text:
# 10.3 Superposition and interference  (Page 3/12)

 Page 3 / 12

The lowest frequency, called the fundamental frequency    , is thus for the longest wavelength, which is seen to be ${\lambda }_{1}=2L$ . Therefore, the fundamental frequency is ${f}_{1}={v}_{\text{w}}/{\lambda }_{1}={v}_{\text{w}}/2L$ . In this case, the overtones    or harmonics are multiples of the fundamental frequency. As seen in [link] , the first harmonic can easily be calculated since ${\lambda }_{2}=L$ . Thus, ${f}_{2}={v}_{\text{w}}/{\lambda }_{2}={v}_{\text{w}}/2L={2f}_{1}$ . Similarly, ${f}_{3}={3f}_{1}$ , and so on. All of these frequencies can be changed by adjusting the tension in the string. The greater the tension, the greater ${v}_{\text{w}}$ is and the higher the frequencies. This observation is familiar to anyone who has ever observed a string instrument being tuned. We will see in later chapters that standing waves are crucial to many resonance phenomena, such as in sounding boxes on string instruments.

## Beats

Striking two adjacent keys on a piano produces a warbling combination usually considered to be unpleasant. The superposition of two waves of similar but not identical frequencies is the culprit. Another example is often noticeable in jet aircraft, particularly the two-engine variety, while taxiing. The combined sound of the engines goes up and down in loudness. This varying loudness happens because the sound waves have similar but not identical frequencies. The discordant warbling of the piano and the fluctuating loudness of the jet engine noise are both due to alternately constructive and destructive interference as the two waves go in and out of phase. [link] illustrates this graphically.

The wave resulting from the superposition of two similar-frequency waves has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats , with a frequency called the beat frequency    . We can determine the beat frequency by adding two waves together mathematically. The result,

${f}_{B}=\mid {f}_{1}-{f}_{2}\mid$

is the beat frequency. The resultant wave has the average frequency of the two superimposed waves, but it also fluctuates in overall amplitude at the beat frequency ${f}_{\text{B}}$ . This result is valid for all types of waves. However, if it is a sound wave, providing the two frequencies are similar, then what we hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.

## Making career connections

Piano tuners use beats routinely in their work. When comparing a note with a tuning fork, they listen for beats and adjust the string until the beats go away (to zero frequency). For example, if the tuning fork has a $\text{256}\phantom{\rule{0.25em}{0ex}}\text{Hz}$ frequency and two beats per second are heard, then the other frequency is either $\text{254}$ or $\text{258}\phantom{\rule{0.25em}{0ex}}\text{Hz}$ . Most keys hit multiple strings, and these strings are actually adjusted until they have nearly the same frequency and give a slow beat for richness. Twelve-string guitars and mandolins are also tuned using beats.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!


text:
# Publishing house / Banach Center Publications / All volumes

## A generalization of the conservation integral

### Volume 43 / 1998

Banach Center Publications 43 (1998), 273-284 DOI: 10.4064/-43-1-273-284

#### Abstract

Starting from the scheme given by Hudson and Parthasarathy [7,11] we extend the conservation integral to the case where the underlying operator does not commute with the time observable. It turns out that there exist two extensions, a left and a right conservation integral. Moreover, Itô's formula demands for a third integral with two integrators. Only the left integral shows similar continuity properties to that derived in [11] used for extending the integral to more than simple integrands. In another approach we extend the previous notions for the integrals to much larger domains of definition and to much more processes, including anticipating ones. Similar to [5,10], we use the Skorohod integral and the Malliavin derivative acting on a symmetric Fock space [3,4]. It appears that this formulation unifies all three integrals in the double integrator one.

#### Authors

• Volkmar Liebscher

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

format: plain


text:
# nLab Poisson vertex algebra

A Poisson vertex algebra is a unital commutative associative algebra with a derivation and a $\lambda$-bracket; it has to satisfy the axioms of a Lie conformal algebra; and the $\lambda$-bracket and the multiplication form a Poisson algebra (i.e. satisfy the Leibniz rule). It is a Poisson analogue of a conformal Lie algebra.

• Haisheng Li, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math. http://arxiv.org/abs/math/0209310
• A. De Sole, V. G. Kac, M.; Wakimoto, On classification of Poisson vertex algebras, Transform. Groups 15 (2010), no. 4, 883–907 MR2012a:17051 doi
• B. Bakalov, A. De Sole, Non-linear Lie conformal algebras with three generators, Selecta Math. (N.S.) 14 (2009), no. 2, 163–198 dpo MR2009m:17024
• Alberto De Sole, Victor G. Kac, Daniele Valeri, Classical W-algebras and Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras, arxiv/1207.6286
• Gaywalee Yamskulna, Vertex Poisson algebras associated with Courant algebroids and their deformations; I, math.QA/0509122
• A. De Sole, Poisson vertex algebras in the theory of Hamiltonian equations, Talk at Algebraic Lie Theory, Newton Inst. U. Cambridge 2009, video

Non-local Poisson vertex algebras are studied with applications to the study of integrability of Hamiltonian PDE-s in

Last revised on February 6, 2013 at 17:20:08. See the history of this page for a list of all contributions to it.


text:
# Asymptotic form of Fourier type integral

1. Jun 18, 2011

### bruno67

I have an integral like

$$F(\lambda)=\int_{-\infty}^\infty e^{i\lambda x} f(x) dx,$$
where $\lambda$ is a real parameter and $f(x)$ is an integrable function of x. I am looking for a method to calculate an approximate form of $F(\lambda)$ for very small $|\lambda|$. Methods like stationary phases or steepest descent can sometimes be used to calculate similar asymptotic expressions for large values of the parameter, but I am not sure how to proceed in case $\lambda$ is small.

Thanks.

2. Jun 18, 2011

### Mute

A couple of ideas:

You could try changing variables. For example, letting $y = \lambda x$ gives

$$F(\lambda) = \frac{1}{\lambda}\int_{-\infty}^{\infty} dy e^{iy} f\left(\frac{y}{\lambda}\right).$$

If you know the large argument asymptotics of f(x), you could then expand f(y/lambda) in an asymptotic series and hopefully the resulting integrals for that are simpler. There is an issue with the y = 0 contribution to the integral - however, since it's just a point I don't think it contributes.

The other option is to expand

$$F(\lambda) \approx F(0) + F'(0)\lambda$$

where

$$F(0) = \int_{-\infty}^\infty dx~f(x)$$
$$F'(0) = i\int_{-\infty}^{\infty} dx~x f(x)$$

This assumes of course that you can do these integrals and that they exist.

3. Jun 19, 2011

### bruno67

Thanks, but the problem with the first suggestion is that f(x) only admits a Taylor expansion for large values of its argument, so the expansion of $f(y/\lambda)$ is not integrable from $-\infty$ to $+\infty$. To give its exact form, f(x) is given by

$$f(x)=\frac{(a^2+x^2)^{-b}}{x+ic},$$
where $a,b,c>0$ and $b<<1$.

Unfortunately the second suggestion doesn't work either, since $x f(x)$ is not integrable.

4. Jun 19, 2011

### Mute

The expansion you would need to do is a large-argument expansion, because y/lambda would be large. It looks like the leading contribution from a large y/lambda expansion would be $(\lambda/y)^{1-2b}$, although wolframalpha claims the Fourier transform of that is zero for 0 < Re(b) < 1/2. Hopefully the leading correction to the asymptotic series for $f(y/\lambda)$ is non-zero and integrable.

Yes, if $F(\lambda)$ is not an analytic function of lambda, this won't work.

5. Jun 19, 2011

### bruno67

I am probably missing something but, $y/\lambda$ is not large on the whole interval of integration: no matter how small $\lambda$ is, there will always be an interval around $y=0$ over which $y/\lambda$ is small. In this particular case, the Taylor expansion of the integrand is only valid for large enough $y/\lambda$; to be exact, for $y/\lambda>\max(a,c)$. Even if we wanted to integrate this expansion, the leading term, which I think is actually $(y/\lambda)^{-1-2b}$, is not integrable over $\mathbb{R}$, due to its behavior near the origin.

6. Jun 19, 2011

### Antiphon

Whether you can use stationary phase/steepest descent depends on the derivatives of f(x) in relation to lambda. You might still be able to use it.

7. Jun 19, 2011

### bruno67

Hi, my f is
$$f(x)=\frac{(a^2+x^2)^{-b}}{x+ic}.$$

8. Jun 19, 2011

### Antiphon

For some rational values of b you might find a closed form solution via complex contour integration.

My gut tells me you are going to see the gamma function and simple confluent hypergeometric functions but only for some b.

Have you got access to a copy of Gradshteyn & Ryzhik? Or Abramowitz & Stegun? Your library should have these, and frankly so should you.

9. Jun 19, 2011

### Antiphon

Also, you could do a Taylor series using the first two or three terms of the exponential, leave f(x) as it is. You will probably find closed form integrals for those, especially in the two references I gave.

10. Jun 19, 2011

### bruno67

b is usually not rational, and I don't think it's possible to solve this integral by contour integration for irrational b. Expanding the exponential in a Taylor series doesn't work, unfortunately (see posts 3-4).


text:
# Office Hours

I have sign-up office hours during the academic year. Sign-up office hours are primarily for research advising. To make an appointment, please use my online calendar:


text:
• https://me.yahoo.com
COST (GBP)
3.90
0.00
0

future value

Time within a period when a payment is made
Controller: CodeCogs

C++

Future Value

 doublefuture_value( double rate int nper double pmt double pv PaymentPoint when )
Evaluate the future value of an investment. This is done by solving the following equations to find the value of fv :

If rate = 0,

$pv&space;+&space;pmt&space;*&space;nper&space;+&space;fv&space;=&space;0$

If rate /= 0,

$pv(1+rate)^{nper}&space;+&space;pmt[1+rate(when)]&space;\frac{(1+rate)^{nper}}{rate}&space;+&space;fv&space;=&space;0$

Where pv is the present value, rate is the interest rate, nper is the number of periods over which to calculate and pmt is the payment made each period.

