text:
# Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse‐Graining

\$52.00 + tax

Choosing balls that best approximate a 3D object is a non‐trivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object ${\mathcal{F}}_{\mathcal{O}}$ defined by a union of n balls with $k balls defining a region ${\mathcal{F}}_{\mathcal{S}}\subset {\mathcal{F}}_{\mathcal{O}}$. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations. The inner approximation problem is reduced to a geometric generalization of weighted max k‐cover, solved with the greedy strategy which achieves the classical $1-1/e$ lower bound. The outer approximation is reduced to exploiting the partition of the boundary of ${\mathcal{F}}_{\mathcal{O}}$ by the Apollonius Voronoi diagram of the balls defining the inner approximation. Implementation‐wise, we present robust software incorporating the calculation of the exact Delaunay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application‐wise, we exhibit accurate coarse‐grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments. Choosing balls which best approximate a 3D object is a nontrivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object ${\mathcal{F}}_{\mathcal{O}}$ defined by a union of n balls with $k balls defining a region ${\mathcal{F}}_{\mathcal{S}}\subset {\mathcal{F}}_{\mathcal{O}}$. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations.
No References
No Citations
No Supplementary Data
No Article Media
No Metrics

Document Type: Research Article

Publication date: September 1, 2014

• Access Key
• Free content
• Partial Free content
• New content
• Open access content
• Partial Open access content
• Subscribed content
• Partial Subscribed content
• Free trial content
X


text:
Project deliverable Open Access

# Improved key process in representing Arctic warming (D3.5)

Davy, Richard; Gao, Yongqi

### Dublin Core Export

<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Davy, Richard</dc:creator>
<dc:creator>Gao, Yongqi</dc:creator>
<dc:date>2019-09-30</dc:date>
<dc:description>Summary:

This Blue-Action task was focused on improving the representation of some of the most important
physical processes which contribute to Arctic warming within the climate models used by the
consortium. The two processes we addressed were the effect on the atmospheric state of the fracturing
of the sea ice cover and turbulence under strongly stable thermal stratification. The creation and
development of
The work done: We first analysed the results of previously performed large eddy simulations which
resolved the turbulence over leads to determine the effect leads have on sensible heat flux from open
water. Because of the effect of three-dimensional structures in the turbulent mixing above leads, the
heat flux coming from leads can be amplified compared to the fluxes one would get from open water
under the same air-sea temperature difference. The amplification effect strongly depends on the width
of the lead, with the largest effect occurring for leads of widths around 1.4 km. We assessed the
functional sensitivity of this amplification effect to key parameters used in the turbulence-resolving
model, including the length scale for the convective boundary layer, which reflects the background
stability in the atmosphere.
We combined this relation between the amplification effect of heat fluxes as a function of lead width
with observed distributions of lead widths. These were taken from the peer-reviewed literature where
The key findings: The presence of leads in sea ice dramatically alters the surface energy balance in the
Arctic. There is a large seasonal cycle to the effect of the presence of leads, because the flux from the
leads depends strongly on the background stability in the atmosphere. In the winter when the
atmosphere is often strongly stably stratified, the leads strongly amplify the surface sensible heat flux
coming from open water. In the summer there is the opposite effect and the generally weaker
atmospheric stability reduces the flux coming from leads.</dc:description>
<dc:description>The Blue-Action project has received funding from the European Union's Horizon 2020
Research and Innovation Programme under Grant Agreement No 727852.</dc:description>
<dc:identifier>https://zenodo.org/record/3559470</dc:identifier>
<dc:identifier>10.5281/zenodo.3559470</dc:identifier>
<dc:identifier>oai:zenodo.org:3559470</dc:identifier>
<dc:relation>info:eu-repo/grantAgreement/EC/H2020/727852/</dc:relation>
<dc:relation>doi:10.5281/zenodo.3559469</dc:relation>
<dc:relation>url:https://zenodo.org/communities/blue-actionh2020</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:title>Improved key process in representing Arctic warming (D3.5)</dc:title>
<dc:type>info:eu-repo/semantics/report</dc:type>
<dc:type>publication-deliverable</dc:type>
</oai_dc:dc>

32
29
views

format: plain


text:
# Power rule for complex functions

I'm reading Stein and Shakarchi and on p. 232, they have for $0 < \alpha < 2$, $z = x\in \mathbb{R}^{-}$ the negative real numbers, that $f'(z) = \alpha |x|^{\alpha-1} e^{i\pi(\alpha-1)}$, and I don't see how they get this, where $f(z) = z^\alpha$

I'm trying to figure out some rules for the complex derivative of $z^\alpha$, where $z, \alpha \in \mathbb{C}$. I don't think it's true in general that it should be $\alpha z^{\alpha - 1}$, as in the case for real functions.

Here's what I have so far. Given any branch of the logarithm, the derivative of $\log(z)$ is $1/z$ by the inverse function theorem. So we wish to compute the derivative of $z^\alpha = e^{\alpha \log z}$. By the chain rule, this is $$\frac{\alpha}{z} e^{\alpha \log z} = \frac{\alpha}{z} |z|^\alpha e^{\alpha i \arg \theta}.$$ Is this the most one can say about the derivative of $z^\alpha$, for arbitrary $z, \alpha$?

