text:
# Identifying an unknown compound(w/ NMR/IR)?

I need some help identifying the following unknown compound. Any suggestions or hints regarding the possible compound would be highly appreciated.

So far I am thinking the broad peak on the IR is an alcohol that somehow mixed with possible sp3 CH bonds.And the peak around 1600 around is I think a sp Carbon-Carbon bond. The compound should have four distinct groups of protons. Beyond this basic analysis, I couldn't really come up with a compound. Please help.

EDIT(07/09/14): I have updated the question with newer NMR(below the first NMR) and IR.

Some other additional info I worked out in the Lab:

look: white powder like solid.

mp: 116.3-119.0

Solubility Water - insoluble NaOH - insoluble Sulfuric Acid - soluble

Classification Tests: Water Insoluble Phenol Test - positive; a dark red precipitate formed. Chromic Acid Test - positive; green precipitate formed. Bromine Test for Unsaturation - positive; Bromine discoloration happened until about 2 ml of Bromine was added. White precipitate formed.

This is my first time doing these tests so please don't consider their results too strictly. My Instructor said IR is much safer way to determine functional groups.

• Any initial guesses based on the information you have so far? – jonsca Jul 2 '14 at 4:07
• This appears to be a homework question, please share your thoughts and attempts towards the solution. You could start at looking, where those protons are, identify which kind of IR band (~3400) it is, last look at the coupling constants. – Martin - マーチン Jul 2 '14 at 4:18
• @jonsca Edited with some ideas but can't think of anything concrete. This is my first time doing this without anything else(in most other cases, I have been given a molecular formula to work with). Any help would be appreciated ! – Pepria Jul 2 '14 at 4:32
• The signal at 7.15 is chloroform, so you only have three signals belonging to your substance – Mad Scientist Jul 2 '14 at 5:48
• Have you got a full display of the NMR spectrum? Specifically, I would like to see a water impurity in your solvent at 1.6ppm, and possibly any -OH peaks which would be likely to broadened significantly, but still visible in CDCl3. Is the baseline of the NMR spectrum flat, or does it contain any underlying exchange-broadened signals (as your IR would indicate)? Can you describe your process for collecting your IR data? I will collate my thoughts on the structure based on NMR, but the IR is somewhat contradictory. Are you sure your solvents were dry? – long Jul 4 '14 at 7:55

Have a look at typical IR bands (other source) and identify them.

- 3424 phenolic $\ce{OH}$ stretch
- 1640 aromatic $\ce{C=C}$ bending or $\ce{C=C}$ alkenyl stretch
- 1264 aromatic $\ce{C-O}$ stretch
- 747 aromatic $\ce{C-H}$ bending

Have a look at you spectrum, identify your residual solvent signal.

7.15 is chloroform $\ce{CDCl3}$

Have a look at the typical $\ce{{}^{1}H-NMR}$ shifts table and identify the protons.

These are aromatic protons

What is the summary of your first conclusions?

It's an aromatic compound with an hydroxyl group.

How many protons?

multiple of 5

Calculate the coupling constants via $$J=\Delta\delta_{\mathrm{ppm}}\cdot\nu_0$$ and identify the splitting pattern. (I can only assume it is a 500MHz spectrometer, hence $\nu_0=500~\mathrm{MHz}$.)
Example: Doublet at 7.55 ppm: $$J=(7.5615-7.5461)\cdot500~\mathrm{MHz}=7.7~\mathrm{Hz}$$ Solutions:

7.55 doublet, $2\ce{H}$, $J=7.7~\mathrm{Hz}$
7.40 doublet of doublet (fake triplet), $2\ce{H}$, $J=7.6~\mathrm{Hz}$, $J=7.5~\mathrm{Hz}$
The last one is tricky because there is one peak not picked
It should be at 7.30 a doublet of doublet, $1\ce{H}$, ${}^3J=7.7~\mathrm{Hz}$, $J=7.6~\mathrm{Hz}$

Disclaimer:
Please read long's answer and follow his thoughts. They make a lot of sense and hint that the compound we are looking at is not the one I am proposing. But due to my lack of current knowledge, I cannot make another suggestion.

Combine all you see and you will get a clue which substance it is.

7.55 ortho
7.40 meta
7.30 para
Phenol.

• The IR spectrum looks rather strange, very few signals compared to a reference spectrum from the SDBS. I also thought it was phenol after a quick look, but the IR confused me. – Mad Scientist Jul 2 '14 at 13:09
• @MadScientist I cannot open the link you posted, nor have I any reference elsewhere. But what other compound could it be? What signals are you missing? – Martin - マーチン Jul 2 '14 at 13:15
• This is a version of the image that should be accessible. I still think it is phenol, there isn't much else it could be. But the IR is weird, which confused me. – Mad Scientist Jul 2 '14 at 13:30
• @MadScientist I totally agree, that looks rather weird compared. Maybe the spectrum given by the OP is just very thin or a different medium - it does not state how it was measured. – Martin - マーチン Jul 2 '14 at 13:56
• @MadScientist With regards to the IR, the unknown was a solid which I dissolved in dichloromethane and use NaCl plates. I washed the plates with methanol but made sure it was fully dry before using the IR machine. – Pepria Jul 2 '14 at 14:25

The data given is this question has changed sufficiently from the original post that it warrants a new answer to the latest (and much more reliable) data, although should be taken in conjunction with Martin's response, and Long's original response. Let's first look at the IR data:

IR Spectrum:

A much better spectrum, and, compared to the original spectrum, serves as a very good example of how easy it is to introduce unwanted water into a sample. Peaks around 3020, 763 and 692 indicate an aromatic. The 763 and 692 are indicative of a mono-substituted benzene ring. There are also some clear aromatic overtones between 1950-1750.

NMR Spectrum:

Still only a partial NMR spectrum, and one that we can only assume is correctly calibrated. Linewidths are better, and significantly, we see a small coupling from the peak at 7.15ppm.

The splitting pattern and peak ratio observed is indicative of a monosubstituted benzene ring (see above); 7.30 (1H, dd, H4), 7.41(2H, , H3) and 7.56(2H, d, H2). Long-range (small couplings) are ignored. We also have a peak at 7.15 (1H, d, H5). Typical coupling in these systems is 6.5-8.5Hz for ortho coupling, 1-3 for meta, and <1 for para. Shifted from benzene (7.34ppm), the substituents come at H2 (+0.22), H3 (+0.07), H4 (-0.04).

The peak at 7.15 could be one of few things. A bis-halide is a possibility, but more likely is an alkene. No real evidence in the IR supports an alkene (although my IR knowledge is limited). However, a symmetrical alkene may be a possibility, as the main -C=C- absorbance would be IR inactive, although Raman active. This gives us two possibilities that come to immediate mind:

cis-stilbene (cis-1,2-Diphenylethylene) m.p. 5-6oC

Additive prediction for the aromatics and alkene proton can be done using many basic NMR texts. I'm using Pretsch, Buhlmann and Badertscher for this case (as it's the only one I have with me at the moment):

• H2 : 7.34+0 = 7.33
• H3 : 7.34-0.07 = 7.27
• H4 : 7.34-0.15 = 7.19
• H5 : 5.25+1.38(Argem)-0.07(Artrans) = 6.56

trans-stilbene (trans-1,2-Diphenylethylene) m.p. 123-125oC

• H2 : 7.34+0.16 = 7.50
• H3 : 7.34+0 = 7.34
• H4 : 7.34-0.15 = 7.19
• H5 : 5.25+1.38(Argem)+0.36(Arcis) = 6.99

As this is clearly a problem session for you to make a final determination of structure, I will leave it for you to make that judgement. however, as a final point, this whole exercise, I hope, serves to demonstrate the importance of preparing good quality samples in order to collect high quality and reliable data. I hope you can see the difference between your original data and final data. This is a message that I continually preach to all those researchers I work with, both at an undergraduate level and postgraduate.

• Thanks a lot. I think I am going to go with trans-1,2-Diphenylethylene whose melting point is fairly close to what I got in lab. I will be attempting to a Bromine derivative test tomorrow after which I should be pretty certain if this is it-but it makes sense. Also I asked a very similar question(chemistry.stackexchange.com/questions/14061/…). I am fairly certain that the compound is benzoic acid but I would be very appreciative if you can confirm my suspicions. Thanks a lot for your help ! – – Pepria Jul 10 '14 at 1:45

This answer aims to build on the general approach that Martin has provided, which overall makes a reasonable summation based on the data provided. IR is not really my specialty, but there is some more information that we can get out of the NMR data which should be helpful, and more reliable (in my opinion) than the IR data. Unfortunately, I am away away from my office for the next week, so cannot provide immediate references to support some statements here, so you'll have to take some things on face value. Let's begin with an overall summary of what data we have:

• A partial 1H NMR spectrum, with only some of the peaks integrated. This is an expanded region of what we can assume to be a 500MHz (based on the export path). The linewidths are broad, and there is no clear source to allow confirmation of correct calibration. We therefore need to make two assessments:

• The calibration is incorrect, and the peak at 7.15, which has no integration, is in fact the residual CHCl3, and all chemical shifts need to adjust downfield (0.09-0.11 depending on what value for CHCl3 in CDCl3 you use; I use 7.26. For simplicity, let's adjust the chemical shifts downfield by +0.1ppm
• The calibration is correct, in which case the peak at 7.15 cannot be discounted, and should therefore have its integral determined. A full display NMR spectrum would be very useful here to look for underlying exchange broadened proton signals. This might occur anywhere from about 2-15ppm, and may be very broad such that they appear as a hump in the baseline, but even in CDCl3, we should see them, and
• An IR spectrum which looks to have been run at pretty low concentration. I did not see your original IR spectrum, and wonder why you needed to redo it. What would be nice to know is whether the ratio of intensities for your absorbance peaks are the same for both IR data sets; particularly did the ratio of the broad stretch at 3422 change with respect to absorbances at 3019, 763 and 692?

• Your sample is a solid, as you mention in one of your comments. This would be a useful peice of information to have from the start. It is soluble in dichloromethane.

IR Spectrum.

