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04Feb08

For any of a million reasons I recently found myself reading Phys. Rev. D 28 1010, Analysis of anomalies in higher space-time dimensions by Frampton and Kephart. The paper derives the chiral anomaly in $d \geq 4$ spacetime dimensions, and includes some neat polygonal Feynman diagrams such as this:

What really got me thinking, however, was a statement the authors made in the introduction to the paper:

We are very hesitant in invoking more than four dimensions since there is no direct experimental motivation. However, a leading candidate for a finite quantum gravity is proposed by the superstring which necessitates $D=10$. If, as appears to us unlikely, any $N=8$ supergravity should be finite in $D=4$, our motivation to consider $D>4$ would be correspondingly weakened.

Another important conceptual issue in considering $D>4$ is whether the higher dimensions are merely a mathematical device […] or are actual physical dimensions […]

So here’s a paper that’s a bit older than I am. It addresses extra dimensions, even though it was written just shortly after Rubakov and Shaposhnikov’s first papers on the brane-world scenario and a whole fifteen years before the ADD scenario made XD models `cool’. Further, it points to motivation from superstring theory, even though it preceeded the first superstring revolution. As such, the paragraph highlights candid opinions on [what turned out to be] nascent `big ideas’ before they were hits.

(This is `cool’ in roughly the same way that listening to a rock band before they were popular is `cool,’ or the way cheering for a basketball player in the NCAA before they make it big in the NBA is `cool’.)

Here are a few of my own notes (corresponding to my numbering above) with the benefit of 20/20 hindsight:

(1) We are very hesitant in invoking more than four dimensions since there is no direct experimental motivation.

To this day we still don’t have any experimental motivation for higher dimensions, but we do have expected signatures that we’re looking for. These include mini black-holes, the CMB power spectrum, and Kaluza-Klein particles. What’s changed since then? In 1998 the ADD brane-world model linked extra dimensions to the hierarchy problem, which theorists had (by then) been using as a motivation for new physics. It turned out that braneworld theories had a rich model-space for theorists who were tired of playing with SUSY. Even without direct experimental motivation, XD models are now common buzzwords for public outreach and nobody is ashamed to mention them as `motivation’ in a seminar talk.

(2) However, a leading candidate for a finite quantum gravity is proposed by the superstring which necessitates $D=10$.

This paper was written just before the Green-Schwarz-led Superstring Revolution of 1984, so I suppose its authors had good noses for research directions. Today string theory is still the leading candidate for quantum gravity, though nobody in the 1980s could have expected the `string debate’ to ever become so (at times) heated or to spill over into popular culture. All the same time, papers today would have instead written this sentence as, “String theory predicts $D=10$, so …”

(3) If, as appears to us unlikely, any $N=8$ supergravity should be finite in $D=4$, our motivation to consider $D>4$ would be correspondingly weakened.

It wasn’t only a year (or two) ago that the finiteness of $N=8$ SUGRA became a hot topic. I haven’t kept up with developments on this front, but the last I heard of it was that there were some strong indications that $N=8$ SUGRA in $D=4$ is indeed finite. However, unlike what these authors seem to think, this has come off as more of a curiosity rather than a reason to abandon string theory. (I don’t quite understand the details of this, but it seems that SUGRA by itself is not a consistent theory of gravity?)

(4) Another important conceptual issue in considering $D>4$ is whether the higher dimensions are merely a mathematical device […] or are actual physical dimensions […]

Again, I’m stepping out of my domain of comfortable understanding, but this touches on (I think) what is now called deconstruction. (Pioneered by Georgi and friends.) The main idea is that one can play with a theory’s gauge group to make the theory mimic behavior in a higher-dimensional lattice. That is to say that the higher dimensions involved are `mathematical devices’ to explain lower-dimensional behavior. Deconstruction was one of the motivations for Little Higgs models.

#### 4 Responses to “Academic hindsight”

1. I am glad they did not say “String theory predicts D=10, so …”.

🙂

Evan

2. I have been following along with S.W. Hawkings theory of black holes and alternating universes. If he is correct in the assumption that universes with black holes will destroy themselves, and the only universe left will be ones that do not contain black holes then the question is how can a universe not have black holes? In theory black holes are made from dying stars and for there to be universes without blackholes it would only mean they contain one or both either they have a complete void of matter or they are comprised of stars, atoms and other particles that defy the laws of physics as we know it making the alternate universe a place of infinite life and energy that never dies. A utopia, or as in the bible a place called heaven where life is everlasting. The implications of this if proven could be the most important finding in the history of mankind. An actual mathematical eqution that proves there is a heaven! He claims that if a human being would go through a black hole into the event horizon they would be destroyed but they would not feel the effect but pass through and he could not understand the implications of this. I say the reason for this anonomly is because humans have one thing that is everlasting and can never be destroyed and that is a soul!

3. I am by trade a HVAC technician who deals with refrigerant gases. If you know about a term called superheat you would understand that certain gases in mixture together boil at extremly low temperatures. I think these gases may be part of the make-up of comets due to the fact that space is a very cold place and in order for gases to boil and evaporate in space they must be comprised of these refrigerants some with boiling points that range from 40 deg. F to -200 deg. F. Think on that one!

4. 4 Jonathan Powell

It all makes sense now….


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Time-like nucleon form factor measurements at PANDA

Abstract : The electromagnetic probe is an excellent tool to investigate the structure of the nucleon. The nearly 4π detector PANDA, to be installed on the FAIR accelerator complex at Darmstadt, will allow to make a precise determination of the electromagnetic form factors of the proton in the time-like (TL) region with unprecedented precision. In the framework of the one-photon exchange, the center of mass unpolarized differential cross section of the reaction $\overline{p} p \rightarrow e^+ e^-$ is a linear combination of the squared moduli of the electric $G^{p}_E$ and magnetic $G^{p}_M$ proton form factors. The precise measurement of the angular distribution over almost full angular range then directly gives these quantities. Only two experiments have provided the ratio $R = {|G^{p}_E|/|G^{p}_M|}$ but with very large statistical uncertainties. Within PANDA, there is a unique opportunity to measure separately the moduli of these two proton form factors $G^{p}_E$ and $G^{p}_M$ in good conditions, up to around q 2 = 14 GeV2 .
Keywords :
Document type :
Journal articles
Domain :

http://hal.in2p3.fr/in2p3-00442391
Contributor : Sophie Heurteau <>
Submitted on : Thursday, January 7, 2010 - 10:47:37 AM
Last modification on : Wednesday, September 16, 2020 - 4:07:56 PM

Citation

M. Sudol. Time-like nucleon form factor measurements at PANDA. Hyperfine Interactions, Springer Verlag, 2009, 194, pp.241-247. ⟨10.1007/s10751-009-0081-y⟩. ⟨in2p3-00442391⟩

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# Question about the math involved in E-field calculations

• B
Sho Kano
When calculating the electric field from a point above a line of charge using coulomb's law, the integral that comes up is of the form $\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$. But if the point we were asked for is right in the middle, the horizontal (cosine) components cancel out, leaving only the vertical components, so we'd have to throw in a sine term to make the expression correct. Why, if we add the sine and cosine components, the result is not equal to the expression without sine and cosines?
Namely,
$\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } \neq { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } sin\theta \right) }^{ 2 }\quad +\quad { \left( \int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } cos\theta \right) }^{ 2 }$

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## Answers and Replies

Mentor
2021 Award
the integral that comes up is of the form ...
You probably should post your reference for this so we can all be clear. This expression seems too simple to me, so I suspect that they are making some important assumptions.

Sho Kano
You probably should post your reference for this so we can all be clear. This expression seems too simple to me, so I suspect that they are making some important assumptions.
Here is an image I found on the web

The calculation (if I replace d with a, and the charges are taken to be positive ones)
$dE\quad =\quad k\lambda \frac { dx }{ { r }^{ 2 } } \\ E\quad =\quad k\lambda \int _{ -a }^{ a }{ \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } } sin\theta \\ sin\theta \quad =\quad \frac { a }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } }$

Mentor
2021 Award
Ok, so you are assuming the point which is on the line bisecting the charge segment and at a distance equal to half the length of the segment.

Sho Kano
Ok, so you are assuming the point which is on the line bisecting the charge segment and at a distance equal to half the length of the segment.
Oh I see, sorry. What I mean is a = -a, but I didn't mean for the distance of the point away from the line to be a too. It doesn't matter though does it?

Mentor
2021 Award
Oh I see, sorry. What I mean is a = -a, but I didn't mean for the distance of the point away from the line to be a too. It doesn't matter though does it?
Well, if d is different from a then d will need to show up in your expression.

Sho Kano
Ok, here is what I have to make it more clear (according to the picture)
$dE\quad =\quad k\lambda \frac { dx }{ { r }^{ 2 } } \\ E\quad =\quad k\lambda \int _{ -a }^{ a }{ \frac { dx }{ { x }^{ 2 }+{ d }^{ 2 } } } sin\theta \\ sin\theta \quad =\quad \frac { d }{ \sqrt { { x }^{ 2 }+{ d }^{ 2 } } }$

Mentor
2021 Award
Yes, that all looks right. The d is inside the integral and the a shows up only in the limits.

Sho Kano
Yes, that all looks right. The d is inside the integral and the a shows up only in the limits.
I guess my question is: If the sine component measures the vertical contribution, and the cosine measures the horizontal, what does $\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$ do?

pixel
I guess my question is: If the sine component measures the vertical contribution, and the cosine measures the horizontal, what does $\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$ do?

It measures the total contribution. When you do the integral, only the vertical contribution survives.

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Sho Kano
It measures the total contribution. When you do the integral, only the vertical contribution survives.
That integral comes out as $\frac { 1 }{ a } arctan(1)\quad -\quad \frac { 1 }{ a } arctan(-1)\quad =\quad \frac { \pi }{ 2a }$, which is not equal to the one with sine theta factored in

Mentor
2021 Award
I guess my question is: If the sine component measures the vertical contribution, and the cosine measures the horizontal, what does $\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$ do?
The integrand measures the magnitude of the E field from one small piece of the charge, but the integral is not physically meaningful.

If you have two force vectors (3,0) and (0,4) then the resultant force is (3,4). The magnitude of the resultant force is 5, not 7. You can write down the sum of the magnitudes, and you can calculate it, but it isn't useful as far as I know.

Sho Kano
Sho Kano
Thanks, that cleared some things up. That integral is not physically meaningful because the fields are not all in the same direction.

