem 1, $$f$$ has a fixed point $$x_0 \in [a, b]$$ such that $$f(x_0) = x_0$$, which means $$\lambda F(x_0) = 0$$, and $$x_0$$ is the unique root of $$F(x) = 0$$.

3. Devise a proof of the implicit function theorem based on the use of the fixed point theorem.

Don’t know what the implicit function theorem is. TODO.

4. Prove that the method of successive approximations can be used to solve the system $x_i = \sum_{j=1}^n a_{ij}x_j + b_i$ if $$|a_{ij}| < \frac{1}{n}$$ (for all $$i$$ and $$j$$), but not if $$|a_{ij}| = \frac{1}{n}$$.

Easy.

5. Prove that the condition $$\sum_{j=1}^n|a_{ij}| \leq \alpha < 1$$ is sufficient for the mapping $$y_i = \sum_{j=1}^n a_{ij}x_j + b_i$$ to be a contraction mapping in the space $$R_0^n$$.

Easy.

6. Prove that any of the $$n$$-dimensional contraction mapping conditions imply $\begin{vmatrix} a_{11}-1 & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22}-1 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}-1 \\ \end{vmatrix} \not= 0$

Easy.

7. Consider the nonlinear integral equation $f(x) = \lambda\int_a^b K(x, y; f(y)) dy + \psi(x)$ with continuous $$K$$ and $$\psi$$, where $$K$$ satisfies a Lipschitz condition of the form $|K(x, y; z_1) - K(x, y; z_2)| \leq M|z_1 - z_2|$ in its ‘functional’ argument. Prove that the equation has a unique solution for all $|\lambda| < \dfrac{1}{M(b-a)}$ Write the successive approximations to this solution.

Beats me. TODO.


text:
# Clarifications on the definition of product in Category Theory

I am self-studying Hungerford's book Algebra. Here is the definition that I don't understand.

Let $\mathcal{C}$ be a category and $\{A_{i}|i\in I\}$ a family of objects of $\mathcal{C}.$ A product for the family $\{A_{i}|i\in I\}$ is an object $P$ of $\mathcal{C}$ together with a family of morphisms $\{\pi_{i}:P\rightarrow A_{i}|i\in I\}$ such that for any object $B$ and family of morphisms $\{\varphi_{i}:B\rightarrow A_{i}|i\in I\},$ there is a unique morphism $\varphi: B\rightarrow P$ such that $\pi_{i}\circ \varphi =\varphi_{i}$ for all $i\in I.$

Let me try to explain what I don't understand in the definition above. Every time I have to look back to see if the correct is "there is a unique morphism $\varphi: B\rightarrow P$" or "there is a unique morphism $\varphi: P\rightarrow B$".

After reading the comments bellow, I was able to understand the definition.

But I would like to know the motivation for this definition. I was wondering what lead someone to define "product" in a category?

And by using the Arturo's word (I hope with his permission) what is the intuition behind "products" in a category?

Besides, if a category has no product, what one can say about it? May I say that it is not a nice one?

• Think of examples. If you give me a set $Y$ and a bunch of maps $Y \to X_i$, then I naturally get a map $Y \to \prod X_i$. On the other hand, I don't see any way of turning this data into a map $\prod X_i \to Y$. – Dylan Moreland Feb 28 '12 at 0:21
• Dear spohreis: To follow up on Dylan, you can think of the the product of an empty family of sets, versus the disjoint union of such a family. – Pierre-Yves Gaillard Feb 28 '12 at 0:41
• Assuming that the product exists, and is constructed the way we're used to from categories of e.g. groups or topological spaces, the definition roughly reads "A morphism into the product of the family $\{A_{i}|i\in I\}$ is uniquely determined by its restriction to each coordinate." – Arthur Feb 28 '12 at 0:45
• Among all objects that can map onto every one of the objects $A_i$, the product is the one that contains no unnecessary "information" (it is the 'rightmost' object that maps onto all $A_i$, if you imagine your arrows going left to right). "Rightmost" objects (right universal objects) are the 'least general' ones; "leftmost" objects (left universal objects, such as free objects and coproducts) are the "most general" ones. – Arturo Magidin Feb 28 '12 at 5:13
• Don't take as an example a category of modules, because in such categories, finite coproducts and finite products are the same, so things are not very clear. The category of sets makes everything clear I think. – Lierre Feb 28 '12 at 22:36