Clearly, the significant signal is the broad peak at 3422, and this is textbook-indicative of an O-H stretch. It's probably a little too high to consider a N-H group of any sort. So, it could be an alcohol or an acid, but we have no C=O peak, so it leaves us with an -OH group. We have absorbances at 3019, 763 and 692; all indicative of an aromatic. More specifically, 763 and 692 are indicative of a mono-substituted benzene ring. As I say though, IR is not really my thing, and that's about all I can get from this spectrum. My biggest concern is the reliability of the OH peak.

NMR Spectrum

The splitting pattern and peak ratio observed is indicative of a monosubstituted benzene ring (see above); 7.30 (1H, dd, H4), 7.39(2H, dd, H3) and 7.55(2H, d, H2). Typical coupling in these systems is 6.5-8.5Hz for ortho coupling, 1-3 for meta, and <1 for para. Looking at the H2 signal at 7.55, we can use our knowledge of coupling constants to determine the frequency of the spectrometer:

7.5615-7.5461 = 0.0154 x Spect.Freq ~ 7.5Hz => 487MHz, so close enough to 500MHz, and confirms our suspicions that it is a 500MHz, as the export path suggests.

So, we can calculate an accurate ortho coupling for H2-H3 to be:

7.5615-7.5461 = 0.0154 x 500 = 7.7Hz.

Now, mono-substituted benzene rings have been extensively studied and are very well understood; chemical shift data has been widely tabulated, and forms the basis for many chemical shift prediction algorithms. Starting with the benzene chemical shift (7.34ppm) as a basis, it is possible to use the shifts of each group to infer some information about the type of substituent. The first thing to look for with this type of system is the order of H2 versus H3 (versus naked benzene). An electron-donating group increases shielding, and the ortho proton (H2) is typically found upfield of the meta proton (H3). Electron withdrawing groups decrease shielding, and H2 typically experiences a downfield shift from benzene, and usually resonates downfield from the meta (H3) proton. For the system you have, H2 is downfield of H3, and this is indicative of an electron-withdrawing group. This is a very strong argument against this system being phenol. Phenol has its H2 protons upfield of H3. This is also what is so confusing about the IR spectrum you have.

So, let's now consider the possible structure for this unknown compound you have. Let's make the assumption that, as a homework/tutorial problem, this is going to be a fairly simple molecule, with a pretty common substituent. Remember we have two scenarios to consider for our NMR.

Scenario 1 (corrected for CHCl3 at 7.26ppm): the substituents come at H2 (+0.21), H3 (+0.05), H4 (-0.04). Looking at Pretsch, Buhlmann and Badertscher, this matches incredibly well for the substituent being a phenyl group [H2 (+0.22), H3 (+0.06), H4 (-0.04)]. This would give the structure biphenyl, a white solid, which has a reported H2-H3 coupling of 7.71Hz. Very strong evidence by NMR, but is not supported by -OH stretch in IR data, although all other IR data is in agreement.

Scenario 2 (spectrum already correctly calibrated): If we assume that the spectrum is correctly calibrated, then the CHCl3 residual peak comes under the H4 signal - probably could be the sharp peak which is the second peak from the right in this group. This means that the peak at 7.15 needs to be considered. By eye, its integral is roughly 1. A singlet of chemical shift of 7.15 is typical of a bis-halide, and so we could consider α,α-dichlorotoluene or α,α-dibromotoluene. Both are sufficiently electron withdrawing to give H2 downfield of H3, and However, the former is definitely a liquid at room temp, and I suspect the latter is also. Therefore, not strong candidates.

I would like to have seen the original IR spectrum, and the full NMR spectrum to have confidence in any prediction. However, if I were just shown the NMR data, I would have confidence in predicting the structure as biphenyl. I hope you can provide the real solution to this eventually.

• What I miss most in the NMR data is a signal for the OH proton that should be there considering the IR data. I usually like solving puzzles like that, but here again it gives me the creeps. I have the strong feeling, that there is missing something important. Anyway, thank you very much for having a look and sharing your thoughts - very helpful indeed. – Martin - マーチン Jul 6 '14 at 12:37
• I have submitted for new NMR and will obtain another IR for the same compound again. Hopefully, I can update the question tomorrow with the new information. – Pepria Jul 8 '14 at 2:28
• @long I updated the question with a brand new NMR(below the old one-I think they are very similar) and new IR reading which doesn't have the OH. Please consider the new spectroscopy. Thank You so much for all the help you have given already ! – Pepria Jul 9 '14 at 15:08
• @PeprikaDesilva - Aaah, much better data sets. Remiss of me when I looked at the NMR before, but most likely trans-stilbene. I will put together another answer when I have a closer look shortly. – long Jul 9 '14 at 21:44


text:
# Delimited Continuations: Easy? or Fundamentally Difficult?

Apologies in advance for the length of this question, but this is as short as I have been able to make it.

## The Prescription

I'm looking at the following tutorial on delimited (or composable) continuations:

http://community.schemewiki.org/?composable-continuations-tutorial

The author(s) propose the following "rewrite rules"

;; RULE 1 ;;
(reset (...A... (shift K E) ...B...))
; -->
(let ((K (lambda (x) (reset (...A... x ...B...)))))
(reset E)) ;; You may call K in here!

;; RULE 2 ;;
(reset E)
; -->
E

This is somewhat backed up by equations 2 and 3 on page 100 of Shan's paper, http://tinyurl.com/q7ebt4a. Qualifications of "somewhat" make up the center of my question below, after some more motivation and examples.

## Motivating Examples

I can rephrase the above in a pair of slogans:

1. The argument of a delimited continuation K replaces the shift in a fresh copy of the original reset
2. The body E of the shift, in a fresh reset, replaces the original reset.

The rules above can be directly written and played with in Mathematica, and that would be grand. Here is a first attempt:

reset[h_[As___, shift[k_, E_], Bs___]] :=
Block[{K = Function[x, reset[h[As, x, Bs]]]},
reset[E /. {k -> K}]]
reset[E_] := E

Here are a few examples pulled from papers:

reset[1 + shift[k, k[42]]]
reset[1 + shift[k, k[k[42]]]]
{a, reset[{b, shift[k, k[k[{1, {c}}}]]}]}
ClearAll[$g$];
reset[1 + shift[k, Function[$g$ = k]]][];
$g$[42]
reset[1 + shift[k, "zork"]]
reset[{1, shift[k, k[20]], shift[k, k[30]]}]

> 43
> 44
> {a, {b, {b, {1, {c}}}}}
> 43
> "zork"
> {1, 20, 30}

This would be really cool for concurrent state machines or lazy streams or reactive patterns and more.

## It May Not Be Right

The problem is that I don't think it's right because it doesn't look any deeper than first level. Ironically, every example I have seen in the literature only has shifts at level 1 of the expression inside the reset, BUT

If I fire up Racket (a modernized Scheme), do (require racket/control) and then

(reset (+ 1 (shift k (k 42)))

> 43

as expected and as consistent with the above. If I do

(reset (+ 1 (* 2 (shift k (k 42)))))

> 85

this looks intuitively right, but doesn't match the prescription above. That prescription, if I read it correctly, should lead to the second rewrite rule and just produce (+ 1 (* 2 (shift k (k 42)))) because it can't "see" the shift at the second level, "inside" the (* 2 (shift ...)) subexpression.

Of course, my Mathematica version matches the prescription:

reset[1 + 2 * shift[k, k[42]]]

> 1 + 2 shift[k, k[42]]

## Is an Easy Solution Possible?

I now think that Shan's paper above is being too glib, with an imaginary context operator that lets shift refer to a reset at higher levels.

Everything else I've been able to find on implementing delimited continuations is vastly larger and would be a lot of very sideways work (http://tinyurl.com/oec83ky, http://lambda-the-ultimate.org/node/4313), requiring one ore more of the following:

2. Scheme's call/cc or idiosyncratic macro system
3. Building your own VM for stack-hacking and caching (http://tinyurl.com/n9kz5fb)
4. Rewriting your whole program into CPS (continuation-passing style), either manually or by writing a metacircular translator, e.g., http://tinyurl.com/bo8zl9c)
5. Building your language up from a tiny lambda calculus in CPS (fig. 1, p. 101 in http://tinyurl.com/q7ebt4a)
6. Throw and Catch, considered desirable to avoid

I have spent a lot of time trying to get the above working with recursive equations (see junk code below), but I am now despairing that it's not easy. A shift at a lower level might refer to a matching reset at a higher level, which must be threaded around the equations in one way or another.

Now, finally to the actual question of this post:

## Am I blindly missing an easy way in Mathematica?

I would like to avoid Throw and Catch in favor of a functional solution, because I have been led to believe that delimited continuations are actual, ordinary functions, unlike continuations, which are jumps that (confusingly) look like functions, and because Throw and Catch are too close to Scheme's call/cc, which I definitely want to avoid (if I don't avoid it, I might as well use Scheme and not Mathematica, after all).

## Broken Junk Code

May have some useful ideas, however.

ClearAll[u, v, r, q, s];
v[cc_, cc_[E_]] := E;
u[cc_, cc_[E_]] := E;
u[cc_, h_[As___]] :=
With[{xs = Select[{As}, Head[#] === cc &]},
If[{} =!= xs,
v[cc, First[xs]],
xs]];
u[cc_, E_] := E;

r[E_] := Module[{cc = Unique["cc"]},
u[cc, q[cc, E]]];

q[cc_, h_[As___, s[k_, E_], Bs___]] :=
Module[{K = x \[Function] r[h[As, x, Bs]]},
r[cc[E /. {k -> K}]]];

q[cc_, h_[As___]] :=
With[{qs = q[cc, #] & /@ {As}},
h[Sequence @@ qs]];

q[cc_, Function[E_]] := Function[E]

q[cc_, E_] := cc[E]

format: plain


text:
# Continuous Family of Norms&Operators

I want to ask the definition of a concept I see around but has not been able to find the definition for. Given a finite dimensional vector bundle $\pi : E \rightarrow K$ over a normed convex vector space K (with isomorphic fibers), what would be the definition of a continuous family of norms {$|.|_{x\in K}$}? I have two guesses but I don't know which one is correct, or if any of it is correct,

1- It is continuous as a map, $|.|: E \rightarrow R$ in the sense that for any sequence $v_n \rightarrow v$ in $E$ with $\pi(v_n)=x_n$, we have $|(v_n)_{x_n}|_{x_n}\rightarrow |(v)_{x}|_{x}$.

