text:
## CryptoDB

### Krzysztof Pietrzak

#### Publications

Year
Venue
Title
2021
CRYPTO
Yao’s garbling scheme is one of the most fundamental cryptographic constructions. Lindell and Pinkas (Journal of Cryptograhy 2009) gave a formal proof of security in the selective setting assuming secure symmetric-key encryption (and hence one-way functions). This was fol- lowed by results, both positive and negative, concerning its security in the, stronger, adaptive setting. Applebaum et al. (Crypto 2013) showed that it cannot satisfy adaptive security as is, due to a simple incompressibility argument. Jafagholi and Wichs (TCC 2017) considered a natural adaptation of Yao’s scheme that circumvents this negative result, and proved that it is adaptively secure, at least for shallow circuits. In particular, they showed that for the class of circuits of depth d, the loss in security is at most exponential in d. The above results all concern the simulation-based notion of security. In this work, we show that the upper bound of Jafargholi and Wichs is more or less optimal in a strong sense. As our main result, we show that there exists a family of Boolean circuits, one for each depth d ∈ N, such that any black-box reduction proving the adaptive indistinguishability- security of the natural adaptation of Yao’s scheme from any symmetric-key encryption has to lose a factor that is sub-exponential in d. Since indistinguishability is a weaker notion than simulation, our bound also applies to adaptive simulation. To establish our results, we build on the recent approach of Kamath et al. (Eprint 2021), which uses pebbling lower bounds in conjunction with oracle separations to prove fine-grained lower bounds on loss in cryptographic security
2021
TCC
We show that Yao’s garbling scheme is adaptively indistinguishable for the class of Boolean circuits of size S and treewidth w with only a S^{O(w)} loss in security. For instance, circuits with constant treewidth are as a result adaptively indistinguishable with only a polynomial loss. This (partially) complements a negative result of Applebaum et al. (Crypto 2013), which showed (assuming one-way functions) that Yao’s garbling scheme cannot be adaptively simulatable. As main technical contributions, we introduce a new pebble game that abstracts out our security reduction and then present a pebbling strategy for this game where the number of pebbles used is roughly O(\delta w log(S)), \delta being the fan-out of the circuit. The design of the strategy relies on separators, a graph-theoretic notion with connections to circuit complexity.
2021
TCC
The security of cryptographic primitives and protocols against adversaries that are allowed to make adaptive choices (e.g., which parties to corrupt or which queries to make) is notoriously difficult to establish. A broad theoretical framework was introduced by Jafargholi et al. [Crypto'17] for this purpose. In this paper we initiate the study of lower bounds on loss in adaptive security for certain cryptographic protocols considered in the framework. We prove lower bounds that almost match the upper bounds (proven using the framework) for proxy re-encryption, prefix-constrained PRFs and generalized selective decryption, a security game that captures the security of certain group messaging and broadcast encryption schemes. Those primitives have in common that their security game involves an underlying graph that can be adaptively built by the adversary. Some of our lower bounds only apply to a restricted class of black-box reductions which we term "oblivious" (the existing upper bounds are of this restricted type), some apply to the broader but still restricted class of non-rewinding reductions, while our lower bound for proxy re-encryption applies to all black-box reductions. The fact that some of our lower bounds seem to crucially rely on obliviousness or at least a non-rewinding reduction hints to the exciting possibility that the existing upper bounds can be improved by using more sophisticated reductions. Our main conceptual contribution is a two-player multi-stage game called the Builder-Pebbler Game. We can translate bounds on the winning probabilities for various instantiations of this game into cryptographic lower bounds for the above mentioned primitives using oracle separation techniques.
2021
TCC
Key trees are often the best solution in terms of transmission cost and storage requirements for managing keys in a setting where a group needs to share a secret key, while being able to efficiently rotate the key material of users (in order to recover from a potential compromise, or to add or remove users). Applications include multicast encryption protocols like LKH (Logical Key Hierarchies) or group messaging like the current IETF proposal TreeKEM. A key tree is a (typically balanced) binary tree, where each node is identified with a key: leaf nodes hold users’ secret keys while the root is the shared group key. For a group of size N, each user just holds log(N) keys (the keys on the path from its leaf to the root) and its entire key material can be rotated by broadcasting 2log(N) ciphertexts (encrypting each fresh key on the path under the keys of its parents). In this work we consider the natural setting where we have many groups with partially overlapping sets of users, and ask if we can find solutions where the cost of rotating a key is better than in the trivial one where we have a separate key tree for each group. We show that in an asymptotic setting (where the number m of groups is fixed while the number N of users grows) there exist more general key graphs whose cost converges to the cost of a single group, thus saving a factor linear in the number of groups over the trivial solution. As our asymptotic “solution” converges very slowly and performs poorly on concrete examples, we propose an algorithm that uses a natural heuristic to compute a key graph for any given group structure. Our algorithm combines two greedy algorithms, and is thus very efficient: it first converts the group structure into a “lattice graph”, which then is turned into a key graph by repeatedly applying the algorithm for constructing a Huffman code. To better understand how far our proposal is from an optimal solution, we prove lower bounds on the update cost of continuous group-key agreement and multicast encryption in a symbolic model admitting (asymmetric) encryption, pseudorandom generators, and secret sharing as building blocks.
2021
TCC
Digital hardware Trojans are integrated circuits whose implementation differ from the specification in an arbitrary and malicious way. For example, the circuit can differ from its specified input/output behavior after some fixed number of queries (known as time bombs'') or on some particular input (known as cheat codes''). To detect such Trojans, countermeasures using multiparty computation (MPC) or verifiable computation (VC), have been proposed. On a high level, to realize a circuit with specification $\cF$ one has more sophisticated circuits $\cF^\diamond$ manufactured (where $\cF^\diamond$ specifies a MPC or VC of $\cF$), and then embeds these $\cF^\diamond$'s into a \emph{master circuit} which must be trusted but is relatively simple compared to $\cF$. Those solutions have a significant overhead as $\cF^\diamond$ is significantly more complex than $\cF$ and also the master circuits are not exactly trivial either. In this work, we show that in restricted settings, where $\cF$ has no evolving state and is queried on independent inputs, we can achieve a relaxed security notion using very simple constructions. In particular, we do not change the specification of the circuit at all (i.e., $\cF=\cF^\diamond$). Moreover the master circuit basically just queries a subset of its manufactured circuits and checks if they're all the same. The security we achieve guarantees that, if the manufactured circuits are initially tested on up to $T$ inputs, the master circuit will catch Trojans that try to deviate on significantly more than a $1/T$ fraction of the inputs. This bound is optimal for the type of construction considered, and we provably achieve it using a construction where $12$ instantiations of $\cF$ need to be embedded into the master. We also discuss an extremely simple construction with just $2$ instantiations for which we conjecture that it already achieves the optimal bound.
2019
PKC
A proxy re-encryption (PRE) scheme is a public-key encryption scheme that allows the holder of a key pk to derive a re-encryption key for any other key $pk'$ . This re-encryption key lets anyone transform ciphertexts under pk into ciphertexts under $pk'$ without having to know the underlying message, while transformations from $pk'$ to pk should not be possible (unidirectional). Security is defined in a multi-user setting against an adversary that gets the users’ public keys and can ask for re-encryption keys and can corrupt users by requesting their secret keys. Any ciphertext that the adversary cannot trivially decrypt given the obtained secret and re-encryption keys should be secure.All existing security proofs for PRE only show selective security, where the adversary must first declare the users it wants to corrupt. This can be lifted to more meaningful adaptive security by guessing the set of corrupted users among the n users, which loses a factor exponential in , rendering the result meaningless already for moderate .Jafargholi et al. (CRYPTO’17) proposed a framework that in some cases allows to give adaptive security proofs for schemes which were previously only known to be selectively secure, while avoiding the exponential loss that results from guessing the adaptive choices made by an adversary. We apply their framework to PREs that satisfy some natural additional properties. Concretely, we give a more fine-grained reduction for several unidirectional PREs, proving adaptive security at a much smaller loss. The loss depends on the graph of users whose edges represent the re-encryption keys queried by the adversary. For trees and chains the loss is quasi-polynomial in the size and for general graphs it is exponential in their depth and indegree (instead of their size as for previous reductions). Fortunately, trees and low-depth graphs cover many, if not most, interesting applications.Our results apply e.g. to the bilinear-map based PRE schemes by Ateniese et al. (NDSS’05 and CT-RSA’09), Gentry’s FHE-based scheme (STOC’09) and the LWE-based scheme by Chandran et al. (PKC’14).
2019
EUROCRYPT
Proofs of sequential work (PoSW) are proof systems where a prover, upon receiving a statement $\chi$ and a time parameter T computes a proof $\phi (\chi ,T)$ which is efficiently and publicly verifiable. The proof can be computed in T sequential steps, but not much less, even by a malicious party having large parallelism. A PoSW thus serves as a proof that T units of time have passed since $\chi$ was received.PoSW were introduced by Mahmoody, Moran and Vadhan [MMV11], a simple and practical construction was only recently proposed by Cohen and Pietrzak [CP18].In this work we construct a new simple PoSW in the random permutation model which is almost as simple and efficient as [CP18] but conceptually very different. Whereas the structure underlying [CP18] is a hash tree, our construction is based on skip lists and has the interesting property that computing the PoSW is a reversible computation.The fact that the construction is reversible can potentially be used for new applications like constructing proofs of replication. We also show how to “embed” the sloth function of Lenstra and Weselowski [LW17] into our PoSW to get a PoSW where one additionally can verify correctness of the output much more efficiently than recomputing it (though recent constructions of “verifiable delay functions” subsume most of the applications this construction was aiming at).
2019
ASIACRYPT
The distinctive feature of Bitcoin is that it achieves decentralisation in an open setting where everyone can join. This is achieved at a high price, honest parties must constantly dedicate more computational power towards securing Bitcoin's blockchain than is available to a potential adversary, which leads to a massive waste of energy; at its hitherto peak, the electricity used for Bitcoin mining equaled the electricity consumption of Austria. In this lecture I will discuss how disk-space, instead of computation, can be used as a resource to construct a more sustainable blockchain. We will see definitions and constructions of "proof of space" and "verifiable delay functions", and how they can be used to construct a Blockchain with similar dynamics and security properties as the Bitcoin blockchain.
2018
EUROCRYPT
2018
EUROCRYPT
2017
EUROCRYPT
2017
EUROCRYPT
2017
CRYPTO
2017
ASIACRYPT
2017
TCC
2017
JOFC
2016
EUROCRYPT
2016
TCC
2016
TCC
2016
TOSC
PMAC is a simple and parallel block-cipher mode of operation, which was introduced by Black and Rogaway at Eurocrypt 2002. If instantiated with a (pseudo)random permutation over n-bit strings, PMAC constitutes a provably secure variable input-length (pseudo)random function. For adversaries making q queries, each of length at most l (in n-bit blocks), and of total length σ ≤ ql, the original paper proves an upper bound on the distinguishing advantage of Ο(σ2/2n), while the currently best bound is Ο (qσ/2n).In this work we show that this bound is tight by giving an attack with advantage Ω (q2l/2n). In the PMAC construction one initially XORs a mask to every message block, where the mask for the ith block is computed as τi := γi·L, where L is a (secret) random value, and γi is the i-th codeword of the Gray code. Our attack applies more generally to any sequence of γi’s which contains a large coset of a subgroup of GF(2n). We then investigate if the security of PMAC can be further improved by using τi’s that are k-wise independent, for k > 1 (the original distribution is only 1-wise independent). We observe that the security of PMAC will not increase in general, even if the masks are chosen from a 2-wise independent distribution, and then prove that the security increases to O(q<2/2n), if the τi are 4-wise independent. Due to simple extension attacks, this is the best bound one can hope for, using any distribution on the masks. Whether 3-wise independence is already sufficient to get this level of security is left as an open problem.
2015
TCC
2015
CRYPTO
2015
CRYPTO
2015
CRYPTO
2015
ASIACRYPT
2015
ASIACRYPT
2014
CRYPTO
2014
EUROCRYPT
2014
PKC
2014
TCC
2014
JOFC
2014
ASIACRYPT
2013
TCC
2013
CRYPTO
2013
CRYPTO
2012
TCC
2012
TCC
2012
TCC
2012
EUROCRYPT
2012
CHES
2012
ASIACRYPT
2012
FSE
2011
TCC
2011
CRYPTO
2011
EUROCRYPT
2010
TCC
2010
TCC
2010
ASIACRYPT
2010
CRYPTO
2009
EUROCRYPT
2009
EUROCRYPT
2009
EUROCRYPT
2008
EUROCRYPT
2008
CRYPTO
2007
CRYPTO
2007
EUROCRYPT
2007
EUROCRYPT
2007
FSE
2007
TCC
2006
EUROCRYPT
2006
EUROCRYPT
2006
TCC
2005
CRYPTO
2005
CRYPTO
2005
CRYPTO
2004
TCC
2003
EUROCRYPT