Calculating the future value of an investment according to a scedule of compound interest rates is done using the following equation:

$\sum_{i=1}^{nper}&space;(pv&space;*&space;schedule[i])$

Where nper is the number of periods (and in this case entries in the shedule array), pv is the principal value and schedule is an array of compound interest rates.

References:

http://www.vni.com/products/imsl/jmsl/v30/api/com/imsl/finance/Finance.html

Example 1

#include <stdio.h>
#include <codecogs/finance/banking/future_value.h>

int main()
{
double d = Finance::Banking::future_value(0.05, 20, -30000, -30000,
Finance::Banking::pp_StartOfPeriod);
printf("30,000 set aside per year\n");
printf("5%% interest compounded yearly\n");
printf("After 20 years, the value of the investment will be: %7.2f\n\n", d);

d = Finance::Banking::future_value(0.00, 7, -50, 0, Finance::Banking::pp_StartOfPeriod);
printf("50 set aside per year\n");
printf("no interest\n");
printf("After 7 years, the value of the investment will be: %.2f\n", d);
return 0;
}
Output:
30,000 set aside per year
5% interest compounded yearly
After 20 years, the value of the investment will be: 1121176.49

50 set aside per year
no interest
After 7 years, the value of the investment will be: 350.00

pp_StartOfPeriod or pp_EndOfPeriod.

Parameters

 rate is the interest rate nper is the number of periods pmt is the amount set aside each period pv is the present value when is the point in each period when the payment is made, either

Returns

a double, the future value of the investment.

Authors

James Warren (May 2005)
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.

Other Documentation

Type	Description

pp_EndOfPeriod	Each payment is made at the end of the period.

pp_StartOfPeriod	Each payment is made at the start of the period.

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.


text:
# Structural Information Learning Machinery: Learning from Observing, Associating, Optimizing, Decoding, and Abstracting

27 Jan 2020  ·  ·

In the present paper, we propose the model of {\it structural information learning machines} (SiLeM for short), leading to a mathematical definition of learning by merging the theories of computation and information. Our model shows that the essence of learning is {\it to gain information}, that to gain information is {\it to eliminate uncertainty} embedded in a data space, and that to eliminate uncertainty of a data space can be reduced to an optimization problem, that is, an {\it information optimization problem}, which can be realized by a general {\it encoding tree method}. The principle and criterion of the structural information learning machines are maximization of {\it decoding information} from the data points observed together with the relationships among the data points, and semantical {\it interpretation} of syntactical {\it essential structure}, respectively. A SiLeM machine learns the laws or rules of nature. It observes the data points of real world, builds the {\it connections} among the observed data and constructs a {\it data space}, for which the principle is to choose the way of connections of data points so that the {\it decoding information} of the data space is maximized, finds the {\it encoding tree} of the data space that minimizes the dynamical uncertainty of the data space, in which the encoding tree is hence referred to as a {\it decoder}, due to the fact that it has already eliminated the maximum amount of uncertainty embedded in the data space, interprets the {\it semantics} of the decoder, an encoding tree, to form a {\it knowledge tree}, extracts the {\it remarkable common features} for both semantical and syntactical features of the modules decoded by a decoder to construct {\it trees of abstractions}, providing the foundations for {\it intuitive reasoning} in the learning when new data are observed.

PDF Abstract

## Code Add Remove Mark official

No code implementations yet. Submit your code now

## Datasets

Add Datasets introduced or used in this paper

## Results from the Paper Add Remove

Submit results from this paper to get state-of-the-art GitHub badges and help the community compare results to other papers.


text:
# help calculating G cost of A* algorithm

This topic is 2767 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I am trying to implement the A* algorithm for pathfinding and having trouble understanding G cost.

Here is a small grid area of node id's from a larger grid.
 52 53 54 55 56 102 103 104 105 106 152 153 154 155 156 202 203 204 205 206 252 253 254 255 256

Horizontal and vertical moves increment 10, diagonal increments 14. I am followig this tutorial ... http://www.policyalm...tarTutorial.htm

Node 103 is my starting node. So the lowest F score above takes me to node 154 from 103. I have calculated the G cost as follows from node 153.
 14 10 14 10 10 14 10 14
Next iterations F score gives me node 205, so I then calculate G cost from node 205 and I get the following:
 28 24 28 24 24 28 24 28
And one more iterations F score gives me node 256, so I calculate G cost from node 256:
 42 38 42 38 38 42 38 42
Am I calculating G cost correctly?

Something I don't understand is after I choose the lowest F score, the G cost of 4 nodes is always the same and therefore I can't choose a lowest G cost at this point.

Any help much appreciated.

##### Share on other sites
You didn't tell us what node you are trying to reach or what your heuristic is, so we can't know what node minimizes F at each stage. The G cost for a particular node should only decrease, so if you find a new way to reach it that takes longer, you don't update it.

If you could post the full 5x5 matrix of G costs after each step, it would be much more clear whether you understand it or not. It should be something like this:
Initially: - - - - - - 0 - - - - - - - - - - - - - - - - - - After expanding 103: 14 10 14 - - 10 0 10 - - 14 10 14 - - - - - - - - - - - - After expanding 154: 14 10 14 - - 10 0 10 28 - 14 10 14 24 - - 28 24 28 - - - - - - ...

##### Share on other sites

You didn't tell us what node you are trying to reach or what your heuristic is, so we can't know what node minimizes F at each stage. The G cost for a particular node should only decrease, so if you find a new way to reach it that takes longer, you don't update it.

So I'm looking at your grid and I don't get something.

After expanding to 154, why does node 2,4 in your example not equal 20. It's 28. But it's only down 2 "10" vertical moves away. Same for node 4,2, it's 2 "10" horizontal moves away. Yet node 4,4 is 28 because it is 2 "14" moves diagonally.

Is G cost only ever calculated 1 time after moving to the current node and just when adding it to the open list? You don't calculate G cost for nodes already on the open list?

##### Share on other sites

If you could post the full 5x5 matrix of G costs after each step, it would be much more clear whether you understand it or not. It should be something like this:

The node in parenthesis is the node I just moved to after finding lowest F score.

My grid is 50x50, and the initial node starts at 3,3 (i.e. node 103)
 - - - - - - - - - - - - - - (0) - - - - - - - - - - - - - - -

Then I calculate G cost for all adjacent nodes to 3,3:
 - - - - - - - 14 10 14 - - - 10 (0) 10 - - - 14 10 14 - - - - - - - -

Lowest F takes me to node 154 (i.e. 4,4). So I move to 4,4, and recalculate G for all adjacent nodes:
 - - - - - - - 14 10 14 - - - 10 28 24 28 - - 14 24 (28) 24 - - - 28 24 28 -
In the above, (4,4) was 14 then I added 14 equalling 28 and compared it to adjacent node 3,3 which is 28. (4,4) was 14 and I added 10 equalling 24 and compared it to adjacent node 4,3 which is 24. I did this for all 8 nodes surrounding the current node. Several nodes are always same. Here there are 4 nodes of 24 which is smaller than 28. Not just one being the better path.

Next lowest F takes me to node 205 (i.e. 5,5), move to 5,5, and recalculate G:
 - - - - - - - 14 10 14 - - - 10 28 24 28 - - 14 24 42 38 42 - - 28 38 (28) 38 - - - 42 38 42
Everytime I recalulate G, I take the G of the current node, and at 10 or 14 depending on directon, and compare it to the adjacent nodes G. Current node is always higher in my case.

This is the top corner of a 50x50 grid. The path being traversed is (3,3), (4x4), (5x5) ... making a diagonal path going downwards and to the right from 3,3. The node I'm trying to reach is 25,47.

I think I'm just not understanding how G is supposed to be calculated after each iteration of lowest F score. I've reread the tutorial over and over, but that part doesn't come clear to me.

##### Share on other sites

You didn't tell us what node you are trying to reach or what your heuristic is, so we can't know what node minimizes F at each stage.

Per the tutorial, my heuristic is ...

H = the estimated movement cost to move from that given square on the grid to the final destination, point B. This is often referred to as the heuristic,
[/quote]

All I do is count blocks vertically and horizontally to come up w/ H cost. And then my F score is the sum of G + H.

##### Share on other sites

After expanding to 154, why does node 2,4 in your example not equal 20. It's 28. But it's only down 2 "10" vertical moves away. Same for node 4,2, it's 2 "10" horizontal moves away. Yet node 4,4 is 28 because it is 2 "14" moves diagonally.

That's because I haven't expanded node 104, so I haven't found the more straight path to get there with a cost of only 20.

Is G cost only ever calculated 1 time after moving to the current node and just when adding it to the open list? You don't calculate G cost for nodes already on the open list?
[/quote]
No, G is calculated when adding the node to the open list and any other time it appears in an expansion, but we always keep the minimum value, not the latest.

##### Share on other sites

[quote name='alvaro' timestamp='1331603111' post='4921538']
You didn't tell us what node you are trying to reach or what your heuristic is, so we can't know what node minimizes F at each stage.

Per the tutorial, my heuristic is ...

H = the estimated movement cost to move from that given square on the grid to the final destination, point B. This is often referred to as the heuristic,
[/quote][/quote]
That's not your heuristic: It's the definition of a heuristic. What function specifically are you using? Euclidean distance is a popular choice...

All I do is count blocks vertically and horizontally to come up w/ H cost. And then my F score is the sum of G + H.
[/quote]
How do you combine your horizontal and vertical counts?

##### Share on other sites

How do you combine your horizontal and vertical counts?

I just add them together and that becomes H.

If I start at 3,3 and the destination is 8,10 ... (8-3=5, 10-3=7) so my H=(5+7)=14. I think the tutorial calls it a manhatton algorithm because it's like counting city blocks, only horiz/vert count.

##### Share on other sites
Yes, that's called the Manhattan distance. That explains why you are exploring diagonal movements first.

Anyway, is it clear now how you handle updates to G?

##### Share on other sites

Yes, that's called the Manhattan distance. That explains why you are exploring diagonal movements first.

Anyway, is it clear now how you handle updates to G?

I won't say its clear, but you've given me much to think about. Let me chew on this a bit. Thanks.

