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 Title: Characterization of polymer-modified electrodes from transient ultramicroelectrode experiments Author(s): Barton-Taylor, Marvin Eric Doctoral Committee Chair(s): Faulkner, Larry R. Department / Program: Chemistry, Analytical Discipline: Chemistry, Analytical Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Chemistry, Analytical Abstract: Reliable diffusion coefficients and concentrations have been obtained simultaneously by fitting chronoamperometric transients of an ultramicroelectrode to the split function derived by Aoki and Osteryoung. An extensive investigation into the effect of noisy data and small bias currents on the diffusion coefficients and concentrations obtained by this analysis is also presented. The sensitivity of the analysis to proper background correction is discussed in the light of previous literature and demonstrated experimentally using the well studied Fe(CN)$\sb6\sp{-3/-4}$ system. This technique is used to measure diffusion coefficients and concentrations of benzoquinone (BQ) in a bulk cross linked/quaternized polyvinyl pyridine (QPVP) matrix. The activation energy for diffusion of BQ in the bulk QPVP matrix is determined from temperature resolved studies and are found to be comparable to those obtain for BQ in thin films of QPVP. The enthalpy of partitioning of the BQ into the polymer matrix is also determined. The use of BQ as a probe to obtain in situ film thickness of a thin film modified electrode, consisting of a spin coated QPVP on a gold or glassy carbon electrode, occupies a significant portion of this work. A novel application of the technique to the determination of diffusion coefficients of mobile electroactive species in solid crystalline material is explored briefly. This technique could be generally useful for the measurements of transport coefficients and partition coefficients of electroactive species in polymer electrolytes such as those currently under investigation for rechargeable lithium batteries.Equations for differential pulse voltammetry (DPV) at an ultramicroelectrode are also derived using the split function and excellent agreement with experimental data for a reversible redox couple is demonstrated. High quality approximations to these equations, which could significantly reduce computation time for analysis, are derived along with their range of applicability. Issue Date: 1992 Type: Text Language: English URI: http://hdl.handle.net/2142/23568 Rights Information: Copyright 1992 Barton-Taylor, Marvin Eric Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9315949 OCLC Identifier: (UMI)AAI9315949


text:
In Chebfun2 we would like to be able to find the roots of a bivariate polynomial system. That is, find $(x,y)$ such that

$$F(x,y) = ( f(x,y),\ g(x,y) ) = 0.$$

A suggestion by John Boyd of the University of Michigan for finding the zeros of a system of polynomials goes by the name of Marching Triangles [5]. This works by determining the zero contours of $f$ and $g$ separately, and to achieve this the algorithm subdivides the domain many small triangles. On each triangle it approximates $f$ (or $g$) by a piecewise linear polynomial, and there are two possible cases:

1. The values of the function at the vertices are all of the same sign.

2. One function value is of a different sign to the others.

If (1) holds then the triangle is marked as not containing a zero contour. If (2) holds, then a linear polynomial $p(x) = ax + by + c$ is formed which interpolates the three vertices and the zero contour of $f$ in that triangle is approximated by the zero line of $p(x)$. Further subdivision is taken in triangles which are deemed to contain a zero contour.

The MATLAB command contourc uses Marching Squares [4], a similar technique using subdivision with squares instead of triangles. This algorithm is employed by contour, contours and contourf and implemented in contourc. One of the Chebfun Examples is based on the contourc command [1].

The Chebfun2 roots command uses the MATLAB contourc command to find the zero contours of a single function $f(x,y)$.

### Zero contours

Let's solve a problem from [2] of finding the solution to $f(x,y) = g(x,y) = 0$:

d = [-4 4 -4 4];
f = chebfun2(@(x,y) 2*y.*cos(y.^2).*cos(2*x)-cos(y),d);
g = chebfun2(@(x,y) 2*sin(y.^2).*sin(2*x)-sin(x),d);

This example was derived from an autonomous system of ODEs. Here are the zero contours computed by Marching Squares.

LW = 'linewidth'; lw = 1.6; MS = 'markersize';
plot(roots(g),'r',LW,lw), hold on, plot(roots(f),'g',LW,lw)

### Solutions to bivariate systems

A different problem is to find the isolated solutions to a bivariate system $f(x,y) = g(x,y) = 0$. The Chebfun2 command roots(f,g) solves this problem by first computing the zero contours of $f$ and $g$ by marching squares, and then using their intersections as initial guesses in a Newton iteration.

For example we can find all the solutions to the example above with the following code:

r = roots([f;g],'ms');
plot(r(:,1),r(:,2),'.k',MS,20), hold off

(Note: The roots command in Chebfun2 uses two different algorithms, one based on a hidden variable resultant method and the other on Marching Squares. The 'ms' flag ensures that the Marching Squares algorithm is employed. For more information type help chebfun2v/roots.m.)

### Trott's curve

Trott's curve an important curve in algebraic geometry. Here we use the command roots(f,g) to compute the intersections between this curve and the zero contours of $f(x,y) = y-x^6$.

trott = @(x,y) 144*(x.^4+y.^4)-225*(x.^2+y.^2) + 350*x.^2.*y.^2+81;
f = chebfun2(trott);
g = chebfun2(@(x,y) y-x.^6);
plot(roots(f),'b',LW,lw), hold on, plot(roots(g),'r',LW,lw), axis equal
r = roots([f;g],'ms');
plot(r(:,1),r(:,2),'k.',MS,20), hold off

### A function with several local minima

The roots command in Chebfun2 can also be used to compute the local extrema of a function $f(x,y)$ because they satisfy $df/dx = df/dy = 0$.

f = chebfun2(@(x,y) (x.^2-y.^3+1/8).*sin(10*x.*y));
r = roots(gradient(f),'ms');             % critical points
plot(roots(diff(f,1,2)),'b',LW,lw)       % plot zero contours of f_x
hold on, plot(roots(diff(f)),'r',LW,lw)  % plot zero contours of f_y
plot(r(:,1),r(:,2),'k.',MS,20)           % plot extrema

Note that we have used the Chebfun2 gradient command, which computes the $x$ and $y$ derivatives of $f$.

The last example can also be found in [3] along with additional information about the rootfinding algorithm in Chebfun2.

### References

1. Chebfun Example fun/ContourFun

2. C. S. Coleman, Gulp!, CODEE Newsletter, 1995.

3. A. Townsend and L. N. Trefethen, An extension of Chebfun to two dimensions, SIAM Journal on Scientific Computing, 35 (2013), C495-C518.

4. Marching Squares, Wikipedia article, http://en.wikipedia.org/wiki/Marching_squares

5. Marching Triangles, Wikipedia article, http://en.wikipedia.org/wiki/Marching_triangles

format: plain


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# Periodic boundary conditions

1. Nov 28, 2005

### StatusX

I'm taking solid state, and again and again we use the periodic boundary conditions, that the wavefunction should be unchanged by displacements of the length of the sample, L (assume 1D for simplicity). The argument was that the surface is so far away that it shouldn't have an effect on the properties in the interior, so we are free to use boundary conditions that are mathematically convenient. But this argument doesn't work for me, because if the surface is so far away as to be irrelevant, why should 'L' enter into things at all? Why should the properties be different deep inside a 1 cm sample from inside an otherwise identical 2 cm sample?

2. Nov 28, 2005

### Physics Monkey

They shouldn't be different! The intensive thermodynamic quantities like density, pressure, etc are indeed independent of the size of your system. Think about it for second, does the Fermi energy depend on the size of your system? What about the density of states per unit volume? It is only when you calculate extensive variables like total energy or total magnetization that the volume of your system enters (as it should). Even when dealing with such extensive quantities, the dependence on volume is trivial: it is always volume times some intensive quantity. For macroscopic systems the dependence on volume is exactly as simple as you would expect.

Of course, things can get interesting when this picture eventually begins to break down.

Last edited: Nov 28, 2005
3. Nov 29, 2005

### marlon

Err, do you know the concept of Bloch Wave functions?

The plane wave wavevector (or Bloch wavevector) k (multiplied by Planck's constant, this is the particle's crystal momentum) is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone.

The Brillouin zone is the Fourier Transform of the Wigner Seitz Cell. This is the smallest primitive unit cell that ALSO exhibits the same symmetry as the crystal itself. Keep in mind that the coordinates in the Brillouin zone are MOMENTA not x,y and z-coordinates. So this length L has a rather special meaning.

marlon

4. Nov 29, 2005

### StatusX

Maybe that's all true, but my point is that the reason I was told, that the surface should not have an effect, is not a good enough reason. I guess the real reason is something like "if the sample is large, there needs to be more allowed k values to give the same intensive properties," but this was never made clear to me.

5. Nov 29, 2005

### marlon

i don't quite understand what you are saying here. Could you clarify, please

thanks

marlon

6. Nov 29, 2005

### nbo10

There can be very strange surface effects. But for most anything you do for classes these effects can be ingored

7. Nov 29, 2005

### StatusX

I mean, these boundary conditions mean the allowed k values are spaced by 2$\pi$/L, and I wasn't sure why such a property should depend on L.

8. Nov 29, 2005

### Physics Monkey

You would expect the spacing to depend on L. You know the set of inequivalent states is always contained in the 1st Brillouin Zone and the volume of the zone is fixed by the lattice parameters i.e. it is independent of system size. But it is perfectly clear that a bigger sample should have more states, right? Since you have to stuff all these extra states in the same volume in k space, your states must be closer together for a larger system.