Also, they claim on p.231 that $z^\alpha = \alpha \int_0^z \zeta^{-\beta} \; d\zeta$, where $\alpha + \beta = 1$, and where the integral is taken along any path in the upper half-plane. They say "In fact, by continuity and Cauchy's theorem, we may take the path of integration to lie in the closure of the upper half-plane". I don't know what they mean by this last statement.

-
In your first second, I don't know what you mean by $z=x\in{\bf R}^-$, and I don't see how anyone gets a formula for $f'(z)$ without saying what $f$ is. – Gerry Myerson Apr 13 '12 at 5:49

It is always true that if $f(z) = z^\alpha$ then $f'(z) = \frac \alpha z z^\alpha$, for any branch (and your calculation verifies this). One is sometimes careful about rewriting this as $\alpha z^{\alpha-1}$ since there are now two different powers and you may need to pay special attention to which branches they separately defined for.
As for what is meant by In fact, by continuity and Cauchy's theorem, we may take the path of integration to lie in the closure of the upper half-plane'' they are just claiming (I believe) that part of the path may lie on the real line (which is the boundary of the upper half plane).
I think all the information you need is in Example 2 on page 210. The key here I think is that the boundary behaviour of $f$ is such that if $x$ travels from $-\infty$ to $0$ on the real line then $f(x)$ travels from $\infty e^{i \alpha \pi}$ to $0$ determined by $arg(z) = \alpha \pi$.
The RHS of the derivative you wrote down (via the chain rule) may exhibit your answer now by considering $z=-x$ and noting that in the final stages of simplification replace $-1$ by $e^{i \pi}$. A few lines of algebra may then present yo with you answer.


text:
A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The Niven numbers take their name from Ivan M. Niven from a paper delivered at a conference on number theory in 1977. All integers between zero and n are Harshad numbers in base n.

Stated mathematically, let X be a positive integer with m digits when written in base n, and let the digits be ai (i = 0, 1, ..., m − 1). (It follows that ai must be either zero or a positive integer up to n − 1.) X can be expressed as

$X=\sum_{i=0}^{m-1} a_i n^i.$

If there exists an integer A such that the following holds, then X is a Harshad number in base n:

$X=A\sum_{i=0}^{m-1} a_i.$

Harshad numbers in base 10 form the sequence:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, ... (sequence A005349 in OEIS)

A number which is a Harshad number in any number base is called an all-Harshad number, or an all-Niven number. There are only four all-Harshad numbers: 1, 2, 4, and 6.

## What numbers can be Harshad numbers?

Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also Harshad numbers. But for the purpose of determining the Harshadness of n, the digits of n can only be added up once and n must be divisible by that sum; otherwise, it is not a Harshad number. For example, 99 is not a Harshad number, since 9 + 9 = 18, and 99 is not divisible by 18.

The base number (and furthermore, its powers) will always be a Harshad number in its own base, since it will be represented as "10" and 1 + 0 = 1.

For a prime number to also be a Harshad number it must be less than or equal to the base number. Otherwise, the digits of the prime will add up to a number that is more than 1 but less than the prime and, obviously, it will not be divisible. For example: 11 is not Harshad in base 10 because the sum of its digits "11" is 1+1=2 and 11 is not divisible by 2, while in hexadecimal the number 11 may be represented as "B", the sum of whose digits is also B and clearly B is divisible by B, ergo it is Harshad in base 16.

Although the sequence of factorials starts with Harshad numbers in base 10, not all factorials are Harshad numbers. 432! is the first that is not.

Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all Harshad numbers in base 10.[1][2] They also constructed infinitely many 20-tuples of consecutive integers that are all Harshad numbers, smallest of which exceed 1044363342786.

The integers starting run of exactly n consecutive integers that are all Harshad numbers (i.e., smallest x such that x, x+1, ..., x+n-1 are Harshad numbers but x-1 and x+n are not) for n = 1,2,... are:

12, 20, 110, 510, 131052, 12751220, 10000095, 2162049150, 124324220, 1, 920067411130599, ... (sequence A060159 in OEIS)

H.G. Grundman extended the Cooper and Kennedy result to show that there are 2b but not 2b+1 consecutive Harshad numbers in base b.[3][2]

In binary, there are infinitely many sequences of four consecutive Harshad numbers; in ternary, there are infinitely many sequences of six consecutive Harshad numbers. Both of these facts were proven by T. Cai in 1996.[2] More generally, there are exist infinitely many 2n-tuples of consecutive Harshad numbers in base n as proved by Wilson in 1997.[4]

In general, such maximal sequences run from N · bk - b to N · bk + (b-1), where b is the base, k is a relatively large power, and N is a constant. Interpolating zeroes into N will not change the sequence of digital sums, so it is possible to convert any solution into a larger one by interpolating a suitable number of zeroes, just as 21 and 201 and 2001 are all Harshad numbers base 10. Thus any solution implies an infinite class of solutions.

## Estimating the density of Harshad numbers

If we let N(x) denote the number of Harshad numbers ≤ x, then for any given ε > 0,

$x^{1-\varepsilon} \ll N(x) \ll \frac{x\log\log x}{\log x}$

as shown by Jean-Marie De Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and Kátai proved that

$N(x)=(c+o(1))\frac{x}{\log x}$

where c = (14/27) log 10 ≈ 1.1939.