• ### Game Developer Survey

We are looking for qualified game developers to participate in a 10-minute online survey. Qualified participants will be offered a \$15 incentive for your time and insights. Click here to start!

• 11
• 23
• 36
• 16
• 75


text:
## Algebraic & Geometric Topology

### Duality and Pro-Spectra

#### Abstract

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories.

#### Article information

Source
Algebr. Geom. Topol., Volume 4, Number 2 (2004), 781-812.

Dates
Received: 7 August 2004
Accepted: 31 August 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882531

Digital Object Identifier
doi:10.2140/agt.2004.4.781

Mathematical Reviews number (MathSciNet)
MR2100680

Zentralblatt MATH identifier
1054.55010

#### Citation

Christensen, J Daniel; Isaksen, Daniel C. Duality and Pro-Spectra. Algebr. Geom. Topol. 4 (2004), no. 2, 781--812. doi:10.2140/agt.2004.4.781. https://projecteuclid.org/euclid.agt/1513882531

#### References

• M. Ando and J. Morava, A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space, in Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999), 11–36, Contemp. Math., 279, Amer. Math. Soc., Providence, RI, 2001.
• G. Z. Arone, K. Bauer, J. D. Christensen, and D. C. Isaksen, A classification of polynomial functors using pro-spectra, in preparation.
• M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, vol. 269, Springer Verlag, 1972.
• M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer Verlag, 1969.
• J. C. Becker and D. H. Gottlieb, A history of duality in algebraic topology, History of topology, North-Holland, Amsterdam, 1999, 725–745.
• A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric Applications of Homotopy Theory, vol. II (Proc. Conf., Evanston, IL, 1977), Lecture Notes in Mathematics, vol. 658, Springer Verlag, 1978, 80–130.
• R. L. Cohen, J. D. S. Jones and G. B. Segal, Floer's infinite dimensional Morse theory and homotopy theory, in The Floer memorial volume, 297–325, Progr. Math., 133, Birkhäuser, Basel, 1995.
• D. Dugger and D. C. Isaksen, Etale homotopy types and sums-of-squares formulas, in preparation.
• W. G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology, North-Holland, 1995, 73–126.
• D. A. Edwards and H. M. Hastings, Cech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, vol. 542, Springer Verlag, 1976.
• A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, 1997.
• P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, 2003.
• M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, 1999.
• M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208.
• D. E. Hurtubise, The Floer homotopy type of height functions on complex Grassmann manifolds, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2493–2505.
• D. C. Isaksen, A model structure for the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001), 2805–2841.
• D. C. Isaksen, Calculating limits and colimits in pro-categories, Fund. Math. 175 (2002), no. 2, 175–194.
• D. C. Isaksen, Etale realization on the $\A^1$-homotopy theory of schemes, Adv. Math. 184 (2004), no. 1, 37–63.
• D. C. Isaksen, Strict model structures for pro-categories, in Algebraic Topology: Categorical Decomposition Techniques (Skye, 2001), Progress in Mathematics, vol. 215, Birkhäuser, 2003, 179–198.
• D. C. Isaksen, Completions of pro-spaces, Math. Zeit., to appear.
• J. P. May, Stable algebraic topology, 1945–1966, History of topology, North-Holland, Amsterdam, 1999, 665–723.
• C. V. Meyer, Approximation filtrante de diagrammes finis par Pro-$C$, Ann. Sci. Math. Québec 4 (1980), no. 1, 35–57.
• D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, Springer Verlag, 1967.
• E. H. Spanier and J. H. C. Whitehead, Duality in homotopy theory, Mathematika 2 (1955), 56–80.
• D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79.


text:
# Depreciation

Depreciation is a group of methods used in project economics that allows for the CAPEX to be spread out over the useful life of the capital. In many cases such as mills and processing plants, the useful life of capital parallels the life of the project. By spreading this expenditure out, future revenue streams are depreciated by subtracting a portion of these capital expenses. The benefits of this can manifest themselves in both tax and accounting purposes.

When analyzing the cash flows and reported results from an operation, including depreciation in the balance sheet can give a better perspective to what the costs/benefits of production actually are instead of lumping all CAPEX to the beginning of a project. The method used varies from company to company and thus careful attention must be paid to each individual balance sheet and income statement.

For taxation purposes, depreciating can reduce the taxable income at an operation as it is subtracted from gross revenue alongside operating expenses. When dealing with taxation however, the maximum depreciation rate for a specific class of asset is prescribed (at least in Canada) by the government. Most machinery at a mining operation will fall under Class 38 with a rate of 30%, and most buildings will fall under Class 1 with a rate of 6%.[1]

## Methods

Simple syntax used in the depreciation methods is defined as such:

IC represents the initial cost of capital
DCi represents the depreciation charge in year i
BVi represents the book value of the asset in year i
n represents the estimated number of periods in the assets’ useful life
r represents the depreciation rate as a decimal

### Straight-line depreciation

Straight-line depreciation is the simplest of methods and is commonly used because of this. In this method, the capital cost of the asset is spread evenly over each year of its useful life according to the prescribed (or selected) depreciation rate.[2]

$DC = \left( {IC - SV} \right)r$
$n = \frac{1}{r}$

This depreciation charge is constant for each year and thus only needs to be calculated once. Since n and r are inversely related, typically a useful life is estimated for an asset and then the depreciation rate follows. A haul truck with an initial cost of $750,000 and a salvage value of$10,000 exhibits the following characteristics over a 5 year useful life (depreciation rate of 20%) if straight-line is used.

 Year Book Value Depreciation Charge 0 $750,000$0 1 $602,000$148,000 2 $454,000$148,000 3 $306,000$148,000 4 $158,000$148,000 5 $10,000$148,000

### Declining balance depreciation

Declining Balance Depreciation (also known as DBD) methods allocates higher depreciation charges to the beginning of the project. This method is used in scenarios in which the asset is more productive in its earlier years and will generate more revenue and thus will have a higher depreciation charge associated with it in these years. This is accomplished by determining the depreciation charge for the year as a product of the previous book value and current depreciation rate.[2]

$B{V_i} = IC{\left( {1 - r} \right)^i}$

Thus every year, the book value becomes a fraction (1 − r) of the previous year’s value.

$D{C_i} = IC{\left( {1 - r} \right)^{i - 1}}r$


text:
# Count the Palindromes [duplicate]

Given a number n, calculate the amount of bases in the range of [2, n) in which b(n) is a Palindrome.

## Example

n = 8 has the base conversions:

2 = 1000
3 = 22
4 = 20
5 = 13
6 = 12
7 = 11

Of which 2 of them, 3 = 22 and 7 = 11 are palindromes. So return 2.

## Clarifications

• For the sake of convenience, Your answer only needs to be correct for inputs 3 - 36 inclusive, as few languages can handle base conversion of >36.
• A single digit number is considered a palindrome.
• This is sequence A135551 on OEIS

## Test Cases

3   -> 1
4   -> 1
5   -> 2
6   -> 1
7   -> 2
8   -> 2
9   -> 2
10  -> 3
11  -> 1
12  -> 2
13  -> 2
14  -> 2
15  -> 3
16  -> 3
17  -> 3
18  -> 3
19  -> 1
20  -> 3
21  -> 4
22  -> 2
23  -> 2
24  -> 4
25  -> 2
26  -> 4
27  -> 3
28  -> 4
29  -> 2
30  -> 3
31  -> 3
32  -> 3
33  -> 3
34  -> 3
35  -> 2
36  -> 5

## marked as duplicate by Mego♦ code-golf StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 31 '17 at 12:15

• @JonathanFrech actually I think Dom is right - once the sequences are aligned (one starts at index 0 the other at index 1), A126071(n) = A135551(n) + 1. This is evident from the facts that (a) n in Unary is 00...0 [palindromic]; (b) for n>1, n in base n is 10 [not palindromic]; and (c) n in base n+1 is 11 (or 1 for n=1) [palindromic]. Also it even states in the formula section of A135551 "a(n) = A126071(n) - 1". – Jonathan Allan Aug 31 '17 at 7:11
• FYI: My latest submission in Pyth shows the ease of porting. – Jonathan Allan Aug 31 '17 at 7:39
• I have voted to reopen this question, because in my honest opinion the challenges (and tasks) are different enough, and porting the answers there might not be the best option (in some languages, at least). – Mr. Xcoder Aug 31 '17 at 8:56
• @Mr.Xcoder, if wrapping something in a loop from 1 to n is not sufficiently trivial a change to be a dupe, what is sufficiently trivial? – Peter Taylor Aug 31 '17 at 11:38
• @Mr.Xcoder, the number of answers an earlier challenge received is nothing to do with whether it's a dupe or not; "the challenge there is quite old (and many languages have evolved since then, other languages were designed in the mean time" is an argument for keeping it open in spite of being a dupe which was also made in this case to which see also this. If you think an obvious dupe should be reopened in spite of being an obvious dupe, take it to meta first. – Peter Taylor Aug 31 '17 at 11:47

# 05AB1E,  10  8 bytes

-2 bytes thanks to Emigna (use "bifurcation" instruction Â to replace duplicate and reverse DR & increment the loop to avoid needing to close it.)

G¹N>вÂQO

Try it online!

### How?

G¹N>вÂQO
G        - for N in range(1, input()+1):
¹       -   1st input
N      -   N (the loop variable)
>     -   increment
в    -   base conversion
Â   -   push reverse
Q  -   equal?
O -   sum
- print top of stack

• You can replace DR with Â. – Emigna Aug 31 '17 at 6:38
• Also, if we know that n can't be a palindrome in base n you could do G¹N>вÂQO. – Emigna Aug 31 '17 at 6:44
• We know it's not a palindrome in base n, but it's always a palindrome in base 1, so unless there is a loop for [2,n) or [2,n]... EDIT oh wait you increment, but then don't close the loop before summing, how's that work?? – Jonathan Allan Aug 31 '17 at 6:47
• Gloops for [1,n), as we incremented N it goes from [2,n] – Emigna Aug 31 '17 at 6:50
• Because we only have the bools we wish to sum on the stack. No residuals. So we can sum each loop iteration. When we get to O, the only things on the stack are our sum so far and the value we wish to add to the sum from this iteration. – Emigna Aug 31 '17 at 6:51

# J, 30 bytes

[:+/i.&.(-&2)(-:|.)@(#.inv)"0]

Try it online!