9. Dec 1, 2005

### Modey3

StatusX,

Think of a 1 dimensional lattice. A small lattice has fewer modes of vibration than a larger lattice. You can prove that to yourself.

Modey3

10. Dec 1, 2005

### armandowww

...if I could express myself too... In one of many proofs of Bloch Theorem, boundary conditions by Born and von Karmann are simply essential. Without them you are not able to execute the Fourier expansion of psi on the "big" crystal... That proof is very important because you can see, once you've done it, how to build up wavefunction u (with the same periodicity of potential) and the appearance of n, the "clever" band index, which explain why dispersion relations are never single-valued.
Moreover, in quantum mechanics periodical boundary conditions are enough to emphasize the hamiltonian hermitian properties... without request meticolously that psi must vanish on crystal surface.
Strictly speaking, in the independent electron approximation, periodicity for your potential is unreal because 10^23 is not the mathematical infinite... but we are physicists... so, please try to satisfy yourself with some kind of trick.:tongue:

11. Dec 2, 2005

### marlon

Well, the answer to that question is in my first post. The reason is : Fourier Transform. This is the way that the Brilluoin zone is defined. The fact that we go from a spatial coordinate base (Wigner Seitz Cell) to a momentum coordinate base (Brillouin Zone) is because of symmetry. Without going into this too deeply, the philosophy is that you try to define a region in which you calculate all interactions going on. This region needs to be as small as possible, containing the lowest possible amount of atoms so that the calculations require less computational effort. Thanks to the crystal symmetry you can define such a small region. You calculate the interactions in a few points (ie k-point sampling) and then, thanks to the symmetry, you automatically know these interactions in other "equivalent" points of the crystal.

marlon

12. Dec 2, 2005

### Galileo

Marlon, I think StatusX wants a good physical argument for the use of periodic boundary conditions. That has nothing to do with Bloch's theorem, which tells you wavefunction properties in a periodic potential. The periodic boundary conditions here are applied without reference to any potential energy function. (Although I assume the context is the free electron fermi gas model in a box)

To StatusX, all I can add is that you need boundary conditions to solve your problem and there are two more or less 'natural' choices:
- The wavefunction is 0 at the ends of the box
- Periodic boundary conditions
Both will give identical number of states, density of states etc, but periodic boundary conditions will give you traveling waves which are more natural to use than standing waves. You want to study things like electron transport after all.

If L is large enough, the allowed values for k will be practically a continuum, so the particle really is free. But you will get a discrete set of eigenstates and is a lot easier to work with. It's just a mathematical trick.

13. Dec 2, 2005

### marlon

Galileo,

I must say i disagree with what you state. One can use periodic boundary conditions because one iterates the unit cell when describing a crystal. In the momentum (crystal momentum) base (after Fourier Transformation) one iterates the Brillouin zone, thanks to the Bloch Theorem. That is why this is so important. If this was not possible, all ab initio calculations would be, let's say, quasi impossible. It is the aspect of symmetry that is central in this discussion.

True, i must admit that it still is not very clear to me what the original poster is actually asking about. Perhaps you could clarify this for me, if you wish

regards
marlon

14. Dec 2, 2005

### Dr Transport

Think of a single Brillouin zone and calculate whatever, the band structure, absorption, density of states, etc..... If we have an ensemble of the zones, why should the properties of an ensemble be any different than a single zone. Periodic boundary conditions ensure that the properties of the material are the same regardless of where you are inside the crystal. All propertiues are calculated with regards to the unit cell, and as Galileo has stated there are two choices, periodic or zero at the boundaries.
If I measure the absorption in a crystal, it should be the same if I have a 1 cm3 sample or if I have a 100 cm3, $$\alpha = \frac{4\pi nk}{\lambda}$$ is the same for either sample.
Now if you are at a surface and calculating surface states, that is another matter all together.

If this doesn't convince you, think about a series of masses around a circle vibrating, if you go around one time you better get the same answer, hence periodic bc's.....

Last edited: Dec 2, 2005
15. Dec 2, 2005

### Physics Monkey

As Galileo noted, Bloch's Theorem applies when the potential is periodic. Now, a finite crystal can't technically supply a periodic potential unless it has periodic boundary conditions. In this technical sense, it may be said that periodic boundary conditions are the natural way to reconcile a finite crystal with Bloch's Theorem. In other words, it wouldn't make mathematical sense to impose hard boundary conditions on the wavefunction when you technically have to assume periodic boundary conditions for the crystal to get Bloch's Theorem in a finite crystal.

Physically, of course, none of this matters. You can still quote Bloch's Theorem as an excellent approximation, put in the size of the crystal by hand, and take whatever boundary conditions you want, and get all the same results.

Just a thought.

16. Dec 3, 2005

### Galileo

True, but it's completely possible (as done in many books) to discuss the (crude) free electron model without reference to atomic structure (since that doesn't matter in this model). So no unit cell, periodic potential crystal momentum etc are mentioned or needed. And yet periodic boundary conditions are imposed.
There is however a lucky relation between the two IIRC, but I`ll have to check my notes on that and I don't have access to them now.

17. Dec 4, 2005

### marlon

Ofcourse, that is completely true.

I was talking about interacting many body physics, though. I assumed this since the OP was talking about periodicity in structure, which you do not have in the case of a free electron gas.

regards
marlon


text:
# I need help solving these rational equations.

• Oct 27th 2009, 01:57 PM
Pre-Calc
I need help solving these rational equations.
first
solve for x
1/x-1 + 1/x =5
• Oct 27th 2009, 02:03 PM
pickslides
Quote:

Originally Posted by Pre-Calc
1/x-1 + 1/x =5

Hey cobber, please consider how you post these problems. For starters pls post 1 problem per thread and show your working to this point. Better we help you learn rather than do the problems for you.

My other concern is the lack of brackets. Why? Well which is your problem?

$\frac{1}{(x-1)} + \frac{1}{x} =5$

or

$\frac{1}{x}-1 + \frac{1}{x} =5$

I'm guessing the first one, but how can I be sure?
• Oct 27th 2009, 02:09 PM
Pre-Calc
the first one is my problem im new to this forum so i didn't no how to put it that way sorry.
• Oct 27th 2009, 02:21 PM
e^(i*pi)
Quote:

Originally Posted by Pre-Calc
the first one is my problem im new to this forum so i didn't no how to put it that way sorry.

First things first, clear up some domain issues by saying $x \in \mathbb{R} \: , \: x \neq 0\,,\,1$

$
\frac{1}{(x-1)} + \frac{1}{x} =5
$

Take $\frac{1}{x}$ from both sides and rewrite $5$ as $\frac{5x}{x}$ (this is allowed because $x \neq 0$)

$\frac{1}{x-1} = \frac{5x-1}{x}$

Cross multiply: $x = (5x-1)(x-1)$

Expand, simplify and solve the quadratic :)

Spoiler:
$5x^2-6x+1 = x \: \rightarrow \: 5x^2-7x+1 = 0$

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$= \frac{7 \pm \sqrt{(-7)^2-(4)(5)(1)}}{(2)(5)} = \frac{7 \pm \sqrt{29}}{10}$
• Oct 27th 2009, 02:27 PM
Pre-Calc
ty.


text:
Algebraic & Geometric Topology

Ideal triangulations of 3-manifolds II; taut and angle structures

Abstract

This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3–manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program SNAPPEA of Weeks, and the program SNAP of Coulson, Goodman, Hodgson and Neumann. Casson has also written a program to find hyperbolic structures on such 3–manifolds, by solving Thurston’s hyperbolic gluing equations for ideal triangulations. In this second paper, we study the question of when a taut ideal triangulation of an irreducible atoroidal 3–manifold admits a family of angle structures. We find a combinatorial obstruction, which gives a necessary and sufficient condition for the existence of angle structures for taut triangulations. The hope is that this result can be further developed to give a proof of the existence of ideal triangulations admitting (complete) hyperbolic metrics. Our main result answers a question of Lackenby. We give simple examples of taut ideal triangulations which do not admit an angle structure. Also we show that for ‘layered’ ideal triangulations of once-punctured torus bundles over the circle, that if the manodromy is pseudo Anosov, then the triangulation admits angle structures if and only if there are no edges of degree 2. Layered triangulations are generalizations of Thurston’s famous triangulation of the Figure–8 knot space. Note that existence of an angle structure easily implies that the 3–manifold has a CAT(0) or relatively word hyperbolic fundamental

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1505-1533.