2- Or, first we choose a basis for each fiber $E_x$ write an isomorphism $T_{xy} :E_x \rightarrow E_y$, between each fibers. And then given a vector $v_x$ in the fiber $E_x$ and a sequence of points $x_k$ in K converging to x, $|T_{x_nx}(v_x)|_{x_n} \rightarrow |v_x|_{x}$

The difference between the two is whether if you fix vector component $v_x$ or now. I could ask the same question about continuous family of operators $T_x :E \rightarrow E$, is it continuity as a map from E to E or should we somehow again identify each fiber like in the second case above.

Thanks

-

One would be far more likely to define such a notion with something like your first definition, though the property people would really care about is that for any continuous section $\xi$ of your vector bundle $E \to X$, $x \mapsto |\xi(x)|$ defines a continuous map $X \to \mathbb{R}$. In particular, for continuity to make sense, all you need is for $X$ to be a topological space.
As for what actually tends to appear in practice, one usually doesn't just want a continuous family of norms but rather a metric, a continuous family $g$ of inner products in the sense that for any continuous sections $\xi$ and $\eta$ of $E \to X$, $x \mapsto g(x)(\xi(x),\eta(x))$ defines a continuous map $X \to \mathbb{R}$ (or $\mathbb{C}$, if $E \to X$ is complex). Again, you simply need $X$ to be a topological space. An example, of course, is the Riemannian metric on a Riemannian manifold $X$, which is a metric on the tangent bundle $TX \to X$, though more generally, in the case of smooth vector bundles over smooth manifolds, one usually wants metrics to be smooth in the same sense.
On behalf of Finsler metrics, I raise an objection to "but rather a metric, a continuous family $g$ of inner products". After all, what is now known as a Finsler metric was the original definition of a metric by Riemann. –  user53153 Feb 2 '13 at 3:59


text:
# Struggling with Newton's Second Law Equations

Tags:
1. Oct 11, 2014

### TehCylons

Hello,

I am a second year undergrad student majoring in Astronomy and Computer Science. I am having a hard time with my physics courses for the following reasons:

1) SmartPhysics is the text we use (if you can call it a text...). Basically it is an online HW and "pre-lecture" system that allows students to watch a 15 minute lecture that covers ALL topics of a unit. The textbook is very short and is verbatum what the pre-lectures give you (with few exceptions). My instructor does not lecture. We are told to watch the pre-lecture and come with questions. However, the pre-lecture does not go through very many examples and therefore leaves me with little understanding of how to stretch and apply the methods that they use to other problems. Basically, I don't know what I don't know until I start my HW.

2) We are often way ahead on HW and pre-lectures vs. the labs/quizzes and few practice problems that we do in class. So, by the time we get around to discussing it the HW is usually due that day or the next.

3) I struggled in my Classical Mechanics course with Newton's Second Law. My FBD's are usually spot on, but I have a little trouble translating those into second law equations and I struggle even more with manipulating the equations (say I have 3 unknowns & 3 equations.... I often go in circles and end up with a = a or a =1). Now, I am in the beginning of Electricity and Magnetism, but we are covering rotational mechanics first.

I love physics, I did great in the introductory physics course for which I had a different text and instructor. I just cannot seem to grasp this style of teaching/learning. I take great notes from the pre-lectures (I spend 1-1.5 hrs on the 15 minute video). I would take great notes in class as well if there was anything to take notes on.

I would love any pointers from people who had similar struggles or websites/books that would be good for someone in my situation.

extra context: I wouldn't say it is the math part that I have a hard time with. I am currently taking multivariate calculus and I am not struggling with that (nor have I with any of my other math classes).

2. Oct 11, 2014

### Staff: Mentor

iHere are Newton's Laws 1, 2, 3 in a nutshell
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html

Newton's 2nd Law - http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#nt2cn

Limitations of N2L - http://hyperphysics.phy-astr.gsu.edu/hbase/limn2.html#ln21

http://hyperphysics.phy-astr.gsu.edu/hbase/limn2.html#ln22

A lot of people find calculus relatively easy, but applying it to understanding physics may be difficult for some.

It might be easy and straightforward to solve for some df(t)/dt, for y'(t) + ay(t) = f(t) or $\int f(t) dt$, but knowing how to apply that when f(t) = m(t), p(t), v(t), F(t), E(t), or combinations thereof, can be daunting at first.

Some physical relationships can be described with first or second order ODEs, but often the physics of a system requires PDEs or coupled PDEs, and sometimes nonlinear PDEs or coupled PDEs, depending on the complexity.

Last edited: Oct 11, 2014
3. Oct 11, 2014

### TehCylons

Thank you so much for your reply and the examples. You are absolutely correct about applications being my issue. It is easy for me to think about what physical forces are being applied and by what, but wrapping my head around exactly how it affects the behavior of a system is another thing entirely. However, with what I want to do it is SO important. So, I really want to get better at it. :)

Again, I really appreciate you taking the time to reply and I will check out your suggestions!

Edit: I just skimmed through them quickly and they are great. That is going to help a lot. Thank you again!


text:
# Precision measurement of the B0s-B¯0s oscillation frequency with the decay B0s → D−sπ+

LHCb Collaboration; et al; Bernet, R; Müller, K; Steinkamp, O; Straumann, U; Vollhardt, A (2013). Precision measurement of the B0s-B¯0s oscillation frequency with the decay B0s → D−sπ+. New Journal of Physics, 15:053021.

## Abstract

A key ingredient to searches for physics beyond the Standard Model in B0s mixing phenomena is the measurement of the B0s– ${{\overline{ {\mathrm {B}}}{}}^0_{\mathrm { s}}}$ oscillation frequency, which is equivalent to the mass difference Δms of the B0s mass eigenstates. Using the world's largest B0s meson sample accumulated in a dataset, corresponding to an integrated luminosity of 1.0 fb−1, collected by the LHCb experiment at the CERN LHC in 2011, a measurement of Δms is presented. A total of about 34 000 B0s → D−sπ+ signal decays are reconstructed, with an average decay time resolution of 44 fs. The oscillation frequency is measured to be Δms = 17.768 ± 0.023 (stat) ± 0.006 (syst) ps−1, which is the most precise measurement to date.

A key ingredient to searches for physics beyond the Standard Model in B0s mixing phenomena is the measurement of the B0s– ${{\overline{ {\mathrm {B}}}{}}^0_{\mathrm { s}}}$ oscillation frequency, which is equivalent to the mass difference Δms of the B0s mass eigenstates. Using the world's largest B0s meson sample accumulated in a dataset, corresponding to an integrated luminosity of 1.0 fb−1, collected by the LHCb experiment at the CERN LHC in 2011, a measurement of Δms is presented. A total of about 34 000 B0s → D−sπ+ signal decays are reconstructed, with an average decay time resolution of 44 fs. The oscillation frequency is measured to be Δms = 17.768 ± 0.023 (stat) ± 0.006 (syst) ps−1, which is the most precise measurement to date.

## Citations

33 citations in Web of Science®
36 citations in Scopus®

## Altmetrics

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English 2013 21 Feb 2014 09:51 05 Apr 2016 17:36 IOP Publishing 1367-2630 Publisher DOI. An embargo period may apply. https://doi.org/10.1088/1367-2630/15/5/053021
Permanent URL: https://doi.org/10.5167/uzh-91738

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# Find the order of a group

Given a $$2*2$$ matrix M, over $$Z_2$$ and the determinant of the matrix is $$1$$. Is there any shorter way of showing that M has $$6$$ elements?

There is already an answer but they check all tuples one by one and then add the terms up to find the final answer.


text:
france

0th

Percentile

France Map

This france database comes from the NUTS III (Tertiary Administrative Units of the European Community) database of the United Nations Environment Programme (UNEP) GRID-Geneva data sets. These were prepared around 1989, and so may be somewhat out of date.

Users of data sets supplied through UNEP/GRID are requested to incorporate in output products and reports acknowledgements to the originator of the data and to the fact that they were acquired through UNEP/GRID. Appropriate wording may be "UNESCO (1987) through UNEP/GRID-Geneva".

Keywords
datasets
Usage
data(franceMapEnv)
format

The data file is merely an assignment to a character string which specifies the name of an environment variable which contains the base location of the binary files used by the map drawing functions. This environment variable (R_MAP_DATA_DIR for the datasets in the maps package) is set at package load time if it does not already exist. Hence setting the environment variable before loading the package can override the default location of the binary datasets.

References

Richard A. Becker, and Allan R. Wilks, "Maps in S", AT&T Bell Laboratories Statistics Research Report [93.2], 1993.

Richard A. Becker, and Allan R. Wilks, "Constructing a Geographical Database", AT&T Bell Laboratories Statistics Research Report [95.2], 1995.

map
map('france', fill = TRUE, col = 1:10)


text:
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text:
# Math Help - SOLVE FOR x please....

e4x+1= 2ex

2. ## Re: SOLVE FOR x please....

Originally Posted by Scoplex
e4x+1= 2ex
Can you solve $3x+1=\log(2)~?$

3. ## Re: SOLVE FOR x please....

x = [log2 - 1]/3 ???

Does this mean that log(2) is equal to 2e?
I would then do the following:
4x+1=x
3x=-1
x=-1/3 ??? Is this then correct?

This logs and stuff get my head cooking sometimes.

4. ## Re: SOLVE FOR x please....

Originally Posted by Scoplex
x = [log2 - 1]/3 ???CORRECT

Does this mean that log(2) is equal to 2e?
What do you mean by that?

The $\log(2)$ is just $\log(2)$, no more, no less.

6. ## Re: SOLVE FOR x please....

Hello, Scoplex!