Straightforward solution featuring a lot of parentheses.

# Explanation

[: +/ i.&.(-&2) (-:|.)@(#.inv)"0 ]
]  Right argument, n
i.&.(-&2)                     Range [2,n)
-&2                        Subtract 2
i.                              Create range from n-2
"0    For each value in the range
(#.inv)      Convert to that base *
(-:|.)              Palindrome check
|.                 Reversed
-:                   Matches self (= is rank 0)
+/                               Sum (i.e. count palindromes)

*#.inv is used since #: expects you to specify the number of digits in the base (it doesn't treat a single argument the way you might expect).

• nice J. I like the i.&.(-&2) trick. – Jonah Aug 31 '17 at 5:45
• @Jonah thanks, it's not mine though -- saw it on the Project Euler forums (guess some people like to obfuscate -- or perhaps even golf -- their answers) – cole Aug 31 '17 at 17:03

# Python 2, 90 bytes

lambda n:sum(h(n,i)==h(n,i)[::-1]for i in range(2,n))
h=lambda v,b:v and[v%b]+h(v/b,b)or[]

An unnamed function taking n that counts the palindromes, which utilises the helper function h which performs the base conversions.

Try it online!

# Jelly, 6 bytes

bḊŒḂ€S

Try it online!

### How?

bḊŒḂ€S - Link: number, n
Ḋ     - dequeue (with implicit range(n) as input) -> [2,3,4,...,n]
b      - convert to base  -> [nb2, nb3, nb4, ..., nbn] (note: nbn is [1,0])
ŒḂ€  - isPalindrome? for €ach  -> list of 1s and 0s
S - sum -> number that are palindomes

# Mathematica, 57 bytes

Tr@Boole@Table[PalindromeQ@IntegerDigits[#,i],{i,2,#-1}]&

# Ruby, 47 bytes

->n{(2...n).count{|i|(s=n.to_s(i))==s.reverse}}

Try it online!

# Pyke, 10 bytes

tFOAbD_q)s

Try it here!

t          -   input - 1
FOAbD_q)  -  for i in ^:
O        -   ^ + 2
Ab      -    base(input, ^)
D_q   -     ^ == reversed(^)
s - sum(^)

• Also 10 bytes: StFAbD_q)s (porting my Pyth solution) – Mr. Xcoder Aug 31 '17 at 8:41

# Pyth, 9 bytes

lf_IjQTtS

Test it here!

# Pyth, 9 bytes

sm_IjQdtS

Try it here!

# How?

• sm_IjQdtS - Full program with implicit input (Q at the end)

• tS - All the integers from 2 to the input (inclusive). I chose inclusive because it is golfier and any number n converted to base n is 10.

• m - Map over the above with a variable d.

• jQd - Convert the input to base d (the current number in the range) as a list.

• _I - The best part of this program: Is invariant under reverse? – This checks if the list is palindromic.

• s - Sum, which acts like "count the number of truthy results" in this case.

# Pyth,  10  9 bytes

Port of irtosiast's submission to "All your base palindromic belong to us".

-1 byte thanks to FryAmTheEggman - use the filter alias # to replace f and do away with the T (which seems also to have been available back then too).

lt_I#jLQS

Try it online!

• I don't know if this existed for the previous question, but you can change f_IT to _I#. – FryAmTheEggman Aug 31 '17 at 18:07
• – Jonathan Allan Aug 31 '17 at 18:57

# RProgN 2, 18 bytes

x=x1-2R³x\Br³]ier+

This is essentially what I used as a reference implementaiton.

## Explained

x=x1-2R³x\Br³]ier+
x=                  # Associate 'x' with the implicit input.
x1-               # Get x - 1
2R             # Get all numbers in the range 2 through x - 1
r        # Replace each element by the function...
³x\B         # Convert x to base i.
r   # Replace each element by another function...
³]ie    # Duplicate, Inverse, Equals. Checks if it's a palindrome.
+  # The sum of. Which counts how many are palindromes.

Try it online!

# Python 2, 187 bytes

def P(n):
p=0
for b in range(2,n):
M=0
while n>=b**M:M+=1
N,I,B=n,M-1,[]
while N:
for f in range(b)[::-1]:
if f*b**I<=N:N-=f*b**I;B+=[f];I-=1;break
p+=0>(B[::-1]==B)*I
print p

Try it online!

A bit on the lengthy side, as there is no built-in used for changing base.

0#b=[]
x#b=mod x b:div x b#b
f n=sum[1|b<-[2..n],n#b==reverse(n#b)]

Try it online! Use by calling f. Works with arbitrary high bases.

# JavaScript (ES6), 77717068 67 bytes

n=>(g=b=>b<n&&([...s=n.toString(b)].reverse().join==s)+g(++b))(2)

(Golfed from 77 down to 68 while waiting for the challenge to be reopened)

## Test it

f=
n=>(g=b=>b<n&&([...s=n.toString(b)].reverse().join==s)+g(++b))(2)
o.innerText=[...Array(34)].map(_=>(""+x).padStart(2)+": "+f(x++),x=3).join\n
<pre id=o>

# Japt-x, 9 8 bytes

ò2@sX êS

## Explanation

             :Implicit input of integer U.
ò2           :Range [2,U]
@          :Map each X
sX        :  Convert U to a base-X string
êS     :  Is it a palindrome?
:Implicitly reduce by addition and output


text:
# An implementation of the Todd-Coxeter algorithm in python3

There are two main strategies for Todd-Coxeter: Felsch and HLT. These are implemented in the following class ToddCoxeter:


text:
`ISRN Mathematical PhysicsVolume 2012 (2012), Article ID 601749, 16 pageshttp://dx.doi.org/10.5402/2012/601749`
Research Article

## Center-Vortex Loops with One Self-Intersection

Institut für Theoretische Physik, Universität Karlsruhe (TH), Kaiserstrasse 12, 76131 Karlsruhe, Germany

Received 7 December 2011; Accepted 1 April 2012

Academic Editors: E. Akhmedov and S. Ketov

Copyright © 2012 Julian Moosmann and Ralf Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the 2D behavior of one-fold self-intersecting, topologically stabilized center-vortex loops in the confining phase of an SU(2) Yang-Mills theory. This coarse-graining is described by curve-shrinking evolution of center-vortex loops immersed in a flat 2D plane driving the renormalization-group flow of an effective “action.” We observe that the system evolves into a highly ordered state at finite noise level, and we speculate that this feature is connected with 2D planar high superconductivity in FeAs systems.

#### 1. Introduction

The idea of a nontrivial ground state being responsible for the emergence of “elementary” particles is a rather old one: already Lord Kelvin proposed that atoms and molecules should be considered knotted lines of vortices representing distortions in a universal medium (or ground state)—the ether [1]. As we know now, the physics of atoms and molecules described in terms of a much more efficient and elegant framework quantum mechanics. The agent responsible for the chemical bond—Lord Kelvin's electron—is considered a spinning point particle in quantum mechanics, and this yields an excellent description of atomic physics, collider physics, and in the bulk of condensed matter physics.

There are, however, theoretical discrepancies with the concept of the electron being a point particle, and there are exceptional experimental situations pointing to the limitations of this concept to describe reality. As for the former, we have the old problem of a diverging classical self-energy not resolved in quantum electrodynamics where the electron mass is introduced as a free parameter whose running with resolution needs an experimental boundary condition. On the other hand, the two-dimensional dynamics of strongly correlated electrons in condensed matter physics signals the relevance of nonlocal effects possibly related to the nontrivial anatomy of the electron becoming relevant in collective phenomena [24]. Also, recent high-temperature plasma experiments indicate unexpected explosive behavior not unlikely related to the mechanism for lepton emergence, see [5] and references therein.

Early papers, mostly investigating on a 4D Euclidean lattice the role of center vortices in forming a confining ground state at low temperatures/resolution [611], are based on the definition of a dual order parameter for quark confinement by [12]. Recent developments in understanding the confining phase of an SU(2) Yang-Mills theory suggest that Lord Kelvin's ideas may actually be realized in Nature, see also [1316]. The authors of [1316] construct a plausible effective low-energy action for the 4D SU(2) Yang-Mills theory with solutions to the associated field equations representing closed confining strings knotted into stable solitons. In the thermodynamic approach of [17] the emergence of magnetic center-vortex loops (CVLs) is related to discontinuous phase changes of a complex order parameter for confinement across the (downward) Hagedorn transition and the fact that no magnetic charges exist where these flux lines could end. Also, it was discussed in [17] how the locations of topologically stabilized self-intersection represent isolated, spinning magnetic charges. (Notice that with respect to the electromagnetic U(1) of the Standard Model there is a dual interpretation of magnetic charges emerging in an SU(2) Yang-Mills theory).

In our previous article [18] we have investigated the sector with self-intersections by considering a resolution-dependent ensemble average. The corresponding weight-functional is defined purely in terms of the planar curves's geometry. The resolution dependence of this geometry, in turn, is determined by a curve-shrinking equation (heat-equation) [19, 20]. The validity of this description of spatial coarse-graining is motivated by considerations relating local curvature with the direction and speed of “motion” of the associated line-segment. The requirement that the partition function over a given ensemble of planar curves is invariant under a change of the resolution then yields the renormalization-group evolution of the weight-functional which is written as the exponential of an “action.” Here the term “action” is slightly misleading since we do not aim at describing the time-evolution of the system by demanding stationarity of the “action” under curve variation. To do the latter, a model, which relates resolution and time (being a macroscopic concept associated with the measuring apparatus), needs to be introduced. We thus regard resolution over time as the more fundamental quantity to describe certain subatomic systems. Our observation is that the effective “action” exhibits a transition towards dilational invariance after a finite, critical decrease of resolution. On average, CVLs with are shrunk to circular points for a resolution less than the critical value which de facto removes them from the spectrum and thus generates an asymptotic mass gap. (CVLs with are massive [17, 21]). Knowing the evolution of the weight-functional, one is in a position to compute the resolution dependence of “observables” as ensemble averages of the associated (nonlocal or local) “operators.” As for the evolution of the initially sharp center-of-mass position, we observe a spread of the variance with decreasing resolution saturating at a finite value. This is similar to the unitary free-particle evolution of a position eigenstate in quantum mechanics.