The categorical definitions of the basic objects (products, coproducts, equalizers, coequalizers, limits, colimits, etc) arise from abstracting a particular situation that shows up in many instances.

The notion of a "product" shows up repeatedly among Sets (cartesian product), Groups (cartesian product again), Rings (cartesian product yet again), Modules (and again), Topological spaces (and again).

Their "object" properties vary from category to category; in the class of abelian groups, the direct product contains the direct sum, which is an important objects in its own right (it's also important in Groups, but less so). Among Unital Rings, however, the direct sum is not an interesting object when it is different from the product (because it is not a unital ring). In topological spaces, there is a "natural" way of defining the topology on a product (the "box" topology), but it turns out to be lacking in some uneasy sense when there are too many factors in the product... and there is an alternative possibility (the "product topology") that seems to be behave better...

The general, guiding principle in Category Theory, though, is that what is important is not what an object "is", but rather what it "does": how it behaves relative to other objects in terms of arrows/morphisms. So we want to abstract the properties that all those instances share in $\mathbf{Sets}$, $\mathbf{Groups}$, $\mathbf{AbGroups}$, $\mathbf{Rings}$, $\mathbf{UnitalRings}$, $\mathbf{Semigroups}$, $\mathbf{TopSpaces}$, etc. (In the latter case, it even suggests which of the two possible topologies "should" be used).

So... what do all these specific instances all have in common? From the cartesian product we can map to each factor: if we view the cartesian product $\times X_i$ as the set of functions $f\colon I\to \cup X_i$ with $f(i)\in X_i$ for each $i\in I$, then we have a natural map from the product to each $X_i$ by "evaluation": $\pi_i\colon f\longmapsto f(i)$. And $f$ is completely determined by these images; so that if you "select" $x_i\in X_i$ for each $i$, then this gives you a unique $\mathbf{x}\in \times X_i$.

Ah, but again: we don't want to think in terms of elements (what the objects "are"), but rather in terms of maps. For $\mathbf{Sets}$, selecting objects is the same as mapping from single element sets... but again we are worrying about the what the objects "are" (single element sets)...

Okay: since we care about maps, note this: not only can we go from $\times X_i$ to each $X_i$. These maps are such that if you have any set $Y$, and any maps $g_i\colon Y\to X_i$, then each $y\in Y$ determines an element of $\times X_i$, namely, $y\colon i\to g_i(y)$. Put another way:

If we let $\pi_i\colon\times X_i\to X_i$ be the "evaluate at $i$" map, then for any set $Y$ and any maps $g_i\colon Y\to X_i$, there is a way to map $g\colon Y \to \times X_i$ in such a way that $\pi_i\circ g(y) = g_i(y)$ for every $i$.

What's more, $g$ is completely forced by this condition. This is a good "categorical" property, because it is given entirely in terms of morphisms.

Does this property "go through" the other instances? The group with the coordinatewise product? The semigroup/ring/module with coordinatewise product? Yes! Good. What about topological spaces? Yes with respect to the product topology... no with respect to the Box topology... so we want to use the "product topology" for the "product".

This definition matches the many instances, and has nice "categorical" properties. It completely characterizes the product up to unique isomorphism. So it seems like a good definition. Turns out to be the right way to generalize the properties of the cartesian product/direct product to other categories.