Dates
Accepted: 13 June 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796487

Digital Object Identifier
doi:10.2140/agt.2005.5.1505

Mathematical Reviews number (MathSciNet)
MR2186107

Zentralblatt MATH identifier
1096.57018

Citation

Kang, Ensil; Rubinstein, J Hyam. Ideal triangulations of 3-manifolds II; taut and angle structures. Algebr. Geom. Topol. 5 (2005), no. 4, 1505--1533. doi:10.2140/agt.2005.5.1505. https://projecteuclid.org/euclid.agt/1513796487

References

• I Agol, Bounds on exceptional Dehn filling, \gtref4200015431449
• B Burton, E Kang, J H Rubinstein, Ideal triangulations of $3$–manifolds III; Taut structures in low census manifolds, in preparation
• P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$–manifolds, Math. Comp. 68 (1999) 321–332
• D Coulson, O A Goodman, C D Hodgson, W D Neumann, Computing arithmetic invariants of $3$–manifolds, Experiment. Math. 9 (2000) 127–152
• M Freedman, J Hass, P Scott, Least area incompressible surfaces in $3$–manifolds, Invent. Math. 71 (1983) 609–642
• F Gueritaud, On canonical triangulations of the mapping tori over the punctured torus.
• W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245–375
• W Haken, Some results on surfaces in $3$–manifolds, from: “Studies in Modern Topology”, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.) (1968) 39–98
• J Hempel, $3$–Manifolds, Princeton University Press, Princeton, N. J. (1976)
• W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society, Providence, RI (1980)
• W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
• W Jaco, J H Rubinstein, $1$–efficient triangulations of $3$–manifolds, in preparation
• E Kang, Normal surfaces in knot complements, PhD thesis, University of Connecticut (1999)
• E Kang, Normal surfaces in non-compact 3–manifolds, J. Aust. Math. Soc. 78 (2005) 305–321
• E Kang, J H Rubinstein, Ideal triangulations of 3–manifolds I; spun normal surface theory, from: “Proceedings of the Casson Fest (Arkansas and Texas 2003)”, \gtmref7200410235265
• M Lackenby, Taut ideal triangulations of $3$–manifolds, \gtref4200012369395
• M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
• E E Moise, Affine structures in $3$–manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952) 96–114
• J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$–dimensional manifolds, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 1–20
• J Simon, Compactification of covering spaces of compact $3$–manifolds, Michigan Math. J. 23 (1976) 245–256 (1977)
• M Stocking, Almost normal surfaces in $3$–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171–207
• W Thurston, Three-dimensional geometry and topology, Vol 1, (Silvio Levy, editor), Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ (1997)
• S Tillmann, Degenerations and normal surface theory, PhD thesis, University of Melbourne (2002)
• J L Tollefson, Normal surface $Q$–theory, Pacific J. Math. 183 (1998) 359–374


text:
# Why is caustic soda (NaOH) used to clean beer bottles?

I see in many blogs in the internet that cleaning bottles with $\ce{NaOH}$ is very effective. I also realize that industries use caustic soda to clean bottles. Why is caustic soda very effective at cleaning bottles? What makes it so special from other solvents which made it preferable for cleaning?

• Most organic chemistry lab use $\ce{KOH + i-PrOH}$ to clean dirty glassware. – Mathew Mahindaratne Apr 2 '18 at 21:14

Caustics such as $\ce{NaOH}$ and $\ce{KOH}$ are effective in converting insoluble fatty acids in organic fats and oils into soluble soaps in a proccess called saponification. The caustics also react with cell membranes, DNA and RNA, destroying undesired yeasts, bacteria and other microorganisms.

That said, caustics react with the glass itself, removing the surface and etching the glass. Use will weaken and damage containers.

Use caution! Caustics also destroy skin and clothing, and are particularly damaging to the eye.

• Did you mean "Caustics such as $\ce{NaOH}$ and $\ce{KOH}$ are effective" instead of "Caustics such as $\ce{NaOH}$ and $\ce{NaOH}$ are effective" ? – Avyansh Katiyar Apr 3 '18 at 3:31
• @ Avnish Kabaj , Thanks! So much for copy-&-paste. – DrMoishe Pippik Apr 4 '18 at 0:02
• Slightly etching the glass is actually a benefit if you really, really want to clean the surface of a vessel. That's why chemists often clean their lab glassware this way. – matt_black Apr 4 '18 at 10:25
• @matt_black, true, caustics are effective cleaners. However, getting rid of the last bit of alkali (or sodium silicate that forms) can be a problem for some uses. Also, I've seen caustic reagent destroy glass bottles when left too long. – DrMoishe Pippik Apr 4 '18 at 16:23

The caustics also react with glass, cell membranes, DNA and RNA, destroying undesired yeasts, bacteria and other microorganisms.

• Hi Baraka, could you elaborate on your answer? – Felipe S. S. Schneider Aug 26 '19 at 18:05
• during the cleanness of bottle in bottle washer in industry it need strong and effective disinfectant to ensure all contamination are removed when deal with contamination its about stains, bacteria, cell membranes and others undesired material, so that caustic is strong and more effective cause it react with glass effective and also removed simple and also have the ability to destroy all sources of microorganism that its my cause to select caustic to best plus coast – Baraka Magambo Aug 29 '19 at 13:25


text:
# Revision history [back]

There is the object IndependentSets in sage.graphs.independent_sets that can help you (as the complement of an independent set is a vertex cover). {{{ sage: from sage.graphs.independent_sets import IndependentSets sage: G = graphs.PetersenGraph() sage: V = set(G.vertices()) sage: for I in IndependentSets(G, maximal=True): ....: print(V.difference(I)) set([1, 3, 4, 5, 7, 8, 9]) set([1, 3, 4, 5, 6, 7]) set([1, 2, 4, 5, 8, 9]) ... set([0, 1, 2, 4, 7, 8, 9]) set([0, 1, 2, 3, 5, 8, 9]) }}} To add further restriction, you can just filter. If it is not efficient enough, you will have to implement your own backtracker.

There is the object IndependentSets in sage.graphs.independent_sets that can help you (as the complement of an independent set is a vertex cover). {{{ cover).

sage: from sage.graphs.independent_sets import IndependentSets
sage: G = graphs.PetersenGraph()
sage: V = set(G.vertices())
sage: for I in IndependentSets(G, maximal=True):
....:     print(V.difference(I))
set([1, 3, 4, 5, 7, 8, 9])
set([1, 3, 4, 5, 6, 7])
set([1, 2, 4, 5, 8, 9])
...
set([0, 1, 2, 4, 7, 8, 9])
set([0, 1, 2, 3, 5, 8, 9])
}}}

To add further restriction, you can just filter. If it is not efficient enough, you will have to implement your own backtracker.


text:
@Jhannybean

@AaronAndyson

@ganeshie8

4. IrishBoy123

$$y = 2(n-2)^3$$ $$(\dfrac{y}{2})^\frac{1}{3} = n-2$$ etc

how did you do that

6. IrishBoy123

$$f(n) = 2(n-2)^3$$ $$y = 2(n-2)^3$$ yes?

7. IrishBoy123

$$\frac{y}{2} = (n-2)^3$$ ok?

why would you do y/2, isn't it |dw:1443341707529:dw|

9. IrishBoy123

$\dfrac{y}{2} = \dfrac{2(n-2)^3}{2}$

10. IrishBoy123

you want the inverse, is that right, ie $$n = n(y)$$

11. IrishBoy123

$\dfrac{y}{2} = \dfrac{2(n-2)^3}{2} \implies \dfrac{y}{2} = (n-2)^3$

f(n) = 2(n-2)^3

we need to inverse this function

14. IrishBoy123

yes, indeed label the $$f(n)$$ as y just because it makes it easier to write out. then we need to find and equation in form $$n = n(y) = f^{-1}(n)$$ so $$y = 2(n-2)^3$$ $$\frac{y}{2} = (n-2)^3$$ like so.

but isn't it like n= 2(y-2)^3

i like this way

@Miracrown

@insa

19. IrishBoy123

ok let's go that way $$n= 2(y-2)^3$$ $$\frac{n}{2} = (y-2)^3$$

alright, next

21. IrishBoy123

$(\frac{n}{2})^{\frac{1}{3}} = y-2$

22. IrishBoy123

yes?

wait wait, where did 1/3 come from

24. IrishBoy123

take cube root of each side

why cant it be |dw:1443342930062:dw|

26. IrishBoy123

$$a^3 = b$$ $$a = b^{\frac{1}{3}} = \sqrt[3]{b}$$ same thing

$\frac{ \sqrt[3]{n} }{ 2}$

lets go this way if you dont mind

29. IrishBoy123

if that way, than $\large (\frac{n}{2})^{\frac{1}{3}} = \sqrt[3]{\frac{n}{2}}= y-2$

yup, then $\frac{ \sqrt[3]{n} }{ 2 } + 2$

31. IrishBoy123

be careful the cube root must include the 1/2 its $y=\sqrt[3]{\frac{ n} { 2 }} + 2$ NOT $y=\frac{ \sqrt[3]{n} }{ 2 } + 2$

yeah

so is that it

34. IrishBoy123

good!

36. IrishBoy123

shoot but if it's a new question, new thread is best

find the inverse of g(x) = 1/x - 2

i know that x = 1/y - 2

@IrishBoy123

40. IrishBoy123

you like to do it this way $x = \frac{1}{y} - 2$ $\frac{1}{y} = x + 2$ $y = \frac{1}{x+2}$

and the inverse of -4x + 1 is $\frac{ x-1 }{ 4 }$ right

42. IrishBoy123

|dw:1443344105564:dw|


text:
# D/dx y^2 help

1. Jun 29, 2013

### canoluk2003

Hi all!
In questions having implicit functions,
this expression -> "d/dx y^2" often appears in the calculation process.
I use the chain rule to convert it to 2y x dy/dx
But why can the chain rule be used here?
I actually don't understand at all..
Any help is appreciated.
Thanks guys!

2. Jun 29, 2013

### Tenshou

your calculation is correct? what is the problem?

okay so, lets lets set $f = y ^2$ this means if you take the gradient of it you get $df = f_x dx + f_y dy$ this is the chain rule, no? since the derivative of x is zero then you can say $f_y = 2y$ plugging this in you get $df = 2y dy$ since we have established that x was zero, but to get your answer all you have to do is "take" the rate of change of x, since that is the parameter you wish to "constrain" your self to and get $\frac{df}{dx} = 2y \frac{dy}{dx}$

3. Jun 29, 2013

### HallsofIvy

This is NOT correct! (Unless that middle "x" indicates multiplication- a very bad notation when x is also used as a variable!) The derivative "d/dx y^2" is 2y dy/dx.