The purpose of the present paper is to extend the procedure of [18] to the case of . We now have a singled-out point on the curve: the location of the self-intersection where practically the entire mass of the soliton resides [17]. Setting the Yang-Mills scale of the SU(2) theory equal to the electron mass keV, which in turn determines the mass of the intersection point, we interpret this soliton as an electron or a positron [5, 21]. In the presence of a static electric or magnetic background field it is physically possible to lift the two-fold degeneracy w.r.t. the two possible directions of center-flux: the soliton exhibits a two-fold spin degeneracy. Notice that as long as both wings of center flux are of finite size the position of the intersection point can be shifted at almost no cost of energy. In particular, if the inner angle between in- and outgoing center-flux at the intersection is sufficiently small, then a motion of points on the vortex line directed perpendicular to the bisecting line of the angle easily generates a velocity of the intersection point which exceeds the speed of light, see Figure 1. Recall that the path-integral formulation of quantum mechanics admits such superluminal motion in the sense that the according trajectories sizably contribute to transition amplitudes.

Figure 1: Points on the center flux lines moving oppositely on a line perpendicular to the bisecting line of the angle with velocity modulus . For sufficiently small the velocity modulus of the intersection point is superluminal: .

The paper is organized as follows. In Section 2 we discuss the physics associated with the emergence of topologically stabilized CVLs with intersection number , and how their spatial 2D coarse-graining is captured by a curve-shrinking flow. Some mathematical results on the properties of this flow for immersed curves, which are relevant for our subsequent numerical analysis, are briefly discussed. Also, we repeat our discussion in [18] of how the renormalization-group flow of an effective “action” is driven by the curve-shrinking evolution of the members of a given ensemble of curves. In Section 3 we explain our numerical analysis concerning the computation of the effective “action,” the variance of the location of the self-intersection, and the entropy associated with a given ensemble. Finally, in Section 4 we summarize our results and interpret them in view of certain 2D layered, quasimetallic systems exhibiting high- superconductivity.

#### 2. Conceptual Framework

##### 2.1. Self-Intersecting Center-Vortex Loops

The transition from the non-self-intersecting to the self-intersecting CVL sector is by twisting of non-self-interesecting curves. The emergence of a localized (anti)monopole in the process is due to its capture by oppositely directed center fluxes in the intersection core (eye of the storm). By a rotation of the left half-plane in Figure 2(a) by an angle of , see Figure 2(b), each wing of the CVLs forms a closed flux loop by itself thereby introducing equally directed center fluxes at the intersection point. This does not allow for an isolation of a single, spinning (anti)monopole in the core of the intersection and thus is topologically equivalent to the untwisted case Figure 2(a). However, another rotation of the left-most half-plane in Figure 2(c) introduces an intermediate loop which by shrinking is capable of isolating a spinning (anti)monopole due to oppositely directed center fluxes. Notice that in the last stage of such a shrinking process (short distances between the cores of the flux lines), propagating dual gauge modes are available. (On large distances these modes are infinitely massive which is characteristic of the confining phase): There is repulsion due to Biot-Savart which needs to be overcome. This necessitates an investment of energy manifesting itself in terms of the mass of the isolated (anti)monopole (eye of the storm). Alternatively, the emergence of an isolated (anti)monopole is possible by a simple pinching of the untwisted curve, again having to overcome local repulsion in the final stage of this process.

Figure 2: (Topological) transition from the sector (a), (b), (c) to the sector (d) by twisting and subsequent capture of a magnetic (anti)monopole in the core of the final intersection. Arrows indicate the direction of center flux.

For the analysis performed in the present work we solely regard the situation depicted in Figure 2(d) and thus no longer need to discuss the direction of center flux within a given curve segment. This is not relevant for the process of a spatial coarse-graining microscopically described by the same curve-shrinking flow as applied to sector with [18].

##### 2.2. Euclidean Curve Shrinking Flow

The restriction of evolution of a CVL with self-intersection number to the plane is an essential constraint on generality if we aim at a fundamental description of the effectively quantum mechanical behavior of a charged lepton (electron, muon, tau-lepton). An electron bound inside a hydrogen atom certainly “moves” in 3D and quantum mechanics describing it as a spinning, relativistic point particle is in accurate agreement with experiment. The quantum mechanically incompletely understood condensed-matter physics of strongly correlated 2D electrons, however, is a possibly fertile application field of our restriction of curve evolution to the plane, see Section 4.

Notice that by immersing an SU(2) CVL with finite core size and mass of the dual gauge field into a flat 2D surface at , , a hypothetic observer measuring a positive (negative) curvature of a segment of the vortex line experiences more (less) negative pressure in the intermediate vicinity of this curve segment leading to its motion towards (away from) the observer, see Figure 3.

Figure 3: Highly space-resolved snapshot of a CVL segment. The pressure in the region pointed to by the normal vector is more negative than the pressure thus leading to a motion of the segment along .

The (inward directed) speed of a point in the core of the vortex will be a monotonic function of the curvature at this point. On average, this shrinks the CVL. Alternatively, one may globally consider the limit , , that is, the confining phase of an SU(2) Yang-Mills theory, but now take into account the effects of an environment which locally relaxes this limit (by collisions) and thus also induces curve shrinking. This situation is described by the following equation for a flow in the (dimensionless) parameter : where is arc length, is a point on the CVL in the plane, and is a string tension effectively expressing the distortions induced by the environment. After a rescaling, , , (2.1) assumes the following form where is a (dimensionless) curve parameter, the (inward-pointing) Euclidean unit normal, the scalar curvature, defined as , and denotes the Euclidean scalar product of the vectors and . In the following we resort to a slight abuse of notation by using the same symbol for the functional dependence on or .

We now consider curves with one self-intersection, that is, , in the sense of the stable situation of Figure 2(d). This situation was mathematically analysed in [22]. Since the direction of center flux is inessential for the shrinking process we may actually treat this situation in a way as depicted in Figure 2(b), where the curve is defined to be a smooth immersion into the plane with exactly one double point and a total rotation number zero, . Here the (dimensionless) curve length is given by the smooth integration . Notice that this is topologically distinct from the case Figure 2(d) where one encounters a nonvanishing rotation number which is not smoothly deformable to zero.

In the case a smooth, embedded curve shrinks to a circular point under the flow for [19, 20]. That is, the isoperimetric ratio approaches from above. The curve in situation Figure 2(b) separates the plane into three disjoint areas two of which are finite and denoted by and . We understand by the finite, critical value of where either or or both vanish. This corresponds to a singularity encountered and thus terminates the flow.

Recall that in the case the rate of area change is a constant, . This is no longer true for . However, we have that Also, for we have in comparison to the case the more relaxed constraint that .

In contrast to the case the isoperimetric ratio for the case is bounded for if and only if . Notice that the case physically is extremely fine-tuned.

##### 2.3. Effective “Action”

We now wish to interpret curve-shrinking as a Wilsonian renormalization-group flow taking place in the  CVL sector in the sense defined in Section 2.2. A partition function, defined as a statistical average (according to a suitably defined weight) over  CVLs, is to be left invariant under a decrease of the resolution determined by the flow parameter . Notice that, physically, is interpreted as a strictly monotonic decreasing (dimensionless) function of a ratio where () are mass scales associated with an actual (initial) resolution applied to the system. The role of can also be played by the finite temperature of a reservoir coupled to the system.

To devise a geometric ansatz for the effective “action” , which is a functional of the curve representable in terms of integrals over local densities in (reparametrization invariance), the following reflection on symmetries is in order. (i) Scaling symmetry : for , implying at fixed , the “action” should be invariant under further finite rescalings (decoupling of the fixed length scales and ), (ii) Euclidean point symmetry of the plane (rotations, translations, and reflections about a given axis): Sufficient but not necessary for this is a representation of in terms of integrals over scalar densities w.r.t. these symmetries. That is, the “action” density should be expressible as a series involving products of Euclidean scalar products of , or constancy. However, an exceptional scalar integral over a nonscalar density can be devised. Consider the area , calculated as The density in (2.5) is not a scalar under translations.

We now resort to a factorization ansatz as where in addition to Euclidean point symmetry () is (is not) invariant under . In principle, infinitely many operators can be defined to contribute to . Since the evolution homogenizes the curvature except for a small vicinity of the intersection point higher derivatives of w.r.t. should not be of importance. We expect this to be true also for Euclidean scalar products involving higher derivatives . To yield conformally invariant expressions such integrals need to be multiplied by powers of and/or or the inverse of integrals involving lower derivatives. At this stage, we are not capable of constraining the expansion in derivatives by additional physical or mathematical arguments. To be pragmatic, we simply set equal to the isoperimetric ratio: We conceive the nonconformal factor in as a formal Taylor expansion in inverse powers of or due to the property of conformal invariance for .

Since we regard the renormalization-group evolution of the effective “action” as induced by the flow of an ensemble of curves, where the evolution of each member is dictated by (2.2), we allow for an explicit dependence of the coefficient of the lowest nontrivial power or . In principle, this sums up the contribution to of certain higher-power operators which do not exhibit an explicit dependence. Hence we make the following ansatz: The initial value is determined from a physical boundary condition such as the mean length at .

##### 2.4. Geometric Partition Function

Let us now numerically investigate the effective “action” resulting from a partition function w.r.t. a nontrivial ensemble . The latter is defined as the average over the ensemble . Let us denote by an ensemble consisting of curves where is obtained from by adding a new curve . The effective “action” in (2.6) (when associated with the ensemble we will denote it by ) is determined by the function , compare with (2.8), whose flow follows from the requirement of -independence of : This is an implicit, first-order ordinary differential equation for which needs to be supplemented with an initial condition . A natural initial condition is to demand that the quantity coincides with the algebraic mean defined as From a value for follows. We also have considered a modified factor While, due to (2.13), the ansatz for the geometric effective "action" in (2.6) thus is profoundly different for such a modification of physical results such as the evolution of the variance of the intersection agree remarkably well, see Section 3.