## Nivenmorphic numbers

A Nivenmorphic number or Harshadmorphic number for a given number base is an integer t such that there exists some Harshad number N whose digit sum is t, and t, written in that base, terminates N written in the same base.

For example, 18 is a Nivenmorphic number for base 10:

 16218 is a Harshad number
16218 has 18 as digit sum
18 terminates 16218

Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.

Bloem (2005) defines a multiple Harshad number as a Harshad number that, when divided by the sum of its digits, produces another Harshad number. He states that 6804 is "MHN-3" on the grounds that

$\begin{array}{l} 6804/18=378\\ 378/18=21\\ 21/3=7 \end{array}$

and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008·1010, which is smaller, is also MHN-12. In general, 1008·10n is MHN-(n+2).

## References

1. ^ Cooper, Curtis; Kennedy, Robert E. (1993). "On consecutive Niven numbers". Fibonacci Q. 31 (2): 146–151. ISSN 0015-0517. Zbl 0776.11003.
2. ^ a b c Sándor & Crstici (2004) p.382
3. ^
4. ^ Wilson, Brad (1997). "Construction of 2n Consecutive n-Niven Numbers". Fibonacci Q. 35: 122–128. ISSN 0015-0517.
• E. Bloem 2005/2006. Harshad numbers. Journal of Recreational Mathematics, 34(2): 128
• Jean-Marie De Koninck and Nicolas Doyon, On the number of Niven numbers up to x, Fibonacci Quarterly Volume 41.5 (November 2003), 431–440
• Jean-Marie De Koninck, Nicolas Doyon and I. Katái, On the counting function for the Niven numbers, Acta Arithmetica 106 (2003), 265–275
• Sandro Boscaro, Nivenmorphic Integers, Journal of Recreational Mathematics 28, 3 (1996 - 1997): 201–205
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. ISBN 1-4020-2546-7. Zbl 1079.11001.


text:
# zbMATH — the first resource for mathematics

Fractional Brownian motion and the Markov property. (English) Zbl 0921.60067
Fractional Brownian motion is used as a model of signal or noise in various domains: geophysical data, communication processes, etc. It belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite-dimensional Markov process. Increments of this process depend on Hurst parameter $$H\in (0,1)$$: they are independent if and only if $$H=1/2$$ (Brownian motion), they are negative correlated if $$H<1/2$$, and if $$H>1/2$$ they are positive correlated. The authors give an efficient algorithm to approximate this process. Also, an ergodic theorem which applies to functionals of the type $$\int_{0}^{t}\varphi(V_{h}(s))ds$$, where $$V_{h}(s)=\int_{0}^{s}h(s-u)dB_{u}$$ is the finite long memory part with $$B$$ being a real Brownian motion, is given.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60G15 Gaussian processes 65C99 Probabilistic methods, stochastic differential equations 60F17 Functional limit theorems; invariance principles
Full Text:


text:
Chapter 17, Problem 98CWP

### Chemistry

9th Edition
Steven S. Zumdahl
ISBN: 9781133611097

Chapter
Section

### Chemistry

9th Edition
Steven S. Zumdahl
ISBN: 9781133611097
Textbook Problem

# Consider the reaction: H 2 S ( g ) + SO 2 ( g ) → 3 S ( g ) + 2 H 2 O ( l ) for which ∆H is −233 kJ and ∆S is −424 J/K.a. Calculate the free energy change for the reaction (∆G) at 393 K.b. Assuming ∆H and ∆S do not depend on temperature, at what temperatures is this reaction spontaneous?

(a)

Interpretation Introduction

Interpretation: The free energy change of the given reaction and the temperature at which the reaction becomes spontaneous is to be calculated.

Concept Interpretation: The free energy change of the given reaction is calculated by using the formula,

ΔG=ΔHTΔS

To determine: The free energy change of the given reaction.

Explanation

Explanation

Given

The balanced chemical equation for the given reaction is,

H2S(g)+SO2(g)3S(g)+2H2O(l)

The enthalpy change of the reaction is 233kJ(=233×103J) .

The entropy change of the reaction is 424J/K .

The temperature of the reaction is 373K .

The free energy change of the given reaction is calculated by using the formula,

ΔG=ΔHTΔS

Where,

• ΔS is the entropy change of the process

(b)

Interpretation Introduction

Interpretation: The free energy change of the given reaction and the temperature at which the reaction becomes spontaneous is to be calculated.