It's pretty straight forward. $u(y)= y^2$ where y(x) is a function of x.
The chain rule says that
$$\frac{du}{dx}= \frac{du}{dy}\frac{dy}{dx}$$

Or course, since $u= y^2$, $du/dy= 2y$ so that
$$\frac{du}{dx}= \frac{du}{dy}\frac{dy}{dx}= 2y \frac{dy}{dx}$$

It's that simple.


text: Speaker : Professor Shin-Yao Jow (University of Pennsylvania) Effective cone of the space of maps from $P^1$ to a Grassmannian 2012-03-06 (Tue) 11:00 - 12:00 Seminar Room 617, Institute of Mathematics (NTU Campus) We will describe the effective cone of the Quot scheme which compactifies the space of maps from $P^1$ to a Grassmannian. This space has seen some applications in enumerative geometry and mathematical physics, and determining its effective cone is a first step in understanding its birational geometry.


text:
# lbfgs.m

Calculates an approximation to the Newton-Raphson search direction using past gradients to build a serviceable substitute to a Hessian. The Hessian matrix is never explicitly formed or inverted. This function is the implementation from section 4 of http://dx.doi.org/10.1090/S0025-5718-1980-0572855-7

## Syntax

    direction=lbfgs(dx_hist,dg_hist,g)

## Arguments

   dx_hist         - history of x increments, a stack
of column vectors, from the latest
to the earliest

dg_hist         - history of gradient increments,
a stack of column vectors, from
the latest to the earliest

## Output

   direction       - LBFGS approximation to the
search direction

## Notes

The L-BFGS algorithm is the default of fminnewton.m, and is a good mix of computational efficiency and fast convergence.


text:
# GCAGS Transactions

Abstract

Gulf Coast Association of Geological Societies Transactions
Vol. 45 (1995), Pages 1-1

# Abstract: Spindletop to Deep Water

James M. Coleman

In 1859, "Colonel" Drake drilled an onshore well to a depth of 69.5 feet; it produced 35 barrels of oil per day. Some 135 years later, drillings and completions of wells are routinely conducted at subsea depths of 20,000 feet and in water depths of 8,000 feet. What a tribute to engineering technology and to development of exploration concepts! Drake and Bissell of the Pennsylvania Rock Oil Company spent the huge sum of $2,490 to complete the first well; today, development of a well in the deep waters of the Gulf of Mexico requires expenditures of several millions of dollars. The quest for petroleum commenced with Drake's discovery, but the "well that changed the world and gave birth to the modern oil industry" was the discovery at "Spindletop" in 1901, atop a salt dome near Beaumont, Texas. While the crew was encountering drilling problems at a depth of 1,020 feet, "...there was a deafening roar and the well erupted like a volcano - - mud, gas, oil, and rocks shooting hundreds of feet into the air;" the well gushed 100,000 barrels of oil a day, and by the end of the year, 400 such gushers were producing oil. By 1903 Gulf Coast salt domes were producing 20 percent of the nation's oil. By 1907, Oklahoma was the nation's top oil-producing state. Even as early as the 1900's, however, there were dire predictions that, "the U. S. was running out of oil." Initial exploration concentrated around salt domes, those regions that had topographic expression or extremely strong gravity anomalies. In 1913 Charles Gould, founder of the University of Oklahoma geology department, presented a paper at an international conference stating that most of the big oil pools were located beneath anticlines, and the rush to locate anticlines was on. A few years later, Wallace Pratt (Humble Oil and Refining) changed exploration strategies by showing that major petroleum reserves could be found not only in association with salt domes and anticlines but also where oil was trapped against faults. With this strategy, he convinced Humble to drill a well on the downthrown side of a fault and discovered one of the largest fields in Texas, the Mexia field. From that date onward, drilling activity increased significantly and major advances were made, such as the development of the Hughes rotary bit and highly precise magnetometers and the initiation of geophysical prospecting. Development of the first downhole electric log by Schlumberger in 1927 increased the geologist's capability of locating potential oil and gas traps and aided in correlation of strata from one location to the next. The alternating sands and shales encountered in the wells were thought simply to be the result of major transgressions and regressions associated with structural activity, and little attention was given to stratigraphic sequences or environmental settings of the major producing horizons. By the mid-1930, most of the major structures onshore in the Gulf of Mexico had been exploited, many deeper wells were encountering shales with few sands, and again predictions were made that "most of the hydrocarbons in the Gulf Coast had been produced." By 1933 however, crude techniques were available for obtaining offshore geophysical surveys, and in 1934 the Texas Company drilled a well one mile from the shoreline, off Louisiana, from a submersible. The offshore industry was born. In 1947 the first bottom-supported platform was installed in 18 feet of water 12 miles offshore--at a cost of$230,000. Complex subsurface structure and stratigraphy as revealed by marine seismic and driller's logs made correlations extremely difficult; responding to this challenge, micropaleontologists began to develop major biostratigraphic schemes, many of which have stood the test of time. Offshore activity increased at a frantic pace, and drilling continued to move farther offshore. The structural integrity of the platforms allowed them to be emplaced in deeper water depths and multiple deeper penetrating wells to be drilled.

By the mid-1950s, 92 offshore platforms had been installed to water depths of nearly 100 feet; by the end of the 1960's, some 500 platforms had been installed in water depths to 350 feet. Significant production in the offshore waters of the Gulf of Mexico had become a reality, and some 200 million barrels of oil and 575 trillion cubic feet of gas had been produced by the end of 1950's.

Offshore seismic improved significantly, and major structures were continually being mapped to the edge of the continental shelf. Because many of the major structures in the relatively shallow waters of the continental shelf had been defined and wells were being drilled deeper, exploration geologists cultivated better definition of the regional stratigraphy and depositional settings of the producing sands. Most of the geologic models being utilized were based on relatively shallow-water settings (i.e., deltaic, fluvial channels, beaches, and shallow-marine settings). As wells continued to be drilled to greater depths and platforms emplaced in deeper waters, many of the geologic models developed for shallow-water producing sequences became inadequate, and more attention was placed on development of deep water environmental models. By the beginning of the 1970s, some 12,500 wells were producing on the continental shelf of the Gulf of Mexico, and development costs of these facilities had approached some $13 billion dollars. Production in the deeper waters of the Gulf of Mexico began in 1979 with the installation of the Cognac platform in slightly over 1,000 feet of water and at a cost in excess of$250 million. Today, platforms are being installed in water depths in excess of 3,000 feet, and plans are evolving for drilling wells in water depths approaching 8,000 feet. The offshore waters of the Gulf of Mexico have indeed proved to be a major hydrocarbon province, and some 4,900 platforms have been installed since the initial one in 1934 in less than 10 feet of water.

In less than 100 years, from "Spindletop" in 1901 to the present production in the deep waters of the Gulf of Mexico, hydrocarbon production in the Gulf of Mexico basin has been a story of unprecedented technological advancements and development of explorational concepts. In Louisiana alone some 1,165,000 producing wells have been drilled and some 25.2 billion barrels of oil and 214 trillion cubic feet of gas have been produced.

End_of_Record - Last_Page 1---------

## ACKNOWLEDGMENTS AND ASSOCIATED FOOTNOTES

Coastal Studies Institute, Louisiana State University, Baton Rouge, LA 70803

Copyright © 1999 by The Gulf Coast Association of Geological Societies


text:
# Reference request: complexity of $k$-partite $k$-SAT

Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT:

given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal from each group. Is $\phi$ satisfiable?

There is an almost straightforward reduction from $k$-SAT to $k$-partite $k$-SAT that takes formula on $n$ variables and produces formula on $kn$ variables. This proves that this problem is exponentially hard under $ETH$.

On the other hand there a very simple $O^*(2^{(1-\frac{2}{k})n})$ time algorithm for this problem: branch over all variables except 2 groups. You will end up with $2$-SAT problem, that can be solved in polynomial time.

As we don't know algorithms with this kind of savings for $k$-SAT, I'd be surprised if there is a simple reduction from $k$-SAT to $k$-partite $k$-SAT that preserves number of variables.