#### 3. Results of Simulation

##### 3.1. Preparation of Ensembles

Similar as in [18] we normalize all curves to have the same initial area , and, since we are now interested in the position of the intersection where the (anti)monopole, is localized, we have applied a translation to each curves in the ensembles such that the location of the intersections initially coincides with the origin.

Since the critical value of the flow parameter varies from curve to curve, we order the members of the maximal-size ensemble into subensembles such that . The types of ensembles obtained in this way are referred to as -ordered. We also have performed all simulations with ensembles whose members are picked randomly from and have obtained strikingly similar results for ensemble averages of “observables” using and for the evolution to the left of .

The maximal-size ensemble at is depicted in Figure 4 with the universal choice . The curves in Figure 4 are arranged in a -ordered way. We have . In Figure 5 the evolution of an initial curve under curve shrinking is shown from two viewpoints. The flow is started at and stopped at a value of shortly below . In Figure 6 the flow of the intersection points, corresponding to the initial curves depicted in Figure 4, is shown. The search for solutions to the second-order partial differential equation (PDE) (2.2) subject to periodic boundary conditions in the curve parameter, , and for the initial conditions depicted in Figure 4 was performed numerically using the method of lines. That is, the PDE was discretized on a uniform grid in the parameter yielding a semidiscrete problem in terms of a system of ordinary differential equations (ODEs) in which was solved using Mathematica. Figure 5 indicates why this technique is called the numerical method of lines. As one can also see from Figure 5, a set of discrete points on the curve, although remaining equidistant in , may evolve under the flow such that the spatial distances between next-neighbours-points fall below the numerical precision. Numerically, the flow then encounters a singularity (not to confused with the earlier mentioned nonfictitious singularities). To recognize such a situation automatically, (2.4) was exploited: the evolution was stopped as soon as a sizable deviation occured from what (2.4) predicts. The configuration obtained at this point in was fitted in such a way that a new discretization in yielded well-separated points to restart the methods of lines. (2.4) was also used as an indicator for the final singularity at where or or both vanish.

Figure 4: Initial curves contributing to the ensemble . The intersection points coincide with the origin, and all curves have the same area . By definition is -ordered.
Figure 5: Plot of the evolution of an CVL (curve 12 of Figure 4) under (2.2). The thick central line indicates the trajectory of the intersection point which coincides with the origin at .
Figure 6: Flow of the intersection points for the initial curves depicted in Figure 4.
##### 3.2. Renormalization-Group Invariance of Partition Function

For all ensembles , the dependence of the coefficient in (2.8) roughly behaves like a square root , where is the weakly ensemble-dependent minimal resolution. For the modified “action” the coefficient is well approximated by a linear function . Again, denotes a weakly ensemble-dependent minimal resolution. For -ordered ensembles, the results for for the “actions” (2.8) and (2.13) are shown in Figures 7 and 8, respectively. The results for ensembles do not differ sizably from those presented in Figures 7 and 8.

Figure 7: The squares of the coefficients entering the ansatz for effective “action” of (2.6) specializing to (2.8) for -ordered ensembles up to .
Figure 8: The coefficient entering the ansatz for the effective “action” of (2.6) specializing to (2.13) for -ordered ensembles up to .
##### 3.3. Variance of Location of Self-Intersection

The mean intersection over the ensemble is defined as where is the location of self-intersection (intersection point) of curve at . The scalar statistical deviation of over the ensemble is defined as where and similarly for the coordinate . In Figure 9 plots of are shown when evaluated over the ensembles subject to the “action” and the initial condition . In Figure 10 the according plots of are depicted as obtained with the “action” and subject to the initial condition . Relaxing the constraint of -ordering () does not entail a qualitative change of the results.

Figure 9: Plots of for the -ordered ensembles with . We have employed the ansatz for the “action” .
Figure 10: Plots of for the -ordered ensembles with . We have employed the ansatz for the “action” .

The results presented in Figures 9 and 10 are unexpected since in the sector the variance of the “center-of-mass” saturates rapidly to finite values. In contrast, for the sector the variance of the location of the self-intersection initially increases, reaches a maximum, and decreases to zero at a finite value of . This is readily confirmed by the evaluation of the entropy, see Section 3.4.

##### 3.4. Evolution of Entropy

Let us now evaluate the flow of entropy defined as where is given by (2.6). In Figure 11 plots are shown for . when evaluated with the “action” for -ordered ensembles of size . These graphs look very much alike to the ones generated using the “action” . Notice the continuous approach to zero at finite values of . This implies that order emerges spontaneously in the system with decreasing resolution: starting at a finite value of , a particular member of is singled out by its weight approaching unity. Judging from our results for the sector [18], this behavior is highly unexpected. Therefore the nontrivial topology of induces qualitative differences to the coarse-graining process.

Figure 11: Flow of the entropies for -ordered ensembles of size when evaluated with the “action” . The situation does not change qualitatively if the “action” is used.

#### 4. Summary, Interpretation of Results, and Conclusion

In this paper we have investigated the spatial coarse-graining of CVLs, immersed in a flat 2D plane, of an SU(2) Yang-Mills theory being in its confining phase. The focus was on the sector with one topologically stabilized self-intersection (existence of an isolated magnetic charge at its location, ). We have analysed this coarse-graining process in terms of curve ensembles generated by evolving an initial situation under the curve-shrinking flow [19, 20, 22]. The idea here is to suppose that curve shrinking in the parameter represents an exact coarse-graining of a given initial state and to reconstruct the associated ensemble-weight of the statistical approach (exponential of effective “action”) by demanding invariance of the corresponding partition function under the flow in (renormalization-group evolution). Notice that is related to a physical resolution, applied to probing the system, in a strictly monotonic decreasing manner. This resolution may be associated with a local momentum transfer exerted by an observer or a globally defined temperature of an environment. The functional dependence of on these physical parameters depends on the given experimental situation. It is, however, reasonable to assume that finite values of universally correspond to finite values of these physical parameters.

In Sections 3.3 and 3.4 we have obtained the unexpected result that a statistical ensemble of renormalization-group evolved curves spontaneously orders itself in the sense that, starting from finite values of , only a particular member of the ensemble survives the process of 2D spatial coarse-graining. That is, the entropy attributed to the ensemble is practically zero for sufficiently large values of . For the location of self-intersection (charge of an electron) this means that no dissipation of energy, provided by the environment, can be mediated by the monopole situated within the core of the intersection if the resolution falls below a critical, finite value. This result must drastically depend on the two-dimensionality of space and the fact that we consider the sector with , compare with [18].

The recently discovered, unconventional systems do not appear to exhibit an explicit, strong correlation between the electrons contained in their theoretically suggested, 2D-superconducting layers, see [23] and references therein. If the two-dimensional behavior of noninteracting electrons, subject to an environment represented by the parameter , indeed is described by the coarse-graining process investigated in the present work, then the sudden decrease of entropy that we observe at a finite value of should ultimately be connected to this particular kind of high- superconductivity. Here is a monotonically decreasing function of temperature.

#### Acknowledgment

The authors would like to thank Francesco Giacosa and Markus Schwarz for useful conversations.

#### References

1. W. T. Kelvin and P. G. Tait, Treatise on Natural Philosophy, vol. 2, Cambridge University Press, 1867.
2. J. G. Bednorz and K. A. Müller, “Possible high ${T}_{c}$ superconductivity in the Ba-La-Cu-O system,” Zeitschrift für Physik B Condensed Matter, vol. 64, no. 2, pp. 189–193, 1986.
3. P. W. Anderson, “Present status of the theory of high ${T}_{c}$ cuprates,” http://arxiv.org/abs/cond-mat/0510053.
4. P. W. Anderson, “Twenty years of talking past each other: the theory of high ${T}_{c}$,” Physica C, vol. 460-462, pp. 3–6, 2007.
5. F. Giacosa, R. Hofmann, and M. Schwarz, “Explosive Z pinch,” Modern Physics Letters A, vol. 21, no. 36, pp. 2709–2715, 2006.
6. G. Mack and V. B. Petkova, “Comparison of lattice gauge theories with gauge groups ${Z}_{2}$ and $\mathrm{SU}\left(2\right)$,” Annals of Physics, vol. 123, no. 2, pp. 442–467, 1979.
7. G. Mack, “Predictions of a theory of quark confinement,” Physical Review Letters, vol. 45, no. 17, pp. 1378–1381, 1980.
8. G. Mack and V. B. Petkova, “Sufficient condition for confinement of static quarks by a vortex condensation mechanism,” Annals of Physics, vol. 125, no. 1, pp. 117–134, 1980.
9. G. Mack and E. Pietarinen, “Monopoles, vortices and confinement,” Nuclear Physics B, vol. 205, no. 2, pp. 141–167, 1982.
10. H. B. Nielsen and P. Olesen, “A quantum liquid model for the QCD vacuum. Gauge and rotational invariance of domained and quantized homogeneous color fields,” Nuclear Physics B, vol. 160, no. 2, pp. 380–396, 1979.
11. E. Tomboulis, “'t Hooft loop in $\mathrm{SU}\left(2\right)$ lattice gauge theories,” Physical Review D, vol. 23, no. 10, pp. 2371–2383, 1981.
12. G. 't Hooft, “On the phase transition towards permanent quark confinement,” Nuclear Physics B, vol. 138, no. 1, pp. 1–25, 1978.
13. L. Faddeev and Antti J. Niemi, “Stable knot-like structures in classical field theory,” Nature, vol. 387, no. 6628, pp. 58–61, 1997.
14. L. Faddeev and A. J. Niemi, “Partially dual variables in $\mathrm{SU}\left(2\right)$ Yang-Mills theory,” Physical Review Letters, vol. 82, no. 8, pp. 1624–1627, 1999.
15. L. Faddeev and A. J. Niemi, “Aspects of electric and magnetic variables in $\mathrm{SU}\left(2\right)$ Yang-Mills theory,” Physics Letters. B, vol. 525, no. 1-2, pp. 195–200, 2002.
16. L. Faddeev and A. J. Niemi, “Spin-charge separation, conformal covariance and the $\mathrm{SU}\left(2\right)$ Yang-Mills theory,” Nuclear Physics B, vol. 776, no. 1-2, pp. 38–65, 2007.
17. R. Hofmann, “Nonperturbative approach to Yang-Mills thermodynamics,” International Journal of Modern Physics A, vol. 20, no. 18, pp. 4123–4216, 2005, Erratum vol. 21, pp. 6515–6523, 2006.
18. J. Moosmann and R. Hofmann, “Evolving center-vortex loops,” http://arxiv.org/abs/0804.3527.
19. M. Gage and R. S. Hamilton, “The heat equation shrinking convex plane curves,” Journal of Differential Geometry, vol. 23, no. 1, pp. 69–96, 1986.
20. M. A. Grayson, “The heat equation shrinks embedded plane curves to round points,” Journal of Differential Geometry, vol. 26, no. 2, pp. 285–314, 1987.
21. R. Hofmann, “Yang-Mills thermodynamics,” http://arxiv.org/abs/0710.0962.
22. M. A. Grayson, “The shape of a figure-eight under the curve shortening flow,” Inventiones Mathematicae, vol. 96, no. 1, pp. 177–180, 1989.
23. H.-H. Klauss and B. Büchner, “Neuer Goldrausch in der Supraleitung,” Physik Journal, vol. 7, p. 18, 2008.