Mentor
2021 Award
Yes, exactly

pixel
I guess my question is: If the sine component measures the vertical contribution, and the cosine measures the horizontal, what does $\int { \frac { dx }{ { x }^{ 2 }+{ a }^{ 2 } } }$ do?
It measures the total contribution. When you do the integral, only the vertical contribution survives.

I see now what you were asking. Yes, the integrand is the magnitude of the E-field, including horizontal and vertical components, from a small element on the line. The integral would be adding those contributions without taking the direction of the components into account. An easy way to see this is If d = 0. We'd expect a zero net field at the point of interest, because of equal contributions from the left and right side, but the integral would give a nonzero result, so not meaningful as already mentioned.


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# Homework Help: Impulse-Diffy eq.

1. Dec 14, 2009

### polarbears

1. The problem statement, all variables and given/known data
$$y''+y=\delta (t-2\pi )cos(t)$$
$$y(0)=0,y'(0)=1$$

2. Relevant equations

3. The attempt at a solution
The left side is (s^2+1)Y(s)-1=RHS

My problem is the fact that cosine is being multiplied by the delta function. I put it in the form of an intergral but I dont know what to do from there.

2. Dec 15, 2009

### HallsofIvy

Well, good! Delta functions usually make integrals trivial. What integral did you get?

3. Dec 15, 2009

### annoymage

hmm, i dont know what is this question, may i know, what topic should i study for this question?

4. Dec 15, 2009

### polarbears

Differential Equations -Laplace transforms

OHHHH waitttt does the delta function just determine the bound of my intergral?


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TL;DR

Here we are with TASK #1 from The Weekly Challenge #147. Enjoy!

# The challenge

Write a script to generate first 20 left-truncatable prime numbers in base 10.

In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading left digit is successively removed, then all resulting numbers are primes.

Example

9137 is one such left-truncatable prime since 9137, 137, 37 and 7
are all prime numbers.

# The questions

I have two questions:

1. are single-digit prime numbers considered left-truncatable? I would say no, because there is no “resulting” number after chopping off the first digit. Anyway, my first intuition about what $0!$ should be failed me, so who knows? This will be my working assumption, though.

2. why stop so low at 20? We’re not even reaching the example!

While we’re at it, a small personal reflection too.

I enjoy these challenges, in their unique shape of being sometimes quite vague and relying heavily on examples. I think our fine host is a fox disguised as a lamb, throwing us puzzles that sometimes give us that distinct wtf?!? moment that often arises when we’re dealing with customers. It’s a stimulus to go beyond, ask questions, gather more requirements, try to see things from different perspectives, make assumptions… in summary, architecting a solution.

Thanks manwar.

# The solution

OK, let’s get this started. I initially thought about a constructive approach, which ended up with a good amount of code and being sufficiently complicated to leave me with a question… is it even correct?

I mean, I know that the constructive approach (more on this in a minute, promised!), but is my implementation right? No clue.

So I also implemented a dumber, brute-force exclusive approach based on iterating over all odd values and filtering stuff out if the required condition does not apply. As a comparison. As a test. As a reference.

You know what? In the end, the original constructive approach was correct since the beginning, while the “simpler” exclusive approach had a bug!

Anyway, I thought of baking the choice between them in the code:

#!/usr/bin/env raku
use v6;

sub MAIN (Int:D $n = 20, :$exclusive = False) {
$exclusive ?? exclusive($n) !! constructive($n); }  Let’s start with the constructive approach. We’re aiming for a nth-left-truncatable($n) function that will give us the n-th number that applies, so we just have to print all values from the first to the twentieth:

sub constructive ($n) { put nth-left-truncatable($_) for 1 .. $n }  As a matter of fact, it’s the underlying function that is constructive. The main insight here is that, for a number to be left-truncatable, the left truncation must either be a single-digit prime number, or it must be left-truncatable itself. Hence, it’s easy to find new candidates for left-truncatable numbers: just start from the ones you have and try adding all possible digits from 1 to 9 in front of them. If we end up with a prime… it’s also left-truncatable. sub nth-left-truncatable ($nth) {
state @cache = (10..99).grep({ .is-prime && .substr(1, 1).is-prime });
state $prefix = 1; state$first-id = 0;
state $next-first-id = @cache.elems; state$id = $first-id; while @cache <$nth { # find moar!
my $candidate = ($prefix ~ @cache[$id++]).Int; @cache.push($candidate) if $candidate.is-prime; if$candidate.chars == @cache[$id].chars { # toppled over! if$prefix < 9 {
++$prefix; } else {$prefix = 1;
($first-id,$next-first-id) = ($next-first-id,$id);
}
$id =$first-id; # just reset the cursor
}
}
return @cache[$nth - 1]; }  The implementation aims at giving something similar to a lazy thing. If we already have what we need in the @cache… well, it’s a good day! Let’s return it and go take an ice cream. Well, if you’re in Australia, anyway, otherwise a hot chocolate will do for me, thanks. Otherwise, we go looking for more lef-truncatable numbers until we have enough to cover the request. To do so, then, we keep some state variables so that we remember where we stopped in our previous search and we can restart from there. The @cache is pre-warmed with all the two-digits left-truncatable numbers. This is necessary because we’re excluding single-digit primes from the lot, so we use the two-digits ones as a starting point. OK, time for the exclusive approach. Here we have a is-left-truncatable($n) test function, which tells us whether a given input $n is indeed left-truncatable or just something else. This is used to print values as we find them, until we hit the limit: sub exclusive (Int:D$n is copy = 20) {
my $i = 9; while$n > 0 {
next unless is-left-truncatable($i =$i + 2);
$i.put; --$n;
}
}

The implementation actually uses the same insight as before: the characteristic of being left-truncatable is inherently recursive, so why not?

sub is-left-truncatable ($n) { return False if$n < 10 || $n ~~ /0/; return False unless$n.is-prime;
state %cache;
if %cache{$n}:!exists { my$truncated = $n.substr(1); return$truncated.is-prime if $truncated < 10; %cache{$n} = is-left-truncatable($truncated); } return %cache{$n};
}

Just as we’re at it, I decided to throw some %cache just because memoization is so cool. Totally overkill and unneeded in this case, isn’t overengineering so funny?!?

For the Perl counterpart I decided to go only with the constructive alternative. At the end of the day, it’s the one I personally like more, and I already had plenty of references to compare to.

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

say nth_left_truncatable($_) for 1 .. shift // 20; sub nth_left_truncatable ($nth) {
state $cache = [ grep { is_prime($_) && is_prime(substr $_, 1) } 10 .. 99 ]; state$prefix = 1;
state $first_id = 0; state$next_first_id = $cache->@*; state$id = $first_id; while ($cache->@* < $nth) { my$candidate = $prefix .$cache->[$id++]; push$cache->@*, $candidate if is_prime($candidate);
if (length($candidate) == length($cache->[$id])) { # toppled over! if ($prefix < 9) {
++$prefix; } else {$prefix = 1;
($first_id,$next_first_id) = ($next_first_id,$id);
}
$id =$first_id; # just reset the cursor
}
}
return $cache->[$nth - 1];
}

sub is_prime { # https://en.wikipedia.org/wiki/Primality_test
return if $_[0] < 2; return 1 if$_[0] <= 3;
return unless ($_[0] % 2) && ($_[0] % 3);
for (my $i = 6 - 1;$i * $i <=$_[0]; $i += 6) { return unless ($_[0] % $i) && ($_[0] % (\$i + 2));
}
return 1;
}

Alas, we lack a primality test in the language here, so we have to code one. This is taken from Wikipedia, there’s probably better stuff around but this does the job in decent time for our purposes.

OK, enough for this post… stay safe everybody!


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# Second countability and products of Borel $\sigma$-algebras

We know that the Borel $\sigma$-algebra of the Cartesian product space (with the product topology) of two topological spaces is equal to the product of the Borel $\sigma$-algebras of the factor spaces. (The product $\sigma$-algebra can be defined via pullbacks of projection maps...)

When one upgrades the above statement to a product of a countable family of topological spaces, the analagous result, namely that the Borel $\sigma$-algebra is the the product Borel $\sigma$-algebra, is conditioned by the topological spaces being second countable. Why?

My question is this: how and why does second countability make its appearance when we upgrade the finite product to countable product? (Second countability means that the base is countable, not just locally...)

My difficulty is that I do not see how suddenly second countability is important when we pass from finite to countable products.

-


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# Schedule for: 17w5099 - Mathematical Analysis of Biological Interaction Networks

