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# MediaPlayersvideos 1.1

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MediaPlayersvideos 1.1 contains of the executables below. They take 105.96 KB (108504 bytes) on disk.

• Uninstall.exe (105.96 KB)

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# how to find the ratio of circumference and the ratio of area of two circles?

question 1: if the lengths of the radii of two circles are 1 and 4 respectively, hat is the ratio of their circumference?of their areas?

question 2: the areas of two circles are 9 cm(squared) and 16 cm, squared .what is the ratio of the radii?

Relevance

RATIO OF THEIR CIRCUMFERENCE :

Circumference of a circle is " 2 pi r "

Substitute thr radius in the formula .

r1 = 1

r2 = 4

2 pi r1

-----------

2 pi r2

= > 1 / 4

RATIO OF THEIR AREAS :

1/ 16

2 ) 9 / 16 IS THE RATIO OF THEIR AREAS

HOPE THIS HELPS................

Circumference of Circle = 2 x radii x pi

Given: 1 and 4

And since the formula for Circumference does not alter for a Circle, your ratio for Circumference is:

1 to 4 or 1/4 or 0.25 depending upon what your teacher wants.

Given: 1 and 4:

It would be 1 x 1 = 1 and 4 x 4 = 16

Your ratio would be: 1 to 16 or 1/16 or 0.0625

As for Given: 9 cm squared and 16 cm squared

You need to square root these which will give you:

3 and 4.

Your ratio is: 3 to 4 or 3/4 or 0.75

• Anonymous
5 years ago

RE:

how to find the ratio of circumference and the ratio of area of two circles?

question 1: if the lengths of the radii of two circles are 1 and 4 respectively, hat is the ratio of their circumference?of their areas?

question 2: the areas of two circles are 9 cm(squared) and 16 cm, squared .what is the ratio of the radii?

Source(s): find ratio circumference ratio area circles: https://tr.im/7BjQF

this gives you ratio of circumference as 1:4.

Area = pi(r^2)

this gives ratio 1:16.

Let the radii of both circles be, r and R. where:

R = 4r (given the 1:4 ratio)

Now find the circumference of each circle:

C = 2πR . . . and . . . c = 2πr

Since R = 4r, substitute it in:

C = 2πR

C = 2π(4r)

C = 8πr

Now since r/R = 1/4, find the ratio of c/C

2πr / 8πr = 1/4

So the circumference is also a 1:4 ratio.

Now do the same with the Area:

A = πR² . . . and a = πr²

A = π(4r)²

A = 16πr²

find the ratio a/A:

πr² / 16πr² = 1/16

So the ratio of areas are 1:16.

#2 works this backwards, you've given the areas and need to find the ratio of radii:

A = πR² . . . and . . .a = πr²

16 = πR² . . . and . . .9 = πr²

16/π = R² . . . and . . . 9/π = r²

4/√π = R . . . and . . . 3/√π = r

calculate r/R

(3/√π) / (4/√π)

(3/√π) * (√π /4)

3/4

The ratio is 3:4.

• Anonymous
5 years ago

A1/A2 = 6(pi)m^2 / 150(pi)m^2 = 1/25 A1/A2 = 1/25 = 1/5^2 since Area is proportional to the radius and likewise, Circumference is proportional to the radius of the circle. If A1?A2 = 1/25 (=1/5^2) then So, c1/c2 = 1/5 <= ANS

• 7 years ago

Find the ratio of the circumferences of two circles, radii 10cm and 15cm.

format: plain


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# Does a fluid's temperature affect the way light passes through it?

For example, if I were to supercool water would it's refractive index still be 1.33 or would it be 1.31, the same as water-based ice even though it's still in liquid form?

-

Yes, refractive index is usually temperature dependent. I'm guessing here that the index of supercooled water would be different than room temperature water, but also different from ice, because the density is different in each case.

-
Yes, index of refraction in general depends on temperature. (Note that it also depends on wavelength.) My copy of the CRC Handbook gives the following for liquid water using a He-Ne laser with $\lambda = 632.80$ nm:
$n = 1.31744$ at 100 C, $n = 1.33306$ at 0 C, with steps of 10 C in between.


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# Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs.

What is the expected number of coin tosses?

• Yet another copy and paste from Brilliant.org: brilliant.org/i/5rCgJ3 – Erick Wong Apr 17 '13 at 5:49
• @ErickWong: Is this a recent problem on brilliant.org? – robjohn Apr 17 '13 at 8:48

Let $e$ be the expected number of tosses. It is clear that $e$ is finite.

Start tossing. If we get a tail immediately (probability $\frac{1}{2}$) then the expected number is $e+1$. If we get a head then a tail (probability $\frac{1}{4}$), then the expected number is $e+2$. Continue $\dots$. If we get $4$ heads then a tail, the expected number is $e+5$. Finally, if our first $5$ tosses are heads, then the expected number is $5$. Thus $$e=\frac{1}{2}(e+1)+\frac{1}{4}(e+2)+\frac{1}{8}(e+3)+\frac{1}{16}(e+4)+\frac{1}{32}(e+5)+\frac{1}{32}(5).$$ Solve this linear equation for $e$. We get $e=62$.

• It is clear that e is finite, but how can you show it properly though ? Thanks. – Dark Jul 3 '15 at 17:39
• If one wants, let $X$ be the number of tosses. Then $\Pr(X=n)\le (1/2)^{n-5}$. So $E(X)\le \sum n (1/2)^{n-5}$, a convergent series. – André Nicolas Jul 3 '15 at 17:58
• The same method obviously generalizes to give $e_n$, the expected number of tosses to get $n$ consecutive heads ($n \ge 1$): $$e_n=\frac{1}{2}(e_n+1)+\frac{1}{4}(e_n+2)+\frac{1}{8}(e_n+3)+\frac{1}{16}(e_n+4)+\cdots +\frac{1}{2^n}(e_n+n)+\frac{1}{2^n}(n),$$ the solution of which is easily found to be $$e_n = 2(2^n - 1).$$ – r.e.s. Jul 19 '15 at 23:57
• Why are TT, TTT not considered? – Jaydev Jul 24 '17 at 1:24
• @Jaydev TT and TTT both are covered by the case "if we get a tail immediately". – David K Oct 6 '17 at 13:13

Lets calculate it for $n$ consecutive tosses the expected number of tosses needed.

Lets denote $E_n$ for $n$ consecutive heads. Now if we get one more head after $E_{n-1}$, then we have $n$ consecutive heads or if it is a tail then again we have to repeat the procedure.

So for the two scenarios:

1. $E_{n-1}+1$
2. $E_{n}{+1}$ ($1$ for a tail)

So, $E_n=\frac12(E_{n-1} +1)+\frac12(E_{n-1}+ E_n+ 1)$, so $E_n= 2E_{n-1}+2$.

We have the general recurrence relation. Define $f(n)=E_n+2$ with $f(0)=2$. So,

\begin{align} f(n)&=2f(n-1) \\ \implies f(n)&=2^{n+1} \end{align}

Therefore, $E_n = 2^{n+1}-2 = 2(2^n-1)$

For $n=5$, it will give us $2(2^5-1)=62$.

• Amazing solution, thank you – Jaydev Jul 24 '17 at 1:30

Here is a generating function approach.

Consider the following toss strings, probabilities, and terms

$$\color{#00A000}{ \begin{array}{llc} T&\frac12&\qquad\frac12x\\ HT&\frac14&\qquad\frac14x^2\\ HHT&\frac18&\qquad\frac18x^3\\ HHHT&\frac1{16}&\qquad\frac1{16}x^4\\ HHHHT&\frac1{32}&\qquad\frac1{32}x^5\\ \color{#C00000}{HHHHH}&\color{#C00000}{\frac1{32}}&\color{#C00000}{\qquad\frac1{32}x^5} \end{array} }$$ Each term has the probability as its coefficient and the length of the string as its exponent.

Possible outcomes are any combination of the green strings followed by the red string. We get the generating function of the probability of ending after $n$ tosses to be \begin{align} f(x)&=\sum_{k=0}^\infty\left(\frac12x+\frac14x^2+\frac18x^3+\frac1{16}x^4+\frac1{32}x^5\right)^k\frac1{32}x^5\\ &=\frac{\frac1{32}x^5}{1-\left(\frac12x+\frac14x^2+\frac18x^3+\frac1{16}x^4+\frac1{32}x^5\right)}\\ &=\frac{\frac1{32}x^5}{1-\frac{\frac12x-\frac1{64}x^6}{1-\frac12x}}\\ &=\frac{\frac1{32}x^5-\frac1{64}x^6}{1-x+\frac1{64}x^6} \end{align} The average duration is then \begin{align} f'(1) &=\left.\frac{\left(\frac5{32}x^4-\frac6{64}x^5\right)\left(1-x+\frac1{64}x^6\right)-\left(\frac1{32}x^5-\frac1{64}x^6\right)\left(-1+\frac6{64}x^5\right)}{\left(1-x+\frac1{64}x^6\right)^2}\right|_{\large x=1}\\ &=\frac{\frac4{64}\frac1{64}+\frac1{64}\frac{58}{64}}{\left(\frac1{64}\right)^2}\\[12pt] &=62 \end{align}