text:
1 of

• Views: 228

#### Description

Testing Attention Restoration in a Virtual Reality Driving Simulator

## Introduction

Increasingly, research has demonstrated that road characteristics can impact both driving behavior and the activities the driver will undertake once arrived to destination (Antonson et al., 2009; Keay et al., 2009; Calvi, 2015; Murphy and Greene, 2016; Cassarino and Murphy, 2018). Urban roads present higher road complexity than rural roads and can impose more cognitive demands on the driver (Murphy and Greene, 2016). Higher cognitive demands translate not only into less safe driving, but also into poorer cognitive performance after the drive (Murphy and Greene, 2016). These findings support the idea that environmental situations that are perceptually complex (e.g., presenting visual clutter) engage more attentional resources and are thus more cognitively fatiguing (Lavie et al., 2004). Complementing this hypothesis, encouraging evidence suggests that the impact of road characteristics on drivers’ attentional resources may depend on the presence of natural elements. A review on roadside features and driving safety (Wilde, 2010) suggested that green scenery on the road can have restorative effects on attention. Similarly, a recent review indicated that roadside vegetation can reduce drivers’ stress and frustration (Van Treese et al., 2017). This research is informed by Attention Restoration Theory (ART, Kaplan and Kaplan, 1989; Kaplan, 1995); ART suggests that natural environments are more restorative for attention than urban settings because engaging bottom-up involuntary attention (defined as “soft fascination”) while “freeing-up” top-down directed attentional resources (Kaplan and Berman, 2010; Bratman et al., 2012; Berto, 2014). Several behavioral studies support ART by showing that even a brief exposure to natural vs. urban settings, either through walking or seeing images, can relieve from the attentional fatigue caused by a cognitive task completed prior to the environmental exposure (Hartig et al., 2003; Berto, 2005; Berman et al., 2008). Supportive evidence has come from neuroimaging as well (Martínez-Soto et al., 2013; Bratman et al., 2015; Chen et al., 2016), although recent systematic reviews have shown that restorative effects are small (de Keijzer et al., 2016; Ohly et al., 2016).

While most studies on attention restoration have used walking as a form of real-life exposure to nature, very little is known about cognitive restoration in relation to driving, which requires monitoring of the road, and therefore a certain level of attentional engagement. If nature engages bottom-up attention only, one should expect that driving in roads with natural elements, such as rural roads, should be associated with less attentional fatigue than driving on urban roads. In line with this hypothesis, an experimental study used a pre- and post- design where participants were mentally stressed before being exposed to video-tapes of either highway vegetation or roads with man-made material of 5-min duration (Cackowski and Nasar, 2003); an assessment of mental stress after exposure found higher tolerance to frustration in participants who viewed natural rather than urban roads. However, no studies to our knowledge have tested ART while driving using the classic experimental paradigm described above (i.e., changes in attention).

In the present study, we used a simulated driving paradigm to test the hypothesis that driving in an urban or rural virtual environment would differentially impact on attentional fatigue after completing a demanding cognitive task (Sustained Attention to Response Task or SART, Robertson et al., 1997), as shown in previous studies on nature and sustained attention (Berto, 2005). The SART is a measure of attentional capacity as well as the ability to inhibit unwanted responses for a prolonged time; it has been used in previous investigations of ART as a way to mentally fatigue participants before exposure to natural or urban scenes and to measure attention restoration after exposure (Berto, 2005). In the present study, a pre-post design was employed, whereby participants performed the SART task before and after either a rural or urban drive. Assuming that rural roads are more restorative (i.e., less cognitively demanding) than urban roads due to the presence of green, we hypothesized that driving through a rural rather than urban environment after having been mentally fatigued would be associated with improvements in attentional performance at the end the drive. Given the very high usage of cars in our society, investigating the impact of road nature on drivers’ cognitive abilities has important implications for enhancing our understanding of how road characteristics influence cognitive performance.

## Materials and Methods

### Participants

In line with Berto (2005), a total of 38 participants (Mean age = 22.1, SD = 3.43; 44% female) were recruited through convenience sampling among undergraduate and graduate students at University College Cork, Ireland. Participants were randomly assigned to an urban or rural environmental exposure (n = 19 in each group). Half of the participants (n = 19) were fully licensed drivers with an average of 5.5 years of driving experience (SD = 3.24), whereas the other half (n = 19) included individuals with no full license and mean driving experience of 2.3 years (SD = 3.81). All participants read and signed a consent form prior to data collection in accordance with the Declaration of Helsinki. Ethical approval for the study was received by the School of Applied Psychology Ethics Committee, University College Cork. All participants read an information sheet briefing on the aims of the study and all were asked to read and sign a written consent form prior to participation in the study. No vulnerable populations were included in the study.

### Design

A 2 × 2 mixed between-within design was employed, with the participants’ performance at SART, (assessed pre- vs. post- exposure to virtual reality environments in a full vehicle driving simulator) as the within-subjects factor; and Environment type (urban vs. rural) as the between-subjects factor.

### Material and Apparatus

#### Sustained Attention to Response Task (SART)

The SART is an experimental paradigm used to measure sustained attention (Robertson et al., 1997). In this task, participants viewed a random sequence of digits (1–9) appearing on the central projector screen of the simulator, while sitting in the vehicle (see Figure 1A). A computer keyboard was placed on the participant’s lap and they were asked to press the spacebar as quickly as possibly at the appearance of each digit, except for the digit three. The numbers appeared on the center screen of the simulator. The task was programmed in E-Prime 2.0 software.

Figure 1. (A) Sustained Attention to Response Task (SART) paradigm: A sequence of digits was viewed by the participants while sitting in the vehicle; digits appeared in the central project screen of the simulator. Participants completed the SART before and after driving in the simulated environment, consisting of a computerized graphic depicting a rural (B) or urban (C) scenario.

Before the experimental block there were 18 practice trials so that participants were accustomed to the task and apparatus. The experimental block consisted of 252 trials (28 of each digit between one and nine) presented in one of five semi-randomly assigned fonts in the range of 12–29 centimeters. In the test trial, the target stimulus (i.e., the number 3) appeared 28 times, while the remaining 224 digits were non-lures. Digits appeared on the screen every 1,125 ms, for the duration of 200 ms, followed by a 900 ms mask, which was a diagonal cross contained within a 29-centimeter ring. Both the digits and the mask were white against a black background. Instructions on how to complete the task were showed on the computer screen prior to the appearance of both the practice and the test trial.

#### Virtual Reality Environments

The driving simulator we utilized is considered to be a medium-level driving simulator (not high as it is not placed on a steward platform). It consists of a full-size Volkswagen Polo vehicle with manual transmission which has all vehicle controls available including functional speedometer and tachometer. The vehicle faces 3 large projection screens and has LCD screens in the wing mirrors and a rear projection screen which can be viewed through the rear-view mirror. The vehicle is equipped with 7.1 Dolby surround sound, enabling the creation of a more immersive environment with engine sounds as well as noises from other road users such as beeping or harsh braking. The simulator is housed in a dark, cool room with black-out blinds, black walls and a fan to provide airflow. In the adjacent control room, the experimenter can monitor the participants’ progress. The simulator uses STISIM 400W software (STISIMdrive.com) which allows for flexible programming various driving environments.

A computerized graphic rural drive and an urban drive were designed for the study, as shown in Figures 1B,C. The rural drive (Figure 1B) presents a road surrounded by trees (isolated or in groupings) and fields; while the urban drive presents the same kind of road surrounded by some pedestrians and buildings either low rise commercial or tall commercial or residential buildings (Figure 1C). All drives were designed to minimize the occurrence of features which are known to cause simulator sickness, such as curves or sudden stops (Classen et al., 2011). Similar to previous studies on ART, the scenarios were pilot tested for perceived pleasantness and restorative potential with a separate group of participants utilizing the Attention Restoration Scale (Hartig et al., 1997). Participants were assessed for motion sickness through a questionnaire before and after the drive, closely monitored for any signs of sickness, offered regular breaks and reminded that they could withdraw from the study at any point.

Driving behavior was recorded in terms of average speed, standard deviation from the average speed, lane position, and mistakes (including, number of occurrences of red-light tickets, speed excess, collisions, and road lane excursions).

### Procedure

Participants were first introduced to the vehicle and its controls. They were then given a 10-min practice drive to become accustomed to the responsiveness of the vehicle. This practice drive consisted of a mix between urban and rural environments. Once participants were comfortable the practice drive was stopped, and they were presented with the first SART task. Once they completed the SART task they were randomly assigned to drive in the rural or urban environment. During the drive, which lasted 10 min, participants were asked to maintain a speed of approximately 60 Km/h. The duration of the test drive was based on Berto (2005), who found that 10 min of viewing images of natural scenes was enough for participants to experience restoration. Also, previous studies have demonstrated differences in cognitive performance between drivers exposed to different scenarios after a 10–12 min test drive (Murphy and Greene, 2016). This duration was also chosen to avoid potential discomfort for the participants. The participants were then asked to complete the SART again (Session 2), after which they filled a short demographic questionnaire.