Arriving in Banff, Alberta on Sunday, June 4 and departing Friday June 9, 2017
Sunday, June 4
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (Corbett Hall Lounge (CH 2110))
Monday, June 5
07:00 - 08:45 Breakfast
Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
08:45 - 09:00 Introduction and Welcome by BIRS Station Manager (TCPL 201)
09:00 - 09:30 David Anderson: Overview talk (TCPL 201)
09:30 - 10:15 Gheorghe Craciun: Overview talk (TCPL 201)
10:15 - 11:00 Coffee Break (TCPL Foyer)
11:00 - 11:30 Daniele Cappelletti: Stochastically Modeled Reaction Networks with Absolute Concentration Robustness
Deterministic reaction networks equipped with a kinetics are termed absolute concentration robust (ACR) if one or more species have the same value at any positive steady state of the system. This property is particular interesting from a biological point of view, since it implies that the expression of certain molecules at a stationary regime does not change under modification of the total mass of the system. Structural conditions implying absolute concentration robustness have been found for mass action models. Surprisingly, under the same structural conditions it has been shown that the associated stochastically modeled systems undergo an extinction event with probability 1. This leads to a discrepancy between the deterministic model, where some species have always the same value at equilibrium, and the stochastic model, where the same species eventually absorb the total mass of the system. I will present a result which solves the discrepancy between the two models at typical time frames. The result implies that, under certain conditions, the averages of the ACR species counts tend to their ACR equilibria. Specifically, this holds up to any fixed time point, when the total number of molecules tends to infinity in an appropriate multiscale fashion. Finally, I will show by examples that absolute concentration robustness does not necessarily imply an extinction event for the stochastically modeled system, contrary to what could be believed.
(TCPL 201)
11:30 - 12:00 Badal Joshi: Graphically balanced equilibria and stationary measures of reaction networks
The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models. Correspondingly, in stochastic models these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz identified stationary distributions of a complex balanced network; meanwhile Cappelletti and Wiuf developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure and reaction vector balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. We develop the idea of decomposing both deterministic and stochastic systems into so-called factor systems'' and we establish the correspondence between factors of a complex balanced deterministic system and those of the corresponding stochastic system. Furthermore, in spirit of earlier work by Joshi, we give additional conditions for when detailed balance of Markov chain theory implies detailed balance of reaction network theory. This is joint work with Daniele Cappelletti.
(TCPL 201)
12:00 - 13:00 Lunch (Vistas Dining Room)
13:00 - 14:00 Guided Tour of The Banff Centre
Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.
(Corbett Hall Lounge (CH 2110))
14:00 - 14:15 Group Photo
Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!
(TCPL Foyer)
14:15 - 14:30 Coffee Break (TCPL Foyer)
14:30 - 15:15 Alicia Dickenstein: Overview talk - Algebraic tools in the study of reaction systems
I will highlight different algebraic tools for the analysis of reaction networks taken with mass-action kinetics. I will briefly describe some joint recent results with Mercedes Pérez Millán about structural properties of many common biochemical networks. I will also briefly introduce ongoing results joint with Frédéric Bihan and Magalí Giaroli on new conditions to describe multistationarity parameters, based on ideas from real algebraic geometry.
(TCPL 201)
15:15 - 16:00 Anne Shiu: Overview talk - Algebraic methods for analyzing bistability and oscillations in reaction systems
This talk highlights algebraic techniques for analyzing reaction networks taken with mass-action kinetics. Specifically, we describe how steady-state parametrizations arising from elimination (usually of intermediates) can be harnessed to analyze bistability and oscillations. These techniques allow us to investigate the dynamics of certain biological signaling networks, namely, multisite phosphorylation networks: the emergence of bistability in a network underlying ERK regulation, and the capacity for oscillations in a mixed-mechanism processive/distributive phosphorylation network.
(TCPL 201)
16:00 - 18:00 Breakout session (TCPL 201)
18:00 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
19:30 - 20:30 Open problem session (TCPL 201)
Tuesday, June 6
07:00 - 08:30 Breakfast (Vistas Dining Room)
08:30 - 09:15 Anne Condon: Overview talk - An introduction to molecular programming with Stochastic CRNs
Stochastic Chemical Reaction Networks (CRNs) can be viewed as programs whose instructions are reactions; these instructions execute asynchronously and in parallel to produce a number of output molecules that is a function of the initial counts of molecular species in a well-mixed solution. For example, the simple program "X + Y --> Z + Z", executing in a solution that initially contains copies of two molecular species X and Y, eventually produces a number of copies of molecule Z that is exactly twice the minimum of the initial counts of X and Y, thereby computing 2min{#X,#Y}. How fast does this program run, as a function of the initial species counts (assuming fixed conditions such as volume)? Are there faster programs that produce the same output? More generally, what can and cannot be computed by CRN programs? These questions are attracting much attention in light of significant success in "compiling" CRN programs into real molecular controllers that can sense and respond to conditions in a chemical environment. A beautiful emerging theory of computing with CRNs is providing sharp answers to such questions. The theory and underlying computing models have their roots partly in distributed computing, where population protocols and Petri nets - essentially CRNs in disguise - shed light on the computing power of massively parallel systems of distributed computing agents, interacting asynchronously. In this talk I'll introduce some stochastic CRN computing models, as well as results on their computational power that are due to Angluin, Aspnes, Doty, Soloveichik and others, along with open questions and directions for future work.
(TCPL 201)
09:15 - 10:00 Erik Winfree: Overview talk (TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:15 Yiannis Kaznessis: Overview talk - Closure Scheme for Chemical Master Equations
(TCPL 201)
11:15 - 12:00 Eduardo Sontag: Overview talk - Dynamic response phenotypes in systems biology: Scale-invariance and monotone I/O systems
Among the central questions in systems biology are those of understanding the roles of, and interactions among, signal transduction pathways and feedback loops. This talk focuses on “dynamic phenotypes” characterized by input/output responses to external inputs in addressing such issues, using fold-change detection and monotone architectures as case studies. An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled. This talk will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic nondegeneracy conditions. The theorem allows one to make experimentally testable predictions, and the presentation will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. The talk will also include some speculative remarks about the role of the shape of transient responses in immune system self/other recognition and in cancer immunotherapy, as well as a brief discussion of how control-theoretic structures such as differential positivity (monotonicity) have been experimentally employed together with experimental data in order to elucidating mechanisms for stress responses and chemosensing.
(TCPL 201)
12:00 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:30 Breakout session (TCPL 201)
14:30 - 15:00 Coffee Break (TCPL Foyer)
15:00 - 16:00 Breakout session (TCPL 201)
16:00 - 16:30 Robert Brijder: Sufficient Conditions for the Eventual Dying of Reactions in Discrete Chemical Reaction Networks
We consider chemical reaction networks operating on discrete state spaces, i.e., where we keep track of molecule counts. We give a sufficient syntactic condition on chemical reaction networks for the impossibility of certain reactions to take place in the long term. This result is a statement about the reachability relation and is therefore independent of stochastics. As such, it can equivalently be formulated in terms of Petri nets, which is a well-studied model of concurrency.
(TCPL 201)
16:30 - 17:00 Robert Johnson: Formal Verification of Chemical Reaction Network Equivalence: A Bisimulation Approach
The Chemical Reaction Network (CRN) model is a language designed to describe the behavior of chemical or biological molecules. Determining whether, in a given semantics, two CRNs have the same behavior is an interesting problem both in itself and for its uses in practice. Such practical uses that have been demonstrated include understanding biological systems by comparison to simple, well-understood CRNs, and verifying that physical implementations of abstract CRNs correctly implement their intended specifications. We defined a concept of CRN equivalence based on bisimulation as explored in concurrent systems, and explored its implications for CRNs in the low-copy-number semantics. We then explored algorithms to check whether two CRNs satisfy this concept of equivalence, and the computational complexity of that task. I will present this definition and our results, and place them in context with other concepts and methods to check CRN equivalence. I will also touch on the uses of this area of theory in practical molecular programming.
(TCPL 201)
17:00 - 17:30 Nicolette Meshkat: Using algebraic matroids and avoiding differential algebra in identifiability, observability, and indistinguishability
Algebraic matroids can be used to determine all the algebraic dependency relationships among a set of polynomials, without actually calculating those corresponding polynomial relationships. I'll discuss the application of algebraic matroids in three areas of model analysis: identifiability, observability, and indistinguishability. We'll see that algebraic matroids can be particularly useful in the areas of observability and indistinguishability, especially for large nonlinear models.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
19:30 - 20:30 Open problem session (TCPL 201)
Wednesday, June 7
07:00 - 08:30 Breakfast (Vistas Dining Room)
08:30 - 09:00 Atsushi Mochizuki: Observation and Control of Complex Nonlinear Systems Based on Network Structures
Modern biology provides large networks describing regulations between bio-molecules. It is widely believed that dynamics of molecular activities based on such regulatory networks are the origin of biological functions. However, the information on the networks is not sufficient to determine the resulting dynamics. In this study we present a mathematical theory to provide an important aspect of dynamics from information of regulatory linkages alone. We show that feedback vertex set (FVS) of a regulatory network is a set of determining nodes of dynamics [1]. It assures that i) any long-term dynamical behavior of whole system, such as steady states, periodic oscillations or quasi-periodic oscillations, can be identified by measurements of a subset of molecules in the network, and that ii) the subset is determined from the regulatory linkage alone. The theory also claims that iii) dynamical behavior of whole system can be switched from one attractor to others just by controlling dynamics of FVS. We applied our theory to a real biological network to verify our prediction by collaborating with experimental biologists [2]. We analyzed a regulatory network of ascidian embryo including 90 genes, and responsible for cell differentiation in the early development. We identified five genes in minimum FVS of the network. The exhaustive artificial activation/inhibition of five genes showed that the system could be controlled just by controlling activities of five genes in FVS. [1] Mochizuki, A., Fiedler, B. et al. J. Theor. Biol., 335, 130-146. [2] Kobayashi, K., Maeda, K., Tokuoka, M., Mochizuki, A. and Satou, Y. In Review.
(TCPL 201)
09:00 - 09:30 Ankit Gupta: Numerical estimation of the stationary solution of the chemical master equation
The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the Markov chain that represents the stochastic reaction dynamics. Often this state-space is infinite in size, making it difficult to obtain the exact solutions of the CME. Hence these solutions need to be estimated by stochastic simulations or by solving an approximate CME over a finite truncated state-space. In many applications, one is interested in finding the stationary solution of the CME which corresponds to the stationary probability distribution of the underlying Markov chain. When the state-space is infinite, this stationary distribution satisfies an infinite-dimensional linear-algebraic system which cannot be directly solved. In this talk we will discuss how the stationary distribution can be accurately estimated by solving a finite linear-algebraic system that provides the stationary distribution of a suitably constructed Markov chain over a truncated state-space. We will argue that under certain stability conditions, like the irreducibility of the infinite state-space and exponential ergodicity of the reaction dynamics, the obtained approximations are guaranteed to converge to the exact stationary solution as the truncated state-space expands to the full state-space. We will explain how these stability conditions can be computationally checked for reaction networks and also describe how the approximation errors can be quantified. The presented ideas will be illustrated through many examples.
(TCPL 201)
09:30 - 10:00 Matthew Johnston: Network Translation and Absolute Concentration Robustness
The steady state property of absolute concentration robustness (ACR) has recently gained attention in the literature. A species is said to be ACR if its attains the same value at all positive steady states, i.e. the steady state value does not depend upon initial concentrations. This mathematical property suggests evolved structure in biochemical reaction networks which is capable of maintaining narrow ranges for certain reactants in the face of fluctuating external circumstances. In this talk, I will introduce a new method for establishing ACR in mass action systems which utilizes the recently developed theory of network translation. In addition to expanding the scope of systems which can be guaranteed to permit ACR, our method typically allows the steady state ACR value to be computed directly from a reaction network known as a translated chemical reaction network.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:00 Heinz Koeppl: Biochemical networks in random environments: modeling and inference
Recent measurements of cellular processes on the single-cell level revealed significant cell-to-cell variability even for cells that are genetically identical and share the same growth conditions. The magnitude of this variation goes beyond what can be expected just based on the stochasticity of the specific process under study. The remaining source of variation is believed to stem from the random environment in which the considered cellular process is embedded. In this talk, I will present our efforts to model biochemical networks embedded in random environments and give the corresponding master equations for them. I will also touch upon the problem of statistical inference for such models and show how they can then be used to dissect and quantify the sources of cell-to-cell variability based on measured single-cell data.
(TCPL 201)
11:00 - 11:30 Mercedes Perez Millan: Checking multistationarity in MESSI systems
We introduce a method for simplifying the study of the steady states of biological systems that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, which we call MESSI systems. By explicit elimination of intermediate complexes, we define an important subclass of MESSI systems with toric steady states. For these systems we describe an algorithm that determines whether the system has the capacity for multistationarity, and when it does, it shows two positive steady states and reaction rate constants that witness multistationarity. We illustrate these results in the two-layer phosphorylation cascade system. This is joint work with Alicia Dickenstein.
(TCPL 201)
11:30 - 12:00 Lea Popovic: Rare events calculations for chemical reaction systems
This talk is an attempt to give a brief overview of the theory for probabilities of rare events in chemical reaction networks. I will consider what the existing theory for large deviations of Markov processes enables us to say about stochastic models for CRNs. For these Markov chain models we can describe the expected behaviour of their paths and their variability over time, even for models that involve multiple-scales. Large deviation theory further describes the events not captured by such law of large number or central limit theorem results, as they have exponentially small probabilities in terms of the scaling parameter. For models on multiple scales this theory becomes technically much more challenging, but some calculations can be made. These results can also be very useful in designing efficient algorithms for simulating rare events in CRNs.
(TCPL 201)
12:00 - 13:30 Lunch (Vistas Dining Room)
13:30 - 17:30 Free Afternoon (Banff National Park)
17:30 - 19:30 Dinner (Vistas Dining Room)
Thursday, June 8
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:30 Grzegorz Rempala: Law of large numbers for the SIR process on a random graph
Stochastic SIR-type epidemic processes on random graphs are a special class of interaction networks that have become of interest lately for modeling contact-type epidemics (Ebola, HIV, etc). I will discuss a particular case of the SIR epidemic evolving on a configuration model random graph with given degree distribution. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based SIR process and is useful in building a "network-free" SIR Markov hybrid model for epidemic parameters inference.
(TCPL 201)
09:30 - 10:00 Alan Rendall: Multiple steady states in models for the Calvin cycle of photosynthesis
In the literature there is a variety of models for the Calvin cycle of photosynthesis and a number of statements, based on computer calculations, about the number and stability of their steady states. Here I will discuss some progress in obtaining rigorous results on this issue. It has been shown that in a simple model with five unknowns there are parameters for which there are two positive steady states, one of them stable, and a special case where there is a continuum of steady states. A more detailed model, due to Pettersson and Ryde-Pettersson, with fifteen unknowns turns out to be a system of differential-algebraic equations (DAE), rather than a system of ODE. It has been shown that for certain parameters it has two positive steady states. The proof is related to the method of elementary flux modes. For a closely related model, due to Poolman, there are three positive steady states. The difference between the numbers of steady states in these two models is related to the biological phenomenon of overload breakdown, in which too much demand for sugar causes the system supplying it to collapse.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:00 Carsten Conradi: Multistationarity in biochemical reaction networks
Multistationarity has been recognized as an important feature of dynamical systems originating in Biology. As a consequence of the usually very high parameter uncertainty numerical analysis often fails to establish multistationarity. Hence techniques allowing the analytic computation of parameter values where a given system exhibits multistationarity are desirable. I present conditions in form of polynomial systems that allow to determine parameter vectors where a mass action system exhibits multistationarity.
(TCPL 201)
11:00 - 11:30 Stefan Mueller: Sign conditions in chemical reaction network theory
We give an overview of the use of sign conditions in chemical reaction network theory (in several collaborations). To begin with, we study chemical reaction networks with generalized mass-action kinetics (power-law kinetics). Existence and uniqueness of complex-balancing equilibria (in every stoichiometric class) are determined by sign vectors of the stoichiometric and kinetic-order subspaces. In fact, existence and uniqueness correspond to surjectivity and injectivity of generalized polynomial maps (with full rank), and our results can be interpreted in terms of real algebraic geometry (multivariate Descartes rule of signs). The right hand side of a generalized mass-action system is also a generalized polynomial map (not necessarily with full rank), and its injectivity can be characterized in terms of sign conditions. Clearly, injectivity is sufficient to preclude multiple equilibria. We extend this approach to even more general kinetics (e.g. monotone kinetics). Moreover, (local and global) stability of complex-balancing equilibria (in Lotka-Volterra schemes or planar S-systems) can be characterized in terms of signs of Jacobian matrices. For these systems, we also study limit cycles and the related center problem. Finally, we develop algorithms to compute sign vector conditions (for large systems) and to enumerate sign vectors of minimal and maximal support, respectively.
(TCPL 201)
11:30 - 12:00 Maya Mincheva: Interplay between diffusion and delay
A common feature of pattern formation in both space and time is the destabilization of a stable equilibrium solution of an ordinary differential equation by adding diffusion or delay, or both. Here we study linear stability of general reaction-diffusion systems with off-diagonal time delays. We show that a delay-stable system cannot be destabilized by diffusion, and that a diffusion stable system is also stable with respect to delay, if the diffusion is sufficiently fast. A system with direct negative feedback which is strongly stable with respect to diffusion can be destabilized by off-diagonal delay. This is joint work with Peter Hinow, University of Wisconsin-Milwaukee.
(TCPL 201)
12:00 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:30 Breakout session (TCPL 201)
14:30 - 15:00 Coffee Break (TCPL Foyer)
15:00 - 16:00 Breakout session (TCPL 201)
16:00 - 16:30 Casian Pantea: Inheritance of bistability in mass action reaction networks
We focus on the question of bistability, or existence of multiple (stable) positive equilibria, a dynamical property that underlies important cellular processes, and a recurring theme in recent work on reaction networks. Namely, we consider the question: "when can we conclude that a network admits multiple stable positive equilibria based on analysis of its subnetworks?” We identify a number of operations on reaction networks that preserve bistability as we build up the network, and we illustrate the power of this approach on the much-studied Huang-Ferrell MAPK cascade. This work can be thought of as a step towards a theory of “motifs”, a central theme in systems biology.
(TCPL 201)
16:30 - 17:00 Jinsu Kim Kim: Lyapunov functions and Tiers for ergodicity and mixing times of stochastic reaction networks
The Lyapunov-Foster criteria is one of the well known methods for verifying ergodicity of Markov processes. The "Standard Lyapunov function” has played a significant role in the analysis of deterministic reaction networks, especially in regards to deficiency 0 networks. I will provide conditions on reaction networks that guarantee the Standard Lyapunov function satisfies the Lyapunov-Foster criteria. The analysis utilizes hierarchies of intensities and rate functions (Tier structures). I will also provide some special classes of reaction networks satisfying the conditions, thus proving networks in those classes are positive recurrent. I will also provide some results pertaining to the mixing times of reaction networks.
(TCPL 201)
17:00 - 17:30 Jim Brunner: Robust permanence of polynomial dynamical systems
A “permanent” dynamical system is one whose positive solutions stay bounded away from zero and infinity. The permanence property has important applications in biochemistry, cell biology, and ecology. Inspired by reaction network theory, we define a class of polynomial dynamical systems called tropically endotactic. We show that these polynomial dynamical systems are permanent, irrespective to the values of (possibly time-dependent) parameters in these systems. These results generalize the permanence of 2D reversible and weakly reversible mass-action systems.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Friday, June 9
07:00 - 08:45 Breakfast (Vistas Dining Room)
08:45 - 10:00 Open time / talks (TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:30 Open time / talks (TCPL 201)
11:30 - 12:00 Checkout by Noon
5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.
(Front Desk - Professional Development Centre)
12:00 - 13:30 Lunch from 11:30 to 13:30 (Vistas Dining Room)