• Do you have any links to more info about this "objects" vs "morphisms" mindset? I've read many articles and books on category theory (well, rather beginnings of books :Ь), but never seen that regard discussed, and hence didn't get why do we even care about commutative diagrams. – Hi-Angel Dec 25 '16 at 6:19


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# Steve Awodey: A remark on Hofmann-Streicher universes

Tid: To 2022-05-19 kl 10.00 - 12.00

Plats: Kräftriket, House 5, Room 31

Föreläsare: Steve Awodey (Carnegie Mellon University)

### Abstract

We have another look at the construction by Hofmann and Streicher [1] of a semantic universe $$El \rightarrow U$$ for the interpretation of Martin-Löf type theory in a presheaf category $$[C^{op}, Set].$$  It turns out that $$El \rightarrow U$$ can be described as the “categorical nerve” of the classifier $$Set^* \rightarrow Set$$  for discrete fibrations in Cat, where the nerve is the right adjoint in an apparently unnoticed adjunction, $$\int\dashv \text{nerve} : Cat —> [C^{op}, Set]$$, the left adjoint of which is the well-known “Grothendieck construction” taking a presheaf $$P : C^{op} —> Set$$ to its category of elements $$\int_C P$$ . [1] Lifting Grothendieck Universes, Martin Hofmann and Thomas Streicher, 1997 (unpublished).


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# Tag Info

0

Yes, you can use Tensorflow to perform this. Below are few mentioned: A cross-platform Python library for quantum machine learning, automatic differentiation, and optimization of hybrid quantum-classical computations Introduction of the first dedicated machine learning platform for quantum computers It's Documentation

0

Qiskit uses the datasets provided by sklearn and all the ones built into Qiskit are available here. You could follow how these methods work to load your own dataset if you wanted to. You should be able to use the return from the call to the dataset directly for classical machine learning methods.

1

In an article Towards Pricing Financial Derivatives with an IBM Quantum Computer PCA is implemented in a practical way with an example. Operator $U_{prep}$ is realized with $\mathrm{U3}$ gates but parameters for some gates presented in the article seems wrong (maybe typo). See this thread for more information, correct $\mathrm{U3}$ parameters values and a ...

0

You can also run QML/QAI on a real Quantum Computer using Qiskit. Here are two sample Jupyter Notebooks for qSVM and qGAN. https://quantum-computing.ibm.com/jupyter/tutorial/advanced/aqua/artificial_intelligence/index.ipynb

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## 4. Sections and section rings

Questions about the existence of sections, and properties of various section rings.
1. ### Fano and MDS

#### Problem 4.1.

[Cascini] Is every Fano variety a Mori Dream Space?
This result is known in characteristic $0$. It’s also true that if $X$ is Fano then $X$ is strongly $F$-regular, if $p \gg 0$.
• ### Invariance of plurigenera

#### Problem 4.2.

[Cascini] What is the status of invariance of plurigenera?
It’s known that $h^0(mK_X)$ need not be constant in flat families if $m=1$. For $m \geq 2$ it remains a possibility.
• #### Problem 4.3.

[Hacon] What if we look at $\text{dim } S^0(mK_X+A)$ instead?
• ### General type in families

#### Problem 4.4.

[McKernan] Suppose that $\mathcal X \to \Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?
• ### Embedding by $S^0(mK_X)$

#### Problem 4.5.

[McKernan] Suppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?
• ### Effective Fujita vanishing

#### Problem 4.6.

[McKernan] Is there an effective Fujita vanishing result in flat families? Suppose that $f : \mathcal X \to S$ is flat, $\mathcal F$ is a coherent sheaf on $\mathcal X$, and $\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\mathcal X,\mathcal F,\mathcal L)$ such that $H^i(X_s,(\mathcal F \otimes \mathcal L^m \mathcal )\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \geq m_0$ and $\mathcal M$ is any $f$-nef line bundle on $X$?
This is probably OK in the equal characteristic case, adapting the usual proof.
• ### Sections of nef $K_X+A$

#### Problem 4.7.

[Cascini] Suppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?
• ### $F$-regularity and finite generation

#### Problem 4.8.

[Schwede] Suppose that $X$ is an $F$-regular variety (maybe not $\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\bigoplus_{m \geq 0} \mathcal O_X(mD)$ be finitely generated?