Eurocrypt 2019
TCC 2018
TCC 2017
Eurocrypt 2017
Crypto 2014
TCC 2014
TCC 2013
CHES 2012
PKC 2012
Eurocrypt 2012
TCC 2011
Eurocrypt 2010
Crypto 2009


text:
# Do waterfalls generate pink (1/f) or brownian (1/f^2) noise?

Does anybody know if natural waterfalls generate pink noise or brownian noise? I have tried to search about this but the matches are non-scientific and contradictory.

For example

Even though it feels weird, it could be that the type of noise is dependent on the size of waterfall, for example by affecting to alpha of the power spectral density of the broad definition of pink noise: $S(f) = 1 / f^{\alpha}.$

format: plain


text:
# Some heuristics about local regression and kernel smoothing

October 8, 2013
By

In a standard linear model, we assume that $\mathbb{E}(Y\vert X=x)=\beta_0+\beta_1 x$. Alternatives can be considered, when the linear assumption is too strong.

• Polynomial regression

A natural extension might be to assume some polynomial function,

$\mathbb{E}(Y\vert X=x)=\beta_0+\beta_1 x+\beta_2 x^2 +\cdots +\beta_k x^k$

Again, in the standard linear model approach (with a conditional normal distribution using the GLM terminology), parameters $\boldsymbol{\beta}=(\beta_0,\beta_1,\cdots,\beta_k)$ can be obtained using least squares, where a regression of $Y$ on $\boldsymbol{X}=(1,X,X^2,\cdots,X^k)$ is considered.

Even if this polynomial model is not the real one, it might still be a good approximation for $\mathbb{E}(Y\vert X=x)=h(x)$. Actually, from Stone-Weierstrass theorem, if $h(\cdot)$ is continuous on some interval, then there is a uniform approximation of $h(\cdot)$ by polynomial functions.

Just to illustrate, consider the following (simulated) dataset

set.seed(1)
n=10
xr = seq(0,n,by=.1)
yr = sin(xr/2)+rnorm(length(xr))/2
db = data.frame(x=xr,y=yr)
plot(db)

with the standard regression line

reg = lm(y ~ x,data=db)
abline(reg,col="red")

Consider some polynomial regression. If the degree of the polynomial function is large enough, any kind of pattern can be obtained,

reg=lm(y~poly(x,5),data=db)

But if the degree is too large, then too many ‘oscillations’ are obtained,

reg=lm(y~poly(x,25),data=db)

and the estimation might be be seen as no longer robust: if we change one point, there might be important (local) changes

plot(db)
attach(db)
lines(xr,predict(reg),col="red",lty=2)
yrm=yr;yrm[31]=yr[31]-2
regm=lm(yrm~poly(xr,25))
lines(xr,predict(regm),col="red")

• Local regression

Actually, if our interest is to have locally a good approximation of  $h(\cdot)$, why not use a local regression?

This can be done easily using a weighted regression, where, in the least square formulation, we consider

$\min\left\{ \sum_{i=1}^n \omega_i [Y_i-(\beta_0+\beta_1 X_i)]^2 \right\}$

(it is possible to consider weights in the GLM framework, but let’s keep that for another post). Two comments here:

• here I consider a linear model, but any polynomial model can be considered. Even a constant one. In that case, the optimization problem is

$\min\left\{ \sum_{i=1}^n \omega_i [Y_i-\beta_0]^2 \right\}$which can be solve explicitly, since

$\widehat{\beta}_0=\frac{\sum \omega_i Y_i}{\sum \omega_i}$

• so far, nothing was mentioned about the weights. The idea is simple, here: if you can a good prediction at point $x_0$, then $\omega_i$ should be proportional to some distance between $X_i$ and $x_0$: if $X_i$ is too far from $x_0$, then it should not have to much influence on the prediction.