text:
# proving continuity

Show 40 post(s) from this thread on one page
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• Dec 1st 2010, 07:37 AM
alexandros
proving continuity
given the function:

f(x,y)= $\frac{xy}{x+y}$ if $(x,y)\neq 0$

f(x,y) =0 if (x,y)=(0,0)

prove whether it is continuous or discintinuous at (0,0) by using the ε-δ definition for continuity
• Dec 1st 2010, 07:52 AM
chisigma
The limit...

$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x y}{x+y}$

... doesn't exist so that the function is discontinous in $(0,0)$...

Kind regards

$\chi$ $\sigma$
• Dec 1st 2010, 08:19 AM
alexandros
Quote:

Originally Posted by chi sigma
The limit...

$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x y}{x+y}$

... doesn't exist so that the function is discontinuous in $(0,0)$...

Kind regards

$\chi$ $\sigma$

Thank you, but if you read my post i asked for a proof of that discontinuity by using ε-δ definition for discontinuity
• Dec 1st 2010, 01:21 PM
chisigma
Quote:

Originally Posted by alexandros
Thank you, but if you read my post i asked for a proof of that discontinuity by using ε-δ definition for discontinuity

The problem is that the so called ' $\varepsilon-\delta$ proof' is adequate to demostrate the continuity and not the discontinuity of a function... in this particular case the limit...

$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x y}{x+y}$ (1)

... is 'almost always' 0... almost because when is $x+y=0$ the (1) is not defined...

Kind regards

$\chi$ $\sigma$
• Dec 1st 2010, 03:09 PM
alexandros
How,then do we otherwise prove that the function is discontinuous at (0,0)
• Dec 1st 2010, 03:35 PM
nerdo
Quote:

Originally Posted by alexandros
How,then do we otherwise prove that the function is discontinuous at (0,0)

Here is away:

Choose (Xn,Yn)=(1/n,1/n) then (Xn,Yn)-> (0,0) but F(Xn,Yn)=1/2 so the function is discontinuous at (0,0)
• Dec 1st 2010, 03:37 PM
Plato
Quote:

Originally Posted by alexandros
How,then do we otherwise prove that the function is discontinuous at (0,0)

Can you show that $\displaystyle\lim _{(x,y) \to (0,0)} \frac{{xy}}{{x + y}}$ does not exist?

In order for $f(x,y)$ to be continuous at $(0,0)$ that limit must exist.
• Dec 1st 2010, 03:40 PM
nerdo
Quote:

Originally Posted by nerdo
Here is away:

Choose (Xn,Yn)=(1/n,1/n) then (Xn,Yn)-> (0,0) but F(Xn,Yn)=1/2 so the function is discontinuous at (0,0)

I think this is by far the best way to show that this discontinuous
• Dec 1st 2010, 03:42 PM
Plato
Quote:

Originally Posted by nerdo
I think this is by far the best way to show that this discontinuous

Why do you this there is such a notion as there is best way?
• Dec 1st 2010, 03:52 PM
nerdo
Quote:

Originally Posted by Plato
Why do you this there is such a notion as there is best way?