So I have following question:

• Is there an algorithm with running time $O^*(2^{(1-\epsilon)n})$, where $\epsilon$ is independent on $k$?
• If answer for previous question is no, do we know that complexity goes to $2^n$ as $k$ goes to infinity under $SETH$?
Claim: If there exists an $\epsilon > 0$ such that for every $k'$, $k'$-partite $k'$-SAT can be solved in $2^{n(1-\epsilon)}$ time, then SETH fails.
Proof: Suppose such an algorithm exists. We give an algorithm that, for every $k$ solves $k$-SAT in time $2^{n(1-\epsilon/2)}$. Consider a $k$-SAT instance with $n$ variables, apply the sparsification lemma so that it produces $2^{n(\epsilon/2)}$ instances where, in every instance, every variable occurs in at most $(k/\epsilon)^{O(k/\epsilon)}$ clauses. Each of these instances are $(k/\epsilon)^{O(k/\epsilon)}$-partite in the sense that one can find a coloring of the variables into $k' = (k/\epsilon)^{O(k/\epsilon)}$ colors such that no clause contains variables with the same color. If you want clauses of size exactly $k'$ you can replace clauses of size $k$ by $2^{k'-k}$ clauses of size $k'$. Running the $2^{n(1-\epsilon)}$ time algorithm on each of the $2^{n(\epsilon/2)}$ instances of $k'$-partite $k'$-SAT gives an algorithm for $k$-SAT with running time $2^{n(1-\epsilon/2)}$ as claimed.
Note that the above proof essentially reduces $k$-SAT to $k^k$-partite $k^k$-SAT. So an algorithm with running time $2^{n(1-\frac{\log \log k}{\log k})}$ could still be possible, if very surprising.


text: Given an $n$-tuple of positive real numbers $\alpha$ we consider the hyperpolygon space $X(\alpha)$, the hyperkähler quotient analogue to the Kähler moduli space of polygons in $\mathbb{R}^3$. There exists an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo an elementary transformation in the sense of Mukai as $\alpha$ crosses a wall in the space of its admissible values. We describe the resulting changes in the core of $X(\alpha)$ as well as the changes in the nilpotent cone of the corresponding moduli spaces of PHBs. If time permits, we will explain how to obtain explicit formulas for the computation of the intersection numbers of the core components of $X(\alpha)$ and of the nilpotent cone components of the corresponding moduli spaces of PHBs. As a final application, we describe the cohomology ring structure of these moduli spaces of PHBs and of the components of their nilpotent cone. This is joint work with Leonor Godinho.


text:
# Importance of separability vs. second-countability

For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which can be made for the importance of separability.

Let me subsume the situation: Both notions are intended to guarantee smallness known from classical spaces, from geometry and analysis. Second-countability is a stronger condition, but for metrisable spaces both conditions are equivalent—the word “separable” seems to be more popular in these cases (for example in functional analysis and descriptive set theory). However, under most weaker conditions than metrisability, second-countability is not guaranteed by separability (and in fact metrisability is implied by second-countability and regularity, that is Urysohn’s metrisation theorem): For example the space $[0,1]^{\mathbb{R}}$ with the product topology is separable, but not second-countable, although it is compact and even a product of compact Lie-groups (are there nicer spaces?). There are even locally euclidean, separable spaces which are not second-countable, as required in the usual definition of a topological manifold (see this question). In the locally compact case second-countability implies $\sigma$-compactness, which is useful for integration theory, and the space $X$ is second-countable if and only if the space $C_0(X)$ of continuous numerical functions vanishing at infinity is second-countable. For metrisability there are no analoga. For locally compact groups the second-countability is equivalent to the second-countability of $L^2$ with respect to the Haar measure (it should also hold more generally for certain non-degenerate Borel measures on general locally compact spaces).

Some classical analytic methods using sequences can be used for second-countable spaces: For second-countable spaces compactness is equivalent to countable compactness and sequential compactness. In first-countable Hausdorff spaces you can choose convergent subsequences from every convergent net. Especially in first-countable topological vector spaces (or abelian groups) the convergence of the net of all finite partial sums of a set of vectors is equivalent to the unconditional convergence of a series (the series converges independently of the order). In the separable function space $\mathbb{R}^\mathbb{R}$ this does not work.

Some more general points: Second-countability imposes a strict smallness condition (the cardinality of the topology and in the Hausdorff case the cardinality of the space must be at most the cardinality of the continuum), while separable Hausdorff spaces might consist of $\beth_2$ points. Separability has the advantage of being preserved under continuous images–however, it has the big disadvantage of not being preserved under taking subspaces.

Do you know of any important theorems/theories where separability is crucial—not second-countability? Which generalisations of important concepts from classical analysis only depend on separability? Is the popularity of the word/concept of “separability” just due to the special case of metric spaces? I have even seen some authors using the word “separable” instead of “second-countable” (which sounds reasonable, since “second-countable” sounds cumbersome).

• This 1935 reference also uses this convention: ncbi.nlm.nih.gov/pmc/articles/PMC1076540/pdf/pnas01754-0040.pdf May 18, 2013 at 19:06
• Ahem. $\:$ If countable choice then second countability is a stronger condition than separability. $\;\;$
– user5810
May 19, 2013 at 6:16
• This might not be the right decade to make this comment, but separability seems to be just the poor man's version of overtness, something Paul Taylor has been pointing out. If this is indeed the case, then separability would indeed be a suboptimal notion. May 19, 2013 at 7:07
• And this would be Paul Taylor: paultaylor.eu The photo is a bit blurry, but that's how I percieve Paul most of the time anyway. May 19, 2013 at 12:42
• Thank you, Andrej, but you took the photo! I am a long way from being able to discuss separability and second countability, but for a published introduction to overtness see A Lambda Calculus for Real Analysis www.paultaylor.eu/ASD/lamcra or if you are keen to see an incomplete attempt to introduce the idea to "ordinary mathematicians" you could look at Overt Subspaces of $R^n$ www.paultaylor.eu/drafts/overtrn.pdf . I am not sure that the nLab account is correct. May 19, 2013 at 14:59

An arbitrary product of separable spaces satisfies Suslin´s condition (i.e. any disjoint family of open sets is countable). I find this result remarkable since separability is not preserved under (large) products while Suslin´s condition might or might not be preserved under (even finite) products, depending on the underlying axioms of set theory.

• That is indeed remarkable. Although it is a little bit exotic, Suslin’s condition is a well-known, classical property of the real numbers and its finite products and the theorem you mentioned features separability prominently. I think this is closest to those kinds of theorems I was looking for. May 24, 2013 at 23:12

Separability can be used to study the Stone-Cech compactification of a countable discrete space. Recall that if $X$ is a discrete space, then the Stone-Cech compactification $\beta X$ of $X$ is precisely the set of ultrafilters on $X$. We can therefore use separability to prove facts about ultrafilters without mentioning ultrafilters.

First, we use separability to observe that $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$. Since $\beta\mathbb{N}\subseteq P(P(\mathbb{N}))$ as the set of ultrafilters on $\mathbb{N}$, we have $|\beta\mathbb{N}|\leq|P(P(\mathbb{N}))|=2^{2^{\aleph_{0}}}$. To prove the other direction, let $I$ be a set of cardinality continuum. Then since the product of continuumly many separable spaces is separable, the product space $\{0,1\}^{I}$ is separable. Therefore let $A\subseteq\{0,1\}^{I}$ be a countable dense subset. Then there is a surjective function $f:\mathbb{N}\rightarrow A$. Therefore the function $f$ extends to a continuous function $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$. Since the image $\overline{f}[\beta\mathbb{N}]$ is a compact set, the set $\overline{f}[\beta\mathbb{N}]$ is a closed subset of $\{0,1\}^{I}$, so $\overline{f}[\beta\mathbb{N}]=\{0,1\}^{I}$ since $A\subseteq\overline{f}[\beta\mathbb{N}]$. Since $\overline{f}:\beta\mathbb{N}\rightarrow\{0,1\}^{I}$ is surjective, we have $2^{2^{\aleph_{0}}}=|\{0,1\}^{I}|\leq|\beta\mathbb{N}|$, so $|\beta\mathbb{N}|=2^{2^{\aleph_{0}}}$.

Separability may also be used to prove facts about the Rudin-Keisler ordering. The Rudin-Keisler ordering is the preordering $\leq_{RK}$ on the class of ultrafilters where if $\mathcal{U}\in\beta X,\mathcal{V}\in\beta Y$ are ultrafilters, then $\mathcal{U}\leq_{RK}\mathcal{V}$ if there is a continuous $f:\beta Y\rightarrow\beta X$ with $f(\mathcal{V})=\mathcal{U}$ and $f[Y]\subseteq X$. The motivation for the notion of the Rudin-Keisler ordering is that the Rudin-Keisler ordering measures the size of an ultrapower. In particular, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathcal{A}^{\mathcal{U}}$ is elementarily embeddable in $\mathcal{A}^{\mathcal{V}}$ for each first order structure $\mathcal{A}$.

The Rudin-Keisler ordering is a pre-ordering on $\beta\mathbb{N}$. One can use separability to show that every subset of $\beta\mathbb{N}$ of size at most continuum has an upper bound in $\beta\mathbb{N}$. Assume that $I$ is an index set of cardinality at most continuum and $x_{i}\in\beta\mathbb{N}$ for $i\in I$. Then $\mathbb{N}^{I}$ is separable since the product of at most continuumly many separable spaces is separable, so there is a countable dense subset $A\subseteq\mathbb{N}^{I}$. Therefore let $f:\mathbb{N}\rightarrow A$ be a surjective function. Then $f$ extends to a unique continuous function $\overline{f}:\beta\mathbb{N}\rightarrow(\beta\mathbb{N})^{I}$. The function $\overline{f}$ is clearly surjective, so there is some $x\in\beta\mathbb{N}$ with $\overline{f}(x)=(x_{i})_{i\in I}$. Therefore if $\pi_{i}:(\beta\mathbb{N})^{I}\rightarrow\beta\mathbb{N}$ is the projection mapping, then $\pi_{i}\overline{f}(x)=x_{i}$ for $i\in I$, so $x_{i}\leq_{RK}x$ for $i\in I$.

It should be noted that there are very similar proofs of the above two results using independent sets and independent partitions (the proofs using independent sets and independent partitions are essentially the same proof. See Andreas Blass's comment below). Furthermore, the two above results can be generalized to larger cardinals with the same proofs. In particular, if $X$ is a discrete space, then $|\beta X|=2^{2^{|X|}}$. Furthermore, every subset of $\beta X$ of cardinality at most $2^{|X|}$ has an upper bound in $\beta X$. To prove these facts, one uses a generalization of the notion of separability called the density and the generalized proof is very similar to the original proof. If I remember correctly, the book The Theory of Ultrafilters by Comfort and Negrepontis also gives generalizations of these facts to large cardinals such as compact cardinals.