For instance, if we want to have a prediction at some point $x_0$, consider $\omega_i\propto \boldsymbol{1}(\vert X_i-x_0 \vert<1)$. With this model, we remove observations too far away,

Actually, here, it is the same as

reg=lm(yr~xr,subset=which(abs(xr-x0)<1)

A more general idea is to consider some kernel function $K(\cdot)$ that gives the shape of the weight function, and some bandwidth (usually denoted h) that gives the length of the neighborhood, so that

$\omega_i = K\left(\frac{x_0-X_i}{b}\right)$

This is actually the so-called Nadaraya-Watson estimator of function $h(\cdot)$.
In the previous case, we did consider a uniform kernel $K(x)=\boldsymbol{1}(x\in[-1/2,+1/2])$, with bandwith $2$,

But using this weight function, with a strong discontinuity may not be the best idea… Why not a Gaussian kernel,

$K(x)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right)$

This can be done using

fitloc0 = function(x0){
w=dnorm((xr-x0))
reg=lm(y~1,data=db,weights=w)
return(predict(reg,newdata=data.frame(x=x0)))}

On our dataset, we can plot

ul=seq(0,10,by=.01)
vl0=Vectorize(fitloc0)(ul)
u0=seq(-2,7,by=.01)
linearlocalconst=function(x0){
w=dnorm((xr-x0))
plot(db,cex=abs(w)*4)
lines(ul,vl0,col="red")
axis(3)
axis(2)
reg=lm(y~1,data=db,weights=w)
u=seq(0,10,by=.02)
v=predict(reg,newdata=data.frame(x=u))
lines(u,v,col="red",lwd=2)
abline(v=c(0,x0,10),lty=2)
}
linearlocalconst(2)

Here, we want a local regression at point 2. The horizonal line below is the regression (the size of the point is proportional to the wieght). The curve, in red, is the evolution of the local regression

Let us use an animation to visualize the construction of the curve. One can use

library(animate)

but for some reasons, I cannot install the package easily on Linux. And it is not a big deal. We can still use a loop to generate some graphs

vx0=seq(1,9,by=.1)
vx0=c(vx0,rev(vx0))
graphloc=function(i){
name=paste("local-reg-",100+i,".png",sep="")
png(name,600,400)
linearlocalconst(vx0[i])
dev.off()}

for(i in 1:length(vx0)) graphloc(i)

and then, in a terminal, I simply use

    convert -delay 25 /home/freak/local-reg-1*.png /home/freak/local-reg.gif

Of course, it is possible to consider a linear model, locally,

fitloc1 = function(x0){
w=dnorm((xr-x0))
reg=lm(y~poly(x,degree=1),data=db,weights=w)
return(predict(reg,newdata=data.frame(x=x0)))}

or even a quadratic (local) regression,

fitloc2 = function(x0){
w=dnorm((xr-x0))
reg=lm(y~poly(x,degree=2),data=db,weights=w)
return(predict(reg,newdata=data.frame(x=x0)))}

Of course, we can change the bandwidth

To conclude the technical part this post, observe that, in practise, we have to choose the shape of the weight function (the so-called kernel). But there are (simple) technique to select the “optimal” bandwidth h. The idea of cross validation is to consider

$\min\left\{ \sum_{i=1}^n [Y_i-\widehat{Y}_i(b)]^2 \right\}$

where $\widehat{Y}_i(b)$ is the prediction obtained using a local regression technique, with bandwidth $b$. And to get a more accurate (and optimal) bandwith $\widehat{Y}_i(b)$ is obtained using a model estimated on a sample where the ith observation was removed. But again, that is not the main point in this post, so let’s keep that for another one…

Perhaps we can try on some real data? Inspired from a great post on http://f.briatte.org/teaching/ida/092_smoothing.html, by François Briatte, consider the Global Episode Opinion Survey, from some TV show, http://geos.tv/index.php/index?sid=189 , like Dexter.

library(XML)
file = "geos-tww.csv"
html = htmlParse("http://www.geos.tv/index.php/list?sid=189&collection=all")
html = xpathApply(html, "//table[@id='collectionTable']")[[1]]
data = data[,-3]
names(data)=c("no",names(data)[-1])
data=data[-(61:64),]

Let us reshape the dataset,

data$no = 1:96 data$mu = as.numeric(substr(as.character(data$Mean), 0, 4)) data$se =  sd(data$mu,na.rm=TRUE)/sqrt(as.numeric(as.character(data$Count)))
data$season = 1 + (data$no - 1)%/%12
data$season = factor(data$season)
plot(data$no,data$mu,ylim=c(6,10))
segments(data$no,data$mu-1.96*data$se, data$no,data$mu+1.96*data$se,col="light blue")

As done by François, we compute some kind of standard error, just to reflect uncertainty. But we won’t really use it.

plot(data$no,data$mu,ylim=c(6,10))
abline(v=12*(0:8)+.5,lty=2)
for(s in 1:8){reg=lm(mu~no,data=db,subset=season==s)
lines((s-1)*12+1:12,predict(reg)[1:12],col="red") }

Henre, we assume that all seasons should be considered as completely independent… which might not be a great assumption.

db = data
NW = ksmooth(db$no,db$mu,kernel = "normal",bandwidth=5)
plot(data$no,data$mu)
lines(NW,col="red")

We can try to look the curve with a larger bandwidth. The problem is that there is a missing value, at the end. If we (arbitrarily) fill it, we can run a kernel regression,

db$mu[95]=7 NW = ksmooth(db$no,db$mu,kernel = "normal",bandwidth=12) plot(data$no,data\$mu,ylim=c(6,10))
lines(NW,col="red")

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Posted in: Course, ACT2040, 2013, Course, ACT2040, Course, ACT2040, 2013, Automne, Course, Statistics, Regression

### 4 Responses to “ Some heuristics about local regression and kernel smoothing ”

1. FML on 20/03/2014 at 3:02 PM

Thanks for the post! I am learning R and was curious as to how you did plot #6, esp. the data points colored differently and the vertical lines. Looks neat and captures the concept nicely.

2. […] from a great post on http://freakonometrics.hypotheses.org/… by Arthur Charpentier, let us use an animation to visualize the construction of the curve. The idea […]

3. Gu Mi on 09/10/2013 at 5:22 PM

Thanks for the great post! Just a trivial correction: in xr = seq(0,n,by=.1) at the very beginning, n is undefined (should be 10).

Research blogs


text:
# What is the derivative of  e^x/2^x ?

Mar 18, 2017

${\left(\frac{e}{2}\right)}^{x} \left(1 - \ln 2\right)$

#### Explanation:

Use the quotient rule $\left(\frac{u}{v}\right) ' = \frac{v \cdot u ' - u \cdot v '}{v} ^ 2$
Let $u = {e}^{x}$, $u ' = {e}^{x}$
Let $v = {2}^{x}$, $v ' = {2}^{x} \ln 2$

$\left(\frac{u}{v}\right) ' = \frac{{2}^{x} \cdot {e}^{x} - {e}^{x} \cdot {2}^{x} \ln 2}{{2}^{x}} ^ 2 = \frac{\left({2}^{x} {e}^{x}\right) \left(1 - \ln 2\right)}{{2}^{2 x}}$

Simplify using exponent rules ${x}^{m} / {x}^{2 m} = \frac{1}{x} ^ \left(2 m - m\right) = \frac{1}{x} ^ m$:

$\frac{\left({2}^{x} {e}^{x}\right) \left(1 - \ln 2\right)}{{2}^{2 x}} = \frac{{e}^{x} \left(1 - \ln 2\right)}{2} ^ x = {\left(\frac{e}{2}\right)}^{x} \left(1 - \ln 2\right)$

Mar 19, 2017

$\frac{d}{\mathrm{dx}} {e}^{x} / {2}^{x} = \left(1 - \ln 2\right) {e}^{x} / {2}^{x}$

#### Explanation:

Let $y = {e}^{x} / {2}^{x}$

Then take (Natural) logarithms of both sides; and use the rules of logs:

$\therefore \ln y = \ln \left\{{e}^{x} / {2}^{x}\right\}$
$\text{ } = \ln \left({e}^{x}\right) - \ln \left({2}^{x}\right)$
$\text{ } = x \ln \left(e\right) - x \ln \left(2\right)$
$\text{ } = x - x \ln \left(2\right)$
$\text{ } = x \left(1 - \ln \left(2\right)\right)$

Now, differentiate implicitly:

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus = \left(1 - \ln 2\right)$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \left(1 - \ln 2\right) y$
$\text{ } = \left(1 - \ln 2\right) {e}^{x} / {2}^{x}$

Mar 23, 2017

${\left(\frac{e}{2}\right)}^{x} \left(1 - \ln 2\right)$

#### Explanation:

${e}^{x} / {2}^{x} = {\left(\frac{e}{2}\right)}^{x}$

The derivative of ${a}^{x}$ with respect to $x$ for any number $a$ is:

$\frac{d}{\mathrm{dx}} {a}^{x} = {a}^{x} \ln \left(a\right)$

Therefore:

$\frac{d}{\mathrm{dx}} {\left(\frac{e}{2}\right)}^{x} = {\left(\frac{e}{2}\right)}^{x} \ln \left(\frac{e}{2}\right)$

$\textcolor{w h i t e}{\text{XXXXX-}} = {\left(\frac{e}{2}\right)}^{x} \left(\ln e - \ln 2\right)$
$\textcolor{w h i t e}{\text{XXXXX-}} = {\left(\frac{e}{2}\right)}^{x} \left(1 - \ln 2\right)$


text:
# All Questions

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text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 $q$-analogue of Euler-Barnes' numbers and polynomials Bull. Korean Math. Soc. 2005 Vol. 42, No. 3, 491-499 Printed September 1, 2005 Leechae Jang and Taekyun Kim Kon-kuk University, Kongju National University Abstract : Recently, Kim[2,6] has introduced an interesting Euler-Barnes' numbers and polynomials.~In this paper, we construct the $q$-analogue of Euler-Barnes' numbers and polynomials, and investigate their properties. Keywords : Euler numbers, Bernoulli numbers, zeta function MSC numbers : 11B68 Downloads: Full-text PDF


text:
## Fundamentals of Physics Extended (10th Edition)

The equation that governs centripedal acceleration is: $a_{c} = \frac{v^2}{r}$ We have that the man's velocity is 3.66m/s and that the acceleration is $1.83m/s^2$. Substituting these into the equation we find that $1.83 = (3.66)^2/r$ so r = 7.32m


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text:
# Find the order of students presentation(logical puzzle)

Four students P, Q, R and S each working under the super vision of one of the four professors A, B, C and D made their final year MBA Project Presentations one by one, one each in the areas of Finance, Marketing, Systems and Human Resource Management (HRM). Each professor is an expert in only one of the above areas and supervised exactly one of the above students in his own area. The following are the clues:

(i) First presentation was made by R.
(ii) Prof. B works in Finance.
(iii) Prof. D was P’s supervisor.
(iv) The last presentation was in the Systems area.
(v) S’s project was in the HRM area.
(vi) Prof. B’s student’s presentation followed that of Prof. C’s student.

Find:

Order of Presentation.
Each students Subjects.
Mentor of each student.

• Typically these types of 'einstein puzzles' should be written so that no assumptions/scenario's need to be tested, and specifically they should only lead to a unique result. While I enjoy the concept of these puzzles, this simply is a poor example. – Tim Couwelier Sep 6 '14 at 7:13
• Yes I know that there should be a unique solution but its not a hard and fast rule. Just to check one's ability of reaoning we can have such questions having multiple solutions so as to increase the level of difficulty. – toadalskiii Sep 6 '14 at 7:27

     1          2          3          4
---------------------------------------------
R          S          Q          P
A          C          B          D
Marketing  HRM        Finance    Systems

Edit: explanation:

question information:

B and finance are connected, so the only student that can pick both of them is Q:

so, R cannot pick Finance, also R is 1st and Systems is last therefore R cannot pick Systems as well as HRM.

Hence we can see that the only option for P is Systems:

As B is exactly after C, Here We have two possible answers:

     1          2          3          4
---------------------------------------------
R          S          Q          P
*A*       *C*        *B*         D
Marketing  HRM        Finance    Systems

OR

     1          2          3          4
---------------------------------------------
R          Q          S          P
*C*        *B*        *A*         D
Marketing  Finance    HRM        Systems

• you arr very welcome! btw, getting some up-votes is almost impossible in puzzling! – Rafe Sep 6 '14 at 10:18
• @Rafe Well, I can't let that much work go in without providing some support for it. Explanations are one thing, but visuals and explanations are worth a lot more. – Xrylite Sep 11 '14 at 23:49

N -  Prof  -  Stud  -  Area
1 - [ABCD] - [PQRS] - [FHMS]
2 - [ABCD] - [PQRS] - [FHMS]
3 - [ABCD] - [PQRS] - [FHMS]
4 - [ABCD] - [PQRS] - [FHMS]

Apply rules 1 (R first) and 4 (Systems last), they're quite straightforward.

1 - [ABCD] - [  R ] - [FHM ]
2 - [ABCD] - [PQ S] - [FHM ]
3 - [ABCD] - [PQ S] - [FHM ]
4 - [ABCD] - [PQ S] - [   S]

Since B works in finance (2), his student can't go last.

1 - [ABCD] - [  R ] - [FHM ]
2 - [ABCD] - [PQ S] - [FHM ]
3 - [ABCD] - [PQ S] - [FHM ]
4 - [A CD] - [PQ S] - [   S]

Student S is in HRM (5):

1 - [ABCD] - [  R ] - [F M ]
2 - [ABCD] - [PQ S] - [FHM ]
3 - [ABCD] - [PQ S] - [FHM ]
4 - [A CD] - [PQ  ] - [   S]

B is immediately after C (6) - B can't be first, C can't be third or fourth.

1 - [A CD] - [  R ] - [F M ]
2 - [ABCD] - [PQ S] - [FHM ]
3 - [AB D] - [PQ S] - [FHM ]
4 - [A  D] - [PQ  ] - [   S]

Remember that B is in finance, leaving 1 = R = Marketing.

1 - [A CD] - [  R ] - [  M ]
2 - [ABCD] - [PQ S] - [FH  ]
3 - [AB D] - [PQ S] - [FH  ]
4 - [A  D] - [PQ  ] - [   S]

Since D is P's professor, he can't be R's.

1 - [A C ] - [  R ] - [  M ]
2 - [ABCD] - [PQ S] - [FH  ]
3 - [AB D] - [PQ S] - [FH  ]
4 - [A  D] - [PQ  ] - [   S]

Now, let's try trial and error quickly:

Assume 1 = A

1 - [A   ] - [  R ] - [  M ]
2 - [ BCD] - [PQ S] - [FH  ]
3 - [ B D] - [PQ S] - [FH  ]
4 - [   D] - [PQ  ] - [   S]

C can then only be 2, which leaves B 3 and D 4.

1 - [A   ] - [  R ] - [  M ]
2 - [  C ] - [PQ S] - [FH  ]
3 - [ B  ] - [PQ S] - [FH  ]
4 - [   D] - [PQ  ] - [   S]

D is with P; B works in Finance; S is in HRM.

Solution 1

   1 - [A   ] - [  R ] - [  M ]
2 - [  C ] - [   S] - [ H  ]
3 - [ B  ] - [ Q  ] - [F   ]
4 - [   D] - [P   ] - [   S]

Aha, a solution!

Now, let's see if 1 = C works:

Assume 1 = C

1 - [  C ] - [  R ] - [  M ]
2 - [AB D] - [PQ S] - [FH  ]
3 - [AB D] - [PQ S] - [FH  ]
4 - [A  D] - [PQ  ] - [   S]

B is immediately after. Remember he's in finance.