I think personally if one used the theorem below, the question is trivial:

Attachment 19924
• Dec 1st 2010, 03:55 PM
Plato
Quote:

Originally Posted by nerdo
I think personally if one used the theorem below, the question is trivial:

But using a theorem does not mean the learner understands to basic concept.
A proof is a proof is a proof.
• Dec 1st 2010, 03:59 PM
nerdo
Oh ok i see what you mean, i guess you are right.
• Dec 1st 2010, 04:32 PM
alexandros
Quote:

Originally Posted by nerdo
Here is away:

Choose (Xn,Yn)=(1/n,1/n) then (Xn,Yn)-> (0,0) but F(Xn,Yn)=1/2 so the function is discontinuous at (0,0)

But F(Xn,Yn)= 1/2n and not 1/2 ??
• Dec 1st 2010, 05:12 PM
alexandros
Quote:

Originally Posted by Plato
Can you show that $\displaystyle\lim _{(x,y) \to (0,0)} \frac{{xy}}{{x + y}}$ does not exist?

In order for $f(x,y)$ to be continuous at $(0,0)$ that limit must exist.

To show that : $\displaystyle\lim _{(x,y) \to (0,0)} \frac{{xy}}{{x + y}}$ does not exist?

We must show that: for all real Nos m there exists an ε>0 such for all δ>0 there exists a pair (x,y) such that : |x|<δ and |y|<δ and $|\frac{xy}{x+y}-m|\geq\epsilon$

However to show discontinuity at (0,0) we must show that:

There exists ε>0 such that for all δ>0 there exists (x,y) such that:|x|<δ and |y|<δ and $|\frac{xy}{x+y}|\geq\epsilon$

Two comletely different things
• Dec 2nd 2010, 12:29 PM
SammyS
proving continuity
Quote:

Originally Posted by alexandros
given the function:

f(x,y)= $\frac{xy}{x+y}$ if $(x,y)\neq 0$

f(x,y) =0 if (x,y)=(0,0)

prove whether it is continuous or discontinuous at (0,0) by using the ε-δ definition for continuity

I realize that you need an ε-δ
proof, but first let's find a sequence that shows that $f$ is discontinuous at (0,0). This may help with your ε-δ proof.

The problem with
$f(x,y)$ is that it is undefined along the line $y=-x$, except at (0,0). Let's make a sequence $(x_n, y_n)$ which approaches (0,0) along a path that approaches $y=-x$. One such path is $y=-x+ax^2$. Define $\displaystyle x_n = {1\over n}$, then $\displaystyle y_n=-{1\over n}+{a\over {n^2}}$, so $\displaystyle f(x_n,y_n)= {{\left({1\over n}\right)\left(-{1\over n}+{a\over {n^2}}\right)} \over{\left({1\over n}\right)+\left(-{1\over n}+{a\over {n^2}}\right)}}={{-{1\over {n^2}}+{a\over {n^3}}} \over{a\over {n^2}}} = -{1\over a}+ {1\over n}\to -{1\over a} \text{ as } n \to \infty$.

Since
$f(0,0)=0$, we have that $f$ is not continuous at (0,0).

Rather than showing that : $\displaystyle\lim _{(x,y) \to (0,0)} \frac{{xy}}{{x + y}}$ does not exist, just show that it's not 0. Modify your statement to have m=0.
Given any $\delta>0$, let $x={\delta\over2}$ (may have to tweak this a little). Let $y=-x-x^2=-{\delta\over2}-{{\delta^2}\over{4}}$.
Then $\displaystyle f(x,y)= {{\left({\delta\over 2}\right)\left(-{\delta\over2}-{{\delta^2}\over{4}}\right)} \over{\left({\delta\over 2}\right)+\left(-{\delta\over2}-{{\delta^2}\over{4}}\right)}}=1+{\delta\over 2}$


text:
## GM Expands E85 Effort into Pennsylvania

##### 10 July 2006

General Motors announced that it has facilitated plans for the addition of three new E85 fueling sites in the Pittsburgh area, through a collaborative partnership with VeraSun Energy Corporation, Sheetz, Inc. and the State of Pennsylvania.

GM has been the catalyst behind several similar collaborative partnerships across the country and began an initiative with VeraSun in February as part of a broader, national campaign to boost the use and awareness of E85—85% ethanol and 15% gasoline—in the United States.

VeraSun Energy, the nation’s second-largest ethanol producer, will supply its VE85 brand of E85 to select Sheetz stores in the Pittsburgh area beginning this week and expanding to two additional stores by later this year.

The announcement marks the fourth collaboration in four states that GM and VeraSun have entered with a third-party retailer since the companies began working together to promote awareness of E85 ethanol-based fuel. The first initiative, launched last May in Sioux Falls, S.D., promoted VE85 pumps in seven prominent local stations. In the Chicago metro area, what began as a 20-location effort in February has already expanded to include 32 sites, and the initial four outlets in Minneapolis-St. Paul have grown to 16 in the last four months.

General Motors says that it will continue to spend significant advertising and promotion dollars to help generate awareness of E85 flexible fuel vehicles—and where consumers can purchase E85 ethanol—with consumer marketing initiatives and dealer outreach.

Since May 2005, GM has announced partnerships with states and fuel providers around the country to locate, to date, over 160 new E85 ethanol fueling locations.

I cannot help but wonder what the general public makes of this. Do they honestly think that they will save money with E85? A gas guzzler is still a gas guzzler on E85.

I predict that this push by GM for flex-fuel vehicles will only last a year or two. Maybe GM will be in chapter 11 by then.

I cannot help but wonder what the general public makes of this. Do they honestly think that they will save money with E85? A gas guzzler is still a gas guzzler on E85.

People choose their car every 3 to 10 years. People choose their petrol every 3 to 10 days. If they can get more miles per dollar on E85 than on gasoline -- regardless of just how many miles that is -- then they will choose to do so. Furthermore, shouldn't we be encouraging them to do just that?

The amount of oil necessary to move a car running E85 1 mile is less than the amount of oil necessary to move a car running gasoline 1 mile. Not by much, mind you -- but by a little. Still, anything Americans do to consume less oil -- even by a little -- is a good thing. It's good for domestic policy, foreign policy, environmental policy, and economic policy.

If GM's actions tickle more E85 supply and more E85 demand, then they've done something that benefits tUSA.

I cannot understand all of this push towards "environmentally friendly" E-85 fuel.
EVERYONE knows that it radically increases the fuel consumption of any vehicle it's put it. For example, the E.P.A. has stated that a Nissan 4X4 Titan truck which normally gets around 14 m.p.g. city and 18 m.p.g.highway will see the former drop to around 10 m.p.g. and the latter to 13 m.p.g.

Consider a gallon of E85. That 0.15 gallons of ethanol content requires the energy of 0.3 gallons of gasoline to make it! Next, if this vehicle is driven on the highway it consumes 38% more fuel!
1) Net conventional gasoline input: 1 gallon
2) Net gasoline input for E85:
1.38(0.85 gal gasoline+ 2(.15))= 1.587 gallons of
gasoline consumed!

Hey greenies: Put regular gas in your car and you'll burn 37% less gasoline overall. Isn't that going to cut down on all of that evil carbon dioxide you keep complaining about?

The 16 to 18 year old A.P. Physics students I teach can do this sort of research and math pretty easily. Why is it our so-called energy experts who work for the Government cannot see this; why is there such a poor grasp of the Laws of Thermodynamics?

I think 2 words pretty much sum up our national collective knowledge in this area:

Imperial Rome!

Dave, E85 is 85 percent alcohol. Yes, gets much worse mileage.

The pols don't do thermodynamics, they do campaign contributions and votes.  Ethanol gets lots of farm votes and ADM dollars.

In politics, money talks, thermo walks.  If only the Constitution allowed laws and regulations to be challenged in court if the findings of "fact" beind them were wrong...

Dave,

There's also the geopolitical issue to consider. It's not only about fuel economy and economics, though those are very important considerations. Reducing the amount of money we send to Middle East nations who would rather see us dead is a good idea. I'm not saying we should pursue ethanol at all costs as a result, but that even if it's slightly suboptimal in terms of economy, there are other positive reasons for pursuing alternatives to oil-based gasoline.

I agree that right now the economics of ethanol don't make sense even in the era of $70 oil, but economies of scale could change that, both in terms of E85 production and of engines that are better suited to burning it. And unfortunately we can't just turn on a switch and suddenly have all the infrastructure in place as soon as economics favor E85 (if that happens in the future), so some of it needs to be built out ahead of time. heres some simple math for you guys... gas vehicle gets 20 mpg and on E85 gets 14.6 (27% lower) 20 mpg * 110000 btu/gal gasoline = 5500 btu/mile 14.6 mpg e85 * 78000 btu/gal e85 = 5300 btu/mile result e85 take the same heat per mile as gas... do your homework and see through the BS and misinformation. Dave. While I think Ethanol is a bad idea, too, where did you get your EROIE of (.5). Plus, E85 requires .85 gallon of ethanol, not (.15). rfh. What is the relevance of btu/mile in this discussion. The fact remains that ethanol has a much lower energy content than gasoline. If ethanol got even worse miles per gallon and had an even lower btu content, you would still end up with comparable btus per mile. You can take this scenario all the way down to 1 mile per gallon. Dave: 1. As Bud points out, a gallon of E85 has 0.85 gallons of ethanol in it, not 0.15. 2. Ethanol does not require 2x its volume in gasoline to produce. A statement to that effect is a simple untruth. The amount of energy required to produce a gallon of finished ethanol depends on the feedstock and exact production method used. Even in current US production, the economics of the situation make clear that large amounts of gasoline are not used. Ethanol currently trades, on the CBOT spot market, at$3.50 per gallon. (FN1). Gasoline trades on its NYMEX spot market at around $2.20 per gallon. Other major petroleum based fuels trade at a similar$2/gal range. If ethanol really required two gallons of gasoline to produce, it would cost upwards of $5.50 per gallon ($4.40 for the petroleum, $1 for the corn, etc.). It does not cost that much, therefore it is not consuming that much gasoline to produce. In fact, ethanol requires very little *gasoline* or petroleum input at all. It *does* require other fossil fuel inputs, however. Coal is most commonly used to supply energy for the distillation process, natural gas is used to create fertilizer, and yes, some diesel is used by the tractors to plant and harvest the corn, and trucks or railroads to transport the final products. In spite of all that energy input, ethanol production from corn is still regarded as moderately energy positive. (FN3). That is, more energy overall comes out of the system than is put into it from manmade sources. The "gain" comes from the sun, as the growning plants capture energy by photosynthesis, and eventually yield it up as useable ethanol or byproducts. Sugarcane ethanol -- typical in Brazil -- is highly energy positive at present, and cellulostic ethanol, produced using recently developed enzymes, has finally become an economic reality, at least at the pilot-plant scale. (FN4). 3. Tim makes a point which I have also agreed with for some time now. Groundwork has to be put in place in advance if we hope to take advantage of expected advances in ethanol production. It is a classic chicken-and-egg problem -- that's why FFVs have been running for years on gasoline (and getting CAFE credits) while lacking of an E85 infrastructure, and that's why E85 pumps are being installed across the midwest even while we go through a temporary ethanol supply crunch caused by the switchover from MTBE. I disagree with his assessment, though, that conventional ethanol does not make economic sense at the moment.$70 oil means $2.20 gasoline (before taxes), and with the typical production costs of ethanol in the$2.00 range (before tax credits), that's in the realm of economic sense. Burning coal is a minus, reduced foreign oil and increased domestic cash flow is a plus. Moderate positive energy balance is a plus. Cleaner combustion exhaust is a plus. Weigh it out and take your pick.