• Could you elaborate briefly on why the derivative gives the expected number of flips? – Austin Mohr Aug 20 '13 at 2:37
• @AustinMohr: If $f(x)$ is the generating function of the probability $p_n$ of the ending after $n$ tosses $$f(x)=\sum_{n=0}^\infty p_nx^n$$ then, because the probability of lasting an infinite number of tosses is $0$, we have \begin{align} f(1) &=\sum_{n=0}^\infty p_n\\ &=1 \end{align} Furthermore, \begin{align} f'(1) &=\sum_{n=0}^\infty n\,p_n\\ &=\mathrm{E}(n) \end{align} – robjohn Aug 20 '13 at 5:13
• Do you know how to find the distribution (or expectation and variance) for the number of tosses until either 5 consecutive heads or 5 consecutive tails? (Or 5 consecutive equal results from rolling dice.) Is there a question on math.se about this? – ShreevatsaR Dec 15 '15 at 14:17
• I found an answer using martingales here: quora.com/… but I'm curious if there is a generating functions way (also about the distribution, say variance or number of trials until 90% probability of seeing what we want). – ShreevatsaR Dec 15 '15 at 14:31
• @ShreevatsaR: The generating function for the probability of ending on $n$ tosses for that problem is similar: $$\frac{\frac1{16}x^5}{1-\left(\frac12x+\frac14x^2 +\frac18x^3+\frac1{16}x^4\right)}$$ From that, we can compute the expectation and variance. Looking at the coefficients of the series for the generating function we can also find out that in $66$ tosses, we will have a $90.0761\%$ chance of seeing $5$ heads or $5$ tails in a row. – robjohn Dec 15 '15 at 17:32

This problem is solvable with the next step conditioning method. Let $\mu_k$ denote the mean number of tosses until 5 consecutive heads occurs, given that $k$ consecutive heads just occured. Obviously $\mu_5=0$. Conditioning on the outcome of the next coin throw: $$\mu_k = 1 + \frac{1}{2} \mu_{k+1} + \frac{1}{2} \mu_0$$ Solving the resulting linear system:

In[28]:= Solve[Table[mu[k] == 1 + 1/2 mu[k + 1] + mu[0]/2, {k, 0, 4}],
Table[mu[k], {k, 0, 4}]] /. mu[5] -> 0

Out[28]= {{mu[0] -> 62, mu[1] -> 60, mu[2] -> 56, mu[3] -> 48,
mu[4] -> 32}}

Hence the expected number of coin flips $\mu_0$ equals 62.

• What tool did you use for solving? – pushpen.paul May 11 '15 at 17:19
• @pushpen.paul I used Mathematica – Sasha May 11 '15 at 17:20
• Can you please explain the original equation? – BOS Sep 20 '16 at 15:58
• @BOS Since $\mu_k$ is the conditional expectation, consider the next coin toss. Because a new toss was made, we add 1, in the next state, with equal probabilities we either get next head, in which case we gonna get $k+1$ heads, hence $\mu_{k+1}$, or the tail, in which case we break the streak of consecutive heads, hence $\mu_0$. – Sasha Sep 23 '16 at 3:03
• @Sasha This is the Markov way of solving it, right? This seems most intuitive to me of all methods presented here. – user3496060 Dec 22 '17 at 8:06

The question can be generalized to what is the expected number of tosses before we get x heads.Let's call this E(x). We can easily derive a recursive formula for E(x). Now, there are a total of two possibilities, first is that we fail to get the xth consecutive heads in xth attempt and second, we succeed. Probability of success is 1/(2^x) and probability of failure is 1-(1/(2^x)).

Now, if we were to fail to get xth consecutive heads in xth toss (i.e. case 1), the we will have to use a total of (E(x)+1) moves, because one move has been wasted.

On the other hand if we were to succeed in getting xth consecutive head in xth toss (i.e. case 2), the total moves is E(x-1)+1 , because we now take one move more than that was required to get x-1 consecutive heads.

So,

E(x) = P(failure) * (E(x)+1) + P(success) * (E(x-1)+1)
E(x) = [1-(1/(2^x))] * (E(x)+1) + [1/(2^x)] * (E(x-1)+1)

Also E(0) = 0 , because expected number of tosses to get 0 heads is zero, duh

now,

E(1) = (1-0.5) * (E(1)+1) + (0.5) * (E(0)+1) => E(1) = 2

E(2) = (1-0.125) * (E(1)+1) + (0.125) * (E(1)+1) => E(2) = 6

Similarly,

E(3) = 14

E(4) = 30

E(5) = 62

I would simplify the problem as follows:

Let $e$ = Expected number of flips until $5$ consecutive $H$, i.e., $E[5H]$
Let $f$ = Expected number of flips until $5$ consecutive $H$ when we have seen one $H$, i.e., $E[5H|H]$
Let $g$ = Expected number of flips until $5$ consecutive $H$ when we have seen two $H$, i.e., $E[5H|2H]$
Let $h$ = Expected number of flips until $5$ consecutive $H$ when we have seen three $H$, i.e., $E[5H|3H]$
Let $i$ = Expected number of flips until $5$ consecutive $H$ when we have seen four $H$, i.e., $E[5H|4H]$

Now Start flipping coin, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ then expected number of flips until 5 consecutive $H$ is $(f+1)$. Alternatively if $T$, we wasted 1 flip and expected number is still $(e+1)$ $$e=\frac12(e+1)+\frac12(f+1)\;$$

We now need $f$ to solve above to get $e$. Now we start with 1 $H$ and seeking 4 more $H$ to get total 5 $H$. Again, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ (total $2H$ so far) then expected number of flips until 5 consecutive $H$ is $(g+1)$. Alternatively if $T$, we wasted this flip and expected number is back to $(e+1)$

$$f=\frac12(g+1)+\frac12(e+1)\;$$

Continuing this way... $$g=\frac12(h+1)+\frac12(e+1)\;$$ $$h=\frac12(i+1)+\frac12(e+1)\;$$

Finally, Now we have 4 $H$ and seeking last $H$ to get total 5 $H$. Still, there is $\frac{1}{2}$ probability of getting $H$ or $T$. So if we get $H$ (total $5H$) then we need just $1$ flip. Alternatively if $T$ is observed, we wasted this flip and expected number is back to $(e+1)$ $$i=\frac12(1)+\frac12(e+1)\;$$

Solving these equations, $e=62$, $f=60$, $g=56$, $h=48$, $i=32$
This solution offers some insight into conditional expectations of number of flips needed till 5 consecutive $H$ given 1, 2, 3 and 4 consecutive $H$.

Expectation for getting n consecutive heads is : 2*(2^n-1). Thus for 5 heads it is = 62.

• How did you get the formula? – pushpen.paul May 11 '15 at 18:25

No one else seems to have suggested the following approach. Suppose we keep flipping a coin until we get five heads in a row. Define a "run" as either five consecutive heads or a tails flip plus the preceding streak of heads flips. (A run could be a single tails flip.) The number of coin flips is equal to the number of runs with at least four heads ($R_{4+}$), plus the number of runs with at least three heads ($R_{3+}$), and so on down to the number of runs with at least zero heads ($R_{0+}$). We can see this by expanding the terms:

$R_{0+}$ = # runs with 0 heads + # runs with 1 head + ... + # runs with 5 heads

$R_{1+}$ = # runs with 1 head + # runs with 2 heads + ... + # runs with 5 heads

...

$R_{4+}$ = # runs with 4 heads + # runs with 5 heads

# flips = # flips in runs with 0 heads + # flips in runs with 1 head + ... + # flips in runs with 5 heads

# flips in runs with 0 heads = # runs with 0 heads

# flips in runs with 1 head = 2 x # runs with 1 head

...

# flips in runs with 4 heads = 5 x # runs with 4 heads

# flips in runs with 5 heads = 5 x # runs with 5 heads

By linearity of expectation, the expected number of coin flips is $E(R_{0+}) + E(R_{1+}) + \ldots + E(R_{4+})$. $E(R_{4+})$ is $2 E(R_{5+}) = 2$, because one half of the time we flip at least four heads in a row, we go on to flip five heads in a row, i.e. the following coin flip is heads. In other words, in expectation, it takes two runs that start with four heads to achieve one run of five heads. Likewise, $E(R_{3+}) = 2 E(R_{4+}) = 4$, $E(R_{2+}) = 2 E(R_{3+}) = 8$, $E(R_{1+}) = 2 E(R_{2+}) = 16$, and $E(R_{0+}) = 2 E(R_{1+}) = 32$. The expected number of coin flips is $32 + 16 + 8 + 4 + 2 = 62$.

More generally, given a biased coin that comes up heads $p$ portion of the time, the expected number of flips to get $n$ heads in a row is $\frac{1}{p} + \frac{1}{p^2} + \ldots + \frac{1}{p^n} = \frac{1 - p^n}{p^n(1 - p)}$.

We can solve this without equations. Ask the following (auxiliary) question: how many flips you need to get either $N$ heads or $N$ tails. Then to get $N$ heads only you need twice as many flips. Start with the question of how many flips you need to get either $H$ or $T$. The answer is $$x = \frac12 (1) + \frac12 (1) = 1.$$ The reason is that there is $\frac12$ probability to get $H$, and after $1$ flip you are done. The same for $T$. To get only one $H$ you then need two flips. OK. Now we ask what it takes to get $HH$ or $TT$. The result is $$x=\frac12(1+2) + \frac12(1+2).$$ The number $2$ appears because, say you flip $H$ first, then you need on average $2$ flips to get another $H$, as we learned earlier. The same for $T$. So you need $3$ flips to get $TT$ or $HH$, and you need $6$ flips to get $HH$ only. And so on. You need $\frac12 (1+6) + \frac12(1+6) = 7$ flips to get either $HHH$ or $TTT$, and $14$ to get $HHH$ only. If you need $HHHH$ or $TTTT$, then flip $\frac12(1+14) + \frac12(1+14) = 15$ times, or $30$ times to get just $HHHH$. The sequence is $1, 3, 7, 15, \ldots$ to get either heads or tails. The formula is easy to extract: you need $2^N-1$ flips to get either $N$ heads or $N$ tails, or $2^{N+1}-2$ to get $N$ heads only. If $N=5$ we get the answer: $62$.