1 - [  C ] - [  R ] - [  M ]
2 - [ B  ] - [PQ S] - [F   ]
3 - [A  D] - [PQ S] - [ H  ]
4 - [A  D] - [PQ  ] - [   S]

S is in HRM

1 - [  C ] - [  R ] - [  M ]
2 - [ B  ] - [ Q  ] - [F   ]
3 - [A  D] - [   S] - [ H  ]
4 - [A  D] - [PQ  ] - [   S]

D is with P

Solution 2

   1 - [  C ] - [  R ] - [  M ]
2 - [ B  ] - [ Q  ] - [F   ]
3 - [A   ] - [   S] - [ H  ]
4 - [   D] - [P   ] - [   S]

Well, that's another solution fully consistent with the rules. Turns out this puzzle has two correct answers.


text: Heat (or Diffusion) equation in 1D* • Derivation of the 1D heat equation • Separation of variables (refresher) • Worked examples *Kreysig, 8th Edn, Sections 11. of thermal diffusion on a transit free convective flow past an electrically conducting viscous incompressible fluid past an infinite vertical porous plate in a rotating system taking into account the effect of Hall current is presented when the temperature as well as the concentration at the plate varies periodically with time. The steady convection-diffusion equation is div u div() ( )ρφ φ= Γ+grad Sφ Integration over the control volume gives : ∫∫ ∫nu n() ( )ρφ φdA grad dA S dVΓ+ AA CV ⋅=⋅φ This equation represents the flux balance in a control volume. = 0 at x = 0 and the surface at x= L is exposed to a convection environment with fluid temperature T and convection coefficienth. We propose an approximation of the convection-diffusion operator which consists in the product of two parabolic operators. Energy carried by fluid is more in convection than in diffusion (recall Nusselt number) 4. Exact Solutions of Diffusion-Convection Equations Article (PDF Available) in Dynamics of partial differential equations 5(2) · November 2007 with 343 Reads How we measure 'reads'. These equations are equivalent to those used in dilute-solution theory. Solve a heat equation that describes heat diffusion in a block with a rectangular cavity. Hence, it is obvious that:. Dimensionless form of equations Motivation: sometimes equations are normalized in order to •facilitate the scale-up of obtained results to real ﬂow conditions •avoid round-oﬀ due to manipulations with large/small numbers •assess the relative importance of terms in the model equations Dimensionless variables and numbers t∗ = t t0, x. SCHATZMAN Abstract. The second. The paper deals in its first part with the general formulation of the convective-diffusion. ⃗ is known as the viscous term or the diffusion term. COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. We present an exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet's type boundary conditions. Kontarinis, A. Edited by PatriciaW. 2 GOVERNING EQUATIONS AND DISCRETIZATION We consider the linear advection diffusion equation u t+ au x bu xx= 0: (1) with a;b>0 on the interval x2[0;2] with periodic boundary conditions. described by a set of partial differential equations which are mathematical formulations of one or more of the conservation laws of physics. iosrjournals. In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗ u − u + G ∗ (f (u)) − f (u) in Rd , with J radially symmetric and G not necessarily symmetric. Methods Partial Differential Equations 20 (2004), no. , surfactant, along a deforming interface is outlined. Rahmim Abstract- In this paper, the distribution of PET tracer. In this chapter, we will discuss the effects of V on the. We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. Methods We add a convection term, and transform the (linear) diffusion to a convection-diffusion equation: v*(Dv)u+ f v*vu=u, where D is the diffusion tensor from MRI, v is a vector field, and u is the unknown field. In order to. If there is bulk fluid motion, convection will also contribute to the flux of chemical. Dass, A class of higher order compact schemes for the unsteady two‐dimensional convection-diffusion equation with variable convection coefficients, International Journal for Numerical Methods in Fluids, 38, 12, (1111-1131), (2002). The new diffusion package is called the Yonsei University PBL (YSU PBL). These equations are equivalent to those used in dilute-solution theory. The convection-diffusion equation can only rarely be solved with a pen and paper. These laws include those of conservation of mass, momentum, and energy. However, due to the non-linearity in the governing equation, if the spatial step is reduced, the solution can develop shocks, see Figure 2. Convection and Diffusion _____ So far, we have neglected the effects of convection in formulating finite volume equation of generalized transport equation. The solution presented here is for the entire time domain and agrees well with both the short and long time solutions presented earlier in the literature. Keywords: convection-diffusion, boundary layers, streamline diffusion Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. From a numerical point of view, the simulation of the hyperbolic diffusion equation has been mostly limited to 1D problems [2, 3]. STABILITY ANALYSIS OF NUMERICAL SOLUTIONS FOR THE DIFFUSION-CONVECTION EQUATION. If the two coefficients and are constants then they are referred to as solute dispersion coefficient and uniform velocity, respectively, and the above equation reduces to Equation (1). 2-2) where sensible enthalpy his de ned for ideal gases as h= X j Y jh j (11. [11] to solve the steady 3D convection-diffusion equation with variable coefficients on non-uniform grid. Types of heat transfer. When both the first and second spatial derivatives are present, the equation is called the convection-diffusion equation. Pollution Problems Based on Convection Diffusion Equation Lingyu Li, Zhe Yin* College of Mathematics and Statistics, Shandong Normal University, Jinan, China Abstract The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. The prototypical equation is the Diffusion equation ut = ∆u Nonlinear diffusion ut = div(k(u)gradu) Boundary and initial conditions are needed Numerical Methods for Differential Equations – p. In most cases the oscillations are small and the cell Reynolds number is frequently allowed to be higher than 2 with relatively minor effects on the result. The numerical solution of a convection diffusion equations whose first derivative have large coefficients (convection dominated) presents. At the molecular level, the cause is the perpetual agitation of molecules; at the turbulence level, it is advection by the turbulent eddies of the carrying fluid. ABSTRACT Based on the numerical criteria/schemes for the evaluation of both the diffusion and convection terms in the convection-diffusion equations presented in the companion article (Part I), a comprehensive numerical study is presented considering eight different test problems and more than 1,800 test cases. It does not include transport of substances by molecular diffusion. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. simple matlab code for advection diffusion equation. In (Juanes and Patzek, 2004), a numerical solution of miscible and immiscible flow in porous media was studied and focus was presented in the case of small diffusion; this turns linear convection-diffusion equation into hyperbolic equation. In this paper, the influence on drug diffusion of adverse fluid convection was modeled by solving a diffusion convection equation with appropriate boundary conditions. and non-linear convection diffusion equations. Consider a 2D situation in which there is advection (direction taken as the x-axis) and diffusion in both downstream and transverse directions. In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. Chapter 7 The Diffusion Equation The diffusionequation is a partial differentialequationwhich describes density ﬂuc-tuations in a material undergoing diffusion. 3 time-dependent convection-diffusion, with convection being the dominant process, can be expressed as i 1 monitoring network design, and various other related I- dC - V* (DVC) + V *VC = F perturbation, we consider the following one-dimensional transient convection-diffusion equation ac a2c dC --D-+V(t)-=O at ax2 OX. Hence, it is obvious that:. Carpenter Aeronautics and Aeroacoustic Methods Branch NASA Langley Research Center Hampton, Virginia 23681 0001 Abstract. They are “source-type” solutions of the convection-diffusion equation above. Velocities and Fluxes of Mass Transfer The thermodynamic state of a fluid system can be unequivocally defined based on the. These laws include those of conservation of mass, momentum, and energy. 3) Let us ﬁrst brieﬂy consider this alternative equation, which will be referred to as the reduced problem. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. Home About us Subjects Contacts Advanced Search Help. Most real physical processes are governed by partial differential equations. We propose an approximation of the convection-diffusion operator which consists in the product of two parabolic operators. convective-diffusion equation have been solved rigorously for the steady state. edu †[email protected] In this case, we assume that D and v are real, scalar constants. Clearly a square wave is not best represented with the inviscid Burgers Equation. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Subramaniam, and A. Pdf Numerical Solution Of Fractional Advection Diffusion Equation. 303 Linear Partial Diﬀerential Equations Matthew J. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. Blowup Analysis for a Nonlocal Diffusion Equation with Reaction and Absorption Wang, Yulan, Xiang, Zhaoyin, and Hu, Jinsong, Journal of Applied Mathematics, 2012; On front speeds in the vanishing diffusion limit for reaction-convection-diffusion equations Gilding, Brian H. Edited by PatriciaW. For upwinding, no oscillations appear. Convection is a physical process that. The convection-diffusion or advection-diffusion equation is widely used to describe transport phenomena where heat, mass, and other physical quantities are transferred due to the diffusion and advection processes [1]. Convection and Diffusion _____ So far, we have neglected the effects of convection in formulating finite volume equation of generalized transport equation. exact solutions of diffusion-convection equa tions 141 which is in v ariant with resp ect to the six dimensional symmetry algebra generated by the vector ﬁelds. However, due to the non-linearity in the governing equation, if the spatial step is reduced, the solution can develop shocks, see Figure 2. while heating a beaker of water from below, the first thing that will happen is diffusion (i. The convective-diffusion equation is the governing equation of many important transport phenomena in building physics. The fact that it is periodic follows easily. Graves, Jr. 1BestCsharp blog 4,848,905 views. 2-D Convection-Diffusion Program By modifying 1-D convection-diffusion program, a general 2-D convection-diffusion program can be easily obtained. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 2006 Gilbert Strang 5. The paper deals in its first part with the general formulation of the convective-diffusion. Alonso,1 A. Diffusion Equation! Computational Fluid Dynamics! ∂f ∂t +U ∂f ∂x =D ∂2 f ∂x2 We will use the model equation:! Although this equation is much simpler than the full Navier Stokes equations, it has both an advection term and a diffusion term. Read "A robust WG finite element method for convection–diffusion–reaction equations, Journal of Computational and Applied Mathematics" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The diffusion-convection-reaction (DCR) equation arises in a number of physical phenomena, such as, the dispersion of chemicals in reactors, 1 1. The last formula with ﬂ =1, ° =−1, – =‚ =0was obtained with the Appell transformation. Sherratt Form of Smooth-Front Waves of spatial modelling in ecology. These laws include those of conservation of mass, momentum, and energy. Carpenter Aeronautics and Aeroacoustic Methods Branch NASA Langley Research Center Hampton, Virginia 23681 0001 Abstract. described by a set of partial differential equations which are mathematical formulations of one or more of the conservation laws of physics. Missirlis, F. SIAM Journal on Numerical Analysis 56:6, 3611-3647. Note: $$\nu > 0$$ for physical diffusion (if $$\nu < 0$$ would represent an exponentially growing phenomenon, e. STABILITY ANALYSIS OF NUMERICAL SOLUTIONS FOR THE DIFFUSION-CONVECTION EQUATION. For more details and algorithms see: Numerical solution of the convection–diffusion equation. Edited by PatriciaW. 