### Statistical Analyses

Participants’ performance at the SART was analyzed in terms of d-prime (d’: a measure of signal detection sensitivity, calculated as the standardized difference (z-scores) between the proportion of correct responses on non-lures minus the proportion of incorrect responses on lures), overall mean accuracy (proportion of correct responses on lures and non-lures), mean accuracy on non-lures (pressing the bar), accuracy on lures (not pressing the bar when number three appears), reaction times (in milliseconds) of correct responses (related to pressing the bar in the presence of a non-lure), and inverse efficiency, a measure of speed-accuracy trade-off calculated as the ratio of reaction times over accuracy on non-lures (Bruyer and Brysbaert, 2011). Comparisons between the two exposure groups in terms of gender were conducted using Chi-square test and potential differences in age and driving experience were investigated via an independent samples t-test. These comparisons were carried out to decide whether demographic status or driving experience should be included in the subsequent analyses as covariates. A 2 × 2 mixed-design ANOVA was conducted with Environment (rural vs. urban) as the between-subjects factor, and SART (pre- vs. post-drive) as the within-subjects factor to investigate effects of environmental exposure on changes in attentional performance pre- and post-drive. Post hoc comparisons were conducted via t-test statistics. Comparisons between exposure groups in terms of driving behavior were assessed via independent t-test. In addition, potential effects of driving on attention were tested through a 2 (SART session) × 2 (environmental exposure) × 2 (driving vs. passenger condition) ANOVA with Driving (driver or passenger) and Environment (urban vs. rural) as the between-subject factors, and SART (pre- vs. post-drive) as the within-subjects factor. We conducted a test of normality on the ANOVA unstandardized residuals as well as the Levene’s test of homogeneity (see Supplementary File 1); for measures that did not appear to meet the assumptions of normality, we conducted the analyses using non-parametric tests and found no differences in results (see Supplementary File 1).

## Results

### Environmental Exposure Effects on Attention

The two exposure groups (n = 19 in each group) did not differ significantly in terms of gender (

${{\mathrm{\chi }}}_{{\mathrm{1}}}^{{\mathrm{2}}}$

= 0.11, p = 0.74), age (t36 = -0.42, p = 0.67) or driving experience (t36 = 0.16, p = 0.87).

Table 1. Interaction between environmental exposure and SART session – driving sample.

There was a main effect of environmental exposure for the measure of d’ (F1,36 = 4.18, p = 0.048, μ2 = 0.11), with participants in the rural exposure group (M = 1.26, SD = 1.07) showing overall higher sensitivity (i.e., better performance) than the urban exposure group (M = 0.62, SD = 0.84). There was also a main effect of environmental exposure for the measure of accuracy on lures (F1,36 = 4.61, p = 0.04, μ2 = 0.11), with participants in the rural group (M = 0.64, SD = 0.25) being overall more accurate than those in the urban group (M = 0.48, SD = 0.21). In both cases, however, the size of the effect was small.

We found that the driving behavior of two exposure groups did not differ significantly for any of the measures of interest: average speed (t35 = 0.21, p = 0.84), standard deviation from average speed (t35 = 0.61, p = 0.55), average lane position (t36 = 0.03, p = 0.97), standard deviation from average lane position (t36 = -1.71, p = 0.09), speed excess (t36 = 0.45, p = 0.65), or lane excursions (t36 = 1.67, p = 0.11).

### Testing for the Effect of Driving

As an additional check on our study, we conducted a control study whereby we recreated the same situation, but participants were not required to drive. This was included so that the act of driving could be dissociated from viewing motion. We initially recruited 24 participants (12 in the rural condition and 11 in the urban group); however, eight participants (four in each condition) did not complete the driving scenario due to motion sickness or unwillingness, leaving a final sample of 15 participants (Mean age = 31.26, SD = 6.69; 53.3%% female); these completed the SART before and after a 10-min exposure to the virtual environment (n = 8 rural vs. n = 7 urban road) while seating in the driver’s seat but not driving.

We ran a 2 (SART session) × 2 (environmental exposure) ANOVA for this group with exposure (urban vs. rural) as the between-subject factors, and SART (pre- vs. post-drive) as the within-subjects factor. As shown in Table 2, no interactions emerged for any of the measures of interest.

Table 2. Interaction between environmental exposure and SART session – control sample.

Similarly, no main effects of environmental exposure emerged. A main effect of session was noted for total accuracy (F1,13 = 5.22, p = 0.04, μ2 = 0.27) with an overall small improvement from baseline (M = 0.63, SD = 0.16) to post-exposure (M = 0.68, SD = 0.16).

We then pooled together the data (N = 53) from the two samples (driving, n = 38; non-driving, n = 15), and ran a 2 (SART session) × 2 (environmental exposure) × 2 (driving vs. passenger condition) ANOVA with Driving (driver or passenger) and Environment (urban vs. rural) as the between-subject factors, and SART (pre- vs. post-drive) as the within-subjects factor. As the driving group was older than the non-driving group (t51 = 6.59, p = 0.000, Cohen’s d = 1.72), we included age as a covariate in the ANOVA.

Controlling for age, we found no significant interactions (not shown); a main effect of driving condition emerged for all measures except inverse efficiency (d’: F1,48 = 49.93, p = 0.000, μ2 = 0.48; total accuracy: F1,48 = 65.41, p = 0.000, μ2 = 0.52; accuracy on lures: F1,48 = 6.85, p = 0.01, μ2 = 0.12; accuracy on non-lures: F1,48 = 50.02, p = 0.000, μ2 = 0.45; reaction times: F1,48 = 31.18, p = 0.000, μ2 = 0.38). Specifically, participants who drove were significantly more accurate and slower at the SART than those in the control group (i.e., not driving) both before and after exposure, and independent of exposure condition.

Only in the case of accuracy on non-lures, a main effect of exposure also emerged (F1,48 = 5.34, p = 0.03, μ2 = 0.05), with participants exposed to the rural environment being overall more accurate (M = 0.92, SD = 0.11) than those exposed to the urban environment (M = 0.88, SD = 0.21); however, the effect size of environmental exposure was smaller than that of driving.

## Discussion

We investigated whether potential differences in driving behavior could have influenced these results, however the two exposure groups drove with similar speed (as requested, with few infractions) and accuracy. We also tested whether the driving task could influence the effect of being exposed to urban or rural environments by re-running the experiment in a sample of participants who seated in the car but did not drive, as a further control condition. No restorative effects were noted in this group either, while a small practice effect was found. When comparing the two driving groups (driving vs. passenger), we found that participants who drove were more accurate but slower at the SART than those who were passengers, showing more conservative performance in both sessions. These differences did not appear to depend on sample characteristics such as age. Notably, the passenger group showed an overall improvement in accuracy independent of exposure, which might indicate a practice effect and possibly that the virtual immersion served as a resting interval for both exposure groups (i.e., not driving might have led the participants to not engage enough with the virtual environment to generate restorative effects). However, this interpretation of the results needs to be considered with caution, as the differences between driving conditions at baseline did not depend on the type of environmental exposure and might be related either to a selection bias which we were unable to capture or to the potential effects of the different types of instructions provided to participants at the beginning of the experiment (i.e., one group was asked to drive while the other sat in the car, and this might have created different expectations as well as different levels of engagement with the experiment).

One could argue that completing the SART was an easy task for our sample, and therefore did not cause attentional fatigue. However, the performance of our sample was worse than that of Berto (2005), who used the same cognitive task in a sample of similar age. While this comparison supports the idea that our participants were mentally fatigued by the SART, future studies might assess other measures of mental fatigue other than the SART.

A limitation of our study is the realism of the urban and rural drives, which are clearly a simulation and therefore less rich than real natural scenes in terms of soft fascination features. Nonetheless, these scenarios were pilot tested for perceived pleasantness and restorative potential with a separate group of participants. When considering the study by Berto (2005), which also utilized the SART, it is important to note two critical differences: Firstly, the restorative scenes used in the driving simulator were not photographs of real environments. It is possible that a removal from a realistic scene does not provide scenery-related restoration. Secondly, in all situations the scenery was moving (to give the sense that the car was in motion). This is unlike previous studies where participants viewed static images for 10 min. Linked to this, it is possible that the short duration of the test drive (10 min) might have been insufficient to generate restorative effects; however, restorative effects of nature have been demonstrated after short exposures (Berto, 2005), and previous studies using simulated drives have shown effects on cognitive performance for durations similar to that of our study (Murphy and Greene, 2016).

While the small sample size of each subgroup, as well as the imbalanced number of participants in the two driving conditions, limited the power of our analyses, it is worth noting that the effect sizes were very small; in addition, previous studies on attention restoration have shown effects with samples comparable to the present study.

In light of our results, the present study shows that driving or being a passenger in a simulated drive with no particularly challenging situations does not overall determine a different load on attention after the drive. Of course, a different scenario could be envisaged whereby the drives are very demanding, for e.g., an urban drive with pedestrians suddenly crossing the road, however, such a scenario would differ substantially from the traditional ART paradigms (e.g., observation of scenes). Therefore, our study contributes to the current knowledge about cognitive restoration and natural settings by indicating that attention restoration may not occur when the individual is on a moving vehicle, therefore, potentially less engaged in soft fascination.

## Data Availability Statement

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

## Author Contributions

MC, JC, and AS were major contributors in writing the manuscript. MC, MM, YE, DG, JC, and AS designed the study. MM, YE, and DG conducted the data collection and participated in the data analysis. MC, JC, and AS were major contributors in the data analysis. All authors read and approved the final manuscript.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## References

Information:

The purpose of our website is only to help students to assist, guide and aware them regarding material available. Moreover, it is necessary for you to take the permission if you want to reproduce or commercial purpose.

*All the rights reserved by Developer and Translator.

Did you find an inaccuracy? We work hard to provide accurate and scientifically reliable information. If you have found an error of any kind, please let us know.