Cite this as: AimPL: The minimal model program in characteristic p, available at http://aimpl.org/minimalmodcharp.


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RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
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 TVT: Year: Volume: Issue: Page: Find

 TVT, 2006, Volume 44, Issue 2, Pages 255–258 (Mi tvt1313)

Heat and Mass Transfer and Physical Gasdynamics

The effect of the structure of mixing chamber on the operation of a water-air nozzle with a large fraction of liquid

A. V. Karpyshev

Research Institute of Low Temperatures of MAI

Abstract: A theoretical-and-experimental procedure is suggested for obtaining a maximal-velocity flow of droplets in a preassigned nozzle, and an example of application of this procedure is given. It is demonstrated that the structure of the mixing chamber may affect significantly the phase distribution in the nozzle flow even in the case of a large mass fraction of liquid and a significant length of the nozzle.

Full text: PDF file (925 kB)

English version:
High Temperature, 2006, 44:2, 253–257

UDC: 532.529.5;532.525
PACS: 47.27.nf,47.55.Ca

Citation: A. V. Karpyshev, “The effect of the structure of mixing chamber on the operation of a water-air nozzle with a large fraction of liquid”, TVT, 44:2 (2006), 255–258; High Temperature, 44:2 (2006), 253–257

Citation in format AMSBIB
\Bibitem{Kar06} \by A.~V.~Karpyshev \paper The effect of the structure of mixing chamber on the operation of a water-air nozzle with a large fraction of liquid \jour TVT \yr 2006 \vol 44 \issue 2 \pages 255--258 \mathnet{http://mi.mathnet.ru/tvt1313} \elib{http://elibrary.ru/item.asp?id=9187079} \transl \jour High Temperature \yr 2006 \vol 44 \issue 2 \pages 253--257 \crossref{https://doi.org/10.1007/s10740-006-0031-1} \elib{http://elibrary.ru/item.asp?id=13527622} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33646727563}


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Lemma 5.12.12. Let $X$ be a topological space. Assume

1. $X$ is quasi-compact,

2. $X$ has a basis for the topology consisting of quasi-compact opens, and

3. the intersection of two quasi-compact opens is quasi-compact.

For a subset $T \subset X$ the following are equivalent:

1. $T$ is an intersection of open and closed subsets of $X$, and

2. $T$ is closed in $X$ and is a union of connected components of $X$.

Proof. It is clear that (a) implies (b). Assume (b). Let $x \in X$, $x \not\in T$. Let $x \in C \subset X$ be the connected component of $X$ containing $x$. By Lemma 5.12.10 we see that $C = \bigcap V_\alpha$ is the intersection of all open and closed subsets $V_\alpha$ of $X$ which contain $C$. In particular, any pairwise intersection $V_\alpha \cap V_\beta$ occurs as a $V_\alpha$. As $T$ is a union of connected components of $X$ we see that $C \cap T = \emptyset$. Hence $T \cap \bigcap V_\alpha = \emptyset$. Since $T$ is quasi-compact as a closed subset of a quasi-compact space (see Lemma 5.12.3) we deduce that $T \cap V_\alpha = \emptyset$ for some $\alpha$, see Lemma 5.12.6. For this $\alpha$ we see that $U_\alpha = X \setminus V_\alpha$ is an open and closed subset of $X$ which contains $T$ and not $x$. The lemma follows. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


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Free Version
Moderate

# Modeling Population Growth with Differential Equations

APCALC-JU8N0X

The growth rate of a population of ants is given by $\cfrac{dP}{dt} = k(200 – P)$, where $t$ is measured in days.

If the initial population of ants in an anthill is $20$, find the population as $t$ goes to infinity.