$\text{Solve for }x\!:\;\;e^{4x+1} \:=\: 2e^x$

Take logs: . $\ln\left(e^{4x+1}\right) \:=\:\ln(2e^x)$

. . . . . . $(4x+1)\ln(e) \:=\:\ln(2) + \ln\left(e^x\right)$

. . . . . . $(4x+1)\ln(e) \:=\:\ln(2) + x\ln(e)$ . . . Note that $\ln(e) = 1$

n . . . . . . . . . $4x + 1 \:=\:\ln(2) + x$

n . . . . . . . . . $3x + 1 \:=\:\ln(2)$

. . . . . . . . . . . . . $3x \:=\:\ln(2) - 1$

. . . . . . . . . . . . . . $x \:=\:\dfrac{\ln(2) - 1}{3}$


text:
EVALUATION OF ECOLOGICAL CONCRETE USING MULTI-CRITERIA ECOLOGICAL INDEX AND PERFORMANCE INDEX APPROACH

## Publications

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#### Architecture, Civil Engineering, Environment

Silesian University of Technology

Subject: Architecture, Civil Engineering, Engineering, Environmental

ISSN: 1899-0142

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VOLUME 12 , ISSUE 1 (May 2019) > List of articles

### EVALUATION OF ECOLOGICAL CONCRETE USING MULTI-CRITERIA ECOLOGICAL INDEX AND PERFORMANCE INDEX APPROACH

Citation Information : Architecture, Civil Engineering, Environment. Volume 12, Issue 1, Pages 97-107, DOI: https://doi.org/10.21307/ACEE-2019-009

License : (CC-BY-NC-ND 4.0)

Received Date : 15-May-2018 / Accepted: 15-November-2018 / Published Online: 20-May-2019

### ARTICLE

#### ABSTRACT

This paper proposes a new method of rational and quantitative assessment of ecological concrete in terms of the ecological impact and engineering performance. The concrete mix is evaluated through the multi-criteria Ecological Index (EI) and Performance Index (PI) approach. The EI accounts for the impact of the concrete on environment including the carbon emission and raw materials usage, whereas the PI accounts for the engineering performance of the concrete such as compressive strength and water sorptivity. Depending on the applications of the concrete, different criteria may be chosen for the evaluation. Concrete mixes reported in the literature comprising different types of cement, supplementary cementitious materials and aggregates are analyzed to illustrate the applicability of the proposed multi-criteria assessment method. It is shown that the proposed method is able to effectively reflect the concurrent ecological impact and engineering performance of concrete mixes, and hence facilitate rational design of ecological concrete to suit practical engineering applications.

## 1. INTRODUCTION

The idea of optimized usage of raw materials and reduction of greenhouse gas emissions was adopted by numerous researchers all over the world [1, 2, 3, 4]. Sustainable development is encompassed in the philosophy of modern urbanization planning [5, 6, 7]. In concrete industry, this idea is becoming increasingly well aware of. Development of energy efficient technologies in concrete production is often based on the use of suitable valorised binding materials, whose production is associated with lower energy consumption, such as fly ash, blast-furnace slag, ceramic dust, and so on [8, 9, 10, 11]. The cement industry has been using waste materials over many years for the production of CEM II and CEM III cements, whose market share has been increasing recently from year to year [12, 13, 14]. Alternative fillers such as scrap tires or sludge ash from sewage treatment plants have been introduced [15, 16, 17]. The use of cements with additives often does not cause any deterioration of the properties of concrete, instead it can improve various properties such as resistance to chemical attack and lower heat generation during hardening [13, 14, 18]. The use of recycled additives including fly ashes and other waste materials in pre-cast and ready-mix concrete is becoming increasingly common. Attempts are made to evaluate and appraise the implemented solutions and proposed environmentally friendly concretes. Rating systems for assessing the environmental performance of construction projects have been introduced. There are many formal sustainability rating systems for buildings in worldwide use today, with LEED (Leadership in Energy and Environmental Design) in the USA and BREEAM (Building Research Establishment Environmental Assessment Method) in the UK probably being the most well-known and widely adopted [19, 20]. However, in the majority of real cases, the concrete is usually assessed very roughly. Under various sustainability rating systems, users are encouraged to use cement blended with additives, or an additive in the form of waste in the concrete mix, and/or recycled coarse aggregate, in order to obtain scores or points to improve the overall positive assessment of the construction projects.

A number of researchers have proposed analytical methods for assessing the environmental performance of concrete based on one or more criteria [20, 21, 22, 23]. Some of the methods are quite simple to apply and are focused on determining the total carbon dioxide emission during the production of concrete components. Since carbon dioxide is a major greenhouse gas generated by artificial sources, the environmental friendliness of concrete is commonly represented by the CO2 emission [12]. Another method is to determine the impact of a building on the environment throughout its lifetime [24]. This approach is more complicated because it requires detailed data on: i) Global warming (CO2 emission); ii) Ozone depletion; iii) Acidification of soil and water; iv) Eutrophication; v) Photochemical ozone creation; and vi) Depletion of abiotic resources and fossil fuels (energy consumption) [24]. Information on how waste management is handled at the end of life of the building should also be known. The lack of such data results in the necessity of adopting less precise assumptions and reduces the reliability of the calculations and analyzes. Many different life-cycle analysis (LCA) calculation options are used and new methodologies and solutions are still being proposed recently [25, 26, 27, 28].

Most assessment methods only take into account the environmental impact, but not the engineering performance of the concrete being evaluated. The proposed method is supposed to be much simpler to apply than LCA analyzes, but at the same time it can take into account both the impact on the environment and engineering performance based on two equivalent criteria encompassed in the Ecological Index (EI) and Performance Index (PI), as explained below. The eco-friendliness of the concrete can be expressed in terms of the EI, whereas the engineering performance can be expressed in terms of the PI, where both EI and PI are calculated on the basis of selected concrete properties that are important for the given application and exposure class. This will allow to assess more comprehensively the impact of concrete on the environment and to connect it with the engineering performance of concrete. Concrete with better mechanical properties and functional parameters is usually a more durable material, and thus requires fewer repair or maintenance actions during the life cycle of the structure. This means reducing the amount of energy and materials consumption throughout the service life, which is directly related to the lowering of CO2 emission. On the other hand, the use of low-quality waste materials and decreasing the amount of binder could potentially cause, apart from a positive environmental impact, a lowering of concrete quality or performance possibly to an unacceptable level. The proposed method will allow finding the right compromise between “traditional” high-quality concrete with comparatively large amount of CEM I cement, crushed natural aggregates, additives and appropriate dose of admixtures, and “doctrinally ecological” concrete whose quality might be at the low side or even below the acceptable level.

In the initial postulation of the method and for preliminary assessment of concrete, it is possible to use the data collected by the authors. Having cumulated more accurate data corresponding to the materials used, it is possible to refine the calculation by making use of the addition data to further enhance the accuracy of the concrete assessment.

## 2. DATA AND METHOD OF CALCULATION

### 2.1. Assumptions and values adopted for calculations

A method based on multi-criteria, three-stage assessment of the impact of a concrete mix on the environment and the engineering performance of the resulting concrete has been proposed. The proposed method is principally intended to enable the rational and balanced design of concrete mix as the optimised environmentally-friendly solution to meet the specific application requirements. Basically, the impact on the environment is represented by way of the Ecological Index (EI), and the engineering performance is represented by way of the Performance Index (PI). During the assessment process, the calculated EI would be compared to the calculated PI. In the first stage, the CO2 emission and the amount of non-renewable raw materials used was determined for valuation of EI. To calculate CO2 emissions, “the individual integration method” described in [29, 30] was used. This method allows the inclusion of emissions during the production and transport of concrete components and the production of a concrete mix. Taking cement manufacturing as an example, detailed data values depend on many factors and may vary among individual cement manufacturers [31]. A similar situation occurs for other concrete ingredients and their transport. This necessitates reasonable assumptions of certain values in the calculations. The values of emission factor reported in the literature and the data for use in the calculations are summarized in Table 1.

##### Table 1.

CO2 emission factors of concrete ingredient materials

The assumed values are based on experience and latest literature data. It is noted that the exact data values from industrial production are difficult to obtain, because the exact values are dependent on a number of production parameters such as the machinery and equipment configurations, types of fuel used, manufacturing processes, and supply-chain to the production facilities. These parameters would vary from factory to factory, and would not be systematically disseminated by individual factories to third parties. Besides, there would be batch-to-batch variations of the raw materials characteristics and production parameters. Nevertheless, the purpose of the research is to develop the evaluation methodology and demonstrate its applications. When applying the proposed method, if more accurate data are available, such data can be used for evaluation of the concrete. For the calculations carried out with the proposed method, data on the compositions and properties of concretes presented in previous publications were used [11, 34, 35, 36]. The concrete mix proportions are presented in Table 2 (the unit of the concrete mix proportions is in kg/m3).

##### Table 2.

Concrete mix proportions

It was assumed that CEM I Portland cement contains 100% clinker (in order to simplify calculations, the components constituting less than 5% of the cement by mass were omitted; nevertheless, if reliable data of a component material are available, the data can be incorporated in the calculation to improve the accuracy). In the case of Portland composite cement and Portland blast-furnace slag cement, the maximum possible amount of additive and the remaining amount of clinker were estimated with reference to the product specifications. For example, CEM II/B-V contains approximately 35% ash and 65% clinker by mass, and CEM III/A contains approximately 65% slag and 35% clinker by mass. The total emission was determined taking into account the percentage (mass) share and unit emissions attributed to individual cement components. Carbon emission during the transport of components was assessed according to the data given in the literature [29, 30]. When similar modes of transport are adopted, the variation of data values would be small.

The raw materials factor is a raw materials usage parameter calculated for each concrete component as the product of the consumption of natural raw materials needed to produce 1 kg of this component and an additional parameter denoted as “rarity ratio”. This additional parameter expresses the availability of a given natural resource and the need to conserve it. It was assumed that the rarity ratio should take values from 0 in the case of industrial waste in surplus to 1 in the case of raw materials being depleted. With such assumptions, an arbitrary value of 0.5 was assumed for river sand due to its high availability and at the same time, the need to prevent over-dredging to preserve the fresh water habitat, whereas an arbitrary value of 1.0 was assumed for natural gravels. In the case of crushed granite or basalt aggregates, a rarity ratio of 0.75 was assumed. However, due to the fact that the production of aggregates of suitable size by crushing also results in fractions with smaller granulation, therefore it was assumed that the production of 1 kg of crushed aggregates requires 1.25 kg of rock.