Use Markov chains. The nice part of Markov Chains is that they can be applied to a huge class of similar problems with relatively little thought (it's almost formulaic in application). It's also the most intuitive way to handle these problems.

(in Matlab code notation below)

%% setup full transition matrix with states from zero heads to 5 heads

T = $[ones(5,1)*.5,eye(5)*.5];$

$T = [T;zeros(1,6)]$

%%Take subset "Q" comprised of just transient states (5 heads is absorbing state) $Q = T(1:end-1,1:end-1);$

$M = inv(eye(5)-Q)$

absorbing Markov Chain has a similar example as this question BTW...

ans =

62
60
56
48
32

Where each row is the expected number of steps before being absorbed when starting in that transient state (0 through 4 heads, top to bottom).

A recursive programming solution is also possible. Below is the solution in Python

# Expected number of tosses to get n heads
def Expectation(n):
if n == 0:
return 0

return 2**n + Expectation(n-1)

>>> Expectation(1)
2
>>> Expectation(2)
6
>>> Expectation(3)
14
>>> Expectation(5)
62
>>> Expectation(10)
2046


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text:
# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

Article in press

Authors:

Z. Stanić

Title:

Some properties of the eigenvalues of the net Laplacian matrix of a signed graph

Source:

Discussiones Mathematicae Graph Theory

Received: 2019-09-30, Revised: 2020-03-10, Accepted: 2020-03-10, https://doi.org/10.7151/dmgt.2314

Abstract:

Given a signed graph $\dot{G}$, let $A_{\dot{G}}$ and $D^{\pm}_{\dot{G}}$ denote its standard adjacency matrix and the diagonal matrix of vertex net-degrees, respectively. The net Laplacian matrix of $\dot{G}$ is defined to be $N_{\dot{G}}=D^{\pm}_{\dot{G}}-A_{\dot{G}}$. In this study we give some properties of the eigenvalues of $N_{\dot{G}}$. In particular, we consider their behaviour under some edge perturbations, establish some relations between them and the eigenvalues of the standard Laplacian matrix and give some lower and upper bounds for the largest eigenvalue of $N_{\dot{G}}$.

Keywords:

(net) Laplacian matrix, edge perturbations, largest eigenvalue, net-degree


text:
This is an old version, view current version.

## 3.1 Line-Based Comments

Any characters on a line following two forward slashes (//) is ignored along with the slashes. These may be used, for example, to document variables,

data {
int<lower=0> N;  // number of observations
real y[N];  // observations
}


text:
# Why does a^x = e^(x(lna))

by jessepye
Tags: exlna
 P: 418 Why does a^x = e^(x(lna)) EDIT You got it before I typed this I think this is right, I'm just trying to remember it off the top of my head as my textbook is in school. Let the value of $a^{x}$ be equal to $y$ $a^{x} = y$ Take natural log of both sides $ln(a^{x}) = ln(y)$ Then we can bring the exponent out of the bracket $x * ln(a) = ln(y)$ Then we put both sides as the power of e to cancel the ln on the right $e^{x * ln(a)} = e^{ln(y)}$ $e^{x * ln(a)} = y$ Then since $a^{x} = y$ we sub that in for y and get $e^{x * ln(a)} = a^{x}$
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 Another way to see the same thing is to note that $aln(x)= ln(x^a)$ so that $e^{xln(a)}= e^{ln(a^x)}$. Then, because "$f(x)= e^x$" and "$g(x)= ln(x)$" are inverse functions, $e^{ln(a^x)}= a^x$.


text:
# His ring and her digits

A young lady named Thea decided to change the traditional codes and asked her boyfriend to marry her at the Eiffel Tower.

As he loves puzzles, she said while showing him this text, "rightly choose two consonants that appear twice in this puzzle, select the right digits among the following list and you'll get a present from me <3."

• 00
• 01
• 10
• 14
• 41
• 62
• 72

He found the right digits then received his wedding ring. How did he succeed?

• I think you have a typo "she said while showing her this text". Sep 29 at 16:25

I believe the correct digits are:

$$0$$ and $$4$$

Reasoning:

The puzzle places emphasis on the right column by using rightly and right. Additionally, I believe that the mention of Eiffel Tower and consommes are intentional. Consommes is a French word that translates consumes in English.

Using , we can evaluate if any matching numbers in the right column, when added together (e.g. consume each other), create a new number. $$0$$ and $$0$$ result in zero, so they stay broken apart. $$1$$ and $$1$$ becomes $$2$$ while $$2$$ and $$2$$ become $$4$$. After this process, we're left with $$0$$, $$0$$, $$2$$, $$4$$ and $$4$$.

Out of these, only $$0$$ and $$4$$ occur twice.

Note: I intentionally didn't apply multiple passes, which would break this answer.

• I +1 for the reasoning but actually "consommes" is a typo I made >< The right word is consonants
– JKHA
Sep 29 at 16:52


text:
$$E$$-optimal designs for second-order response surface models.(English)Zbl 1310.62097

Summary: $$E$$-optimal experimental designs for a second-order response surface model with $$k\geq 1$$ predictors are investigated. If the design space is the $$k$$-dimensional unit cube, Z. Galil and J. Kiefer [J. Stat. Plann. Inference 1, 121–132 (1977; Zbl 0381.62062)] determined optimal designs in a restricted class of designs (defined by the multiplicity of the minimal eigenvalue) and stated their universal optimality as a conjecture. In this paper, we prove this claim and show that these designs are in fact $$E$$-optimal in the class of all approximate designs. Moreover, if the design space is the unit ball, $$E$$-optimal designs have not been found so far and we also provide a complete solution to this optimal design problem. {
} The main difficulty in the construction of $$E$$-optimal designs for the second-order response surface model consists in the fact that for the multiplicity of the minimum eigenvalue of the “optimal information matrix” is larger than one (in contrast to the case $$k=1$$) and as a consequence the corresponding optimality criterion is not differentiable at the optimal solution. These difficulties are solved by considering nonlinear Chebyshev approximation problems, which arise from a corresponding equivalence theorem. The extremal polynomials which solve these Chebyshev problems are constructed explicitly leading to a complete solution of the corresponding $$E$$-optimal design problems.

MSC:

 62K20 Response surface designs 62K05 Optimal statistical designs 41A50 Best approximation, Chebyshev systems

Zbl 0381.62062
Full Text:

References:

 [1] Anderson-Cook, C. M., Borror, C. M. and Montgomery, D. C. (2009). Response surface design evaluation and comparison. J. Statist. Plann. Inference 139 629-641. · Zbl 1149.62063 [2] Cheng, C.-S. (1987). An application of the Kiefer-Wolfowitz equivalence theorem to a problem in Hadamard transform optics. Ann. Statist. 15 1593-1603. · Zbl 0629.62075 [3] Denisov, V. I. and Popov, A. A. (1976). $$A-E$$-optimal and orthogonal designs of experiments for polynomial models. Preprint. Scientific Council on a Complex Problem “Cybernetic,” Academy of Science (in Russian). Moscow, Russia. [4] Dette, H. (1993). A note on $$E$$-optimal designs for weighted polynomial regression. Ann. Statist. 21 767-771. · Zbl 0787.62076 [5] Dette, H. and Haines, L. M. (1994). $$E$$-optimal designs for linear and nonlinear models with two parameters. Biometrika 81 739-754. · Zbl 0816.62055 [6] Dette, H. and Röder, I. (1997). Optimal discrimination designs for multifactor experiments. Ann. Statist. 25 1161-1175. · Zbl 0888.62077 [7] Dette, H. and Studden, W. J. (1993). Geometry of $$E$$-optimality. Ann. Statist. 21 416-433. · Zbl 0780.62057 [8] Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer ordering of simplex designs for second-degree mixture models with four or more ingredients. Ann. Statist. 28 578-590. · Zbl 1105.62363 [9] Draper, N. R. and Pukelsheim, F. (2003). Canonical reduction of second-order fitted models subject to linear restrictions. Statist. Probab. Lett. 63 401-410. · Zbl 1116.62331 [10] Farrell, R. H., Kiefer, J. and Walbran, A. (1967). Optimum multivariate designs. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability ( Berkeley , Calif. , 1965 / 66) 113-138. Univ. California Press, Berkeley, CA. · Zbl 0193.17101 [11] Galil, Z. and Kiefer, J. (1977a). Comparison of design for quadratic regression on cubes. J. Statist. Plann. Inference 1 121-132. · Zbl 0381.62062 [12] Galil, Z. and Kiefer, J. (1977b). Comparison of rotatable designs for regression on balls. I. Quadratic. J. Statist. Plann. Inference 1 27-40. · Zbl 0394.62058 [13] Golikova, T. I. and Pantchenko, L. A. (1977). Continuous $$A$$ and $$Q$$-optimal second order designs on a cube. In Regression Experiments ( Design and Analysis ) ( in Russian ) (V. V. Nalimov, ed.) 71-84. Moscow Univ., Moscow. · Zbl 0409.62057 [14] Jennrich, R. I. (1969). Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40 633-643. · Zbl 0193.47201 [15] Kiefer, J. (1959). Optimum experimental designs. J. Roy. Statist. Soc. Ser. B 21 272-319. · Zbl 0108.15303 [16] Kiefer, J. (1961a). Optimum experimental designs. V. With applications to systematic and rotatable designs. In Proc. 4 th Berkeley Sympos. Math. Statist. and Prob. , Vol. I 381-405. Univ. California Press, Berkeley, CA. · Zbl 0134.36606 [17] Kiefer, J. (1961b). Optimum designs in regression problems. II. Ann. Math. Statist. 32 298-325. · Zbl 0099.13502 [18] Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849-879. · Zbl 0291.62093 [19] Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271-294. · Zbl 0090.11404 [20] Kôno, K. (1962). Optimum design for quadratic regression on $$k$$-cube. Mem. Fac. Sci. , Kyushu Univ. , Ser. A , Math. 16 114-122. · Zbl 0156.39906 [21] Laptev, V. N. (1974). Some problems relating to construction of regression experimental designs by means of computer. Ph.D. thesis, Novosibirsk State Technical Univ., Russia. [22] Lim, Y. B. and Studden, W. J. (1988). Efficient $$D_s$$-optimal designs for multivariate polynomial regression on the $$q$$-cube. Ann. Statist. 16 1225-1240. · Zbl 0664.62075 [23] Melas, V. B. (1982). A duality theorem and $$E$$-optimality (translated from Russian). Industrial Laboratory 48 295-296. [24] Melas, V. B. (2006). Functional Approach to Optimal Experimental Design . Springer, New York. · Zbl 1094.62086 [25] Myers, R. H., Montgomery, D. C. and Anderson-Cook, C. M. (2009). Response Surface Methodology : Process and Product Optimization Using Designed Experiments , 3rd ed. Wiley, Hoboken, NJ. · Zbl 1269.62066 [26] Pázman, A. (1986). Foundations of Optimum Experimental Design . Reidel, Dordrecht. · Zbl 0588.62117 [27] Pesotchinsky, L. L. (1975). $$D$$-optimum and quasi-$$D$$-optimum second-order designs on a cube. Biometrika 62 335-340. · Zbl 0309.62054 [28] Pukelsheim, F. (2006). Optimal Design of Experiments . SIAM, Philadelphia, PA. · Zbl 1101.62063 [29] Pukelsheim, F. and Studden, W. J. (1993). $$E$$-optimal designs for polynomial regression. Ann. Statist. 21 402-415. · Zbl 0787.62075 [30] Rafajlowicz, E. and Myszka, W. (1988). Optimum experimental design for a regression on a hypercube-generalization of Hoel’s result. Ann. Inst. Statist. Math. 40 821-827. · Zbl 0698.62072 [31] Silvey, S. D. (1980). Optimal Design . Chapman & Hall, London. · Zbl 0468.62070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Abstract