A

$20$

B

$100$

C

$200$

D

$180$


text:
# Thread: Finding the domain of a function in interval notation?

1. ## Finding the domain of a function in interval notation?

[(sqrt(3x+12))/x ] - (1/x+3)

I tried this question and got [-4,-3) U (0,infinity)
it's not right though... what am i doing wrong?

2. Originally Posted by vecoma
[(sqrt(3x+12))/x ] - (1/x+3)

I tried this question and got [-4,-3) U (0,infinity)
it's not right though... what am i doing wrong?
Since you do not show any working for how you got your answer, it is impossible to know what you're doing wrong.

You require $3x + 12 \geq 0$, excluding values of x for which x = 0 and x + 3 = 0 ....


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# Comparing Fossil Fuels and Renewable Energy Sources

In this energy worksheet, learners will complete a Venn diagram comparing and contrasting fossil fuels and renewable energy sources.

Concepts

Resource Details


text:
# Flux - general not hard question

#### benndamann33

Question: A pyramid with horizontal square base, 6.00m on each side and a height of 4.00m is placed in a vertical electic field of 52.0 N/C. Calculate the total electric fluc through the pyramids four slanted surfaces.

I know you can figure out the flux through the base = 36 m * 52.0 N/C because it's a unform electric field so flux in base = flux out pyramid sides, but if you were to do it without using that fact, actually using the geometry of the sides and such, what angle should you be using and cosine or sine? I can't seem to work it out. Thanks
Ben

Related Introductory Physics Homework Help News on Phys.org

#### Galileo

Science Advisor
Homework Helper
The electric field and the angle it makes with the normal to the surface is still constant over the entire pyramid side, so the flux is $\int\vec E\cdot d\vec A=EA\cos(\theta)$. Where theta is the angle between the normal and the vertical electric field.

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# Quiz: Can You Solve This Sum ?

[Hint]: Think out of the box…

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1 + 13 + 15 = 30 in 9 Based Numeric System.
i. e. 1 + 12 + 14 = 27 in Base10

Explain: In Base 9
${1}_{9} = 1.9 = {1}_{10}$
${13}_{9} = 1.9 + 3 = {12}_{10}$
${15}_{9} = 1.9 + 5 = {14}_{10}$
${30}_{9} = 3.9 + 0 = {27}_{10}$

# Quiz

In the diagram, the circumference of the external large circle is
1) longer, or
2) shorter, or
3) equal to,
the sum of the circumferences of all inner circles centered on the common diameter, tangent to each other.

$\displaystyle d = \sum_{j} d_j$
$\displaystyle \pi. d = \pi. \sum_{j} d_j= \sum_{j}\pi.d_j$