Production of 1 kg of CEM I cement (100% clinker assumed) requires about 1.6 kg of raw materials [37]. The authors assumed the rarity ratio to be 0.5 because natural raw materials for cement production are available in large quantities in Poland and China (the authors’ origin places). In the calculations, raw materials usage is equal 0.8 for CEM I cement, 0.52 for CEM II/B cement and 0.28 for CEM III/A cement.

Data which facilitate the valuation of rarity ratio for individual raw materials are not widely available in the literature. Depending on the availability of raw aggregate materials in the given area, values of these ratios may differ considerably. Data on aggregates are more diverse in different geographical areas of China due to the reason of significant regional variations of types and availabilities of natural aggregate sources. Therefore, the rarity ratios used in this study are based on data of natural aggregate resources available in Poland or regions with similar settings [38]. The proposed method allows the use of specific rarity ratio values of individual raw materials depending on their availability in a given country or locality. When calculating the consumption of natural (non-renewable) resources for a concrete mix, the product of the mass content of a given component and its raw materials factor was summed to give the total raw materials usage. The total CO2 emission was taken as the sum of CO2 emissions of the ingredient materials and their transportation. Since the transportation of materials involves burning of fossil fuel and hence carbon emission, from the ecological viewpoint, usage of locally available ingredient materials would be more preferable than materials from a long distance. The calculations for a sample concrete are shown in Table 3.

##### Table 3.

Sample calculations of CO2 emissions and raw materials usage

In the first stage of the concrete assessment, the ecological impact of concrete is considered. Normalized CO2 emission values and raw materials consumption are calculated. Normalization consisted of dividing the given parameters by the respective reference values, which are determined based on the high quality concrete containing 480 kg/m3 of cement CEM I 42.5, crushed aggregate, superplasticizer and metakaolin as an additive (the concrete has approximately 490 kg CO2 emission per cubic metre and 2000 kg/m3 of raw materials usage) [36]. Calculations for reference values are presented in Table 4.

##### Table 4.

Calculations of CO2 emissions and raw materials usage for reference values

### 2.2. Calculation of Ecological Index

The value of EI is evaluated as the square root of the sum of normalized total emission and normalized total raw materials usage, multiplied by square root of 0.5 so that a concrete mix with reference values of CO2 emission and raw materials usage will give an EI value of 1. A lower EI value means less environmental burden is associated with the concrete, i.e. the concrete is more environmentally friendly. The calculation of EI is given by Equation (1), in which EM represents the total emission in kg/m3 and RM represents the total raw materials usage in kg/m3. The relevant values are multiplied by the respective weighting coefficients, whose values were taken as: wEM = 0.5 and wRM = 0.5 in the present study.

##### (1)
$EI=EM490kg/m3×wEM+RM2000kg/m3×wRM$

### 2.3. Calculation of Performance Index

In the second stage of the concrete assessment, the engineering performance of concrete is concerned. The value of PI is evaluated on the basis of the sum of normalized values of selected concrete properties. Meanwhile, the 28-day compressive strength and water sorptivity are chosen for the calculation of PI. It should be noted that the proposed method is flexible and other properties of concrete may be chosen to suit the application needs. As a reference value, the value of 28-day strength is taken as equal to 60 MPa and the sorptivity is taken as equal to 0.120 cm/h0.5. The value of 60 MPa is often the limit of average strength of high quality concretes, and the value of 0.120 cm/h0.5 is regarded as the limit of “good” class of impermeability when evaluating concrete according to the criteria proposed in [36]. With the assumed base values, the 28-day strength of the concrete under assessment and its sorptivity were compared respectively. The relevant quotients from normalization are multiplied by the respective weighting coefficients, whose values were taken as: wfcm = 0.6 and wS = 0.4 in the present study. The sum of weighting coefficients should be equal to unity so that a concrete mix with reference values of selected properties will give a PI value of 1. The calculation of PI is given by Equation (2).

##### (2)
$PI=fcm60MPa×wfcm+0.120cm/h0.5S×wS$

As far as the durability aspect of a reinforced concrete structure is concerned, in lieu of sorptivity, other properties related to the potential durability of concrete can be accepted. For certain applications where additional properties of the concrete are required, PI can be calculated on the basis of more than two properties and the assignment of the respective weighting coefficients depends on the preferred properties of the concrete. For example, when high resistance to chemical aggression or cyclic freeze-thawing is required, properties related to resistance to these impacts may be associated with larger weighting coefficients than strength or sorptivity. In this case, specific tests need to be conducted to obtain the test results of the properties for calculation. A higher PI value means better concrete properties compared to the reference values.

In the third stage of the concrete assessment, the concurrent ecological impact and engineering performance of concrete is concerned. The concurrent EI and PI achieved are appraised by a Gross Ecological and Performance Indicator (GEPI), as given by Equation (3):

##### (3)
$GEPI=EI2+1PI2$

For a concrete with reference values of ecological parameters and properties, its EI and PI are both equal to 1, and the GEPI will be equal to (2)0.5. In the expression, both EI and 1/PI have smaller values when the concrete is ecologically favourable and has desirable engineering properties, and vice versa. Therefore, when designing a concrete mix in practice, a low GEPI is aimed for concrete with favourable concurrent EI and PI, while a high GEPI should be avoided.

The following should be noted in the interpretation of EI and PI. Since the valuation of EI is dependent on the local ingredient materials and production practice in a given geographical area, the possibilities to directly compare concrete mixes produced in different countries would be limited. On the other hand, since the valuation of PI is dependent on the chosen concrete properties in connection with the application requirements, the significance to directly compare concrete mixes for fulfilling different requirements would be limited. Though it is still possible to make an approximate and qualitative comparison across concrete mixes based on different settings, highly valuable results would be yielded from comparison across concrete mixes for fulfilling any given set of requirements.

## 3. RESULTS AND DISCUSSIONS

In the following, concrete mixes reported in the literature [11, 34, 35, 36] as listed in Table 2 are assessed and compared using the proposed multi-criteria EI and PI approach. The concrete mixes were divided into distinct groups. Figure 1 presents the calculations of EI results for the concrete mixes. Figure 2 shows the results of EI and PI calculations. In the graph of 1/PI plotted against EI, the points characterizing the most favourable concrete mixes in terms of ecology and engineering properties are located nearest to the origin of the coordinate system. This provides a means of graphical visualisation of the concurrently achieved EI and PI with the aid of the GEPI. Numerical results of EI, PI and GEPI calculations for the groups of concrete analyzed are presented in Table 5.

##### Figure 1.

Ecological Index results

##### Figure 2.

Ecological Index plotted against reciprocal of Performance Index

##### Table 5.

EI and PI values of concrete mixes

The results of individual groups of concrete mixes are discussed in detail below.

Concrete I: This group of concrete contained different cement types in the amount of 300 kg/m3, with addition of Class F fly ash from coal combustion in the Kozienice Power Plant in the amount of 300 kg/m3 and metakaolin in the amount of 60 kg/m3. Natural sand and recycled coarse aggregates (RCA) were employed. The EI for concrete mix REC5 with CEM III/A 42.5N cement is similar to the values for Concrete III group containing blast-furnace slag cement CEM III/A 42.5N. The use of other cements types results in higher CO2 emissions and less favourable EI. PI values of all concrete mixes are similar and much lower than in the case of Concrete II NA and II RCA groups, with respectively natural aggregates (NA) and RCA.

Concrete II NA and II RCA: Eight concrete mixtures were included, where ordinary Portland Cement CEM I 32.5 (400 kg/m3 and w/c = 0.5 in reference concrete; and 300 kg/m3 of cement + 200 kg/m3 of fly ash and water/binder ratio = 0.4 in the other concrete mixes) and supplementary cementitious materials including Class C fly ash (FA), silica fume (SF) and metakaolin (MK) were used. The contents of FA, SF and MK were respectively 50%, 10% and 10% of the cement mass in the reference concretes (mix I and I R respectively). In the concrete mix series I to IV, natural aggregate fractions of 0–2 mm (river sand) and 2–16 mm (gravel) were used. Recycled aggregate was applied in concrete mix series I R to IV R.

The change from NA to RCA would alter the consumption of raw materials in the concrete mix design. It should be noted that the water absorbed by RCA was excluded from the calculation of w/c ratio, which should reflect the quantity of free water. This resulted in the reduction of the effective w/c ratio in concrete with RCA and improvement of its properties. With the use of recycled aggregates, the EI of concrete with RCA is obviously more favourable than that of concrete with NA, whereas their PI is at a similar level. The addition of MK and SF leads to a positive impact on the PI of concrete with RCA.

Concrete III: This group of concrete contained blast-furnace slag cement CEM III/A 42.5N in the amount of 300 kg/m3, with addition of Class F fly ash from coal combustion in the Kozienice Power Plant in the amount of 200 kg/m3. Individual mixtures were modified with other types of reactive supplementary cementitious additives including SF, MK, Centrilit NC (amorphous aluminosilicate), and fluidized fly ash. Natural sand and RCA were employed. Among all groups of concrete, this concrete group provides the most favourable EI results. The values of PI vary and depend on the additive used. The least favourable PI is obtained when using fluidized fly ash, whereas the most favourable PI is obtained when Centrilit NC was added in the amount of 10% by mass of cement.

The above results illustrate the favourable effects of Portland blast-furnace slag cement and higher strength cement on the concurrently achievable EI and PI, as well as the beneficial effects of using recycled coarse aggregates and supplementary cementitious materials. It should be noted that the superior rating of concrete with CEM III/A and RCA in terms of the PI values is dependent on the choice of PI criteria. In other words, a change in the PI criteria may alter the rating. For instance, if the assessment is based on the frost resistance criterion, the same concrete mixtures might show a different rating.