Concept Interpretation: The free energy change of the given reaction is calculated by using the formula,

ΔG=ΔHTΔS

To determine: The temperature at which the given reaction is spontaneous.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started


text:
# Final questions

What does a node represent
Click the card to flip 👆
1 / 73
Terms in this set (73)
5. Thermohaline circulation is best described as:
a) Cold, oxygen rich waters moving across the ocean bottoms where they remain
b) A worldwide system of rising and sinking water masses due to density changes caused by changes in temperature and salinity.
c) A slow circulation (on the order of a thousand years for one cycle).
d) B and C
e) All of the above
6. Is the oxygen a limiting factor for organisms in the deep ocean?
a) No, as oxygen diffuses from the epipelagic zone
b) Yes, but only in areas where primary producers (e.g., oxygen-producing
bacteria from deep trenches) cannot sustain efficient production of oxygen
c) No, as the rates of oxygen consumption in deep-water environments are low compared to the supply of waters from polar regions
d) Depends on the proximity to the thermocline


text:
1. ## derivatives

f(x)=ax^5 when x<f
f(x)=bx-4 when x>=f

for f(x) to be differentiable when x=f what should a and b be?

tysm

2. ## Re: derivatives

f(x)=ax^5 when x<f
f(x)=bx-4 when x>=f
let's call the x-value of interest $c$ instead of $f$, so as not to confuse $f$ and $y = f(x)$ ...

first, the function, $f(x)$, has to be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \implies ac^5 = bc-4$

second, the function, $f'(x)$, must also be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x) \implies 5ac^4 = b$

can you finish?

3. ## Re: derivatives

Originally Posted by skeeter
let's call the x-value of interest $c$ instead of $f$, so as not to confuse $f$ and $y = f(x)$ ...

first, the function, $f(x)$, has to be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \implies ac^5 = bc-4$

second, the function, $f'(x)$, must also be continuous at $\displaystyle x=c \implies \lim_{x \to c^-} f'(x) = \lim_{x \to c^+} f'(x) \implies 5ac^4 = b$

can you finish?
should the answer be in terms of f ?

4. ## Re: derivatives

Originally Posted by DiscreteMathHelp
should the answer be in terms of f ?
yes ... change $c$ to $f$ when you are done.


text:
# American Institute of Mathematical Sciences

• Previous Article
Electrical-thermal analytical modeling of monopolar RF thermal ablation of biological tissues: determining the circumstances under which tissue temperature reaches a steady state
• MBE Home
• This Issue
• Next Article
Seasonality and the effectiveness of mass vaccination
2016, 13(2): 261-279. doi: 10.3934/mbe.2015002

## Structured populations with diffusion and Feller conditions

 1 Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland 2 Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Received  February 2015 Revised  September 2015 Published  November 2015

We prove a weak maximum principle for structured population models with dynamic boundary conditions. We establish existence and positivity of solutions of these models and investigate the asymptotic behaviour of solutions. In particular, we analyse so called size profile.
Citation: Agnieszka Bartłomiejczyk, Henryk Leszczyński. Structured populations with diffusion and Feller conditions. Mathematical Biosciences & Engineering, 2016, 13 (2) : 261-279. doi: 10.3934/mbe.2015002
##### References:

show all references

##### References:
 [1] Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391 [2] József Z. Farkas, Thomas Hagen. Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1825-1839. doi: 10.3934/cpaa.2009.8.1825 [3] Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637 [4] Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032 [5] H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183 [6] Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015 [7] Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109 [8] Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891 [9] Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041 [10] Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015 [11] Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233 [12] Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327 [13] Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719 [14] Keith E. Howard. A size structured model of cell dwarfism. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 471-484. doi: 10.3934/dcdsb.2001.1.471 [15] Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289 [16] L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203 [17] József Z. Farkas, Thomas Hagen. Asymptotic behavior of size-structured populations via juvenile-adult interaction. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 249-266. doi: 10.3934/dcdsb.2008.9.249 [18] Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563 [19] Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427 [20] Hongjie Dong, Dong Li. On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437

2018 Impact Factor: 1.313


text:
Revisiting the Dionysian
Thoughts on Nietzsche, Greek drama and Wagner

# Revisiting the Dionysian

sheps

Joined: Dec 15, 2008

Total Topics: 110
Total Posts: 8006
1 of 1 people found this post helpful
Posted Dec 31, 2011 - 8:57 AM:
Subject: Revisiting the Dionysian
Ever since Nietzsche used this term, it has been inseperable from two enormously important epochs of world culture: pre-Socratic Greece, and post-Enlightenment Germany. In The Birth of Tragedy Nietzsche famously asserts, with youthful confidence and a shade of awe, that Wagner lies at the end of a "mighty solar orbit" in terms of German music, and is the possible returning Messiah of the ancient aesthetic school: he is the great man, climbing upon the shoulders of Schiller, Beethoven, Goethe and Winckelmann to rediscover the forgotten land of Hellenistic genius.

Nietzsche demands, as the good classical scholar, that we know our Greek tragedy, and that we know what to expect from Greek drama. Modern critics might well wish to see Aeschylus' Oresteia as a grand political drama, with overtones of modern phenomena like feminism and Marxism, but this isn't (according to N.) the point of the work. The Agamemnon, especially, is a terrifying, pessimistic play; whilst Homer has Odysseus return as a triumphant hero to a loyal wife, the greatest ruler Greece had ever seen is infamously slaughtered defenceless in his bath. The crafty and devious anti-hero of Sophocles' Philoctetes gets the good ending and the girl, whilst the last act of the conqueror of Troy and hope of Greece is to utter a horrifying shriek of agony and despair off-stage.