I hope the above results clear up any confusion about the importance of separability in non-metrizable spaces.

• Concerning "the Rudin-Keisler ordering measures the size of an ultrafilter": Did you mean "the size of an ultrapower"? May 19, 2013 at 18:40
• I'd go a bit further than "there are very similar proofs ... using independent sets and independent partitions." I think of the existence of continuum many independent partitions of $\mathbb N$ and the separability of a product of continuum many separable spaces as being essentially the same theorem. Each is deducible easily from the other. Being more combinatorial than topological, I tend to view the former (and its generalizations to higher cardinals) as the main point, and to view separability as a nice way to make it look topological for those whose tastes differ from mine. May 19, 2013 at 18:45
• Yes. I did mean the size of an ultrapower. May 19, 2013 at 19:31
• It is a very nice and interesting answer, but I accepted Ramiro’s answer, because it describes more that kind of property I was looking for. Thanks. Jun 5, 2013 at 20:24

I second the idea that second countability is more fundamental than separability --- topologies are defined in terms of open sets, not points, and second countability is the natural "countability" condition on the family of open sets. It just says that the topology is countably generated.

Important theorems where separability is crucial: the basic example is that a continuous image of any separable space is separable. Even a quotient of a second countable space need not be second countable.

Is the popularity of the word/concept of "separability" just due to the special case of metric spaces? Yes, I think so. But that's a pretty important special case! I just finished writing a book on measure theory and functional analysis, and I found that by restricting attention to separable Banach spaces and their duals I was able to get by just fine without mentioning generalized convergence (nets/filters). For instance, the weak* topology is metrizable on the unit ball of the dual of a separable Banach space, which is good enough for most purposes by the Krein-Smulian theorem.

• What makes the simple fact that images of separable spaces are separable an important theorem? May 18, 2013 at 19:31
• @Martin That was indeed not that kind of theorems I was looking for (a property does not get important by being well-behaved when taking images). But I think it is worth mentioning (I have also mentioned it in my question). May 18, 2013 at 19:50
• @Martin: do you wish to tell me that this simple fact is not important? May 18, 2013 at 20:50
• No, I was asking an honest question. I agree that it is a useful fact, but it seems more related to continuity than separability per se: It is a characteristic property of continuous functions that the image of the closure is contained in the closure of the image. Thus, dense sets are mapped to dense sets. What makes the instantiation of this observation to countable dense sets particularly noteworthy and important? May 19, 2013 at 3:53
• It sounds like you are asking why separability is important. Speaking from my area of expertise, I could tell you any number of basic theorems about separable C*-algebras which fail in the nonseparable case. May 19, 2013 at 6:08

Let me elaborate a little on your observation:

Separability has the advantage of being preserved under continuous images–however, it has the big disadvantage of not being preserved under taking subspaces.

Even hereditary separability does not imply second-countability (the Sorgenfrey Line would be a counterexample). It is however consistent with the axioms of set theory that every hereditarily separable space enjoys the following weakening of second-countability: every open cover has a countable subcover. This is known as the Lindelof property.

THEOREM (Todorcevic) Assume the Proper Forcing Axiom. Then every hereditarily separable space is Lindelof (and hence hereditarily Lindelof).

Whether hereditarily separable implies hereditarily Lindelof is however independent of the axioms of set theory. For instance, assuming as little as the Continuum Hypothesis there is a counterexample which even has very good extra topological properties (it's compact, first-countable and homogeneous). See:

de la Vega, Ramiro; Kunen, Kenneth, A compact homogeneous S-space., Topology Appl. 136, No. 1-3, 123-127 (2004). ZBL1048.54013.

So suppose that a topological space $$X$$ is separable (or that it at least contains a dense subset which is bijective to an ordinal). Then it is selective, which in this context means that if $$(U_α)_{α \in A}$$ is a family of open subsets of $$X$$, then $$\prod_{α \in A} U_α \neq \emptyset.$$ This is the axiom of choice for open subsets of $$X$$, which is required for the implication second-countability $$\Rightarrow$$ separability.

• By the way, in the above case selectivity follows almost immediately from the well-foundedness of the given ordinal. Jan 11 at 5:39


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15 Jan 2009, 12:16
Congrats to everyone. I will definitely be at the GBR and I am 95% sure I will be in Ann Arbor next fall. I look forward to meeting some of you in April as well as next fall!

Good luck to those on the waitlist and maverickMBA, I am sorry but my slot is taken.
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15 Jan 2009, 12:18
Also, I am in Syracuse, NY so Ann Arbor is going to be like a Hawaiian vacation for me.
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15 Jan 2009, 16:06
filmcity wrote:
coolmayank123 wrote:
...applicants who apply nd dont withdraw to ross r people who didnt get through nd know very well that they dont stand a chance in H/S/W/CHICAGO/KELLOGG/MIT..so Ross is the best bet they have because its definitely the best tier 2 school.

Let me get this straight...

Are you implying that Ross admits/current students are losers who can't get an admit at schools like Booth and Kellogg? Are you really posting this on "Admitted Ross Class of 2011" thread? What a way to ruin the party.

People, dont fret over Mayank's comments!!

He has build quite a reputation for himself in BW forum as well for extremely abominable comments ( remember Rankingsgod!!!). Fortunately, he is smart enough to leave enough footprints ( IIT Grad, First job in BCG, working in assett mgmt now etc. etc). I wonder, how many people would love to work with him in future...

Is he the same guy who has lost his sanity after Chicago admit? I guess he has not been able to digest his accomplishment. He keeps on posting and deleting his messages all over, at least defend what you post!
And as ncprasad says there are tons of other reasons for choosing schools. Two days back I met this guy who is choosing Duke over Kellogg. I, amongst other reasons, chose Duke because my spouse has a stable job in Durham. Ofcourse in the world of coolmayank, both of us are losers !!
Ross is an excellent school and nobody's comment can change that.
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15 Jan 2009, 16:22
matty0212 wrote:
ryguy904 wrote:
And on another note - similar to the weather poll question that I posted, it "feels like" -5 degrees in Ann Arbor right now. Holy crap!

http://www.weather.com/weather/local/US ... centsearch

And this is why I'm glad I grew up near Chicago. I'm not going to say I'm immune to the cold, but I bet I'm going to fare better than those comming from sunny SoCal.

And if you were still in Chicago now you'd be enjoying our -30 degree day.
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15 Jan 2009, 17:15
I just received a phone call from a second year just to congratulate me and see if I had any questions. This was really cool, but when he first said that he was from the admission committee at Ross, the first thing that popped in my head was that they made a mistake like the Kellogg ordeal. My heart skipped a beat but I was fine after I heard the congratulations, etc. I just thought it was sort of funny.
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15 Jan 2009, 17:30
Congrats everyone!! Good luck to those on the waitlist!
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15 Jan 2009, 17:34
dghazer wrote:
I just received a phone call from a second year just to congratulate me and see if I had any questions. This was really cool, but when he first said that he was from the admission committee at Ross, the first thing that popped in my head was that they made a mistake like the Kellogg ordeal. My heart skipped a beat but I was fine after I heard the congratulations, etc. I just thought it was sort of funny.

Uh yeah...I could imagine how that would freak you out!
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16 Jan 2009, 18:31
So, is anyone else going to the admitted student event in San Francisco on 2/5, or any other city for that matter?

Boston - 3/? (TBD)
Chicago - 2/5
LA - 3/? (TBD)
NYC - 2/5
SF - 2/5
DC - 2/4
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16 Jan 2009, 19:35
YihWei wrote:
So, is anyone else going to the admitted student event in San Francisco on 2/5, or any other city for that matter?

Boston - 3/? (TBD)
Chicago - 2/5
LA - 3/? (TBD)
NYC - 2/5
SF - 2/5
DC - 2/4

I'll be at the SF one...

Is it actually set yet?
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16 Jan 2009, 19:41
nomsg7111 wrote:

I'll be at the SF one...

Is it actually set yet?

Sure is!

Le Colonial
20 Cosmo Place
415-931-3600

Thursday 2/5, 7-9PM
Business Casual (I wonder if it's OK to wear my M hat? )
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16 Jan 2009, 19:50
I'm going to miss the reception in DC, at least I'll make GBR.
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17 Jan 2009, 12:25
YihWei wrote:
nomsg7111 wrote:

I'll be at the SF one...

Is it actually set yet?

Sure is!

Le Colonial
20 Cosmo Place
415-931-3600

Thursday 2/5, 7-9PM
Business Casual (I wonder if it's OK to wear my M hat? )

hah...the hat will make it easy to pick you out. Thanks for the info.

I've wanted to try "Le Colonial," I think its upscale Vietnamese food. It was listed on dine about town last year. I hope food is good .
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18 Jan 2009, 00:39
So I guess I'm in it to win it with Ross now that Yale is official.
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19 Jan 2009, 22:40
Hey does anyone know what we will be receiving in the admit folders? Is it real information or just a bunch of Ross fluff?

I know I am planning way ahead, but I kinda want to start locking up an apartment and other stuff. Anyone have an idea of what I can be doing right now to get prepared for next fall?
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19 Jan 2009, 23:50
dghazer wrote:
Hey does anyone know what we will be receiving in the admit folders? Is it real information or just a bunch of Ross fluff?