The problem related to the generation of construction and demolition waste (CDW) is directly associated with its high volume and irregular disposal, especially when such waste is disposed of in landfills. Therefore, it is increasingly important to encourage the reuse of this material, especially in large-scale projects. However, the use of this waste depends on the different constituent materials, which show a wide variability. Within this context, the objective of this work was to study the mechanical performance of construction and demolition waste over time aiming at the use in pavement support layers. Two samples were evaluated, one composed exclusively of construction waste (CDW_1) and the other composed of a soil-waste mixture (S_CDW), with 50% of each material. The tests evaluated the strength of the materials over a curing period from 0 to 224 days. The results clearly indicate an increase in strength and stiffness over time. The CDW_1 samples indicated a more significant increase. The analyses indicated that this behavior occurs due to the effect of self-cementing properties that occur in this type of waste. The results indicate a benefit in terms of strength when using this type of waste either alone or mixed with soil.

Keywords:
Construction and demolition waste; Self-cementing properties; Strength; Stiffness; Pavements

# Resumo

Palavras chave:
Resíduos de construção civil; Propriedades cimentantes; Resistência; Rigidez; Pavimentação

# Introduction

The generation of construction and demolition waste (CDW) has become problematic in urban centers, due to the associated scarcity of disposal sites, the adoption of inefficient, management measures, as well as problems of public sanitation and environmental contamination often caused by irregular disposal. The recycling of waste generated in construction and demolition projects has been encouraged, above all, in order to conserve the large quantities of natural resources typically used in engineering projects.

With the increasing use of CDW for civil engineering applications, there is a need to understand its strength behaviors: a fundamental parameter for justifying its use. Arm (2001)ARM, M. Self-Cementing Properties of Crushed Demolished Concrete in Unbound Layer: results from triaxial test and field test. Waste Management, v. 21, p. 235-239, 2001. observed that an increase in strength occurs with a decrease in permeability in pavement layers that incorporate recycled aggregates. This fact is directly related to probable self-cementing properties of the material, given that its heterogeneity can provide hydration and pozzolanic reactions. The influence on mechanical behavior due to the self-cementing effect of these materials can be evaluated, for example, based on the determination of the change in the resilient modulus and unconfined compressive strength over time (BLANKENAGEL, 2005BLANKENAGEL, B. J. Characterization of Recycled Concrete For Use as Pavement Base Material. Provo, 2005. 66 f. Master Dissertation - Department of Civil and Environmental Engineering Brigham Young University. Provo, 2005.). This self-cementing effect is associated with the physical characteristics and the chemical and mineralogical composition of the waste material, since it is composed of different types of materials that can influence its behavior with respect to its mechanical strength, water absorption and density (GÓMEZ, 2011GÓMEZ, A. M. J. Estudo Experimental de Um Resíduo de Construção e Demolição (RCD) Para Utilização em Pavimentação. Brasília, 2011. 123 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia, Universidade de Brasília, Brasília, 2011.).

Arm (2001)ARM, M. Self-Cementing Properties of Crushed Demolished Concrete in Unbound Layer: results from triaxial test and field test. Waste Management, v. 21, p. 235-239, 2001. compared the behavior of a recycled concrete aggregate and a natural aggregate, normally used in pavement base layers, by means of a cyclic triaxial test. The result of that study found that the recycled aggregate showed a tendency for the resilient modulus to increase over time. On the other hand, the natural aggregates tested with the same method did not show a similar increase. Sangiorgi, Lantieri and Dondi (2014)SANGIORGI, C.; LANTIERI, C.; DONDI, G. Construction and Demolition Waste Recycling: an application for road construction. International Journal of Pavement Engineering, p. 1-8, 2014. evaluated the performance of construction waste in an experimental embankment. The authors confirmed an increase in stiffness over time, attributing this behavior to self-cementing properties, and highlighting the positive mechanical characteristics of CDW when compared with natural aggregates.

Poon, Qiao and Chan (2006)POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006. evaluated the self-cementing effect on the unconfined compressive strength for different fractions of CDW submitted to curing periods of 1 and 7 days. The finer fractions (< 0.15 mm and 0.3 - 0.6 mm) presented greater unconfined compressive strength when compared to the other fractions analyzed. The authors explained that the presence of C2S in the finer fractions indicated that the compound had not yet completed hydration, which may be a possible cause for the greater tendency of these fractions to present self-cementing properties.

The increase in unconfined compressive strength of CDW over time is linked to the self-cementing phenomenon of the material itself. This may be related to the intrinsic properties of the material, such as cement particles that have not yet come into contact with water, which, after crushing and transformation into recycled aggregate, may then have contact with water and incur new hydration reactions, as well as the presence of minerals and chemical compounds found in cement. Thus, the physical, chemical and mineralogical characterization of the CDW is essential for analyzing the cause of the self-cementing effect of the material over time.

The objective of this study was to evaluate the mechanical performance of construction and demolition waste over time as a function of the occurrence of the self-cementing effect of such material. For this, samples composed exclusively of CDW and a soil-CDW mixture (in a 50:50 proportion) were analyzed. The self-cementing effect was evaluated through physical and chemical characterization tests, unconfined compressive strength tests for different curing times, and scanning electron microscopy. The results enabled the evaluation of the behavior of these samples and the effects of the addition of these materials to natural soil.

# Geotechnical characterization of CDW samples

The samples of construction waste (CDW) were collected at the Mauá Recycling Plant, located in the city of Maringá, Paraná, Brazil (Figure 1). In this recycling site the wastes are separated in different mounds according to their composition. In the present work, the selected waste is composed of wastes from concrete, cementitious mortar and soil. The samples were air-dried until reaching hygroscopic moisture content and were subsequently stored in containers protected from contact with other materials. For this research, two different CDW samples were used:

1. waste material passed through a 0.42 mm sieve, referred to as CDW_1; and

2. waste material with particles between 0.01 - 5.0 mm in diameter, referred to as CDW_2.

Figure 1
Recycling site and construction waste

Due to its high granularity and lack of cohesion, sample CDW_2, was mixed with a sandy residual soil (S) collected in the city of Mandaguaçu, Paraná, Brazil. In terms of dry mass, the soil was mixed with the waste material (S_CDW) in the proportion of 50% soil and 50% CDW_2, providing cohesion to the mixture and enabling the evaluation of the strength of this matrix.

The geotechnical characterization tests of the materials were carried out according to the recommendations of the Brazilian Association of Technical Standards (ABNT, 2017ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6458: grãos de pedregulho retidos na peneira de abertura 4,8 mm: determinação da massa específica, da massa específica aparente e da absorção de água. Rio de Janeiro, 2017.): particle density, grain size distribution (ABNT, 2018ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7181: solo: análise granulométrica. Rio de Janeiro, 2018.) and compaction tests (ABNT, 2016aASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7182: solo: ensaio de compactação. Rio de Janeiro, 2016a.). The results presented in Table 1 indicate that the particle density of the CDW is slightly higher than that of the soil. The influence of CDW on the soil matrix in relation to the particle density is relatively low.

Table 1
Results of geotechnical characterization of materials

Figure 2 and Table 1 present the results of the grain size distribution of the studied materials. The results indicate that samples CDW_1 and CDW_2 are basically composed of medium sand and coarse sand, respectively. Due to the low presence of clay and silt fractions, the liquid and plasticity limits could not be determined. As expected, the incorporation of CDW into the soil matrix caused an increase in the gravel fraction and a decrease in the clay fraction (Table 1).