Indeed, after this start to the trilogy, one could be excused for being rather disappointed with what is, undoubtedly, a political ending. In the Eumenides, Aeschylus becomes the first dramatist to commit what is, in my view, a great mistake, and bring the law-court onto the stage. Athena - that asexual goddess born from Zeus' head - banishes pessimism, banishes the vital energy and revenge of the Furies (the conflict between private revenge and public justice is thus alleged to be resolved): it is here that contemporary political and feminist readings come in to their own. The audience is left with a rather toothless ending to what remains the greatest artistic trilogy of Western civilisation. Nietzsche was, perhaps, incorrect to blame Euripides for the triumph of the Apollonian: it is in this play where the bland spirit of rationality first enters into drama.

We must not forget that both Aeschylus and Sophocles - the heroes of Nietzsche's book, in many ways - were both technical innovators in the history of art. Aristotle informs us that both altered the number of actors on stage; Sophocles introduced rudimentary scenery. This ought not distract us from trying to comprehend the significance of the vision of these authors, but it should be born in mind ... especially if we are to compare them with the likes of another great technical innovator in the theatre, Wagner.

Wagner's brilliant Der Ring des Nibelungen is intended, in essence, to be a modern (and German) re-working of Aeschylus' plays. The satyr play is added, of course, and Wagner undoubtedly was trying to realise a "new art", though one which drew heavily from art's origins. Kant had summarily dismissed instrumental music as inferior to other forms of art; though we may scoff at this today, there was undoubtedly some truth in it for certain German thinkers of the 19th Century. There has to have been, else Brahms would have won and Wagner lost: as it was, the man Nietzsche delighted in calling Tartuffian arguably ended up lighting the touch-paper of one of the greatest artistic movements in Western history. Wagner's work represents the final step in 19th Century grandeur and decadence - Britian, with shadows of myth-reconstructers like Walter Scott and France with, well, nothing, could only stand in awe. The music of Wagner's Ring does, at least, come close to uniting the Apollonian and Dionysian, even if the libretto is occasionally facile and pseudo-Homeric. Although we cannot say in absolute fashion that Wagner beats Brahms or Beethoven in terms of musical genius we can confirm that his vision was of a more complete nature; allowing the audience to participate, like an ancient chorus, when during a symphony they can only observe in a rationalistic and ultimately unsatisfying fashion.

The main problem facing the modern audience appears to be that these two epochs have disappeared. Euripides, Socrates and the rationalistic tendency have won: mankind no longer appears to want to marry the Apollonian and Dionysian tendencies in his art. The triumph of the novel, of course, has much to do with this, as does the ultimate failure of the verse drama. Personally, I would be in favour of a new genius taking up the torch from the likes of Wagner, and re-examining art's origins in order to arrive at a new form of art. In a cynical age, however, this is unlikely. Myths, which are so often what is required for this sort of work to arise, are universally mocked and misunderstood. What we can do is enjoy both the formative writers of Western culture, and by doing so preserve their memory, whilst also looking with fondness back at that time in Germany when genuine experimentation was being attempted and when the Dionysian tendency was, to some extent, re-born.

Just some thoughts. At least it's another thread for the unjustly ignored aesthetics section of the forums.
BitterCrank

Joined: Mar 01, 2008
Location: Minneapolis

Total Topics: 196
Total Posts: 9802
Posted Dec 31, 2011 - 10:43 AM:

The New York Metropolitan Opera broadcast Wagner's Götterdämmerung on the radio two or three weeks ago; it was pretty good. As I generally do, I was listening to the opera while doiing something else. Except for some excerpts, Wagner wouldn't rank as highly on my list as he ranks on some people's lists. One has to be at the opera to get the full effect. Today they are doing a new piece, "a baroque pastiche" with material by Handel, Vivaldi, and Rameau. We'll see...

We definitely don't do the Dionesian well in the Anglo-Saxon world -- I'm not sure, exactly, who does. Just going out and getting high, or getting laid, or getting drunk, or merely being torn apart in the forest by the Bacchantae just doesn't cut it. Krista Tippett (onbeing.org) woke me up this morning at 6:00 (#&$*#*@*$) with a discussion among the Dalai Lama, Muslim scholar Seyyed Hossein Nasr, Bishop Katharine Jefferts Schori of the Episcopal Church, and Chief Rabbi Lord Jonathan Sacks of the United Kingdom. Among other things they were talking about sanctifying pleasure. This is about as close as National Public Radio is going to get to Dionysus, I suspect.

One scholar put it thus: The Apollonian is about light (day-time, the sun), far-seeing, straight-arrow, border-respecting thinking, while the Dionysian is about feeling (not so cerebral), border crossing, deviance, and dark (as in night-time, the moon). Maybe masculine-masculine vs. masculine-feminine would be another way of opposing the Appollonian and the Dionysian. (for those not familiar, try Euripides' Bacchae -- it's a pretty good read.) We need not decide we are one or the other. All-Apollo all the time and All Dionysus all the time are both pretty tiresome. I view the Dionysian as something we need to access at times, rather than a state-of-being. I suppose I see the Apollonian as the default condition periodically interupted by things that go "YAHOO!!!" in the night. Some people, not too many, occupy the Dionysian role much more, most quite a bit less. I'm not trying to vere into personality theory here, though aesthetics, personality, and cognition can't be separated if we are going to talk about the Dionysian.