## 4. SUMMARY AND CONCLUSIONS

A method of multi-criteria assessment of concrete in terms of impact on the natural environment and engineering performance has been proposed. The method is principally intended to enable the rational and balanced design of concrete mix as the optimised environmentally-friendly solution to meet the specific application requirements. The impact on environment by the concrete is accounted for via the Ecological Index (EI), and the engineering performance of the concrete is accounted for via the Performance Index (PI). The proposed method enables a quantitative and more rational appraisal of ecological concrete. The calculation procedures for EI and PI have been explained. Dependent on the geographical area and production factors of plants, the carbon emissions of the raw materials and the concrete may vary within certain ranges. The reliability of the EI calculation would be enhanced by having more accurate production data. The compressive strength and sorptivity of concrete have been adopted in this study as the criteria for the PI calculation. Dependent on the applications of the concrete, different criteria and weighting coefficients may be assigned for composing the PI. The selection of criteria allows to cater for the exposure conditions of concrete and project-specific requirements, whereas the selection of weighting coefficients allows to cater for the relative importance of the criteria to meet the project needs. The concurrent ecological impact and engineering performance of concrete is considered via the Gross Ecological and Performance Indicator (GEPI). Examples of EI and PI evaluation comprising concrete mixtures with different types of cement, use of natural resource or recycled coarse aggregates, and addition of different supplementary cementitious materials have been presented.

## References

1. Grin, J., Rotmans, J., & Schot, J. (2010). Transitions to Sustainable Development. Transitions to Sustainable Development: New Directions in the Study of Long Term Transformative Change. doi:10.4324/9780203856598
[CROSSREF]
2. Zuo, J., & Zhao, Z. Y. (2014). Green building research-current status and future agenda: A review. Renewable and Sustainable Energy Reviews, 30, 271–281. doi:10.1016/j.rser.2013.10.021
[CROSSREF]
3. Behera, M., Bhattacharyya, S. K., Minocha, A. K., Deoliya, R., & Maiti, S. (2014). Recycled aggregate from C&D waste & its use in concrete – A breakthrough towards sustainability in construction sector: A review. Construction and Building Materials. doi:10.1016/j.conbuildmat.2014.07.003
4. Węglorz, M. (2014). Selected Aspects of Sustainable Civil Engineering. Architecture Civil Engineering Environment, 7(1), 41–47.
5. Milošević, P. (2012). Sustainable Eco Planning Strategies in East Europe (Case Study of Belgrade). Architecture Civil Engineering Environment, 5(4), 29–42.
6. Pawlikowska-Piechotka, A., & Piechotka, M. (2012). Urban Sustainable Development and Green Agenda Perspective (Case Study in Warsaw). Architecture Civil Engineering Environment, 5(4), 43–52.
7. Słyk, J. (2015). Methodology of Architectural Design And Rules of Cooperation in The Digital Enviroment. Augmented Space as a Field of Research and Alternative Environment for Architectural Creation. Architecture Civil Engineering Environment, 8(4), 11–18.
8. Witkowski, H. (2015). Sustainability of Self-Compacting Concrete. Architecture Civil Engineering Environment, 8(1), 83–88.
9. Pavlík, Z., Fořt, J., Záleská, M., Pavlíková, M., Trník, A., Medved, I., … Černý, R. (2016). Energy-efficient thermal treatment of sewage sludge for its application in blended cements. Journal of Cleaner Production, 112, 409–419. doi:10.1016/j.jclepro.2015.09.072
[CROSSREF]
10. Muhd Norhasri, M. S., Hamidah, M. S., Mohd Fadzil, A., & Megawati, O. (2016). Inclusion of nano metakaolin as additive in ultra high performance concrete (UHPC). Construction and Building Materials, 127, 167–175. doi:10.1016/j.conbuildmat.2016.09.127
[CROSSREF]
11. Kubissa, W., Jaskulski, R., & Reiterman, P. (2017). Ecological Concrete Based on Blast-Furnace Cement with Incorporated Coarse Recycled Concrete Aggregate and Fly Ash Addition. Journal of Renewable Materials, 5(1), 53–61. Doi:10.7569/JRM.2017.634103
[CROSSREF]
12. Gartner, E. (2004). Industrially interesting approaches to “low-CO2” cements. Cement and Concrete Research, 34(9), 1489–1498. Doi:10.1016/j.cemconres.2004.01.021
[CROSSREF]
13. Müller, C. (2006). Environmental and technical aspects of the application of blended cements in concrete. Roads and Bridges – Drogi i Mosty, 5(3), 43–72.
14. Dziuk, D., Giergiczny, Z., & Garbacik, A. (2013). Calcareous fly ash as a main constituent of common cements. Roads and Bridges – Drogi i Mosty, 12(1), 57–69.
15. Mokrzycki, E., & Uliasz- Bocheńczyk, A. (2003). Alternative fuels for the cement industry. Applied Energy, 74(1–2), 95–100. doi:10.1016/S0306-2619(02)00135-6
[CROSSREF]
16. Li, F., & Zhang, W. (2011). Combustion of sewage sludge as alternative fuel for cement industry. Journal Wuhan University of Technology, Materials Science Edition, 26(3), 556–560. doi:10.1007/s11595-011-0267-4
17. Rahman, A., Rasul, M. G., Khan, M. M. K., & Sharma, S. (2013). Impact of Alternative Fuels on the Cement Manufacturing Plant Performance: An Overview. Procedia Engineering, 56, 393–400. doi:10.1016/j.proeng.2013.03.138
[CROSSREF]
18. Dabrowska, M., & Giergiczny, Z. (2013). Chemical resistance of mortars made of cements with calcareous fly ash. Roads and Bridges – Drogi i Mosty, 12(2), 131–146. doi:10.7409/rabdim.013.010
19. Chandratilake, S. R., & Dias, W. P. S. (2013). Sustainability rating systems for buildings: Comparisons and correlations. Energy, 59, 22–28. doi:10.1016/j.energy.2013.07.026
[CROSSREF]
20. Matarneh, R. T. (2017). Development of Sustainable Assessment Method and Design Tool for Existing and Traditional Buildings in Jordan. Architecture Civil Engineering Environment, 10(4), 15–31.
[CROSSREF]
21. Chen, Y., Okudan, G. E., & Riley, D. R. (2010). Sustainable performance criteria for construction method selection in concrete buildings. Automation in Construction, 19(2), 235–244. doi:10.1016/j.autcon.2009.10.004
[CROSSREF]
22. Chen, J. J., Fung, W. W. S., Ng, P. L., & Kwan, A. K. H. (2012). Adding fillers to reduce embodied carbon and embodied energy of concrete. In Twelfth International Conference on Recent Advances in Concrete Technology and Sustainability, Prague (pp. 91–107). Michigan: American Concrete Institute.
23. Zhang, Y. R., Liu, M. H., Xie, H. B., & Wang, Y. F. (2014). Assessment of CO2 emissions and cost in fly ash concrete. In Environment, Energy and Applied Technology: Proceedings of the 2014 International Conference on Frontier of Energy and Environment Engineering (ICFEEE 2014), Taiwan (pp. 327–331). CRC Press.
24. Teixeira, E. R., Mateus, R., Camõesa, A. F., Bragança, L., & Branco, F. G. (2016). Comparative environmental life-cycle analysis of concretes using biomass and coal fly ashes as partial cement replacement material. Journal of Cleaner Production, 112, 2221–2230. doi:10.1016/j.jclepro.2015.09.124
[CROSSREF]
25. Petek Gursel, A., Masanet, E., Horvath, A., & Stadel, A. (2014). Life-cycle inventory analysis of concrete production: A critical review. Cement and Concrete Composites, 51, 38–48. doi:10.1016/j.cemconcomp.2014.03.005
[CROSSREF]
26. Abd Rashid, A. F., & Yusoff, S. (2015). A review of life cycle assessment method for building industry. Renewable and Sustainable Energy Reviews, 45, 244–248. doi:10.1016/j.rser.2015.01.043
[CROSSREF]
27. Lewandowska, A., Noskowiak, A., Pajchrowski, G., & Zarebska, J. (2015). Between full LCA and energy certification methodology - a comparison of six methodological variants of buildings environmental assessment. International Journal of Life Cycle Assessment, 20(1), 9–22. doi:10.1007/s11367-014-0805-3
[CROSSREF]
28. Tait, M. W., & Cheung, W. M. (2016). A comparative cradle-to-gate life cycle assessment of three concrete mix designs. International Journal of Life Cycle Assessment, 21(6), 847–860. doi:10.1007/s11367-016-1045-5
[CROSSREF]
29. Yang, K. H., Song, J. K., & Song, K. I. (2013). Assessment of CO2 reduction of alkali-activated concrete. Journal of Cleaner Production, 39, 265–272. doi:10.1016/j.jclepro.2012.08.001
[CROSSREF]
30. Yang, K. H., Jung, Y. B., Cho, M. S., & Tae, S. H. (2015). Effect of supplementary cementitious materials on reduction of CO2 emissions from concrete. Journal of Cleaner Production, 103, 774–783. doi:10.1016/j.jclepro.2014.03.018
[CROSSREF]
31. Turner, L. K., & Collins, F. G. (2013). Carbon dioxide equivalent (CO2-e) emissions: A comparison between geopolymer and OPC cement concrete. Construction and Building Materials, 43, 125–130. doi:10.1016/j.conbuildmat.2013.01.023
[CROSSREF]
32. Collins, F. (2010). Inclusion of carbonation during the life cycle of built and recycled concrete: Influence on their carbon footprint. International Journal of Life Cycle Assessment, 15(6), 549–556. doi:10.1007/s11367-010-0191-4
[CROSSREF]
33. Cassagnabère, F., Mouret, M., Escadeillas, G., Broilliard, P., & Bertrand, A. (2010). Metakaolin, a solution for the precast industry to limit the clinker content in concrete: Mechanical aspects. Construction and Building Materials, 24(7), 1109–1118. doi:10.1016/j.conbuildmat.2009.12.032
[CROSSREF]
34. Kubissa, W., Jaskulski, R., & Brodnan, M. (2016). Influence of SCM on the Permeability of Concrete with Recycled Aggregate. Periodica Polytechnica Civil Engineering, 60(4), 583–590. doi:http://dx.doi.org/10.3311/PPci.8614
[CROSSREF]
35. Kubissa, W., Simon, T., Jaskulski, R., Reiterman, P., & Supera, M. (2017). Ecological High Performance Concrete. Procedia Engineering, 172, 595–603. doi:10.1016/j.proeng.2017.02.186
[CROSSREF]
36. Kubissa, W. (2016). Sorpcyjność betonu (Sorptivity of concrete). Warszawa: Oficyna Wydawnicza Politechniki Warszawskiej.
37. Woodson, D. D. (2012). Concrete Portable Handbook (1st Edition). Butterworth-Heinemann. doi:10.1016/C2009-0-64403-2
38. Kozioł, W., & Czaja, P. (2010). Rock Mining in Poland – Present Situation, Perspectives. Górnictwo i Geologia, 5(3), 41–58.