Figure 2
Grain size distribution

In the standardized compaction test based on NBR 7182 (ABNT, 2016aASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7182: solo: ensaio de compactação. Rio de Janeiro, 2016a.) different energy levels are established, and in this work standard effort (600 kJ/ m3) was used. Standard effort was imposed by applying 26 blows per layer, and 3 layers were used to compose each point. The dry density (ρd) is the ratio of the solid mass to the total volume, which represents the density when the degree of saturation is equal to zero. The compaction parameters (ρdmax and wop) estimated for the samples are shown in Figure 3 and Table 1.

Figure 3
Compaction curves of the samples

The compaction curve of CDW_2 shows a behavior similar of permeable soils with free drainage. The high volume of voids (Figure 4) facilitates the exit of water when the particles are compressed at a higher density. In this case, the dry density remains constant, regardless of the increase in moisture content. For this type of material, the optimum moisture content corresponds to the point at which the saturation of the voids with water occurs. Therefore, the maximum dry density was 1.540 g/cm3, determined by the average of the values. This value is lower than granular materials, which shows dry density between 1.800 to 2.100 g/cm3. The optimum moisture content was 19.40%, adopted as the maximum possible moisture content to be placed in the sample and compacted without exuding water in the lower part of the compaction cylinder.

Figure 4
Compaction for sample CDW_2

Arulrajah, Piratheepan and Disfani (2013)ARULRAJAH, A.; PIRATHEEPAN, J.; DISFANI, M. M., BO, M. W. Geotechnical and Geoenvironmental Properties of Recycled Construction and Demolition Materials in Pavement Subbase Applications. Journal of Materials in Civil Engineering, v. 25, n. 8, p. 1077-1088, 2013. also observed the same behavior for some CDW samples, i.e., the dry density showed insignificant variation with the increase in the water content. The authors attributed this behavior to the low water absorption of the material and to the presence of the smooth surfaces of the materials.

The results also indicated that for the S_CDW mixture, the compaction behavior was similar to the natural soil behavior. The dry density increased gradually with increasing moisture content until reaching a maximum value, referred to as maximum dry density (ρdmax). For this dry density the moisture content is considered the optimum point (wop). The parameters obtained for the natural soil were maximum dry density and optimum moisture content of 1.872 g/cm3 and 13.20%, respectively, and the degree of saturation equal to 81%. For the S_CDW mixture, the maximum dry density was 1.850 g/cm3, the optimum moisture content equal to 13.20% and the degree of saturation equal to 77%.

The CDW_1 sample presented a curve format similar to the natural soil and the S_CDW mixture. The grain size composition indicates a material less granular than CDW_2 and with the presence of silt, therefore more similar to the grain size distribution of the natural soil. The maximum dry density and optimum moisture content was 1.589 g/cm3 and 19.70%, respectively, and degree of saturation equal to 73%.

# Chemical and mineralogical characterization of the CDW

In order to evaluate the chemical and mineralogical characteristics of the CDW, the following tests were performed: pozzolanic activity, pH and X-ray diffraction (XRD) tests. The pozzolanic activity test was carried out at the Technological Research Institute (IPT) of São Paulo, through the Chapelle Method (ABNT, 2010ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 15895: materiais pozolânicos: determinação do teor de hidróxido de cálcio fixado: método Chapelle modificado. Rio de Janeiro, 2010.). The tests were run on samples CDW_1 and CDW_2. The material was dried at 105 °C until constant mass, then milled until the material was totally passed through the 0.045 mm sieve.

The method is based on the determination of lime consumption related to the amorphous or vitreous phase of the pozzolanic materials. According to NBR 15895 (ABNT, 2010), the additions may be considered pozzolan on being incorporated into the Portland cement composition when the consumption is greater than 436 mg Ca (OH)2/g. Table 2 shows the consumption of calcium hydroxide (Ca (OH)2) found in the samples analyzed. Although the grain size distribution of the materials is different, the results of pozzolanic activity indicated very close values for the samples analyzed. The values of calcium hydroxide consumption were lower than the minimum value established in the standard. The pozzolanic reaction is useful because it converts the relatively soluble calcium hydroxide into C-S-H which increases strength and reduces the permeability of the material.

Table 2
Results of pozzolanic activity and pH

Motta (2005)MOTTA, R. S. Estudo Laboratorial de Agregado Reciclado de Resíduo Sólido da Construção Civil para Aplicação em Pavimentação de Baixo Volume de Tráfego. São Carlos, 2005. 134 f. Dissertação (Mestrado em Engenharia Civil) - Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia de Transportes, São Carlos, 2005. also found values lower than the values established by the standard. However, the author does not rule out the possibility of the CDWs contributing to the increased mechanical strength of the pavement. This is because the minimum consumption index, established in the standard, was defined for the concrete, which presents a very different strength behavior when compared to the new materials that have been developed. Moreover, the selected wastes are composed of wastes from concrete, cementitious mortar and soil. The absence of ceramic materials reduces the pozzolanic activity because the bricks and ceramic materials are responsible to pozzolanic activity with the Ca (OH)2 found in the hydrated cementitious materials.

Gómez (2011)GÓMEZ, A. M. J. Estudo Experimental de Um Resíduo de Construção e Demolição (RCD) Para Utilização em Pavimentação. Brasília, 2011. 123 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia, Universidade de Brasília, Brasília, 2011. evaluated the pozzolanic activity of CDW with Portland cement and also found that the material does not meet the requirements of the standard. However, they stressed that the unconfined compressive strength of the material reached a reasonable value.

The pH test was performed with the material passing through the 2.00 mm sieve, using 20 g of the material in 100 ml of aqueous solution. The pH readings were determined at time intervals of 10, 20, 30, 60, 120, 240, 1440, 2880 and 4320 minutes after the mechanical stirring.

Figure 5 shows the variation in pH as a function of time, where the pH for sample CDW_2 was higher than the value obtained for sample CDW_1. The mean values were 9.72 and 11.03 for samples CDW_1 and CDW_2, respectively. The CDW_1 shows a pH value lower than CDW_2, probably because this sample is composed of small fractions of concrete and cementitious mortar, possibly carbonated. The CDW_2 sample is composed of particles larger than the CDW_1, where non-carbonated fractions can be found or the presence of anhydrous grains.

Figure 5
Data obtained during the pH test for CDW samples

The low performance of the pozzolanic activity of the material may be directly related to absence of amorphous silica, which is from burning of bricks and tiles. As the waste is composed of concrete and cementitious, mortar the most part of the grains of cement are already hydrated. Poon, Qiao and Chan (2006)POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006. also obtained higher pH values for granular CDWs.

According to Gómez (2011)GÓMEZ, A. M. J. Estudo Experimental de Um Resíduo de Construção e Demolição (RCD) Para Utilização em Pavimentação. Brasília, 2011. 123 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia, Universidade de Brasília, Brasília, 2011., who obtained pH equal to 11.2 for CDW samples, a high pH environment could alter the silica minerals. This environment is effective for achieving the stabilization of clays with calcium oxide through pozzolanic reactions where calcium silicate is hydrated, and cementitious materials are formed using the modified silica of the clay minerals. Blankenagel (2005)BLANKENAGEL, B. J. Characterization of Recycled Concrete For Use as Pavement Base Material. Provo, 2005. 66 f. Master Dissertation - Department of Civil and Environmental Engineering Brigham Young University. Provo, 2005. comments that the pozzolanic reaction can only occur at pH levels above 10, because it is in this medium that the silica becomes soluble. Although the pH and pozzolanic activity tests for the S_CDW sample were not performed, these results can contribute with the interpretation of self-cementing waste behavior. This is because lateritic soils are composed of kaolinite, which is soluble in water and allows the reaction of alkali of the anhydrous grains present in the concrete and cementitious mortar of CDW.

To perform the XRD test, the CDW_1 sample was passed through a No. 200 sieve (0.075 mm) and the CDW_2 sample through a No. 100 sieve (0.15 mm). The minerals were identified by means of an X-ray diffractometer, of the brand Bruker model D8 Advance. CuKα (40 kV and 35 mA), with angular range of 0° (2 Theta initial) and 80° (2 Theta final), angular velocity of 1° per minute were used as test parameters. Mineral identification was performed using X'Perth HighScore software.

The diffractograms of the soil samples, CDW_1 and CDW_2, are presented in Figure 6. In the case of the soil, the predominant mineral is quartz coming from basaltic rocks, the presence of hematite was also detected, due to the iron oxides and hydroxides, and of kaolinite which is the predominant clay in lateritic soils, as is the case with this soil. In the case of the waste materials, the diffractograms were similar, where the predominant mineral is calcite, which is characterized as one of the main cement minerals present in sedimentary rocks. The predominant presence of this mineral is consistent since there was a large quantity of materials containing cement, such a concrete, visible in the waste materials. The presence of calcite is due to the process of carbonation of the calcium silicates and the hydroxide of calcium derived from the cement of concretes and cementitious mortars. The presence of quartz from the natural silica found in sands was also confirmed. The results are consistent with results presented by other researchers. Evangelista et al. (2015)EVANGELISTA, L. et al. Physical, Chemical and Mineralogical Properties of Fine Recycled Aggregates Made From Concrete Waste. Construction and building materials, v. 86, p. 178-188, 2015. identified the presence of calcite and quartz in all fractions of fine recycled aggregate (FRA) from concrete. Gómez (2011)GÓMEZ, A. M. J. Estudo Experimental de Um Resíduo de Construção e Demolição (RCD) Para Utilização em Pavimentação. Brasília, 2011. 123 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia, Universidade de Brasília, Brasília, 2011. found quantities of silicates, such as quartz, carbonates, such as calcite, and clay-rich silica and aluminum, such as ilite.