4. As many on this forum like to point out, the best use for ethanol at the moment is probably cutting it into gasoline at a ratio of 10% -- E10. Virtually any car can run on that -- it boosts octane, cleans exhaust, and requires less infrastructure. Since we don't have enough ethanol at present to even constitute 10% of American gasoline volume, purusing such a course should occupy us for some time. However, the point of the programs mentioned in this post is just that -- to go out of the way to build extra infrastructure, in anticipation of the day when you will have more than enough ethanol to cut into gas at a 10% ratio, and will want to start burning E85 widely. Also, ethanol is hard to transport, as it cannot be fed into ordinary long-distance petroleum pipelines due to corrosion problems. Thus, it might make economic sense to start burning E85 in the midwest (where it is produced) before shipping extra ethanol to the coasts in an effort to get all the gasoline up to E10 -- depends on shipping costs. The midwest should therefore prepare to burn E85 even before all of New York, New England, Florida and California are uniformly up to E10. Which they have been doing.

5. Fuel economy -- miles per gallon of product -- while driving an FFV on E85 is now lower than while driving the same car (or more likely pickup truck) on conventional gasoline. That is a function of two things. First is the lower energy content of ethanol. Second is the optimization of current FFV engines towards gasoline and away from ethanol. That allows the first factor to tranlate into lower MPG. An ethanol-optimized engine can theoretically get just about as many miles per gallon of ethanol as a gasoline optimized engine can get from a gallon of gas, despite the lower ethanol energy content, due to ethanol's higher octane-equivalence. (FN5). That allows higher compression ratios and more efficient thermodynamic operation -- an ethanol engine uses more of its input energy for useful work when compared to a similar gasoline engine. Again -- a chicken-and-egg problem. You can't optimize an FFV for ethanol if its owner will use gasoline most of the time, because then it would burn the fuel of choice less efficiently and people would flee it. But ethanol can't make inroads unless its cost-per-mile does not exceed that of gasoline. Hence the short term solution of tax credits and marketing pushes, to get E85 available and established. Once you get past that first hurdle, you can start to offer FFVs optimized for E85 instead of gasoline, and take it from there.

6. All that being said, I'm not convinced that ethanol is the complete solution to all our problems. Neither is hydrogen. But I have become convinced that ethanol is a viable product and a worthwhile partial-solution. It buys us time to improve hybrid and battery technologies, which I think will become a very significant part of the automotive future. It also can be produced at substantial volumes in a somewhat sustainable manner. Its production inputs are amenable to "greening" -- the coal that is burned to run the distilleries can be replaced with green electricity from any given source, when it becomes available. Ethanol widens our energy base, allowing us to turn to domestic sources and alternative foreign suppliers of energy (Brazil, for the moment), instead of the usual middle eastern suspects. It is also "broad based" at the production level. Unlike the highly capital intensive petroleum system, ethanol can be produced economically on a smaller scale by independent operators. Major agribusiness is certainly involved (ADM), but smaller players -- consortia of individual farmers -- are able to compete from bottom to top as well; grow all the corn, build the distillery, market the ethanol directly to blenders and retail chains. That keeps the market a bit more honest and "market like," and less distorted than what we typically see with oil.

FN1: http://www.cbot.com/
FN2: http://www.wtrg.com/daily/gasolineprice.html
FN3: http://www.greencarcongress.com/2006/01/uc_berkeley_stu.html
FN4: http://www.greencarcongress.com/2006/05/goldman_sachs_i.html
FN5: http://www.greencarcongress.com/2004/12/saab_biopower_f.html

t:

The point about BTUs I think is this:

If the cost per gallon is the same, then consumers will pick the fuel with highest miles per gallon. But cost can always be manipulated through taxes.

If cost is not the issue, but energy efficiency is, then ethanol wins. Assume ethanol and gasoline get the same miles/BTUs in a conventional FFV. If ethanol production is net energy positive (which it is), then you get more miles in the end, because you have more BTUs than when you started, and each BTU will take you just as far. If you optimize your FFV for ethanol, then you will get *more* miles per BTU than in an equivalent gasoline-optimized engine. Then you win twice: More BTUs due to energy-positive ethanol production, and more miles per BTU due to optimization. Now you just have to adjust your tax policy to make sure all this gets reflected in the price of each fuel.

Of course, it isn't quite this simple. The energy positive nature of conventional corn ethanol is calculated based not only on the energy content of the ethanol output, but also on the value of the residual protein and oil byproducts, which are often used as animal feed. That's not illegitimate -- animal feed would have to be produced by some other means anyway -- just more complicated.

Also, fossil energy input into ethanol is not gasoline, but coal and natural gas. The former creates more problems than gasoline when burned, the latter less. On the other hand, to use either of those fuels for road transport would require either CNG vehicles (range, weight and infrastructure problems) or xTL converion plants, which consume plenty of energy themselves.

In an important sense, ethanol is a "carrier" fuel. It allows us to turn non-roadworthy energy sources (coal, natural gas, sunshine, possibly wind electricity, if used to run a distillery) into a road-capable liquid fuel. As such, you have to compare it to the alternative methods of creating road-capable liquid fuels, or supplanting such fuels entirely. Those include: Petroleum refining, Fischer-Tropsch xTL, biodiesel, steam locomotives, electric vehicles (plug in cars or scooters, as well as conventional electric streetcars and trains), more efficient use of fuels in conventionally-styled private vehicles (smaller engines/cars, hybrids), public transport and non-motor transport (bike/walk/horse). In many respects, ethanol is not a bad option, compared to what else we have on the table. Not the best in all respects, but not bad. In addition to the environmental implications of each of the above options, you can also get into the whole energy-security, domestic production, foreign wars discussion.

So, the original point: investigating energy efficiency by looking at BTUs is an important yardstick for estimating what other inputs and impacts a likely transportation source is going to have. Most of us regard things like affordability, local air quality, global environmental impact, sustainability, national security and domestic economic development as the real important outcomes in our lives. Miles/BTU is not, for most people, an important end-result, but it is probably a good yardstick to help measure and disentangle some of the complications we face, and clear up some of the choices we have to make. That's the point.

The corporate "drooling" over E85 and its marketing to the exclusion of other, better technologies and smaller displacement motors is troubling. Diesel moves this planet now and for the foreseeable future and biodiesel should be getting at least as much attention. Someone said earlier that "some" diesel is used in production and transport...transport also includes returning the empties on the dreaded "backhaul".

Dang! I stand corrected about what exactly defines E85 fuel.

http://en.wikipedia.org/wiki/E85

Therefore, here are my corrected numbers:

1) Net gasoline input: 1 gallon
2) Net gasoline input for E85:
1.38(0.15 gal gasoline + 2(.85))= 2.553 gallons of gasoline consumed!

I was not only wrong in my first calculations but WAY too optimistic as to what the true economic costs are
for this fuel.

The primary problem we have with these "fuels" is they are so subsidized by the Government that it is very difficult to calculate the true market costs and energy consumption of the manufacturing process. Few people even want to address this. Each interest group such as ADM, environmentalists, et al, are all guilty of producing disinformation to the point we all could discuss this until we're blue in the face and still not have the right data. I believe this is called "junk science".

The way to settle this issue is to find answers for the following points: If ethanol was not subsidized by the Government, would the manufacturers of this product even be in business? I would like to know the value per gallon of all subsidies, plus the added economic cost of increased demand due to the refineries having to use all of this ethanol so as to comply with Government fuel regulations. This I suppose would be a job for forensic economists and accountants. Also, to be fair, what are the true subsidized costs of fossil fuels?

If the free market is not permitted to chose the best products for the least amount of price, then I would say we have a big problem. A very Big Problem.

When I read about the real financial situation of the United States, the simplest question I can ask is this:

Can anyone say Soviet Russia, circa August 1991?

Dave,

I like the free market as much as anyone, and I consider myself a libertarian in that sense...but if we completely wait for the free market to develop alternatives, most likely it would happen at the start of a crisis that will take us many years to respond to. Who knows how much damage that would cause us?

Plus, it is not good for national security and sovereignty to have our economy and way of life effectively being "held hostage" by oil exporting nations, most of which hate us. So I also see development of alternatives to be a national security issue which is within the proper role of the federal government.

That doesn't necessarily mean ethanol will be a big part of it, but it could play at least some part together with other initiatives and alternatives.

btu/mile is the way to normalize between different fuels of differing energy density. so while it looks bad on a mpg basis, E85 is still as energy effeinet as gas on a thermo level. also consider that when E85 usually costs 20-30% less than gas the cost per mile is a wash.

how come nobody says anything about tax breaks/corporate welfare that the oil companies get(exploration cost breaks, etc)?

Dave:

I still don't know where you are getting the idea that a gallon of ethanol requires two gallons of gasoline to produce. Please cite something other than frothy conspiracy theories regarding "hidden subsidies" and the like to prove this point.