### FIGURES & TABLES

Figure 1.

Ecological Index results

Figure 2.

Ecological Index plotted against reciprocal of Performance Index

### REFERENCES

1. Grin, J., Rotmans, J., & Schot, J. (2010). Transitions to Sustainable Development. Transitions to Sustainable Development: New Directions in the Study of Long Term Transformative Change. doi:10.4324/9780203856598
[CROSSREF]
2. Zuo, J., & Zhao, Z. Y. (2014). Green building research-current status and future agenda: A review. Renewable and Sustainable Energy Reviews, 30, 271–281. doi:10.1016/j.rser.2013.10.021
[CROSSREF]
3. Behera, M., Bhattacharyya, S. K., Minocha, A. K., Deoliya, R., & Maiti, S. (2014). Recycled aggregate from C&D waste & its use in concrete – A breakthrough towards sustainability in construction sector: A review. Construction and Building Materials. doi:10.1016/j.conbuildmat.2014.07.003
4. Węglorz, M. (2014). Selected Aspects of Sustainable Civil Engineering. Architecture Civil Engineering Environment, 7(1), 41–47.
5. Milošević, P. (2012). Sustainable Eco Planning Strategies in East Europe (Case Study of Belgrade). Architecture Civil Engineering Environment, 5(4), 29–42.
6. Pawlikowska-Piechotka, A., & Piechotka, M. (2012). Urban Sustainable Development and Green Agenda Perspective (Case Study in Warsaw). Architecture Civil Engineering Environment, 5(4), 43–52.
7. Słyk, J. (2015). Methodology of Architectural Design And Rules of Cooperation in The Digital Enviroment. Augmented Space as a Field of Research and Alternative Environment for Architectural Creation. Architecture Civil Engineering Environment, 8(4), 11–18.
8. Witkowski, H. (2015). Sustainability of Self-Compacting Concrete. Architecture Civil Engineering Environment, 8(1), 83–88.
9. Pavlík, Z., Fořt, J., Záleská, M., Pavlíková, M., Trník, A., Medved, I., … Černý, R. (2016). Energy-efficient thermal treatment of sewage sludge for its application in blended cements. Journal of Cleaner Production, 112, 409–419. doi:10.1016/j.jclepro.2015.09.072
[CROSSREF]
10. Muhd Norhasri, M. S., Hamidah, M. S., Mohd Fadzil, A., & Megawati, O. (2016). Inclusion of nano metakaolin as additive in ultra high performance concrete (UHPC). Construction and Building Materials, 127, 167–175. doi:10.1016/j.conbuildmat.2016.09.127
[CROSSREF]
11. Kubissa, W., Jaskulski, R., & Reiterman, P. (2017). Ecological Concrete Based on Blast-Furnace Cement with Incorporated Coarse Recycled Concrete Aggregate and Fly Ash Addition. Journal of Renewable Materials, 5(1), 53–61. Doi:10.7569/JRM.2017.634103
[CROSSREF]
12. Gartner, E. (2004). Industrially interesting approaches to “low-CO2” cements. Cement and Concrete Research, 34(9), 1489–1498. Doi:10.1016/j.cemconres.2004.01.021
[CROSSREF]
13. Müller, C. (2006). Environmental and technical aspects of the application of blended cements in concrete. Roads and Bridges – Drogi i Mosty, 5(3), 43–72.
14. Dziuk, D., Giergiczny, Z., & Garbacik, A. (2013). Calcareous fly ash as a main constituent of common cements. Roads and Bridges – Drogi i Mosty, 12(1), 57–69.
15. Mokrzycki, E., & Uliasz- Bocheńczyk, A. (2003). Alternative fuels for the cement industry. Applied Energy, 74(1–2), 95–100. doi:10.1016/S0306-2619(02)00135-6
[CROSSREF]
16. Li, F., & Zhang, W. (2011). Combustion of sewage sludge as alternative fuel for cement industry. Journal Wuhan University of Technology, Materials Science Edition, 26(3), 556–560. doi:10.1007/s11595-011-0267-4
17. Rahman, A., Rasul, M. G., Khan, M. M. K., & Sharma, S. (2013). Impact of Alternative Fuels on the Cement Manufacturing Plant Performance: An Overview. Procedia Engineering, 56, 393–400. doi:10.1016/j.proeng.2013.03.138
[CROSSREF]
18. Dabrowska, M., & Giergiczny, Z. (2013). Chemical resistance of mortars made of cements with calcareous fly ash. Roads and Bridges – Drogi i Mosty, 12(2), 131–146. doi:10.7409/rabdim.013.010
19. Chandratilake, S. R., & Dias, W. P. S. (2013). Sustainability rating systems for buildings: Comparisons and correlations. Energy, 59, 22–28. doi:10.1016/j.energy.2013.07.026
[CROSSREF]
20. Matarneh, R. T. (2017). Development of Sustainable Assessment Method and Design Tool for Existing and Traditional Buildings in Jordan. Architecture Civil Engineering Environment, 10(4), 15–31.
[CROSSREF]
21. Chen, Y., Okudan, G. E., & Riley, D. R. (2010). Sustainable performance criteria for construction method selection in concrete buildings. Automation in Construction, 19(2), 235–244. doi:10.1016/j.autcon.2009.10.004
[CROSSREF]
22. Chen, J. J., Fung, W. W. S., Ng, P. L., & Kwan, A. K. H. (2012). Adding fillers to reduce embodied carbon and embodied energy of concrete. In Twelfth International Conference on Recent Advances in Concrete Technology and Sustainability, Prague (pp. 91–107). Michigan: American Concrete Institute.
23. Zhang, Y. R., Liu, M. H., Xie, H. B., & Wang, Y. F. (2014). Assessment of CO2 emissions and cost in fly ash concrete. In Environment, Energy and Applied Technology: Proceedings of the 2014 International Conference on Frontier of Energy and Environment Engineering (ICFEEE 2014), Taiwan (pp. 327–331). CRC Press.
24. Teixeira, E. R., Mateus, R., Camõesa, A. F., Bragança, L., & Branco, F. G. (2016). Comparative environmental life-cycle analysis of concretes using biomass and coal fly ashes as partial cement replacement material. Journal of Cleaner Production, 112, 2221–2230. doi:10.1016/j.jclepro.2015.09.124
[CROSSREF]
25. Petek Gursel, A., Masanet, E., Horvath, A., & Stadel, A. (2014). Life-cycle inventory analysis of concrete production: A critical review. Cement and Concrete Composites, 51, 38–48. doi:10.1016/j.cemconcomp.2014.03.005
[CROSSREF]
26. Abd Rashid, A. F., & Yusoff, S. (2015). A review of life cycle assessment method for building industry. Renewable and Sustainable Energy Reviews, 45, 244–248. doi:10.1016/j.rser.2015.01.043
[CROSSREF]
27. Lewandowska, A., Noskowiak, A., Pajchrowski, G., & Zarebska, J. (2015). Between full LCA and energy certification methodology - a comparison of six methodological variants of buildings environmental assessment. International Journal of Life Cycle Assessment, 20(1), 9–22. doi:10.1007/s11367-014-0805-3
[CROSSREF]
28. Tait, M. W., & Cheung, W. M. (2016). A comparative cradle-to-gate life cycle assessment of three concrete mix designs. International Journal of Life Cycle Assessment, 21(6), 847–860. doi:10.1007/s11367-016-1045-5
[CROSSREF]
29. Yang, K. H., Song, J. K., & Song, K. I. (2013). Assessment of CO2 reduction of alkali-activated concrete. Journal of Cleaner Production, 39, 265–272. doi:10.1016/j.jclepro.2012.08.001
[CROSSREF]
30. Yang, K. H., Jung, Y. B., Cho, M. S., & Tae, S. H. (2015). Effect of supplementary cementitious materials on reduction of CO2 emissions from concrete. Journal of Cleaner Production, 103, 774–783. doi:10.1016/j.jclepro.2014.03.018
[CROSSREF]
31. Turner, L. K., & Collins, F. G. (2013). Carbon dioxide equivalent (CO2-e) emissions: A comparison between geopolymer and OPC cement concrete. Construction and Building Materials, 43, 125–130. doi:10.1016/j.conbuildmat.2013.01.023
[CROSSREF]
32. Collins, F. (2010). Inclusion of carbonation during the life cycle of built and recycled concrete: Influence on their carbon footprint. International Journal of Life Cycle Assessment, 15(6), 549–556. doi:10.1007/s11367-010-0191-4
[CROSSREF]
33. Cassagnabère, F., Mouret, M., Escadeillas, G., Broilliard, P., & Bertrand, A. (2010). Metakaolin, a solution for the precast industry to limit the clinker content in concrete: Mechanical aspects. Construction and Building Materials, 24(7), 1109–1118. doi:10.1016/j.conbuildmat.2009.12.032
[CROSSREF]
34. Kubissa, W., Jaskulski, R., & Brodnan, M. (2016). Influence of SCM on the Permeability of Concrete with Recycled Aggregate. Periodica Polytechnica Civil Engineering, 60(4), 583–590. doi:http://dx.doi.org/10.3311/PPci.8614
[CROSSREF]
35. Kubissa, W., Simon, T., Jaskulski, R., Reiterman, P., & Supera, M. (2017). Ecological High Performance Concrete. Procedia Engineering, 172, 595–603. doi:10.1016/j.proeng.2017.02.186
[CROSSREF]
36. Kubissa, W. (2016). Sorpcyjność betonu (Sorptivity of concrete). Warszawa: Oficyna Wydawnicza Politechniki Warszawskiej.
37. Woodson, D. D. (2012). Concrete Portable Handbook (1st Edition). Butterworth-Heinemann. doi:10.1016/C2009-0-64403-2
38. Kozioł, W., & Czaja, P. (2010). Rock Mining in Poland – Present Situation, Perspectives. Górnictwo i Geologia, 5(3), 41–58.


text:
# Homework Help: Impulse response of recursive DT system

1. Sep 7, 2010

### dionysian

1. The problem statement, all variables and given/known data

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$
Determine the impulse response $$h[n]$$ by calculating the zero-state response with $$x[n] = \delta [n]$$

2. Relevant equations

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$

3. The attempt at a solution

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$
$$y[n] = 1.8\cos (\frac{\pi }{{16}})y[n - 1] - 0.81y[n - 2] + x[n] + 0.5x[n - 1]$$
$$h[n] = 1.8\cos (\frac{\pi }{{16}})y[n - 1] - 0.81y[n - 2] + \delta [n] + 0.5\delta [n - 1]$$
The zero-state response means
$$\begin{array}{l} y[- 1] = 0 \\ y[- 2] = 0 \\ \end{array}$$
So my question is how can I write out the impulse response explicitly in one expression. I think I could calculate it by $$h[0],h[1],h[2].....$$ but I want to write it out as just one expression but the recursive terms$$y[n - 1]$$ and $$y[n - 2]$$ are kind of throwing me off.


text:
Definition:Test Function

Definition

Let $\phi : \R^d \to \C$ be a complex-valued function.