Figure 6
XDR analysis of the soil, CDW_1 and CDW_2

# Methodology for the evaluation of self-cementing properties

The methodology used to evaluate the self-cementing properties of the waste is presented in the flowshart of Figure 7a. Test specimens (Figure 7b) were molded in cylindrical form with a 50 mm diameter and 100 mm height for the CDW_1 and S_CDW samples. The S_CDW mixture was prepared, in terms of dry mass, using a 50:50 ratio (50% soil and 50% CDW_2). The choice of this proportion was made based on research done previously by Silva et al. (2015)SILVA, T. B. et al. Construction and Demolition Waste (CDW) Used in Reinforced Soil Mixtures For Pavement Applications. In: PAN-AMERICAN CONFERENCE ON SOIL MECHANICS AND GEOTECHNICAL ENGINEERING, 15., Buenos Aires, 2015. Proceedigns... Buenos Aires, 2015..

Figure 7
Methodology for evaluation of the self-cementing effect

The compaction of the specimens was performed in 3 layers approximately 35 mm thick with adequate scarification between the layers in order to increase the integration and the adhesion between the superimposed layers. After compaction, the top was leveled so as to obtain a smooth horizontal surface and the specimen was weighed and packed in a plastic bag to be stored in the wet chamber for curing times of 0, 3, 7, 14, 21, 28, 56, 112 and 224 days prior to axial compressive strength tests.

At least three (3) specimens were molded for each of the established cure times. In order to evaluate the effect of the addition of the construction and demolition waste on the soil matrix, 3 specimens of the soil sample were molded. Since the soil used does not have cementitious properties, the specimens molded with this material were not subjected to the curing period and were ruptured on the same molding day (0 days).

The specimens were appropriately controlled in relation to mass, volume and moisture content. After the curing time and before the axial compressive strength tests, the specimens were weighed again and, after rupture, the moisture content was measured. This procedure was carried out to evaluate whether a variation in the moisture content during the curing time occurred, influencing the strength values. In this way, the values of axial compressive strength can be related exclusively to the properties of the materials used.

The parameters established for compaction of the test specimens were the maximum dry density and the optimum moisture content for the curve corresponding to the material used (Table 1 and Figure 3). The moisture content was determined by the ABNT oven method (ABNT, 2016bASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6457: amostras de solo: preparação para ensaios de compactação e ensaios de caracterização. Rio de Janeiro, 2016b.), with a temperature ranging from 105 to 110 °C and a 24 hour period for complete drying of the material.

The dry density (ρd) was determined by Equation (1) and Equation (2). The relationship between the dry density of the specimen (ρd) and the maximum dry density (ρdmax) obtained by the compaction test provides the degree of compaction (DC) (Equation 3). Accordingly, in this research, the specimens were molded with the same degree of compaction, avoiding the influence of this parameter on the strength values. The higher the degree of compaction, the smaller the void volume of the sample and consequently the greater its strength. In this work a degree of compaction was established between 95-100% and the moisture content between ± 2% wop.

Eq. 1 $ρ d = ρ 1 + w$

Where:

ρ(g/cm3) = the density of the test specimen; and

w = the moisture content.

Eq. 2 $ρ = M V$

Where:

M = the mass (g) of the specimen; and

V = the volume.

Eq. 3 $DC = ρ d ρ d max$

The axial compressive strength tests were carried out by applying the load in constant increments where the force values were monitored through a dynamometer ring properly calibrated. In addition, displacement was monitored through a mechanical dial indicator. Figure 8 shows the unconfined compressive strength test.

Figure 8
Unconfined compressive strength test

After the unconfined compression strength tests, parts of the specimens, duly identified, were selected for scanning electron microscopy (SEM) tests. SEM was carried out with the objective of obtaining information about the morphology of the test specimens, evaluating the internal structure in order to observe possible connections and structures formed between particles due to the self-cementing effect of the material.

The tests were carried out in the Laboratory of Microscopy of the Brazilian Center for Physical Research (CBPF) in the city of Rio de Janeiro. For the electron microscopy test, MEV JSM 6460LV, JEOL brand, model EDS NORAN SYSTEM SIX MODEL 200 was used. The prepared samples were inserted inside the scanning electron microscope for image capture.

The void ratio of the specimens (e) were estimated through Equation (4), where volume of void is the sum between air and water and the volume of solid is the volume filled by particles. The water volume was estimated adopting water density as 1g/cm3 and the particle density was obtained in experimental tests (Table 1).

Eq. 4 $e = V v V s$

Where:

Vv = the volume of void (cm3); and

Vs = the volume of solid (cm3). The void ratio is not expressed as a percentage.

The modulus of elasticity (E) or Young's modulus, is a mechanical parameter for evaluating the stiffness of the material, which is considered an intrinsic property of the material since it depends, among other aspects, on the chemical composition and the microstructure. The determination of this parameter can be established through the relationship between the stress (σ) and the deformation (ε) obtained from the curve of the strength of material test (Equetion 5).

Eq. 5 $E = σ ε$

For the tests carried out in this research, the tangent modulus (Et), for a specific point (elastic region), was determined by evaluating the relationship between the stress corresponding to 50% of the rupture stress (σ50%) and the strain (ε50%) for a given point (Figure 9).

Figure 9
Unconfined compressive strength test

# Results and discussion

## Dry density over time

The dry density of the specimens was determined at the time of molding and after the specimen had ruptured. Figure 10 shows the variation for the CDW_1 sample, where the curing period did not alter the dry density of the specimens and consequently the degree of compaction. The results indicate that most of the test specimens were molded with a degree of compaction above 100% and were very close, with a coefficient of variation equal to 1.73% and 1.84% for the molding and rupture conditions, respectively. The relative error was determined with respect to the value of (dmax, obtaining maximum values in modulus equal to 5.73% and 6.29% for the conditions of molding and rupture, respectively.

Figure 10
Dry density versus cure time (CDW_1)

Figure 11 shows the variation in the dry mass for the S_CDW sample, where, in general, the curing period did not change the dry density of the specimens and consequently the degree of compaction. The graph indicates a small increase in dry density for a ruptured specimen at 112 days of cure and a significant decrease in dry density for a ruptured specimen at 224 days of cure. These variations can occur because the specimens are packed in a humid chamber subject to humidity variations. However, since the specimens were generally close to the molding conditions, the packaging in plastic bags was considered to be efficient against variations in moisture content.

Figure 11
Dry density versus cure time (S_CDW)

The results showed a coefficient of variation equal to 1.30% and 1.76% for the molding and rupture conditions, respectively. The relative error was determined in relation to the value of (dmax, obtaining maximum values in modulus equal to 4.05% and 6.43% for the conditions of molding and rupture, respectively.

Therefore, the specimens presented a very close degree of compaction, and the variation in the molding conditions are assumed to have had no effect on the strength values presented below.

## Unconfined Compressive Strength (UCS)

Figure 12 shows the unconfined compressive strength results for the CDW_1 sample, where a well-defined increase in strength over time (Figure 12a) is observed for the sample of compacted waste material under standard effort conditions. The polynomial fit proved to be satisfactory when the average values of unconfined compressive strength were used (Figure 12b).

Figure 12
Unconfined Compressive Strength (CDW_1)

The unconfined compressive strength for the S_CDW mixture was evaluated for the curing period of up to 112 days. Figure 13 shows unconfined compressive strength results for the S_CDW mixture, in which the strength over time also increased as a function of curing time, however, this behavior was not as well-defined (Figure 13) as in the case of the pure CDW sample (Figure 12). The coefficient of variation for the samples tested with a curing time of 35 days was the highest (18.7%) and therefore these results were disregarded for point adjustment (Figure 13b). The results obtained after 21 days of cure indicated a trend of strength stabilization.

Figure 13
Unconfined compressive strength (S_CDW)

The difference between the behavior of the evaluated materials can be attributed to the presence of soil in the matrix, the different grain size distribution, composition of the waste material. The results indicate that samples of CDW, composed exclusively of concrete and cementitious mortar, show a more pronounced rate of increase in strength over time. Poon, Qiao and Chan (2006)POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006. observed that the self-cementing properties of the waste materials are determined by the fine fraction of this material, which represents the active fraction of the CDW. Therefore, the gain in strength can be considered an intrinsic property of the evaluated waste materials, related with the amount of concrete and cementitious mortar present in the sample.

The results indicate that the unconfined compressive strength for the S_CDW mixture shows superior results to the strength of the CDW_1. The addition of soil to the CDW material provided an increasing in the percentage of fine particle (diameter less than 0.002 mm), as can be seen in Table 1. The presence of fine particles provided a better arrangement between the grains because the voids are better filled. Therefore, the specimens had lower void indices than the values observed for the CDW_1 sample, explaining the higher values of unconfined compressive strength.

Figure 14 shows the void ratio (e) for each of the evaluated materials, indicating that the S_CDW samples presented values about 45% lower than the CDW_1 sample values. The void ratio represents the relationship between the volume of voids and the volume of solids, that is, the lower the void ratio the better the particle arrangement and consequently the better the sample strength. Therefore, the addition of soil to the CDW provided a better arrangement between the particles, reducing the void ratio and increasing the strength, confirming the results of this research.

Figure 14
Void ratio as a function of curing time

The increase of resistance values of S_CDW mixture may also be associated with the presence of concrete and cementitious mortar grains.

The results of compaction tests can be used to interpret the behavior of CDW resistance (Table 1 and Figure 3). The S_CDW sample show maximum dry density (1.850 g/cm3) higher than CDW_2 (1.589 g/cm3), and consequently show void ratio lower. Thus, the resistance of S_CDW sample, should be higher than CDW_2 sample, confirming the resistance results. On the other hand, the S_CDW sample show optimum moisture content (13.20%) lower than CDW_2 sample CDW_2 (19.70%), showing that the water volume is lower for this sample and consequently the void ratio is lower. Again, it is noted that resistance results are consistent.