I know I am planning way ahead, but I kinda want to start locking up an apartment and other stuff. Anyone have an idea of what I can be doing right now to get prepared for next fall?

Just got mine tonight. There isn't really anything too substantial in it, although it is a pretty nice binder. Most of the stuff can be found on craigslist or the Ross website. Comes with a T-shirt too...a large...I need a medium...
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Schools: Michigan
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20 Jan 2009, 00:04
nomsg7111 wrote:
dghazer wrote:
Hey does anyone know what we will be receiving in the admit folders? Is it real information or just a bunch of Ross fluff?

I know I am planning way ahead, but I kinda want to start locking up an apartment and other stuff. Anyone have an idea of what I can be doing right now to get prepared for next fall?

Just got mine tonight. There isn't really anything too substantial in it, although it is a pretty nice binder. Most of the stuff can be found on craigslist or the Ross website. Comes with a T-shirt too...a large...I need a medium...

Just looked at the tuition. I'm scared.
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Schools: Ross, Cornell ($), Kellogg, Wharton, Stanford, Berkeley ($\$)
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20 Jan 2009, 09:00
I'll be at the reception in DC in a couple weeks. Hopefully I'll get to meet some of you there.
Also - I did my undergrad at Michigan, so if anyone has any questions I can help with, let me know
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20 Jan 2009, 15:29
Just got the folder today and yeah, nothing much I havent seen online already. I got a large T-shirt as well and I need to trade up the other way. So if anyone out there received a XL or XXL, I will trade you in April...lol.
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20 Jan 2009, 20:15
Just got my admit package today. I can't stop flipping through the binder! Also, lucky me that L is the perfect shirt size!!!

chitchat - when'd you graduate and which school were you in? I was LSA 2003.

Go Blue!
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21 Jan 2009, 20:59
Hi NC,

I always wanted to get in touch with you as I always find your posts very valuable. I am an Indian R1 applicant to R1, but have been waitlisted. Honestly, Ross is my top choice school and I really want to get in there this fall. Do you have any advice for me?

With so many applicants waitlisted, how can I improve my chances of getting admitted?

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LGD 4033 + Ostarine Tabs 20mg$ 46.98 â $64.98. 16 0 obj Once this has been done, we can talk about category of Picard 2-stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2-stacks. Length 3 complexes of abelian sheaves over S placed in degrees [ 2;0] form a 3-category C[ 2;0](S) by adding to the regular morphisms of complexes, the degree -1 and -2 morphisms. ?�:��d��1 ��Q����k�g\�R�)_p��B������n�s-���H��o�@h\��qp���Ŝ�4��� ��������z���k?vqCcׯ��p�v����q�E��P���|��W�˛s=_����j$��i�* 5��B�2�4���տ��H�O�� 41 0 obj << Stacks. Deformation theory of representable morphisms of algebraic stacks 3 where x0: X0 âY0 is ï¬at (we often denote a ï¬at deformation of Xto Y0 simply by x 0: X âY0).The set of ï¬at deformations of Xto Y0 form in a natural way a category (in fact a groupoid; (3.1)), and in section 3 we prove The construction R7!X(R) determines a functor from Ring k to the category of sets. BC is a simplicial set with n-simplices BC n consisting of strings of morphisms a 0!f1 a 1! More concretely this means that a 1-stack on a site SS (or more generally (2,1)-site or eâ¦ 8ÈýµUüX1º¶Ïl)Æ$oHùêñþ_¡í,Û23s. Add to Wishlist. Idea. Stacks. 8 0 obj The first is to investigate a combinatorial aspect of the 2-category of toric algebraic stacks defined in \\cite{I2}. 1m drva.jpg 666 × 624; 49 KB. y review the language of stacks. The column will list new and revised finding aids recently made available online. The second is to give a geometric characterization of toric stacks. (I should just call this setup "the usual"). In fact, there is an equivalence of 2-categories between compactly generated stacks and those classical topological stacks which admit locally compact atlases. Quick View. endobj the 2-category of algebraic stacks over Aï¬ Î± is equivalent to a full sub-2-category of algebraic stacks over Aï¬ Î². Select options. The inclusion Sets â Groupoids (associating to each set the category concerning the invariance of a 2-category of stacks under change of base. This 2-category, called the 2-category of compactly generated stacks, is the analogue of classical topological stacks, but for a different Grothendieck topology. Add to Wishlist. Showing 13â24 of 29 results. 7. 12 0 obj Generally, then, n-stack is a synonym for (n+1)-sheaf, or more restrictively for (n+1,1)-sheaf. Performance Clinics are World's premium quality sports research product suppliers. Frequently Asked Questions; Submit A Ticket; Contact (877) 368-4446; Home; About Us. Stacks. This category has the following 2 subcategories, out of 2 total. Therefore schemes are algebraic spaces, algebraic spaces are DM stacks, and DM stacks are Artin stacks, which is the most general category of algebraic stacks considered. KRATOS (Liquid. A stack A A is called a moduli stack for certain structures, if for any other object X X the groupoid of morphisms X â A X \to A into A A is equivalent to the groupoid of these kinds of structures on X X.. .214 A Weak Colimits and Pseudo-Colimits221 ... opTological and di erentiable stacks are 2-categorical in nature. Quick View; Select options. xڭ]s�6�ݿBo�f,_$�������%�X��L��$�J�*I�u���.�H����lB�b����f���:�vy��u���0�E�j� \�vs�E>_� �]�UKc���/��l�0���v����z� 13 0 obj %���� category), while the notion of pseudo-functor from a category to a 2-category is de ned in [Stacks, Tag 003N] (and see also De nition 2.1 below). 1 sb600 camera right and level with the chips, 1/16 power, through 12'' snoot. Pro-Milnor squares 11 2. 17 0 obj 9 0 obj Stacks. Add to cart. With Alpha 1-andro you will lean out while producing rock hard muscle and increased strength. Quick View. << /S /GoTo /D (section.4) >> stream form a category, morphisms between morphisms of stacks (i.e., natural transformationsÏX: FX â GX satisfyingGf ÏX = ÏY Ff)arewritten as M â 4 * N. Notethatall2-morphismsareinvertible,sinceallmaps inthecategoriesM(X) andN (X) areinvertible. Nootropic Supplements, Stacks. Let 2Pic(S) be the 3-category of Picard 2-stacks where 1-morphisms are additive 2-functors, 2-morphisms are natural 2-transformations, and 3-morphisms are mod-i cations. . SALE. OT��h����������{��9���ܸ�쿸����Ԭ�̆? FIXME. Triggered with Cactus V4s. III.7 The 2-Category of Gerbed E ective Étale Stacks. 5 0 obj Now I have following questions: 2-category as well as of a 2-functor between such 2-categories that respects Picard structure. 1 0 obj Let C be the class of maps RËE =S of Fib(E), where Sranges through the objects of E, E =S is the category of objects of Eover Sand Ris a covering sieve (or, re nement) of S. 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Our products are the highest quality / purity on the market, produced in the most sophisticated and highly regulated US facilities available and tested by World's most trusted quality assurance analysts with quality reports available on request. << /S /GoTo /D [18 0 R /Fit] >> Stacks. Namely, the range of F on objectswillbeasetF(Ob(A)) andthenwemaythinkofF asafunctorbetween Aand the full subcategory of the category of sets whose objects are elements of F(Ob(A)). In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. The rst part is centered around a certain topology on a bred category over a site. In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. . Rick Jardine Stacks and homotopy theory. The base change property 8 1.2. Nisnevich, cdh, and Milnor squares of stacks 12 2.2. endobj The first is to investigate a combinatorial aspect of the 2-category of toric stacks. Stacks in the category S= Sets are just groupoids. The term stack, is a traditional synonym for 2-sheaf or often just (2,1)-sheaf(see there for more details). endobj 40ml/50mg per ml = 2 grams) $97.98. As our archives staff work on an ongoing basis to arrange, preserve, describe and make available to the public the materials under our care, we will spotlight new additions to the website in a regular feature from Out of the Stacks. By de nition groupoid is a category in which all morphisms are isomorphisms. (On the invariance of stacks under change of base) endobj Let Î² > Î±. >'m}��zS?�?��I(��#zA�%ҁT�*)�}p$U�s��98�Z��}��hJ�טv_v-��u���*� v��oO����g#Bo���c�+��S�&+��ٝ���+;.��m&���)2�[}��tq�p������8ƑD � ��}Vm��� A (2,1)-sheaf / stack is equivalently a 1-truncated (â,1)-sheaf/â-stack. Add to Wishlist. Here it is. Stacks. Add to Wishlist. endobj Cloud Hosting & Storage If Xis a k-scheme and R2Ring k, then an R-valued point of Xis a map SpecR!Xin the category of k-schemes. endobj We will give a definition of (strict) $2$-categories as they appear in the setting of stacks. (The fibred topology on a fibred category over a site) I learned that a stack is a contravariant functor $\mathcal{F}:\mathcal{C}^{\operatorname{op}}\to\mathbf{Grpd}$ satisfying a descent property and, categories fibered in groupoids over $\mathcal{C}$ is an intuitive way to think about stacks, where $\mathcal{C}$ is a site (category equipped with a coverage). . Suppose that p : S â Aï¬ Î± is an algebraic stack. Section 2) even though the category of sets is âbigâ. on the vanishing of negative K-groups for this class of stacks. Women in Gamesâ¦ Read More The collection of all R-valued points of Xforms a set X(R). This list may not reflect recent changes (). 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The category of toric stacks - Volume 145 Issue 3 - Isamu Iwanari Before you read this take a look at Section 4.28 and Example 4.30.2.Basically, you take this example and you write out all the rules satisfied by the objects, $1$-morphisms and $2â¦ That is, given two such stacks X and Y and two morphisms f: X !Y and f0: X !Y ; INTRODUCTION 11 (Stacks over stacks in groupoids) Add to Wishlist. Theorem 2.3.2. 2 The model category of D-stacks. Alpha 1-Andro is an advanced product which is more potent than the competition. %PDF-1.5 4 0 obj The Boston Stack Joins Women in Games to Support Local Developers! It is also often used more restrictively as a synonym for (2,1)-sheaf. Select options. . 1 Effect stacking 1.1 Aura stacking 2 Champion statistic stacking 2.1 Stacks additively 2.1.1 Effects that stack additively 2.2 Stacks multiplicatively 2.2.1 Effects that stack multiplicatively 2.3 Special stacking Stacking has two meanings, the one being used can be deduced from the context. Let Ring k denote the category of nitely generated k-algebras. Stacks of lobster traps in Norra Grundsund 8.jpg 4,860 × 3,044; 16.66 MB Stacks of lobster traps in Norra Grundsund 9.jpg 4,779 × 3,058; 14.75 MB Stapel Europoolpaletten.jpg 4,000 × 2,667; 2 MB We establish an equivalence between the 2-category of toric stacks and the 1-category of stacky fans. Strobist: 1 sb600 camera left and high, 1/4 power, 1/2 cto, through speed grid. For this purpose, we obtain two main results. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. Select options. << /S /GoTo /D (section.2) >> Lets Ride Stack$ 175.99 \$ 109.99. Milnor squares of stable â-categories 8 1.1. Introduction This article is a sequel to [15] and it consists of two parts that can be read independently. /Filter /FlateDecode Improve the theorem below and move it to a more appropriate spot. 4.29 2-categories. ... underlying category of sheaves. !f2 a n 1!fn a n Simplicial structure def. This is part of a whole hierarchy of higher categorical generalizations of the notion of sheaf. Use is 2 to 3 capsules a day with 4 hours in between each capsule. STACKS AND SHEAVES OF CATEGORIES, I 655 between them. Home / Shop / Stacks. Even more, one may de ne the 2-category of stacks of groupoids over X ; by St(X ) âSt(Site(G)): Theorem (D.C).
2020 2 category of stacks