Let $K \subseteq \R^d$ be a compact subset.

Let $K$ be the support of $\phi$:

$\map \supp \phi = K$

Let $\phi$ be a smooth function across $K$ with respect to all its variables.

Then $\phi$ is known as a test function.

Also known as

Test functions are also known as bump functions.

Examples

Exponential of $\dfrac 1 {x^2 - 1}$

The graph of the test function. It is smooth everywhere. Outside of its support denoted by dashed lines the function is identically zero. At boundary points it connects smoothly.

Let $\phi : \R \to \R$ be a real function with support on $x \in \closedint {-1} 1$ such that:

$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$

Then $\phi$ is a test function.

Also see

• Results about test functions can be found here.


text:
# hclust

0th

Percentile

##### Hierarchical Clustering

Hierarchical cluster analysis on a set of dissimilarities and methods for analyzing it.

Keywords
multivariate, cluster
##### Usage
hclust(d, method = "complete", members = NULL)# S3 method for hclust
plot(x, labels = NULL, hang = 0.1, check = TRUE,
axes = TRUE, frame.plot = FALSE, ann = TRUE,
main = "Cluster Dendrogram",
sub = NULL, xlab = NULL, ylab = "Height", …)
##### Arguments
d

a dissimilarity structure as produced by dist.

method

the agglomeration method to be used. This should be (an unambiguous abbreviation of) one of "ward.D", "ward.D2", "single", "complete", "average" (= UPGMA), "mcquitty" (= WPGMA), "median" (= WPGMC) or "centroid" (= UPGMC).

members

NULL or a vector with length size of d. See the ‘Details’ section.

x

an object of the type produced by hclust.

hang

The fraction of the plot height by which labels should hang below the rest of the plot. A negative value will cause the labels to hang down from 0.

check

logical indicating if the x object should be checked for validity. This check is not necessary when x is known to be valid such as when it is the direct result of hclust(). The default is check=TRUE, as invalid inputs may crash R due to memory violation in the internal C plotting code.

labels

A character vector of labels for the leaves of the tree. By default the row names or row numbers of the original data are used. If labels = FALSE no labels at all are plotted.

axes, frame.plot, ann

logical flags as in plot.default.

main, sub, xlab, ylab

character strings for title. sub and xlab have a non-NULL default when there's a tree\$call.

Further graphical arguments. E.g., cex controls the size of the labels (if plotted) in the same way as text.

##### Details

This function performs a hierarchical cluster analysis using a set of dissimilarities for the $n$ objects being clustered. Initially, each object is assigned to its own cluster and then the algorithm proceeds iteratively, at each stage joining the two most similar clusters, continuing until there is just a single cluster. At each stage distances between clusters are recomputed by the Lance--Williams dissimilarity update formula according to the particular clustering method being used.

A number of different clustering methods are provided. Ward's minimum variance method aims at finding compact, spherical clusters. The complete linkage method finds similar clusters. The single linkage method (which is closely related to the minimal spanning tree) adopts a ‘friends of friends’ clustering strategy. The other methods can be regarded as aiming for clusters with characteristics somewhere between the single and complete link methods. Note however, that methods "median" and "centroid" are not leading to a monotone distance measure, or equivalently the resulting dendrograms can have so called inversions or reversals which are hard to interpret, but note the trichotomies in Legendre and Legendre (2012).

Two different algorithms are found in the literature for Ward clustering. The one used by option "ward.D" (equivalent to the only Ward option "ward" in R versions $\le$ 3.0.3) does not implement Ward's (1963) clustering criterion, whereas option "ward.D2" implements that criterion (Murtagh and Legendre 2014). With the latter, the dissimilarities are squared before cluster updating. Note that agnes(*, method="ward") corresponds to hclust(*, "ward.D2").

If members != NULL, then d is taken to be a dissimilarity matrix between clusters instead of dissimilarities between singletons and members gives the number of observations per cluster. This way the hierarchical cluster algorithm can be ‘started in the middle of the dendrogram’, e.g., in order to reconstruct the part of the tree above a cut (see examples). Dissimilarities between clusters can be efficiently computed (i.e., without hclust itself) only for a limited number of distance/linkage combinations, the simplest one being squared Euclidean distance and centroid linkage. In this case the dissimilarities between the clusters are the squared Euclidean distances between cluster means.

In hierarchical cluster displays, a decision is needed at each merge to specify which subtree should go on the left and which on the right. Since, for $n$ observations there are $n-1$ merges, there are $2^{(n-1)}$ possible orderings for the leaves in a cluster tree, or dendrogram. The algorithm used in hclust is to order the subtree so that the tighter cluster is on the left (the last, i.e., most recent, merge of the left subtree is at a lower value than the last merge of the right subtree). Single observations are the tightest clusters possible, and merges involving two observations place them in order by their observation sequence number.

##### Value

An object of class hclust which describes the tree produced by the clustering process. The object is a list with components:

merge

an $n-1$ by 2 matrix. Row $i$ of merge describes the merging of clusters at step $i$ of the clustering. If an element $j$ in the row is negative, then observation $-j$ was merged at this stage. If $j$ is positive then the merge was with the cluster formed at the (earlier) stage $j$ of the algorithm. Thus negative entries in merge indicate agglomerations of singletons, and positive entries indicate agglomerations of non-singletons.

height

a set of $n-1$ real values (non-decreasing for ultrametric trees). The clustering height: that is, the value of the criterion associated with the clustering method for the particular agglomeration.

order

a vector giving the permutation of the original observations suitable for plotting, in the sense that a cluster plot using this ordering and matrix merge will not have crossings of the branches.

labels

labels for each of the objects being clustered.

call

the call which produced the result.

method

the cluster method that has been used.

dist.method

the distance that has been used to create d (only returned if the distance object has a "method" attribute).

There are print, plot and identify (see identify.hclust) methods and the rect.hclust() function for hclust objects.

##### Note

Method "centroid" is typically meant to be used with squared Euclidean distances.

##### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988). The New S Language. Wadsworth & Brooks/Cole. (S version.)

Everitt, B. (1974). Cluster Analysis. London: Heinemann Educ. Books.

Hartigan, J.A. (1975). Clustering Algorithms. New York: Wiley.

Sneath, P. H. A. and R. R. Sokal (1973). Numerical Taxonomy. San Francisco: Freeman.

Anderberg, M. R. (1973). Cluster Analysis for Applications. Academic Press: New York.

Gordon, A. D. (1999). Classification. Second Edition. London: Chapman and Hall / CRC

Murtagh, F. (1985). “Multidimensional Clustering Algorithms”, in COMPSTAT Lectures 4. Wuerzburg: Physica-Verlag (for algorithmic details of algorithms used).

McQuitty, L.L. (1966). Similarity Analysis by Reciprocal Pairs for Discrete and Continuous Data. Educational and Psychological Measurement, 26, 825--831. 10.1177/001316446602600402.

Legendre, P. and L. Legendre (2012). Numerical Ecology, 3rd English ed. Amsterdam: Elsevier Science BV.

Murtagh, Fionn and Legendre, Pierre (2014). Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion? Journal of Classification, 31, 274--295. 10.1007/s00357-014-9161-z.

identify.hclust, rect.hclust, cutree, dendrogram, kmeans.

For the Lance--Williams formula and methods that apply it generally, see agnes from package cluster.

• hclust
• plot.hclust
• print.hclust
##### Examples
library(stats) # NOT RUN { require(graphics) ### Example 1: Violent crime rates by US state hc <- hclust(dist(USArrests), "ave") plot(hc) plot(hc, hang = -1) ## Do the same with centroid clustering and *squared* Euclidean distance, ## cut the tree into ten clusters and reconstruct the upper part of the ## tree from the cluster centers. hc <- hclust(dist(USArrests)^2, "cen") memb <- cutree(hc, k = 10) cent <- NULL for(k in 1:10){ cent <- rbind(cent, colMeans(USArrests[memb == k, , drop = FALSE])) } hc1 <- hclust(dist(cent)^2, method = "cen", members = table(memb)) opar <- par(mfrow = c(1, 2)) plot(hc, labels = FALSE, hang = -1, main = "Original Tree") plot(hc1, labels = FALSE, hang = -1, main = "Re-start from 10 clusters") par(opar) ### Example 2: Straight-line distances among 10 US cities ## Compare the results of algorithms "ward.D" and "ward.D2" data(UScitiesD) mds2 <- -cmdscale(UScitiesD) plot(mds2, type="n", axes=FALSE, ann=FALSE) text(mds2, labels=rownames(mds2), xpd = NA) hcity.D <- hclust(UScitiesD, "ward.D") # "wrong" hcity.D2 <- hclust(UScitiesD, "ward.D2") opar <- par(mfrow = c(1, 2)) plot(hcity.D, hang=-1) plot(hcity.D2, hang=-1) par(opar) # }
Documentation reproduced from package stats, version 3.5.0, License: Part of R 3.5.0

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