The variation in the unconfined strength over time can be evaluated through normalized values. In this case, the values of unconfined compressive strength (UCS) were divided by a reference value (UCSref.), which is the value corresponding to the initial time equal to 0 days. This analysis allows evaluating the increase of strength over time. Figure 15 indicates that the CDW_1 sample, composed exclusively of waste material showed an increase of about 4.3 times of the initial strength, when compared with the final strength (224 days). It is important to note that the composition of this sample is waste material fines with a grain size smaller than 0.42 mm. This behavior confirms the observations of Poon, Qiao and Chan (2006)POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006. where the authors emphasize that the self-cementing property of the waste material is governed by the fine fraction of the sample, but also due to the composition of the fraction and the reactive phase presented by these fines.

Figure15
Normalized strength values as a function of curing time

Despite of the S_CDW sample shows values of strength higher than CDW_1, the S_CDW sample did not present the same increase in strength over time. The maximum increase was 2.3 times comparing initial (0 days) and final strength (112 days). It is important to note that for this sample the construction waste percentage is lower than in the CDW_1 sample. Results also show that for the S_CDW sample the increase of strength over time was more pronounced than CDW_1 until 28 days of cure. After this time, the strength values stabilized and the CDW_1 strength values increased significantly. Additionally, the waste material used for the CDW_2 mixture, had larger particles than the particles in the CDW_1 sample, having a much lower fines content. Consequently, a smaller increase in strength was observed after 28 days of cure, with a tendency for values to stabilize in a much shorter curing period than that of the CDW_1 sample.

Parameters as void ratio, density and moisture content allow explaining the resistance behavior of the waste samples. However, the rate of resistance development can only be explained through the composition of the samples, which indicate the self-cementing effect over time.

For the waste samples it was observed that the increasing of the resistance rate development is higher for shorter cure times. This is because there is a large amount of hydraulic phases (anhydrous grains) and pozzolanic active. The presence of concrete and cementitious mortar grains contributes with the resistance behavior.

However, the analyses lead to the conclusion that even the use of the CDW in its original grain size distribution and mixed with soil, provide a considerable increase in the unconfined compressive strength. The feasibility of using this material, in addition to providing a proper disposal option, contributes to the increase in unconfined compressive strength.

## Modulus of Elasticity (E)

Figure 16 shows the mean normalized values of the tangent modulus as a function of curing time, for the CDW_1 and S_CDW samples. The results indicate that the samples provided increased stiffness over time. For the CDW_1 sample, the observed increase in stiffness, relative to the test specimens without curing time, was 20%, 17%, 32%, 34%, 49%, 77%, 94%, 110% and 180% for 3, 7, 14, 21, 28, 35, 56, 112 and 224 days of curing. For the S_CDW sample, the observed increase in stiffness was 21%, 39%, 48%, 55%, 47%, 42% and 52% for 3, 7, 14, 21, 28, 56 and 112 days. Again, the results indicate that the stiffness gain is more pronounced for samples consisting of construction waste alone. However, for the soil-CDW mixture the increase, though in reduced proportions, was also evident.

Figure 16
Normalized values of the tangent modulus (Et)

Similar to the behavior of the unconfined compressive strength, the increase in stiffness for the CDW_1 sample continues to increase up to 224 days of curing, whereas for the mixture (S_CDW) the increase is more pronounced up to 28 days indicating a possible stabilization of the values.

The increase in stiffness over time can be attributed to the presence of non-hydrated cement, again demonstrating the self-cementing behavior characteristic of construction and demolition waste due to the addition of water that provides the occurrence of new reactions in non-inert particles.

Figure 17
Comparison of tangent modulus values (Et)

## Scanning Eléctron Microscopy (SEM)

After the unconfined compressive strength tests, the samples were submitted to an SEM test in order to verify their internal structure. In addition, a dispersive energy spectrometer was used for qualitative determination of the chemical composition of the observed structural elements. The analyses were performed on samples with different curing periods.

Figure 18 shows the micrograph for the CDW_1 sample without cure time. There is no evidence of cement hydration product. Figure 19 shows the micrograph for the CDW_1 sample, with the strength test performed after 35 days of molding, i.e., with a curing period of 35 days. The rupture of the specimens indicated that the strength increased about 200% relative to the uncured sample. This increase may occur because non-inert particles of cement may have reacted causing increased strength.

Figure 18
Micrograph for sample CDW_1 (no cure)

Figure 19
Micrograph for sample CDW_1 (35 days)

The presence of calcite, detected in the sample (Figure 6), can be associated with concrete and cementitious mortar found in CDW. In Figure 19, there is evidence of the formation of structures similar to microcrystals of calcite and ettringite crystals, mainly because the spectral diagram indicates the presence of S. Moreover, it is possible to visualize the presence of microcrystals of calcium carbonate (rhombohedral morphology of crystals). The image also presents an aspect of contamination.

The spectral diagram indicates the presence of Si, associated with the presence of cement, and also to the quartz, present in the rocks. The elements Ca, Al, Mg, S, Si, identified in the sample, are cement hydration products. Therefore, the occurrence of reactions associated with the cement hydration process and consequently the strength gain is clearly evident.

Scrivener (1989)SCRIVENER, K. L. The Microstructures of Concrete. In: SKALNY, J. Materials Science of Concrete. New York: The American Ceramic Society, 1989. points out that calcium hydroxide (CH) is the only cement hydration product that has a defined crystalline structure, forming slightly hexagonal plates. In Figure 20 it is possible to see this type of structure, associated with the presence of ettringite, filling the pores of the soil-CDW matrix. The matrix shown in Figure 20 corresponds to the micrograph for the S-CDW sample, with the strength test performed after 56 days of molding cure. For these samples, the increase in strength was also approximately 200% over the uncured sample.

Figure 20
Micrograph for sample S_CDW (56 days)

The spectral diagram indicates the presence of Ca, Al, Si and Fe, S, which are hydration products of cement and are characterized by the presence of ettringite. Ettringite is detected in the first ages of hydration. As the hydration reactions continue, ettringite turns into monosulfate (AFm). Thus, the presence of ettringite indicates that the reactions are occurring from the anhydrous grains, explaining the resistance gain over time (self-cementing).

In general, the use of the SEM technique helped to identify internal structures characteristic of the cement hydration process, and consequently of the increase in strength observed in the unconfined compressive strength tests. The construction waste promoted new reactions both in matrices that used the pure CDW, with particle size less than 0.42 mm (CDW_1), as well as in matrices that used the most granular construction waste mixed with natural soil (S_CDW). This behavior explains the results of Silva et al. (2015)SILVA, T. B. et al. Construction and Demolition Waste (CDW) Used in Reinforced Soil Mixtures For Pavement Applications. In: PAN-AMERICAN CONFERENCE ON SOIL MECHANICS AND GEOTECHNICAL ENGINEERING, 15., Buenos Aires, 2015. Proceedigns... Buenos Aires, 2015., where soil mixtures with higher CDWs showed higher strength values.

# Final considerations

1. the evaluation of the pozzolanic activity through the Modified Chapelle Method indicated that the material cannot be considered pozzolan, mainly due to the absence of the presence of pozolanic materials found in bricks and tiles. The self-cementing effects, presented in this paper, can be associated with the hydration of anhydrous grains, explaining the resistance gain over time;

2. moreover, the pH values may be attributable to concrete and cementitious mortar fragments are already carbonated. The CDW_1 shows a pH value lower than CDW_2, probably because this sample is composed of small fractions of concrete and cementitious mortar, possibly carbonated. The CDW_2 sample is composed of particles larger than the CDW_1, where non-carbonated fractions can be found or the presence of anhydrous grains;

3. x-ray diffraction indicated the presence of calcite, which is characterized as one of the principal cementitious minerals. The presence of this mineral confirmed the presence of concrete, which was visually observed in the sample. However, it is important to carry out thermal analysis tests for the better characterization of the reactive phases and the low crystallinity;

4. the control of the dry density of the specimens during molding and before rupture allowed strength gain, attributable to variation in this parameter, to be discarded. Increasing the dry density increases the strength of the material because it increases the degree of compaction of the sample. Indeed, the control of this research was efficient and indicated no significant variation in the dry density up to the period of 224 days;

5. unconfined compressive strength results indicated a fairly significant increase in strength over time. It is noteworthy that the samples that used CDW exclusively in their composition showed increasing behavior over time. The samples that used the soil and CDW mixture showed an increase in strength with a tendency toward strength stabilization. This behavior confirmed the observations of Poon, Qiao and Chan (2006)POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006. where the self-cementing properties of the construction waste were found to be associated with the amount of fraction of fine material, related with the amount of concrete and cementitious mortar present in the sample. However, the analyses lead to the conclusion that even the use of CDW in its original grain size distribution mixed with soil, provides a considerable increase in the unconfined compressive strength. Additionally, parameters such as waste mineralogical composition, reactivity of finest particles, density, void ratio and moisture content of samples should be considered;

6. the self-cementing effect of the construction waste was also evaluated by means of the stiffness of the specimens, because the cement is a material, added to soil, that provides an increase of the modulus of elasticity. The results of this study demonstrate an increase in stiffness both in the construction waste sample (CDW_1) as well as in the combined soil-waste sample (S-CDW). Therefore, the self-cementing effect of construction waste is again evident; and

7. the use of scanning electron microscopy proved to be a useful tool for evaluating the development of the internal reactions that occur in the matrix material. Through these analyses it was evident that construction waste promotes strength gain through reactions associated with the presence of cement.

In general, the results presented in this work contribute to the understanding of strength gain in construction and demolition waste. Although the waste is composed of concrete, cementitious mortar and soil showing a non-pozzolanic behavior, the resistance gain over time was observed. This behavior can be associated with the hydration of anhydrous grains or pozzolanic reactions, in case of use pozzolanic cement in concrete and cementitious mortar. In addition, the beneficial effects of soil-CDW mixtures have been proven, encouraging the use of this material as a substitution for natural soil and a reduction in the exploitation of natural soil deposits. It is important to highlight that this work is a contribution to this specific material and further analysis are fundamental to different materials.