John Waters, the Bane of Baltimore, has done some films, like Pink Flamingoes, which are nothing if not Dionysian low-art (How low can he go? Pretty low!). In his earlier oeuvre, it's all deviance and no normal to serve as a backdrop. I llike his stuff. There was a strain of life/performance in gay life, particularly in the later 1960s and 1970s when drag queens and leather boys were becomming visible and sex was experiencing a general disinhibition, while at the same time there was just the right amount of social resistence, that one could find Dionysian invasions of propriety that were "about right." There was definitely a "wall" to play against, and the performance was "off the wall" enough. (My dates represent the Upper Midwestern cultural lag.) A Canadian film, Outrageous, captured this period well -- can't remember the dates. It combined leather, drag, schizophrenia (one character), nuns on roller-skates and so forth.

On a quite-a-bit higher note, there is Carl Orff's 1930s Carmina Burana and Catulli Carmina (I haven't heard the third part of the tryptich). I think it is Dionysian, though some people disagree. Alex Ross [some critic] wrote that "the music itself commits no sins simply by being and remaining popular. That Carmina Burana has appeared in hundreds of films and television commercials is proof that it contains no diabolical message, indeed that it contains no message whatsoever." Well, that's unfair. If you don't read the libretto of course it is going to be content-free. And the music itself encapsulates ecstatic moments. Maybe the piece is too familiar and is not performed often enough by naked symphony orchestras to capture its Dionysianosity.

Could one make a case that there is something a little bit Dionysian in the Mass? A commemoration is made of a god who is killed violently and then eaten... by his followers, like Dionysus? I wouldn't push it too far, but it seems to me there is something to that idea.

BitterCrank

Joined: Mar 01, 2008
Location: Minneapolis

Total Topics: 196
Total Posts: 9802
Posted Dec 31, 2011 - 11:08 AM:

sheps wrote:
The main problem facing the modern audience appears to be that these two epochs have disappeared. Euripides, Socrates and the rationalistic tendency have won: mankind no longer appears to want to marry the Apollonian and Dionysian tendencies in his art. The triumph of the novel, of course, has much to do with this, as does the ultimate failure of the verse drama. Personally, I would be in favour of a new genius taking up the torch from the likes of Wagner, and re-examining art's origins in order to arrive at a new form of art. In a cynical age, however, this is unlikely. Myths, which are so often what is required for this sort of work to arise, are universally mocked and misunderstood. What we can do is enjoy both the formative writers of Western culture, and by doing so preserve their memory, whilst also looking with fondness back at that time in Germany when genuine experimentation was being attempted and when the Dionysian tendency was, to some extent, re-born.

Apollo and Dionysus together yield balance. Separate them and let them go their own way and you get an arid rationalism on the one hand and an over-fertilized, over-watered, over-stimulated swamp on the other hand. Even though we seem to cater to pleasure endlessly the Apollonian is decidedly dominant. Pleasure, per se, isn't Dionysian. A sculpture of Apollo may give great pleasure, stimulating one's aesthetic urges, but that isn't Dionysian. Beethoven or (dare I say it) Wagner, per se, aren't either.

To the extent that the dionysian is border-crossing and god-sacrificing, it is anathema in the buttoned-down arid rationality, if not apollonian, work-a-day world. Disruptive ecstatic movement does not go well with efficient production or sober scholarship. And cultivating the Dionysian inside the confines of the art gallery betrays Apollonianism. Real Dionysian movements break through boundaries, including those of aesthetes and aesthetic managers. It's like having a "permanent, institutionalized revolution." Its just a radical self-contradiction.

ciceronianus
Arbiter of Insects

Joined: Sep 20, 2008
Location: Beyond the compass of change

Total Topics: 99
Total Posts: 6151

Last Blog: "I Will Destroy the Wisdom of the Wise"

Posted Dec 31, 2011 - 11:29 AM:

I think that Nietzsche never understood just what Dionysus meant to the ancients, and doubt that he could. The Dionysian or Bacchic mystery religions were ecstatic cults fueled by intoxicants of various kinds, and the rites were certainly wacky enough, even grisly, but were eventually toned down when Orphism came along, and Dionysian religion like other religions of the time eventually transformed into Christianity. I think the link between Dionysus and Christianity is much closer that Nietzsche liked to think. He takes, of course, a typically romantic view of the ancients, and I think that view was in many respects a kind of fantasy.
sms18
Initiate

Usergroup: Members
Joined: Nov 25, 2011

Total Topics: 0
Total Posts: 35
Posted Dec 31, 2011 - 5:14 PM:

sheps wrote:

Personally, I would be in favour of a new genius taking up the torch from the likes of Wagner, and re-examining art's origins in order to arrive at a new form of art. In a cynical age, however, this is unlikely. Myths, which are so often what is required for this sort of work to arise, are universally mocked and misunderstood. What we can do is enjoy both the formative writers of Western culture, and by doing so preserve their memory, whilst also looking with fondness back at that time in Germany when genuine experimentation was being attempted and when the Dionysian tendency was, to some extent, re-born.

To me it seems that the form of Gesamtkunstwerk that Wagner invented was completely exhausted by Wagner himself, to the point that it was impossible to further develop this form of art, within the scheme of the opera at least. But the flame didn't die out there - 20th cent. Modernism was an era of experimentation, and what springs to my mind especially is Joyce's Ulysses. While the written language is fundamentally Apollonian, Joyce deconstructs and manipulates language to the point where it becomes "musical," and even Dionysian (the Sirens episode epitomizes this). As Joyce was a devout Wagnerian and knew his Nietzsche to an extent, he may have been consciously striving to create his own Apollonian-Dionysian unity.