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# calculate the mass percent composition of carbon in c2h6

Solution for Calculate the mass percent composition of carbon in each the following carbon compounds. #CH_4# b. In chemistry, the formula weight is a quantity computed by multiplying the atomic weight (in atomic mass units) of each element in a chemical formula by the number of atoms of that element present in the formula, then adding all of these products together. The percent composition of carbon in heptane with a chemical formula of C7H16 is 83.90 percent. . CH4 + C2H6 arrow CO2 + H2O 13.43g 64.84g Would I do . Favorite Answer. 1) Assume 100 g of compound is present. If 61.4 g of co2 are produced when 22.7g c2h2 react with sufficient oxygen, what is the percent yield … Ethane has molecular formula C2H6. = 74.91; b. Calculate the mass percent composition of carbon in each the following carbon compounds - Science Mathematics 79.88 \%$c. C2H5Cl, a. Convert grams C6H8O6 to moles or moles C6H8O6 to grams. 2. Percent Composition Calculator. Calculate the percent composition (by mass) of the following hydrocarbon: C 2 H 6 . So, it consists of atoms of different elements. What is the mass percent of carbon in methane, CH4? Answer to Calculate the mass percent composition of carbon in each carbon-containing compound a. CH4 b. C2H6 c. C2H2 d. C2H5Cl B. Percent Composition Calculator is a free online tool that displays the percentage composition for the given chemical formula. This site explains how to find molar mass. CaS. C2H5CI Solution 71E Step-by-step solution Step 1 of 3 The limiting reactant (or limiting reactant) in a chemical reaction is the substance that is totally consumed when the chemical reaction is complete. The formula weight is simply the weight in atomic mass units of all the atoms in a given formula. The percentage mass of nitrogen in one of the oxides is 36.85%. . Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy, Whoops, there might be a typo in your email. = 79.89; c. 92.26; d. … Browse the list of 65.% carbon and 35.% hydrogen. Wasn't, Calculate the mass of carbon in grams, the percent carbon by mass, and the n…, Calculate the mass percent composition of carbon in each carbon-containing c…, Calculate the mass percent composition of carbon in each compound. Molar mass of C6H8O6 = 176.12412 g/mol. 60.% carbon and 40.% hydrogen. The percent composition can be found by dividing the mass of each component by total mass. #CH_4# b. Calculate the mass percent composition of carbon in each the following carbon compounds - Science Mathematics How do you calculate the mass percent composition of carbon in each carbon-containing compound? First, work out the relative formula mass of the compound, using the Ar of Each of the values from the Carbon and Hydrogen. Question: Calculate The Mass Percent Composition Of Carbon In Each Of The Following Carbon Compounds. The molecular mass off carbon is 12 while the given way to 16 point zero photo grams in 200. C2H2d. C2H6c. Using the chemical formula of the compound and the periodic table of elements, we can add up the atomic weights and calculate molecular weight of the substance. Calculate the percent composition of each of the following compounds: C2H6. 1.20 x 10^24. 2) Convert that %N and 100 g to mass N and mass O See, we have two weapons off carbon. We have one carbon and its weight is 12.01 and 2 oxygen. . Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). a. (1 u is equal to 1/12 the mass of one atom of carbon-12) Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol. 1. B. Mass of Hydrogen in 124 g of Ca(C2H3O2)2. It turns out the mask percentage for coven is 92.4 Well, for the last case, the given beat is 64.54 Hence, the mask wasn't age is 37.2. ... What is the formula of the compound between calcium and sulfur that has the percent composition 55.6% (by mass) calcium and 44.4% (by mass) sulfur? C. 70.% carbon and 30.% hydrogen. Answer … a. CH4 b. C2H6 c. C2H2 d. C2H5Cl C4F8, C2H6, C3H4 C8H13O5N C4F8, C2H6, C3H4 C8H13O5N This problem has been solved! All these compounds … Mass Percent Problem . C2H6 4. Ethane has molecular formula C2H6. Give the gift of Numerade. 70.% carbon and 30.% hydrogenD. To calculate percentage composition we will take the example CO2 it stands for Carbon-Di-Oxide. 60.% carbon and 40.% hydrogen. How many grams of Fe2O3 are there in 0.500 mol of Fe2O3? What is the percent composition of carbon and hydrogen in ethane? 1. Calculate the mass percent composition of carbon in each carbon-containing compound. This is not the same as molecular mass, which is the mass of a single molecule of well-defined isotopes. Molecular weight calculation: 12.0107*6 + 1.00794*8 + 15.9994*6 ›› Percent composition by element Calculate the mass percent composition of carbon in each compound. Molecular mass of C2H2 = 26.0 g C2H2. C3H6 3. Molecular weight calculation: 1.00794*2 + 12.0107 + 15.9994*3 âºâº Percent composition by element Enter the percentages of carbon and hydrogen numerically, separated by commas. C2H6 Express … Mass of Nitrogen in 378 g HCN. For bulk stoichiometric calculations, we are usually determining molar mass, which may also be called standard atomic weight or average atomic mass. 1 Answer Ernest Z. Oct 15, 2016 The percentages by mass are a. 37.22$\%\$, cited by the total B off one mole of compound in 200 we can determine the mosque wasn't age of carbon. a. CH4 b. C2H6 c. C2H2 d. C2H5Cl Finding molar mass starts with units of grams per mole (g/mol). A hydrocarbon is a compound that contains mostly carbon and hydrogen. The percent composition is used to describe the percentage of each element in a compound. Acetylene gas, c2h2, burns in oxygen to produce carbon dioxide and water. C. 70.% carbon and 30.% hydrogen. (a) \mathrm{C}_{2} \mathrm{H}_{2} (b) \mathrm{C}_{3} \mathrm{H}_{6} (c) \mathrm{C}_{2} \math… A. Solution: the typical way . 8 years ago. The percent composition is used to describe the percentage of each element in a compound. Calculate the molecular weight D. 75.% carbon and 25.% hydrogen. A mixture of methane (CH4) and ethane (C2H6) of mass 13.43g is completely burned in oxygen. common chemical compounds. What is the identity of the metal M? The answer to “Calculate the mass percent composition of carbon in each compound. Calculate the percent composition, by mass, of carbon in {eq}Na_2CO_3 {/eq}. ›› C2H6 molecular weight. Problem: Ethane has molecular formula C2H6. C2H6O please explain how you get the answer. The percent composition can be found by dividing the mass of each component by total mass. Weights of atoms and isotopes are from … Answer to Calculate the mass percent composition if carbon in each the following carbon compounds. How do you calculate the mass percent composition of carbon in each carbon-containing compound? The percent composition of CO is 42.88 percent carbon by mass and 57.12 percent oxygen. C3H4: Mass % C = 36.033 x 100 / 40.0641 = 89. Formula weights are especially useful in determining the relative weights of reagents and products in a chemical reaction. John. a.) The atomic weights used on this site come from NIST, the National Institute of Standards and Technology. 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