# Acknowledgements

The authors would like to express their appreciation to the CNPq (Process No. 448606 / 2014-3), for their financial support and to the CBPF for providing the infrastructure for the performance of the Scanning Electron Microscopy tests. Furthermore, the authors appreciate the helpful corrections and suggestion from the reviewers.

# References

• ARM, M. Self-Cementing Properties of Crushed Demolished Concrete in Unbound Layer: results from triaxial test and field test. Waste Management, v. 21, p. 235-239, 2001.
• ARULRAJAH, A.; PIRATHEEPAN, J.; DISFANI, M. M., BO, M. W. Geotechnical and Geoenvironmental Properties of Recycled Construction and Demolition Materials in Pavement Subbase Applications. Journal of Materials in Civil Engineering, v. 25, n. 8, p. 1077-1088, 2013.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 15895: materiais pozolânicos: determinação do teor de hidróxido de cálcio fixado: método Chapelle modificado. Rio de Janeiro, 2010.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6457: amostras de solo: preparação para ensaios de compactação e ensaios de caracterização. Rio de Janeiro, 2016b.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 6458: grãos de pedregulho retidos na peneira de abertura 4,8 mm: determinação da massa específica, da massa específica aparente e da absorção de água. Rio de Janeiro, 2017.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7181: solo: análise granulométrica. Rio de Janeiro, 2018.
• ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7182: solo: ensaio de compactação. Rio de Janeiro, 2016a.
• BLANKENAGEL, B. J. Characterization of Recycled Concrete For Use as Pavement Base Material. Provo, 2005. 66 f. Master Dissertation - Department of Civil and Environmental Engineering Brigham Young University. Provo, 2005.
• EVANGELISTA, L. et al Physical, Chemical and Mineralogical Properties of Fine Recycled Aggregates Made From Concrete Waste. Construction and building materials, v. 86, p. 178-188, 2015.
• GÓMEZ, A. M. J. Estudo Experimental de Um Resíduo de Construção e Demolição (RCD) Para Utilização em Pavimentação Brasília, 2011. 123 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia, Universidade de Brasília, Brasília, 2011.
• GRUBBA, D. C. R. P. Estudo do Comportamento Mecânico de Um Agregado Reciclado de Concreto Para Utilização na Construção Rodoviária São Carlos, 2009. 139 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos, 2009.
• MOTTA, R. S. Estudo Laboratorial de Agregado Reciclado de Resíduo Sólido da Construção Civil para Aplicação em Pavimentação de Baixo Volume de Tráfego. São Carlos, 2005. 134 f. Dissertação (Mestrado em Engenharia Civil) - Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia de Transportes, São Carlos, 2005.
• POON, C. S.; QIAO, X. C.; CHAN, D. The Cause and Influence of Self-Cementing Properties of Fine Recycled Concrete Aggregates on the Properties of Unbound Sub-Base. Waste Management and Research, v. 26, p. 1166-1172, 2006.
• SANGIORGI, C.; LANTIERI, C.; DONDI, G. Construction and Demolition Waste Recycling: an application for road construction. International Journal of Pavement Engineering, p. 1-8, 2014.
• SCRIVENER, K. L. The Microstructures of Concrete. In: SKALNY, J. Materials Science of Concrete New York: The American Ceramic Society, 1989.
• SILVA, T. B. et al Construction and Demolition Waste (CDW) Used in Reinforced Soil Mixtures For Pavement Applications. In: PAN-AMERICAN CONFERENCE ON SOIL MECHANICS AND GEOTECHNICAL ENGINEERING, 15., Buenos Aires, 2015. Proceedigns... Buenos Aires, 2015.
• WALLACE, F. S. Estudo Sobre a Aplicação de Agregado Reciclado de Concreto em Construção de Pavimentos. São Carlos, 2011. 116 f. Dissertação (Mestrado em Engenharia Civil) - Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos, 2011.

# Publication Dates

• Publication in this collection
Apr-Jun 2019


text:
# Homework Help: Could given function represent a travelling wave?

1. Apr 23, 2012

### wirefree

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

Verify if given functions could possibly represent a travelling wave?

2. Relevant equations

a) $(x-(v*t))^{2}$

b) $ln({(x+(v*t))/x_{0}})$

c) $1/(x+(v*t))$

3. The attempt at a solution

I suppose that for a function to represent a travelling wave, it must remain finite for all x & t. Hence, since (b) & (c) are infinity at x=t=0, they cannot represent a travelling wave.

Would I be correct to pursue this line of argument? If yes, then how should I proceed with (a)?

Would appreciate assistance.

Best regards,
wirefree

2. Apr 23, 2012

### fortissimo

Check if the given functions satisfy the wave equation: d^2(f)/(dx^2) = v^-2*d^2(f)/(dt^2)

3. Apr 25, 2012

### wirefree

Would I be correct to pursue my line of argument that for a function to represent a travelling wave, it must remain finite for all x & t?

If yes, then how should I proceed with (a) in the initial post?

Would appreciate an answer to my initial question.

Best regards,
wirefree

4. Apr 27, 2012

### wirefree

I am afraid we've not been taught the wave equation yet. The chapter I have finished covers topics incl. displacement relation in a progressive wave, speed of travelling wave, principle of superposition, and a couple of more.

I would greatly appreciate if you could address my query using simpler concepts, such as the one I have mentioned in my earlier post.

Best regards,
wirefree


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## Discussiones Mathematicae Graph Theory

2012 | 32 | 3 | 403-417
Tytuł artykułu

### Iterated neighborhood graphs

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Języki publikacji
EN
Abstrakty
EN
The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph $(V,E_N)$ where $E_N$ = {{a,b} | a ≠ b, {x,a} ∈ E and {x,b} ∈ E for some x ∈ V}. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite.
We present some results concerning the k-iterated neighborhood graph $N^k(G) : = N(N(...N(G)))$ of G. In particular we investigate conditions for G and k such that $N^k(G)$ becomes a complete graph.
Słowa kluczowe
EN
Kategorie tematyczne
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Tom
Numer
Strony
403-417
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Daty
wydano
2012
otrzymano
2011-01-12
poprawiono
2011-07-14
zaakceptowano
2011-07-18
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autor
• Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg, D-09599 Freiberg, Germany
• Institute of Mathematics, University of Lübeck, D-23560 Lübeck, Germany
Bibliografia
• [1] B.D. Acharya and M.N. Vartak, Open neighborhood graphs, Indian Institute of Technology, Department of Mathematics, Research Report No. 7 (Bombay 1973).
• [2] J.W. Boland, R.C. Brigham and R.D. Dutton, Embedding arbitrary graphs in neighborhood graphs, J. Combin. Inform. System Sci. 12 (1987) 101-112.
• [3] R.C. Brigham and R.D. Dutton, On neighborhood graphs, J. Combin. Inform. System Sci. 12 (1987) 75-85.
• [4] R. Diestel, Graph Theory, Second Edition, (Springer, 2000).
• [5] G. Exoo and F. Harary, Step graphs, J. Combin. Inform. System Sci. 5 (1980) 52-53.
• [6] H.J. Greenberg, J.R. Lundgren and J.S. Maybee, The inversion of 2-step graphs, J. Combin. Inform. System Sci. 8 (1983) 33-43.
• [7] S.R. Kim, The competition number and its variants, in: Quo Vadis, Graph Theory?, J. Gimbel, J.W. Kennedy, L.V. Quintas (Eds.), Ann. Discrete Math. 55 (1993) 313-326.
• [8] J.R. Lundgren, Food webs, competition graphs, competition-common enemy graphs and niche graphs, in: Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, F. Roberts (Ed.) (Springer, New York 1989) IMA 17 221–243.
• [9] J.R. Lundgren, S.K. Merz, J.S. Maybee and C.W. Rasmussen, A characterization of graphs with interval two-step graphs, Linear Algebra Appl. 217 (1995) 203-223, doi: 10.1016/0024-3795(94)00173-B.
• [10] J.R. Lundgren, S.K. Merz and C.W. Rasmussen, Chromatic numbers of competition graphs, Linear Algebra Appl. 217 (1995) 225-239, doi: 10.1016/0024-3795(94)00227-5.
• [11] J.R. Lundgren and C. Rasmussen, Two-step graphs of trees, Discrete Math. 119 (1993) 123-139, doi: 10.1016/0012-365X(93)90122-A.
• [12] J.R. Lundgren, C.W. Rasmussen and J.S. Maybee, Interval competition graphs of symmetric digraphs, Discrete Math. 119 (1993) 113-122, doi: 10.1016/0012-365X(93)90121-9.
• [13] M.M. Miller, R.C. Brigham and R.D. Dutton, An equation involving the neighborhood (two step) and line graphs, Ars Combin. 52 (1999) 33-50.
• [14] M. Pfützenreuter, Konkurrenzgraphen von ungerichteten Graphen (Bachelor thesis, University of Lübeck, 2006).
• [15] F.S. Roberts, Competition graphs and phylogeny graphs, in: Graph Theory and Combinatorial Biology, Proceedings of International Colloquium Balatonlelle (1996), Bolyai Society of Mathematical Studies, L. Lovász (Ed.) (Budapest, 1999) 7, 333–362.
• [16] I. Schiermeyer, M. Sonntag and H.-M. Teichert, Structural properties and hamiltonicity of neighborhood graphs, Graphs Combin. 26 (2010) 433-456, doi: 10.1007/s00373-010-0909-x.
• [17] P. Schweitzer (Max-Planck-Institute for Computer Science, Saarbrücken, Germany), unpublished script (2010).
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