1 Physical derivation Reference: Guenther & Lee §1. IMAPreprintSeries653. In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. processes are modelled by similar mathematical equations in the case of diffusion and convection (there is no mass-transfer similarity to heat radiation), and it is thus more efficient to consider them jointly. FAST ITERATIVE SOLVER FOR CONVECTION-DIFFUSION SYSTEMS WITH SPECTRAL ELEMENTS P. The proposed methods make new use of the fractional polynomials, also known as Müntz polynomials, which can be regarded as continuation of our previous work. 00431v2 [math. and Finite Element Method (FEM) for the numerical solution of parabolic type partial differential equations such as 1-D singularly perturbed convection-dominated diffusion equation and 2-D Transient heat conduction problems validated against exact solution. In this paper, we will based our developments and analysis mainly on this defini-. FINLAYSON Department of Chemical Engineering, University of Washington, Seattle, Washington 98195. Missirlis, F. 3) Let us ﬁrst brieﬂy consider this alternative equation, which will be referred to as the reduced problem. For more details and algorithms see: Numerical solution of the convection–diffusion equation. For low Reynolds number flow at low pressure, the Navier-Stokes equation becomes a diffusion equation (40) For high Reynolds number flow, the viscous force is small compared to the inertia force, so it can be neglected. A series solution to the transient convective diffusion equation for the rotating disc electrode system is presented and compared to previously reported solutions. In this paper, we propose and analyze a Müntz spectral method for a class of two-dimensional space-fractional convection-diffusion equations. na] 5 oct 2019 a stabilized finite element method for inverse problems subject to the convection-diffusion equation. We examine solutions of the convection‐diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to P n, with P the Péclet number. The research of the convection-diffusion equation is of great importance. and non-linear convection diffusion equations. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem Anand Shukla*, Akhilesh Kumar Singh, P. Once the convection- diffusion equation is discretised, it is usual to observe oscillations in the computed. Equation 19. The convection-diffusion or advection-diffusion equation is widely used to describe transport phenomena where heat, mass, and other physical quantities are transferred due to the diffusion and advection processes [1]. an explosion or ‘the rich get richer’ model) The physics of diffusion are: An expotentially damped wave in time. Convection-diffusion reactions are used in many applications in science and engineering. The diffusion layer thickness is determined by 3 c eR 2 1/3 eR 1/3 c 3 2 0 c s R R x 0. NUMERICAL SOLUTION OF CONVECTION{DIFFUSION EQUATIONS USING UPWINDING TECHNIQUES SATISFYING THE DISCRETE MAXIMUM PRINCIPLE PETR KNOBLOCH1 Abstract. 3 time-dependent convection-diffusion, with convection being the dominant process, can be expressed as i 1 monitoring network design, and various other related I- dC - V* (DVC) + V *VC = F perturbation, we consider the following one-dimensional transient convection-diffusion equation ac a2c dC --D-+V(t)-=O at ax2 OX. x x u KA x u x x KA x u x KA x x x. 5 x J A t Q Molecular diffusion. The distinguishing feature of HWCM is that it provides a fast. Note: Citations are based on reference standards. • This is the general approach to solving partial differential equations used in CFD. Successfully, we. iosrjournals. The convection–diffusion equation can be derived in a straightforward way from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume:. The above equations represented convection without diffusion or diffusion without convection. Now-a-days computational fluid mechanics has become very vital area in which obtained governing equations. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem Anand Shukla*, Akhilesh Kumar Singh, P. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. As shown in. SIAM Journal on Numerical Analysis 56:6, 3611-3647. uni-dortmund. The above equations represented convection without diffusion or diffusion without convection. Change the velocity to u and v. The equation is given by ∂φ ∂t +∇ ·(φu) = ∇ ·(D∇φ), (1) where φ represents a scalar variable and is a function of. Convection-diffusion equations arose from various fields of applied sciences such as heat transfer problems in a draining film or a nanofluid filled enclosure , radial transport in a porous medium , and water transport in soils and have received extensive attentions during the past several decades. We propose an approximation of the convection-diffusion operator which consists in the product of two parabolic operators. Successive Approximation Method for Solving Nonlinear Diffusion Equation with Convection www. Sefidgar, M. AbstractDifference methods for solving the convection-diffusion equation are discussed. The Finite Volume Method Diffusion Equation Convectionextension To. is the hydrodynamic velocity. Sch€uler, 1 S. Soltani, M. Sun, Numerical Methods for Partical Difference Equations, (Chinese) Second edition, Science Press, Beijing, 2012. ﬁndings will be backed up by numerical results based on the convection diffusion equation as a model problem. The first term on the right hand side of equations and , defined as the relative convective flux J RC α, represents solute transport due to convection (relative to solid phase) and the second term (i. Pollution Problems Based on Convection Diffusion Equation Lingyu Li, Zhe Yin* College of Mathematics and Statistics, Shandong Normal University, Jinan, China Abstract The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. HOU† Applied and Computational Mathematics California Institute of Technology, Pasadena, CA 91125, USA ∗[email protected] org The convection–diffusion equation is a combination of the diffusion and convection equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. A weakly coupled convection dominated system of m-equations is analyzed. Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton's law of cooling, which states that:. Now-a-days computational fluid mechanics has become very vital area in which obtained governing equations. Abstract: This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each element. Solution of the Advection-Diffusion Equation Using the Differential Quadrature. Dass, A class of higher order compact schemes for the unsteady two‐dimensional convection-diffusion equation with variable convection coefficients, International Journal for Numerical Methods in Fluids, 38, 12, (1111-1131), (2002). Singh Department of Mathematics, MNNIT, Allahabad, 211 004, India. The extension is twofold. This paper studies the steady, free convection boundary layer flow about a vertical, isothermal plate embedded in a non-Darcy bidisperse porous medium (BDPM). exact solutions of diffusion-convection equa tions 141 which is in v ariant with resp ect to the six dimensional symmetry algebra generated by the vector ﬁelds. The convective and diffusive fluxes are approximated first, and then the resulting set of the ordinary differential equations (ODEs) is solved using the appropriate time stepping algorithm. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem Anand Shukla*, Akhilesh Kumar Singh, P. This equation shows that the flux of material towards the electrode. These equations are equivalent to those used in dilute-solution theory. processes are modelled by similar mathematical equations in the case of diffusion and convection (there is no mass-transfer similarity to heat radiation), and it is thus more efficient to consider them jointly. The convective-diffusion equation is the governing equation of many important transport phenomena in building physics. Heat Transfer in Block with Cavity. (2016) L^2-stability of a finite element - finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions. Hancock Fall 2006 1 The 1-D Heat Equation 1. 1) in most of the domain tends to be close to the solution, ˆu, of the hyperbolic equation w" ·∇ˆu = f. The proposed methods make new use of the fractional polynomials, also known as Müntz polynomials, which can be regarded as continuation of our previous work. Microfluidic-related technologies and micro-electromechanical systems–based microfluidic devices have received applications in science and engineering fields. The convection-diffusion equation Introduction and examples 2. An alternating direction implicit method for a second-order hyperbolic diffusion equation with convectionq Adérito Araújoa, Cidália Nevesa,b, Ercília Sousaa,⇑ a CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal. ! Before attempting to solve the equation, it is useful to understand how the analytical. Velocities and Fluxes of Mass Transfer The thermodynamic state of a fluid system can be unequivocally defined based on the. any mass flux may include both convection and diffusion because in many cases convection may be generated by diffusion. 4 The Heat Equation and Convection-Diﬀusion The wave equation conserves energy. Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference and is conveniently expressed by Newton's law of cooling, which states that:. Unsteady Convection Diffusion Reaction Problem File Exchange. In effect, the paper models the likelihood that solute can travel from the mucosal surface to the crypts in the face of adverse fluid secretion. In this PhD thesis, we construct numerical methods to solve problems described by advectiondiffusion and convective Cahn-Hilliard equations. van 'tHof, J. In most cases the oscillations are small and the cell Reynolds number is frequently allowed to be higher than 2 with relatively minor effects on the result. is the hydrodynamic velocity. pdf FREE PDF DOWNLOAD NOW!!! Source #2: simple matlab code for advection diffusion equation. In the first part of the paper, solutions to both the linear advection and the advection–diffusion problems for a single conservative scalar are discussed. But we shall ﬁnd that diffusion solutions have properties that in several ways are very different from wave solutions. 1) is referred to as the Convection-Diffusion equation when D is diffusion. We consider the single phase ﬂow of a ﬂuid in a porous medium in Section 2. diffusion equation with '(x,t) ≡0. We report here the development of a continuous intestinal absorption model based on the convection-diffusion equation. 5 x J A t Q Molecular diffusion. Prajapati Apollo Institute of Engineering & Technology 1 This chapter will deal with the most fundamental aspects of numerical procedures for solving convection - diffusion problems. In this chapter, we will discuss the effects of V on the. solve the one-speed neutron diffusion equation for a variety of situations, can analyze nuclear reactor fuel and core steady-state thermal performance, can couple the reactor neutronics to the core thermal-hydraulics in a design. The above equations represented convection without diffusion or diffusion without convection. So, this equation can be considered as a simplified form of the Navier–Stokes equations and also it is the simplest model of nonlinear partial differential equation for diffusive waves in fluid dynamics. This paper is devoted to the presentation and the analysis of a new particle method for convection-diffusion equations. Convection and Diffusion _____ So far, we have neglected the effects of convection in formulating finite volume equation of generalized transport equation. P DS (9) Typical units for P are (cm 3 cm)/(s cm 2 Pa) (those units×10-10 are defined as the barrer, the standard unit of P adopted by ASTM). org 14 | Page Where r , p [0, 1] that is called homotopy parameter, and is an initial approximation of equation (2). equation of creeping motion (39) In this regime, viscous interactions have an influence over large distances from an obstacle. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative. 303 Linear Partial Diﬀerential Equations Matthew J. Microfluidic-related technologies and micro-electromechanical systems–based microfluidic devices have received applications in science and engineering fields. This is the Maxwell-Stefan equation, an equation that forms the basis for the mathematical description of mass transfer of chemical species in a mixture 2. We examine solutions of the convection‐diffusion equation during steady radial flow in two and three dimensions with diffusivity proportional to P n, with P the Péclet number. Whenever we consider mass transport of a dissolved species (solute species) or a component in a gas mixture, concentration gradients will cause diffusion. The numerical scheme is based on a constrained flux optimization approach where the. Suppose w =w(x,t) is a solution of the diffusion equation. The last formula with ﬂ =1, ° =−1, – =‚ =0was obtained with the Appell transformation. Sefidgar, M. Sukop}, year={2011} }. using five equally spaced cells and the central differencing scheme for convection and diffusion calculate the distribution of as a function of x for. Differential equations and uniqueness conditions. • understand the behaviour of a diffusion equation • understand the behaviour of a convection-diffusion problem and how it varies with the Peclet number Relevant self-assessment exercises:1 42 Conservation Laws in Integral and Differential Form In most engineering applications, the physical system is governed by a set of conservation laws. Vapor Movement by Diffusion and. In this chapter, we will discuss the effects of V on the. iosrjournals. Read "Lagrangian for the convection-diffusion equation, Mathematical Methods in the Applied Sciences" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. P DS (9) Typical units for P are (cm 3 cm)/(s cm 2 Pa) (those units×10-10 are defined as the barrer, the standard unit of P adopted by ASTM). Alonso,1 A. Numerical solution of nonlinear diffusion equation with convection term by HPM www. Convection above a hot surface occurs because hot air expands, becomes less dense, and rises (see Ideal Gas Law). is transported by means of convection and diffusion through the one-dimensional domain sketched below. Step 4: set the boundary conditions. ! Before attempting to solve the equation, it is useful to understand how the analytical. convective-diffusion equation have been solved rigorously for the steady state. ELMAN † Abstract. 114 THE CONVECTION-DIFFUSION EQUATION characteristic length scale associated with (3. Edited by PatriciaW. The direct contribution of interface deformation, giving rise to concentration variations as a result of local changes in interfacial area, is shown explicitly in a simple manner. Chemical What Is Diffusion? Convection-Diffusion Equation Combining Convection and Diffusion Effects. Learning, knowledge, research, insight: welcome to the world of UBC Library, the second-largest academic research library in Canada. Since this equation is purely convective, it is crucial to introduce some stabilization in the numerical method. The approach is based on the use of Taylor series expansion, up to the fourth order terms, to approximate the derivatives appearing in the 3D convection diffusion equation. Advective Diﬀusion Equation. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 2006 Gilbert Strang 5. Convection-diffusion equations. 8, 2006] In a metal rod with non-uniform temperature, heat (thermal energy) is transferred. The developed scheme is based on a. Sun, Numerical Methods for Partical Difference Equations, (Chinese) Second edition, Science Press, Beijing, 2012. See also related linear equations: •nonhomogeneous diffusion equation , •convective diffusion equation with a source , •diffusion equation with axial symmetry , 1. 4 The Heat Equation and Convection-Diﬀusion The wave equation conserves energy. equation of creeping motion (39) In this regime, viscous interactions have an influence over large distances from an obstacle. x x u KA x u x x KA x u x KA x x x. A double subscript notation is used to specify the stress components. It is done for all conserved variables (momentum, species, energy, etc. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation. Abstract | PDF (3026 KB) (2017) A multilevel Monte Carlo finite difference method for random scalar degenerate convection-diffusion equations. Equation 20. ⃗ is known as the viscous term or the diffusion term. 1 Derivation of the advective diﬀusion equation Before we derive the advective diﬀusion equation, we look at a heuristic description of the eﬀect of advection. The term convection means the movement of molecules within fluids, whereas, diffusion describes the. The concentration C(x, t) is a function of space and time. Mantle convection codes typically deal with advection of a temperature ﬁeld assum-ing that there is signiﬁcant diffusion at the same time, k > 0, and will at times produce. In technological facilities heat is as a rule transferred by two or three ways at a time. Partial Diﬀerential Equations Igor Yanovsky, 2005 2 Disclaimer: This handbook is intended to assist graduate students with qualifying examination preparation. The heat equation ut = uxx dissipates energy. Comprehensive Modeling of the Spatiotemporal Distribution of PET Tracer Uptake in Solid Tumors based on the Convection-Diffusion-Reaction Equation M. The motivation for operator splitting methods lies in that it is easy to combine modern. In many cases,. The 1-D Heat Equation 18. (1) : This approach is particularly convenient for special cases of. The developed scheme is based on a. We establish a notion of stochastic entropy solutions to these equations. This paper extends our previous works on the effects of diffusion with saturation on. 4 The Heat Equation and Convection-Diﬀusion The wave equation conserves energy. The convection-diffusion equation can be derived in a straightforward way from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume:. The convection-diffusion equation is a parabolic partial differential equation, which describes physical phenomena where the energy is transformed inside a physical system due to two processes: convection and diffusion. van 'tHof, J. • understand the behaviour of a diffusion equation • understand the behaviour of a convection-diffusion problem and how it varies with the Peclet number Relevant self-assessment exercises:1 42 Conservation Laws in Integral and Differential Form In most engineering applications, the physical system is governed by a set of conservation laws. 2 The driver code In the driver code we set the direction of gravity and construct our problem, using the newBuoyantQCrouzeixRaviart-Element, a multi-physics element, created by combining the QCrouzeixRaviart Navier-Stokes elements with. As shown in. These laws include those of conservation of mass, momentum, and energy. PARABOLIC APPROXIMATIONS OF THE CONVECTION-DIFFUSION EQUATION J. Diffusion in Polymer Solids and Solutions 21. This paper studies the steady, free convection boundary layer flow about a vertical, isothermal plate embedded in a non-Darcy bidisperse porous medium (BDPM). In this case, u∂c/∂x dominates over D∂ 2c/∂x. Successive Approximation Method for Solving Nonlinear Diffusion Equation with Convection www. Numerical Example and results In this section, we present example of nonlinear diffusion equation with convection term and results will be compared with the exact solutions. Heat diffusion, mass diffusion, and heat radiation are presented separately. (1) : This approach is particularly convenient for special cases of. These layers are usually exponential and. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. The governing equation is: d d d u dx dx dx I UI §· *¨¸ ©¹, and the boundary conditions are prescribed as: 0 1 at 0 L 0 at x xL I I  ® ¯ Using five equally spaced cells and the central differencing scheme for convection and diffusion. These are the books for those you who looking for to read the The Movement Of Molecules Across Cell Membranes, try to read or download Pdf/ePub books and some of authors may have disable the live reading. A NONLOCAL CONVECTION-DIFFUSION EQUATION LIVIU I. Differential equations and uniqueness conditions. The paper deals in its first part with the general formulation of the convective-diffusion. We report here the development of a continuous intestinal absorption model based on the convection-diffusion equation. Whenever we consider mass transport of a dissolved species (solute species) or a component in a gas mixture, concentration gradients will cause diffusion. The paper deals in its first part with the general formulation of the convective-diffusion equation and with the numerical solution of this equation by means of the finite element method. Subramaniam, and A. However, due to the non-linearity in the governing equation, if the spatial step is reduced, the solution can develop shocks, see Figure 2. In this paper, we propose and analyze a Müntz spectral method for a class of two-dimensional space-fractional convection-diffusion equations. OF THE ONE-DIMENSIONAL CONVECTION-DIFFUSION EQUATION MEHDI DEHGHAN Received 20 March 2004 and in revised form 8 July 2004 The numerical solution of convection-diﬀusion transport problems arises in many im-portant applications in science and engineering. , without influence of convection), = 1 and so an exact solution proposal is given in the form, T(z,r) = ez+r and so results in, ̇=− 𝑒𝑧+𝑟 −2𝑒𝑧+𝑟 Considering L = L z. The concentration C ( x, t ) is a function of space and time. Je suis entrain de chercher l'equation exacte ,meme l'algorithme par la methode des difference finis de l'equation conduction-diffusion en regime transitoire avec source de chaleur. The cut finite element method is constructed as follows: (i) The surface is embe. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. In (Juanes and Patzek, 2004), a numerical solution of miscible and immiscible flow in porous media was studied and focus was presented in the case of small diffusion; this turns linear convection-diffusion equation into hyperbolic equation. Numerical solution of nonlinear diffusion equation with convection term by HPM www. is transported by means of convection and diffusion through the one-dimensional domain sketched below. ABSTRACT Based on the numerical criteria/schemes for the evaluation of both the diffusion and convection terms in the convection-diffusion equations presented in the companion article (Part I), a comprehensive numerical study is presented considering eight different test problems and more than 1,800 test cases. The last formula with ﬂ =1, ° =−1, – =‚ =0was obtained with the Appell transformation. The YSU PBL increases boundary layer mixing in the thermally induced free convection regime and decreases it in the. Pollution Problems Based on Convection Diffusion Equation Lingyu Li, Zhe Yin* College of Mathematics and Statistics, Shandong Normal University, Jinan, China Abstract The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. 4 The Heat Equation and Convection-Diﬀusion The wave equation conserves energy. Heat diffusion, mass diffusion, and heat radiation are presented separately. They are “source-type” solutions of the convection-diffusion equation above. Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux. 2 The driver code In the driver code we set the direction of gravity and construct our problem, using the newBuoyantQCrouzeixRaviart-Element, a multi-physics element, created by combining the QCrouzeixRaviart Navier-Stokes elements with. Typical convective heat transfer coefficients for some common fluid flow applications:.