-NBK

>> how come nobody says anything about tax breaks/corporate welfare that the oil companies get(exploration cost breaks, etc)? <<

For the same reason some people complain about the subsidies mass transit and Amtrak get, while conveniently forgetting about the massive subsidies given to the automotive infrastructure? Yes, gas taxes pay a part of it, but just a part, and the reality is that the auto infrastructure gets far more subsidies than mass transit. Not saying that's good or bad, just noting the inconsistency that I commonly hear.

I'm glad I'm not one of his "A.P. Physics" high school students.

You can talk all day about miles/btu as some sort of efficiency measure but the fact remains that one gets a lot more btus per gallon from gasoline than from ethanol. Unless ethanol costs signficantly less than gasoline, it will not make sense to use it. Miles per btu is hardly an argument in favor of ethanol if the btus from ethanol are more expensive.

Energy efficiency on the so called thermo level does not argue in favor of ethanol if it takes a lot more ethanol to get the same btus per mile. Ethanol is subidized both at the corn production level at the refined product level (twice) and it is still having difficulty competing with gasoline.

As I said before, you could take a fuel which only got you one mile per gallon with a very low btu content and then claim that it was efficient from a btu per mile standpoint. If you keep lowering the numerator and denominator proportionately, you can maintain the same btus per mile and still be getting horrible miles per gallon and be paying paying orders of magnitude more per mile for one's fuel. Hence, for all pracltical purposes, miles per btu mean nothing unless it is put in the context of miles per gallon and dollars per gallon.

While it is true that all forms of energy are being subsidized in some way, it is also true that this is another way that ethanol is benefiting since it requires energy inputs from fuels that are subidized. Thus, it gets subsidized in three ways, once at the corn level, twice at the production level, and third at the input level.

While we will never agree here on the EROEI, it is noted that Pimental still maintains that the EROIE is negative. The primary reason for this is that he counts, for example, the energy required to produce the equipment necessary to produce the product all the way from corn production through distillation and refinement. I still thinks those are valid inputs and should not be ignored by the USDA and industry.

Tim:

Market forces can deal with energy disruptions rather quickly. Consider for example when natural gas and oil prices skyrocketed in the late 1970's (primarily due to monetary inflation at the hands of the Fed), driving up the costs of driving and heating one's home.

People at the time went to driving more fuel efficient vehicles. People went from driving gas hogs that got 12 m.p.g. to cars getting over 30 m.p.g., obviously a personal choice. When petroleum prices begin to go up, consumption will go down, period. We may not be driving S.U.V.s, and we may have to live close to where we work, rather than 30 miles away, but either way, people would adapt because they would have no choice.

Also, around the time folks started to insulate and weather-strip buildings, many people began to heat homes, schools, and businesses with wood and other solid-fuel fired heating equipment that came onto the market fairly quickly which truly saved people a lot of money. Unfortunately, the Clean Air Act of 1991 drove most of the wood stove manufacturers out of business, as it costs about $100,000 per stove just to get E.P.A. certification- a good example of our Government destroying a creative, useful activity. I predict some of these same things will soon occur again, if they haven't started already. Also, when ethanol motorfuel is economically viable, we will see it without need of a government input which does nothing but encourage and support inefficient producers. We still predominately use petroleum-based fuels because it is still cheaper than alternatives. If it is possible to replace gasoline with ethanol, someone will figure it out whose intent is to make money off of it. However, it probably will be much more expensive compared to what we're used to nowadays. One problem about corn-based ethanol that must be faced, according to some sources, is that even if we devote half of our crop to ethanol, it would only replace 6% of current gasoline production! Oh, and you can believe that food costs will go through the roof. If one figures in woody wastes, that rises to a total of 11% of current total gasoline usage. Either way, if indeed we are facing "Peak Oil" our energy consumption will have to go back to a level of something like 1940 America. I'll sum up the situation by paraphrasing the author Howard Kunstler: " Life in America won't be about going places, it will be about staying in one place." t: There are several discrete issues you touch upon in your post. I will try to address them in turn. 1. Consumer price of E85 vs. gasoline: You are right. E85 must cost substantially less per gallon in order to make up for the lower energy content and lower miles per gallon acheived in conventional (non-optimized) FFVs. And, in a good number of instances, it actually does cost less. FN1. However, the need for a lower E85 price is theoretically a temporary state of affairs. If or when FFVs are optimized for higher octane E85, their MPG rating when running on ethanol will go up. 2. Policy Questions -- including energy return estimates and subsidies: This is admittedly a massive and sticky topic. How do you count up all the explicit and hidden subsidies -- and costs -- associated with any given fuel type and manufacturing process? I find it very hard to locate comprehensive and reliable studies of the questions. A few factors which come to mind (based on issue you point out, as well as others): Ethanol: Subsidies on inputs (corn, coal?, other?), subsidies on product ($0.50 tax credit when blended into motor fuel), energy costs, including all inputs (back to the fertilizer and the very machines used to plant the corn); plus what is the actual fair value of all byproducts? Differing production methods: Corn, sugar beet, sugar cane, grain. How efficient is each? What about distilleries run on coal, natural gas, nuclear electricity, renewable electricity? Co-generation? Benefits from cleaner combustion? Positive CO2 balance?

Oil:
Subsidies on inputs (tax breaks for exploration and drilling). Taxes on product -- high enough? Account for all energy inputs, including the cost of all equipment and energy used in exploring, mining, drilling, pumping, piping, transporting, refining, blending, distributing and retailing gasoline? Cost of military presence needed to secure stable international supplies? Macroeconomic costs of sending money abroad? Security costs of funding people who don't like us? Costs of air pollution and global warming?

3. The way I see it, the first matter -- consumer price -- can always be manipulated with relative ease. The government can adjust tax policy with the stroke of a pen to make sure that E85 never costs more than gasoline does on a mile-for-mile basis. That's what current policy is aimed at doing, and does to an imperfect extent.

The question is -- should government do that? Most people on most sides of current ideological fault lines will say "it depends." The answer depends on whether or not the externalities -- costs imposed on society by the consumption of a good, but not reflected in the producer price of that good -- justify a given level of taxation.

For instance, most people agree that taxing motor fuel to pay for the construction and upkeep of roads that cars drive upon is justifiable. Consuming gasoline creates a demand for roads, which is not reflected in the producer's offering price for the product. So we add on a tax to help pay for the roads, which also marginally reduces demand because it makes driving a bit more expensive than it otherwise would be. We could choose not to tax gasoline, and simply pay for roads out of general funds (collected by income tax, general sales tax, property tax, etc.) or out of tolls paid by road users, or yearly registration fees, but in general we think a gas tax is not an unreasonable way to pay for roads.

By the same logic, if we could fairly determine that demand for gasoline required an annual $100 billion in additional military expenditures, above and beyond what we would normally spend, because of our need to regularly intervene in far off places to preserve the stability of the petroleum market, then it would be equally fair to collect that out of a gas tax, rather than out of a general funds tax. Similarly, if consuming petroleum led to$100 billion per year in higher medical costs across society, as air pollution exacerbates diseases and lung problems, then we could collect some of that cost through a gas tax, rather than paying for health care out of general government funds, medicare funds, or private funds.

So the question really is that big: How much is ethanol taxed/subsidized? How much is gasoline taxed/subsidized? What are the true costs -- including all externalities -- of ethanol and gasoline?

And finally -- what is the future potential of both? I don't see gasoline becoming any easier to find, any easier on the environment, or any easier geopolitically. But I do see ways in which ethanol could become a lot better for our country and our globe.

I happen to think that conventional ethanol is not a bad deal, all things considered, but it is honestly not enough to really get excited about. But I also happen to think that ethanol, or other closely related spark-compatible biofuels, have obvious and serious potential for future advances.

I also happen to think that many other things need to be done all along the chain. We should significantly reduce the number of behemoths (SUVs/pickups) on the road -- which we can do probably by pricing in all the externalities *they* create, relative to normal cars. We can promote next-generation solutions that get us away from combustion engines (hybrids/PHEVs/EVs -- heck, maybe even fuel cells). Most importantly -- get people out of cars in increasing numbers, by making the alternatives more practicable. Auto-centric suburbs got built in part because consumers demanded it, but in part because standardized design and zoning criteria were adopted by local governments around the country, making other urban layouts which had (and have) a good deal of consumer demand impossible to build. All the same, we now have to live with what we've got, and we also have to live with the fact that the auto is a legitimate convenience in a wide set of circumstances. We simply have to deal with it rationally.

FN1: http://www.cleanairchoice.org/outdoor/PriceForum.asp

The energy content of pure ethanol is about 70% of that of gasoline, so E85 has about 75% of the energy content. If ou don't care about range on a full tank of gas, follow this simple rule:

If the price of E85 at the pump is less than 75% of the price of gasoline, go for the E85. Otherwise, stick with gasoline unless you a glutton for punishment.

As for the GHG footprint of ethanol: estimates of how many BTUs of fossil fuel it takes to produce one BTU of ethanol from corn kernels (the kind that matters most in the US) do vary. Improvements in agriculture and processing have brought this into mildly postive territory. Figure 0.85 BTUs of fossil fuels for 1 BTU of ethanol. However, those fossil fuel BTUs have to be mined/processed, so it's probably closer to 1. For finished gasoline, figure 1.15 BTUs of crude oil for 1 BTU of finished fuel.

Ergo, driving on E85 produced from domestically grown corn kernels reduces the GHG footprint of your vehicle by 1-(0.85*1 + 0.15*1.15)/1.15 = 11%.

Now compare a full-sized truck with regular car. Let's say the truck gets 12MPG on gasoline, or 19.6 l/100km. CO2 production is 24 g/km per l/100km, so on gasoline the truck causes 1.15 * 19.6 * 24 = 541 g CO2/km. On E85, reduce that by 11% - 482 g CO2/km.

Let's say the car gets 24MPG on gasoline, or 9.8 l/100km, causing 1.15 * 9.8 * 24 = 270 g CO2/km., or 44% LESS than the truck on E85. Of course, you can get cars that get well over 30, even 50MPG combined. The Prius comes in at 104 g CO2/km, i.e. 5 Prius = 1 full-sized pick-up trucks on gasoline.

Conclusion: if you have a legitimate reason for owning a full-size truck (or truck-based SUV), consider running it on E85 if the price is right. If you don't need a big-a** truck, please drive a car instead. With what you save in gas alone (never mind GHG), you can easily afford to rent a larger vehicle for occasional use.

I would only add that a real stickler would try to determine the "real world" difference in economy in his particular FFV when running E85 versus gasoline. Going on the EPA numbers, for instance, suggests that different model FFVs respond slightly differently to that 25% drop in energy content.