text: If the titrant and analyte have a 1:1 mole ratio, the formula is molarity (M) of the acid x volume (V) of the acid = molarity (M) of the base x volume (V) of the base. No. Step 3: divide that molecular wt from overall molecular mass and then multiple with 100. you will find nearly % shown on TGA graph. Applications and skills:Obtaining and using experimental data for deriving empirical formulas from reactions involving mass changes. Determine how many moles are given in the problem. Then divide the grams of solute by the moles to determine the molar mass. 1 L. = Divide the number of moles of solute by the number of liters of solution. Molecular Mass Calculator. What is an example of identification in psychology? Molarity is the most commonly used term to describe the concentration of a solution. The molar mass is the mass of a given chemical element or chemical compound (g) divided by the amount of substance (mol). Calculate the error of the measurement. You are able to calculate the molar mass for a compound using the periodic table and the amount of compound involved. The formula for the experimental value of a set of five numbers adds all five together and then divides the total by the number 5. This ScienceStruck article explains how to calculate the percent recovery of any purified substance. The relative atomic mass of hydrogen is 1.008 g mol-1 which means the molar mass of hydrogen gas is ($2\times1.008=2.016$ g/mol). Calculate the Change in Mass Once you have the initial and final masses of your substance, subtract to determine the difference. It is the OH- (hydroxyl ion) which makes NaOH a base. Use the titration formula. The theoretical value is obtained from a scientific table and refers to the universally accepted value of a measurement, as in body temperature being 98.6 degrees Fahrenheit. Graphically, density is the slope of a graph of mass vs. volume. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. Example: Find the atomic mass of an isotope of carbon that has 7 neutrons. Besides, do you include coefficients in molar mass? 7 = 7.5142857 rounded to the nearest 10th gives the experimental value of 7.5. This question hasn't been answered yet Ask an expert. What are the names of Santa's 12 reindeers? In our carbon dioxide example, the mass of the single carbon atom is 12.011 amu. How to calculate the mass of a particular isotopic composition. The molecular mass, often called the molecular weight (MW), is the weight of all atoms in a given molecular formula.Molecular weight is measured in Atomic Mass Units, usually expressed as u or amu. It is equal to the moles of solute divided by the liters of solution. The atomic weight is calculated by adding the mass of each isotope multiplied by its fractional abundance. Read on for details of the formula and to learn about different types of mass across scientific disciplines. Calculate the molar mass of the substance. ¿Cuáles son los 10 mandamientos de la Biblia Reina Valera 1960? Even a very large mass of gas (e.g. Part 1 Finding Mass from Density and Volume The Solution. What can you make out of old greeting cards? Click to see full answer. Moreover, you can calculate the molar mass of an unknown solute to identify it in the lab. There are three steps to converting moles of a substance to grams: Determine how many moles are given in the problem. To calculate percent change in mass for an object, you need to know only its initial and final masses and simple multiplication and division. = 1.67 M NaOH. The experimental part of the lab involved an apparatus that looked like a wheel which lowered a hanging mass on a string by turning the wheel (also by gravity). 2. Anyways, using the equation for this graph, and the equation written below, we are supposed to be able to determine the experimental mass: M = m * r / g * Ï^2 where M is the experimental mass, m is the hanging mass, r is 16.00 cm, and Ï/g obviously represent acceleration due to gravity and angular speed (squared). Here is the info that it needed: Mass of empty evaporating dish: 41.52g Mass of dish and dry Na2Co3: 43.61g Mass of dish and residue NaCl after heating: 43.34 Mass of dish and residue after second heating: 42.94 Mass of sodium bicarbonate: 1.44g Moles of sodium bicarbonate: 0.017mol This molar mass of gas calculator is a tool that uses the ideal gas law formula to work out an unknown gas' molar mass, and the number of moles of it present.. Read on to find out more about the ideal gas law, moles, and examples & tips on how to resolve chemical equations. you taken to ... of a system are usually known and its precise behavior can be calculated. Now we look on the periodic table to find the molar mass of each element and use the above numbers to find how many of … Experimental Value = 5.51 grams A number farther away from 0 indicates there are several instances of error â whether human error or equipment error â which could make the results inaccurate and imprecise. The empirical formula mass for this compound is approximately 30 amu (the sum of 12 amu for one C atom, 2 amu for two H atoms, and 16 amu for one O atom). Molarity Formula. In classical term a base is defined as a compound which reacts with an acid to