It's hard to say whether this spirit of genuine experimentation is still carried out today, but Wagner definitely isn't forgotten - I hear him all the time in film scores. Can there be a Dionysian rebirth through films? I thought LOTR was pretty good.
sheps

Joined: Dec 15, 2008

Total Topics: 110
Total Posts: 8006
Posted Jan 1, 2012 - 9:21 AM:

BitterCrank wrote:
The New York Metropolitan Opera broadcast Wagner's Götterdämmerung on the radio two or three weeks ago; it was pretty good. As I generally do, I was listening to the opera while doiing something else. Except for some excerpts, Wagner wouldn't rank as highly on my list as he ranks on some people's lists. One has to be at the opera to get the full effect. Today they are doing a new piece, "a baroque pastiche" with material by Handel, Vivaldi, and Rameau. We'll see...

Sounds good. I love Handel's operatic works - Serse's the best - but he just misses out on being my favourite (though he's probably my favourite composer overall) in the department 'cause of the way he seperates the orchestra and the voice so much. It'd be nice to hear a little less counterpoint, but that is asking rather too much from a Baroque composer.

One scholar put it thus: The Apollonian is about light (day-time, the sun), far-seeing, straight-arrow, border-respecting thinking, while the Dionysian is about feeling (not so cerebral), border crossing, deviance, and dark (as in night-time, the moon).

The Furies represent the Dionysian best, for me. Revenge, despair, melancholy, the abyss ... all these things make me think of it, and some of the best music provokes thoughts on these matters. The Greeks in the 6th and 5th century BC were especially good at realising this, I reckon, due to all the wars and domestic troubles they found themselves in. Aeschylus is much more pessimistic about his (traditionally) heroic figures, too: compare Agamemnon with, say, Theseus of the Theban Plays.

Apollo and Dionysus together yield balance. Separate them and let them go their own way and you get an arid rationalism on the one hand and an over-fertilized, over-watered, over-stimulated swamp on the other hand. Even though we seem to cater to pleasure endlessly the Apollonian is decidedly dominant. Pleasure, per se, isn't Dionysian. A sculpture of Apollo may give great pleasure, stimulating one's aesthetic urges, but that isn't Dionysian. Beethoven or (dare I say it) Wagner, per se, aren't either.

That balance is hard to achieve, though, especially in a social setting. That's what makes the Greek playwrights so special: although even Aeschylus had the Apollonian side "win" in the end, with the absolution of Orestes in the law court. Women (the Furies) lose here; their right over children is utterly denied according to the natural philosophy of the day; and the asexual Athena oversees an arguably male-dominated justice. Astonishingly, of course, these plays won or lost based on a sort of clap-o-meter, just going to show what a remarkable society Ancient Greece must have been.

Could one make a case that there is something a little bit Dionysian in the Mass? A commemoration is made of a god who is killed violently and then eaten... by his followers, like Dionysus? I wouldn't push it too far, but it seems to me there is something to that idea.

Oh yeah, for sure! Christianity has its darker sides. As a matter of fact, about the only contemporary classical music which has really grabbed me recently was something I caught at the last Proms on TV: it was a choir singing a Catholic liturgy set to disturbing, minor-key music. Very powerful: Polish, I think, though not by Gorecki. Thanks for the recommendations, btw.

ciceronianus wrote:
I think that Nietzsche never understood just what Dionysus meant to the ancients, and doubt that he could. The Dionysian or Bacchic mystery religions were ecstatic cults fueled by intoxicants of various kinds, and the rites were certainly wacky enough, even grisly, but were eventually toned down when Orphism came along, and Dionysian religion like other religions of the time eventually transformed into Christianity.

But by the time Dionysian tendencies are truly represented to us in the form of art, they're already purified and refined in a sense. We shouldn't forget that Nietzsche really meant Dionysian in an artistic or philosophical sense; not, in all seriousness, the actual revelries of the Dionysian worshippers.

I think the link between Dionysus and Christianity is much closer that Nietzsche liked to think. He takes, of course, a typically romantic view of the ancients, and I think that view was in many respects a kind of fantasy.

I don't know, I think Christianity owes a lot more to Plato than anything pre-Socratic. Do you mean "romantic" with a capital R, by the way? If so, you'll have to clarify and justify that for me.

sms18 wrote:
To me it seems that the form of Gesamtkunstwerk that Wagner invented was completely exhausted by Wagner himself, to the point that it was impossible to further develop this form of art, within the scheme of the opera at least. But the flame didn't die out there - 20th cent. Modernism was an era of experimentation, and what springs to my mind especially is Joyce's Ulysses. While the written language is fundamentally Apollonian, Joyce deconstructs and manipulates language to the point where it becomes "musical," and even Dionysian (the Sirens episode epitomizes this). As Joyce was a devout Wagnerian and knew his Nietzsche to an extent, he may have been consciously striving to create his own Apollonian-Dionysian unity.

I don't really know enough to comment on this, but I didn't know Joyce was a Wagnerian. I'm sure he was influenced by Nietzsche, though: other creative authors who I think of as owing a debt to him are Shaw and Lawrence.