NBK-Boston -

I'd be very surprised if different FFV cars really got radically different MPG ratios on E85 vs. gasoline. There may be some minor differences due to engine construction, but for the purpose of GHG footprint estimates, those are definitely second-order effects compared to the uncertainties in the agriculture and refining overheads.

I suggest yoo look carefully at exactly which numbers EPA is quoting for FFVs: with the CAFE loophole, the aggregate number is based on the notion that owners would fill up on gasoline 50% of the time and on E85 the other 50% (a dubious proposition). However - and here's the kicker - when on E85, only the 15% gasoline are counted as consumed fuel (which is completely false advertising, as consumers have to PAY for the ethanol fraction). As they say: you can fool some of the people all of the time, and those are the ones you want.

The result: wrt CAFE, an FFV truck that actually gets 12 MPG on gasoline and 9 MPG on E85, is classified as if it got 21 MPG on a 50/50 mix, thus evading the gas guzzler tax. The ethanol lobby, the Detroit carmakers and midwestern Congressmen are colluding in this blatant deception.

NGK + Rafael:

I believe you effectively closed discussion on ethanol.
However, I”ll alow myself some additions.

E85 optimized engine will increase thermal efficiency by no more then 5%, which is not nearly enough to compensate for lower volumetric specific energy of E85 vis gasoline. E85 mpg will be always lower then gasoline mpg, thought thermal efficiency of E85 optimized engine will be higher.

IMHO, hidden cost of 100 billion dollars spent yearly on military expeditions applies mostly to imported oil, not pumped here in North America.

The comments to this entry are closed.


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# Proving the reduction formula

I am suppose to find $\int(\ln x)^3$ by using the proof (I have to prove this part first) of

$$\int (\ln x)^n dx = x(\ln x)^n - n \int (\ln x)^{n-1} dx$$

I can not prove it and I do not know how to work with n powers like that, to me it doesn't even look right.

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To prove the last formula (do you see how to apply it?), try integration by parts. The $u$ and $dv$ that work are somewhat surprising; for now let me just say that if your $dv$ is hard to integrate, then you're doing too much work. –  Dylan Moreland Jun 8 '12 at 19:21
I do not understand how to do it with n as a power though, it doesn't make sense. –  user138246 Jun 8 '12 at 19:25
@Jordan: You have seen formulas like $\frac{d}{dx}x^n=nx^{n-1}$? Does that make sense? –  Jonas Meyer Jun 8 '12 at 19:26
Would it be easier to think of the case $n=3$, at first? In essence it is no easier than the general case, but the concreteness might help. –  Dylan Moreland Jun 8 '12 at 19:30
(Joke, sort of): If one is asked to prove an explicit integration formula, one way is to differentiate the "answer." –  André Nicolas Jun 8 '12 at 20:27

We have

$$\mathrm I(n)=\int (\log x)^ndx$$

integrate by parts with

$$dx=du$$ $$\log^n x = v$$

$$x=u$$ $$n (\log x)^{n-1} \frac{1}{x}dx=dv$$

$$\mathrm I(n)=x\log^n x-\int n (\log x)^{n-1}\dfrac{1}{x}x dx$$

$$\mathrm I(n)=x\log^n x-n\int (\log x)^{n-1} dx$$

$$\mathrm I(n)=x\log^n x-n \mathrm I (n-1)$$

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You are using ln and log as the same thing but they are not the same, does this not change the answer? –  user138246 Jun 8 '12 at 19:34
@Jordan Many mathematicians use $\log$ instead of $\ln$ for the base $e$ logarithm. This is probably because it can be considered "the" logarithm. –  Argon Jun 8 '12 at 19:35
I is the integral? I do not really follow what happened in the last step. –  user138246 Jun 8 '12 at 19:42
"$I(n)$" means "that integral, with an n in the exponent". So "$I(n - 1)$" means "that integral, with an $n - 1$ in the exponent"... –  The Chaz 2.0 Jun 8 '12 at 19:52
Oh so it means the integrand to the n power? So with this formula does that mean that the power just reduces by one each time I use the formula? –  user138246 Jun 8 '12 at 19:59

Integration by parts is a useful technique. You can use $u=(\ln x)^n$ and $v'=1$ in the formula $\int uv' dx = uv-\int vu' dx$.

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## Physics 2018 Past Question | WAEC

Study the following past questions and answers for JAMBWAEC  NECO and Post-JAMB. Get prepared with official past questions and answers for upcoming examinations.

41. The circuit contains three ammeters A1, A2 and A3 with readings of A2 and A3 shown as indicated in the diagram. Determine the reading of A1.

• A. 2A
• B. 3A
• C. 5A
• D. 8A

42. Which of the following statements about magnetic lines of force is not correct? They

• A. pass through even non-magnetic materials
• B. run from the north pole to the south pole
• C. are in a state of tension and tend to be shorted
• D. intersect one another

43. In which of the following activities is an induction coil not useful?

• A. Investigation of high voltages
• B. Study of electric discharge
• C. Operation of X-ray tubes
• D. A.C. power transmission

44. In a series R-L-C circuit, R = 10ΩXc= 4Ω and XL = 9Ω. The impedance of the circuit is

• A. 4.1Ω
• B. 5.0Ω
• C. 10.8Ω
• D. 11.2Ω

45. A student uses two compass needles to investigate the magnetic field around a bar magnet. Which of the following diagrams show the correct directions of the needles?

• A. A
• B. B
• C. C
• D. D

46.

Which of the following is emitted in a process of natural radioactivity?

• A. α– particles and $\beta$– particles
• B. α– particles and X-rays
• C. $\beta$– particles and X-rays
• D. $\lambda$– rays and X-rays

47. In which of the following situations is photoelectric effect employed?

I. Automatic devices for switching on light at dusk

II. Automatic doors in departmental stores

III. Automatic devices for switching on air conditioners

IV. Automatic voltage stabilizers

• A. I, II and IV only
• B. I, III and IV only
• C. III and IV only
• D. I and II only

48. Which of the following radiations has the longest wavelength?

• A. Gamma ray
• C. Infrared ray
• D. X-ray

49. In which of the following fields is radioisotopes not used?

• A. Medicine
• B. Oil industry
• C. Agriculture
• D. Weather forecast

50. A nuclide is represented by $$_{32}^{70}B$$. Determine its neutron-proton ratio

• A. 0.5
• B. 0.8
• C. 1.2
• D.1.5

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# Show that $f: \mathbb{R}: \rightarrow \ell^{\infty}$ is not Lebesgue integrable?

Let $$\{I_{n}\}_{n \in \mathbb{N}}$$ be the sequence of real intervals $$[0, 1/2], [1/2, 1], [0, 1/3], [1/3, 2/3]$$, and so on. Let $$f_{n}: \mathbb{R} \rightarrow \mathbb{R}$$ be the indicator function on $$I_{n}$$. Let $$f:\mathbb{R} \rightarrow \ell^{\infty}$$ be $$f(x) = (f_{1}(x), f_{2}(x), ...)$$.

Show that $$f$$ is not Lebesgue measurable.

I'm not sure where to start with this. I tried a contradiction proof, supposing that there exists a sequence of simple measurable functions which converge pointwise to $$f$$ (which would imply that $$f$$ is measurable). However, I'm having trouble reaching a contradiction.

My intuition for this approach is that the most "natural" such sequence, namely $$g_{n}: \mathbb{R} \rightarrow \ell^{\infty}$$ where $$g_{n}(x) = (f_{1}(x), f_{2}(x), ..., f_{n}(x), 0, 0, ...)$$ does not work, because for any $$x \in [0, 1]$$ there are infinitely many intervals $$I_{k}$$ such that $$x \in I_{k}$$, so for any $$g_{n}$$, we can show $$\| g_{n}(x) - f(x) \| = \frac{1}{m} + \frac{1}{m+1} + ... = \infty$$ for some $$m \in \mathbb{N}$$.

However, it is not enough to show that the particular sequence of simple measurable functions $$(g_{n})$$ fails to converge pointwise to $$f$$.

Is there some way to fix this approach? Or should I try something else entirely?

• What is the distinction between $l_{\infty}$ and $\mathbb{R}^{\infty}$? By $\ell^{\infty}$ I just mean the set of all sequence of real numbers indexed by $\mathbb{N}$, which I'm guessing is what is meant by $\mathbb{R}^{\infty}$. This problem is from a very old set of notes (1970s) so maybe the notation is non-standard. I'm a newcomer to measure theory so I could be wrong. – Akhil Jalan Nov 30 '18 at 20:37
• $\ell^\infty$ usually stands for the set of all bounded real sequences. $\mathbb{R}^\infty$ is set of all real sequences, it is also denoted by $\mathbb{R}^{\mathbb{N}}$. – mechanodroid Nov 30 '18 at 21:20

Let $$X\subset \{0,1\}^{\Bbb N}$$ and notice $$X$$ cannot have limit points because a limit point requires a Cauchy sequene within $$X$$, but $$||x-y||=1$$ for distinct $$x,y \in X$$. Hence, $$X$$ is vacuously closed, thus Lebesgue.
This means that $$A=f(V)\subset \{0,1\}^{\Bbb N}$$ is measurable, for $$V\subset [0,1]$$. Clearly, $$V\subset f^{-1}(f(V))$$, suppose $$y \in f^{-1}(f(V))$$ so that $$f(y)=f(x)$$ for $$x\in V$$. This implies $$f_n(y)=f_n(x)$$, which happens only if $$y \in I_n$$ for all $$I_n \in J = \{I \in \{I_n\} : x\in I\}$$, i.e. $$y \in \bigcap_{I\in J}I$$. But $$\text{diam}(I_n)\to 0$$ as $$n\to \infty$$, so $$d(x,y)\to 0$$ shows $$y=x\in V$$ and thus $$f^{-1}(A)=V$$. Set $$V$$ equal to the Vitali set to get the result.
• Clarifying questions: 1. Did you mean to say $\|x-y\| \geq 1$ for distinct $x, y \in X$? 2. When you say $X$ is Lebesgue, do you mean Lebesgue-measurable? And, to double-check, this is because all open/closed sets are part of the Borel $\sigma$-algebra? – Akhil Jalan Dec 1 '18 at 0:53
• @AkhilJalan $x$ and $y$ are binary sequences and $||\cdot||$ is the supremum norm. If they are distinct, they differ at some index where one of them is one and the other is zero, so $||x-y||=1$. – Guacho Perez Dec 1 '18 at 0:54