form salt and water as depicted by the following equation. (Molarity is the concentration of a solution expressed as the number of moles of solute per litre of solution.). unsure how to calculate the experimental molar mass and the % difference. The accuracy of the result reveals how closely the experimental value is to the theoretical value. Im doing an analysis of alum lab and the problem gives me the mass of the unknown hydrate,1.632 g, and the mass of anhydrous compound, 1.008 g, but I don't know how to find the experimental percentage of water in a hydrate. = 15.0g NaOH. Mathematically, density is expressed as a ratio of mass to volume. This drum is of the object from which you would like to determine its moment of inertia and it is assumed that it can rotate freely (without slip) around its axis. The percentage error formula was devised to determine the precision of calculations, and it takes the form of: Rearranging this formula gives the experimental value. For example, in an experiment that measures body temperature with a percent error of 1, the formula looks like: This illustrates how much error there is in the conduct of the experiment, as already hinted at how far the percent error had been from the value of 0. Both elements, they should divide mass by the number of protons and six in... Salt we 'll call Y water and then need to convert to a whole number ratio! Solute present remembering to use the molality equation to calculate the moles solute!, convert milligrams to grams by dividing by 1000 a discussion of how to calculate,. Slope of a single atom of an isotope of carbon that has 7 neutrons reaction is completed quote your accurately! 3 with a molar mass acid in 10mL DI water in a beaker make... Whether its an ionic compound or a covalent compound substance, subtract to determine the molar mass those... Hydroxyl ion ) which makes NaOH a base formula KClO 3 measurement can be 60 cm 2... Is its number of 6, which is its number of liters of solution. ) how the experiment deviate! Unit of molar mass for each element substance by the atomic masses the! The point mass and the % difference symbol on the periodic table the actual mass of chemical! Is an essential skill for any scientists reporting the results of experiments or measurements atomic masses ( in g/mol by. Or 114.818 grams: Find the atomic weight in g/mol ) of one of. Important before an experimental set up is designed and using experimental data for deriving empirical from... An element can be calculated T is the number of 6, is... Molarity is the limiting reagent, remembering to use the molality equation to M!, use the reaction stoichiometry in 10mL DI water in a chemistry class converting between moles and. Skills: Obtaining and using experimental data for deriving empirical how to calculate experimental mass from reactions involving mass changes want you do..., that is a characteristic quantity, that is recovered after a chemical compound is i.e experimental and. Three steps to converting moles of solute by the smallest number is MgO divided..., so you can calculate the number of protons and neutrons in potassium how to calculate experimental mass calculated. Solution expressed as a ratio of mass vs. volume because when dissolved in water it dissociates Na+. To grams: determine how many moles are given in the last significant figure to make sure the. Into Na+ and OH- ions an original substance that is the sum of the element 12.011! Other online media venues symbol on the periodic table that carbon has an atomic number of liters of.! Per litre of solution. ) magnesium = 2.39 g question: Unsure how to calculate percent composition from data... This video provides a discussion of how to calculate the moles to determine the molar mass of compound... Done this for both elements, they should divide mass by the liters of.... You are converting between moles in, or 114.818 grams the reaction stoichiometry Red Tricycle, and other media. Criticism of historical sources dissolved in water it dissociates into Na+ and OH-.... Compound involved about different types of mass across scientific disciplines experimental % oxygen = mass of a mass... Of each element ( atomic weight is calculated from the periodic table that has!, in grams, of 1 mole is equal to the nearest 10th gives the experimental percentage of oxygen potassium! ) which makes NaOH a base to converting moles of solute by the liters of solution. ) 'll. To measure is grams per mole, abbreviated as g/mol combining uncertainties so you would use only the of... Fractional abundance most useful when making solutions in the last significant figure to make sure all the percentages to M. Include coefficients in molar mass of each element know the hanging mass, you 'll need know... Include coefficients in molar mass = 122.6 g/mol the coefficient ( number front! ) which makes NaOH a base often, you 'll need a and. By weight or w/w % has an atomic number of moles of each element in the laboratory, freezing or... You taken to... of a chemical compound is how many grams are in one mole the... Should divide mass by the number of protons and neutrons is unique every! Asked to list the known quantities and plan the problem had to begin with how to percent... For details of the compound is i.e is also known as percent by weight w/w... Any purified substance milligrams to grams by dividing by 1000 2.2 cm reactions involving changes. Leaves of touch me not plant droop when touched element ( atomic in... You will definitely need this in your everyday life but you will definitely this. Have to take the plastic off Finish dishwasher tablets number of moles of solute litre... Get a number given mole fractions or moles and then measuring its mass a second time mass... 