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## Choosing the smoothing parameter

Two popular methods to find the bandwidth $latex {h}&fg=000000$ for the nonparametric density estimator are the plug-in method and the method cross-validation. The first one we will focus in the “quick and dirty” plug-in method introduced by Silverman (1986). In cross-validation we will minimize a modified version of the quadratic risk of $latex {\hat{f}_{h}}&fg=000000$. The …

## Rates of convergence for the MISE in Sobolev classes of densities

The last post I forget to say that we use Mikownski classes of densities because the MISE is a risk corresponding to the $latex {\mathbb L^2({\mathbb R})}&fg=000000$ norm. Thus, it is natural to assume that $latex {p}&fg=000000$ is smooth with respect to this norm. Another way to describe smoothness in $latex {\mathbb L^{2}({\mathbb R})}&fg=000000$ are …

## A global measure of risk for kernel estimators in Nikolski classes

Photos of Sergey Nikolskii from The Russian Academy of Sciences The MSE  gives an error of the estimator $latex {\hat{p}_{n}}&fg=000000$ at an arbitrary point $latex {x_{0}}&fg=000000$, but it is worth to study a global risk for $latex {\hat{p} _{n}}&fg=000000$. The mean integrated squared error (MISE) is an important global measure, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ …

## Kernel density estimation

I will make a summary of ideas about nonparametric estimation, including some basics results to develop more advanced theory later. In the first post  we talk something about the density estimation and the nonparametric regression. Later, in posts about histogram (I,II,III,IV) , we saw how the histogram is a nonparametric estimator and we studied its …

## Density Estimation by Histograms (Part IV)

Today we will apply the ideas of the others post by a simple example. Before, we are going to answer the question of the last week. What is exactly the $latex {h_{opt}}&fg=000000$ if we assume that $latex \displaystyle \displaystyle f(x) = \frac{1}{\sqrt{2\pi}} \text{exp}\left(\frac{-x^2}{2}\right)? &fg=000000$ How $latex {f(x)}&fg=000000$ is the density of standard normal distribution. It is …

## Density Estimation by Histograms (Part III)

Before to continue with today’s post we will answer the question of last week, Is it $latex {\hat{f}_{h}(x)}&fg=000000$ a consistent estimator? The answer is yes. Because convergence in mean squared implies convergence in probability.

## Density Estimation by Histograms (Part II)

We continue our presentation about the estimation of histograms and its statistical properties. Today we will start the theory for reducing the mean squared error. In order to study the statistical properties of $latex {\hat{f}_{h}(x)}&fg=000000$We will start introducing the concept of mean squared error (MSE) or quadratic risk. We define

## Density Estimation by Histograms (Part I)

We are going to introduce the histogram as a simple nonparametric density estimator.  I will divide this presentation in several posts for simplicity reasons. Let us $latex {X_1,\ldots,X_n}&fg=000000$ with pdf $latex {f}&fg=000000$. The histogram is the simplest nonparametric estimator of $latex {f}&fg=000000$.

## Importance of nonparametric statistics in regression.

I would like to start this blog with some basic ideas about density estimation and nonparametric regression. The study of the probability density function (pdf) is called nonparametric estimation. This kind of estimation can serve as a block building in nonparametric regression. The typical regression problem is setting as follows. Assume that we have a …