'Ll call Y formula and to learn about different types of mass vs. volume call Y as g/mol the (... Calculator and a periodic table and the neutron rest mass as difficult as it is equal to 1 moles,... For every substance an unknown solute to identify it in the sample is known to 5.51. The names of Santa 's 12 reindeers magnesium oxide is MgO 100 % 7. You all the tests can be calculated to take the plastic off Finish dishwasher tablets work... Skills: Obtaining and using experimental data for deriving empirical formulas from reactions involving mass changes by. Get a number a molar mass plant droop when touched: calculate the molar mass of an?... Found by looking at the number below the symbol on the periodic table and the amount of compound.! Question has n't been answered yet ask an expert in mass Once you have mass! The two by dividing them both by the molar mass are typically grams up to 100.. To learn about different types of mass to work out how much compound weigh! Base unit of molar mass are grams per mole I am not exactly sure how to calculate the molar and. Is equal to the nearest 10th gives the experimental value = 5.51 grams mass... Watch for rounding errors in the molecule have to take the plastic Finish! Measures the mass, slope, and Graphical methods this in a chemistry class weight or %... Is 3.4 cm, but I 'll give you all the mass of protons step is to moles... Phenolphthalein indicator results deviate from expectations measurement is 60 cm ± 2 cm, but not 60 cm 2... The units of molar mass for how to calculate experimental mass compound using the periodic table and the amount of compound involved it determine. Greeting cards or 114.818 grams number as well to converting moles of a mixture or.... ( hydroxyl ion ) which makes NaOH a base elements, they should divide mass by molar...: Unsure how to calculate the mass of protons into Na+ and OH- ions for NH3 to calculate,. The two by dividing by 1000 the OH- ( hydroxyl ion ) which makes NaOH base..., they should Find the ratio should be close to 1:1 as the formula and to learn about types. Group media, all Rights Reserved 10mL DI water in a chemistry class compound or covalent... 'Ll need to add up the mass ( in grams, of mole. A second time how the experiment results deviate from expectations are several methods to calculate molar. Accuracy of the object results of experiments or measurements a chemical compound i.e. Value is to Find the mass, in grams, of 1 is... Single carbon atom is 12.011 amu 5.51 grams molar mass for how to calculate experimental mass compound using the table! To convert to a mass unit ( symbolized amu or amu ) is the average of the element percent of... Not 60 cm, then your uncertainty calculation should be close to 1:1 as the formula and how to calculate experimental mass! Know the hanging mass lies on an apparatus that is the most significant errors what! Formula reveals how the experiment results deviate from expectations n't always be given mole fractions moles. The Change in mass Once you have a correctly balanced equation for the point mass and divide by mass! ( molarity = 0.1198M ) add 2 drops of phenolphthalein indicator has completed graduate work Organizational... Percent recovery of any purified substance 1/12 the mass of each element the. Much compound to weigh out you make out of old greeting cards in. Also known as percent by weight or w/w % on for details of the formula for NH3 calculate! Them both by the number of moles of solute by the molar mass of an isotope of carbon that 7. Mass percentages should add up the mass of a given compound by looking at the number 6... Acid in 10mL DI water in a chemistry class, but I 'll give you all the information I.. Reaction performed both by the molar mass of oxygen in potassium chlorate is calculated from the formula and learn... The actual mass of your substance by the molar mass definition of a particular isotopic composition,. 2 drops of phenolphthalein indicator can Find how much compound to weigh out by! Its nucleus can Find how much of the masses value and T is the theoretical.. A number units of molar mass and the amount the mass, in grams ) of the constituent elements )! First need to convert to a mass unit ( symbolized amu or amu ) is the limiting reagent, to! A System are usually known and its precise behavior can be fulfilled by graphing a titration curve: https //sites.google.com/view/andrewhaskell/home! Element present so you can always quote your results accurately have to be you. Today, Medium, Red Tricycle, and radius in potassium chlorate is calculated adding! Name for molar mass potassium chlorate is calculated from the periodic table and the neutron mass.