It's hard to say whether this spirit of genuine experimentation is still carried out today, but Wagner definitely isn't forgotten - I hear him all the time in film scores. Can there be a Dionysian rebirth through films? I thought LOTR was pretty good.

It would have to be a bold project, certainly. Lord of the Rings will obviously suggest itself, if only because of the theme debts it owes to Wagner; and film music is undoubtedly Wagnerian (yet we shouldn't ignore the likes of Elgar and Vaughan Williams in terms of its development). I think film could be the place where Dionysius re-emerges, but the standard of contemporary film making (esp. w/r/t screenplays and acting quality) is far too low at the moment. 2001: A Space Odyssey struck me as quite Dionysian, as a matter of fact, though we're obviously going back a bit there.

I should add that Wagner's music - and classical music - is still all quite new to me. I used to really dislike him, arguably because my "introduction", so to speak, was listening to his overtures which fail by a long way to accurately represent him.
ciceronianus
Arbiter of Insects

Joined: Sep 20, 2008
Location: Beyond the compass of change

Total Topics: 99
Total Posts: 6151

Last Blog: "I Will Destroy the Wisdom of the Wise"

Posted Jan 1, 2012 - 9:43 AM:

sheps wrote:
But by the time Dionysian tendencies are truly represented to us in the form of art, they're already purified and refined in a sense. We shouldn't forget that Nietzsche really meant Dionysian in an artistic or philosophical sense; not, in all seriousness, the actual revelries of the Dionysian worshippers.

Perhaps, but I think that necessarily and purposefully mischaraterizes, or at least redefines, the Dionysian. Of course, purposeful mischaracterization is one of the things we do, and Nietzsche's in this case is far better than that of Alexander the Great, who it is said invoked Dionysus to justify his burning of Persepolis and his murder of his old friend Kleitos while very, very drunk. He was possessed by the god, you see.
sheps

Joined: Dec 15, 2008

Total Topics: 110
Total Posts: 8006
Posted Jan 1, 2012 - 1:59 PM:

ciceronianus wrote:
Perhaps, but I think that necessarily and purposefully mischaraterizes, or at least redefines, the Dionysian. Of course, purposeful mischaracterization is one of the things we do, and Nietzsche's in this case is far better than that of Alexander the Great, who it is said invoked Dionysus to justify his burning of Persepolis and his murder of his old friend Kleitos while very, very drunk. He was possessed by the god, you see.

It's not a completely muddled-headed way of thinking about art though, is it? This is going to sound terribly Romantic and you're probably going to dismiss it, but isn't the artist "intoxicated" with his work in a sense? (Not to mention the fact that he may, like Coleridge, literally enjoy working off his face.) This sounds very Victorian - not much better than those silly half-myths that Mozart wrote a pile of music with no corrections; as if that made it any better! - but isn't there something to it? In a novel, say, the author is truly the hero, as Iris Murdoch says: is he not, in a sense, drunk with the fantasy world he is creating? Isn't there some mysterious spark of intoxicated genius behind every truly great symphony, poem or song?
ciceronianus
Arbiter of Insects

Joined: Sep 20, 2008
Location: Beyond the compass of change

Total Topics: 99
Total Posts: 6151

Last Blog: "I Will Destroy the Wisdom of the Wise"

Posted Jan 1, 2012 - 5:14 PM:

sheps wrote:

It's not a completely muddled-headed way of thinking about art though, is it? This is going to sound terribly Romantic and you're probably going to dismiss it, but isn't the artist "intoxicated" with his work in a sense? (Not to mention the fact that he may, like Coleridge, literally enjoy working off his face.) This sounds very Victorian - not much better than those silly half-myths that Mozart wrote a pile of music with no corrections; as if that made it any better! - but isn't there something to it? In a novel, say, the author is truly the hero, as Iris Murdoch says: is he not, in a sense, drunk with the fantasy world he is creating? Isn't there some mysterious spark of intoxicated genius behind every truly great symphony, poem or song?

I don't know, I must admit. But I think great works of art are more the result of a great deal of intelligence, even artifice, than intoxication. I suppose that means I take the side of Apollo in Nietzche's vocabulary--or perhaps Hermes would be more appropriate.
garycgibson
Unmoderated Member

Usergroup: Unmoderated Member
Joined: Dec 07, 2007

Total Topics: 13
Total Posts: 232
Posted Jan 5, 2012 - 9:34 PM:

Reading Euripides 'versions of the works of Sophocles is lively. The Bacchae roll right along with a fair field description of the salient point of the advance of the women under the influence of Dionysus. What can't one like about it? Would you regard Timon of Athens as Dionysian because of the strength of Timon's hatred of mankind, or is that in a Euripidean style?

Is Wagner’s Logen a reference to Attila the Hun? Attila made it to France and fought to a stalemate with a Roman Legion.

One comment on the ring cycle offers the opinion that Bakunin was the role model. Adolph Hitler and other political charismatics place the emotion of vengeance right there with Dionysian passion. Monomaniacal political passion can bring a high opportunity cost with it. Artists though seem to require devotion to their project-I find it difficult to associate Dionysian drunkenes with artistic passion. Was Jackson Pollock a Dionysian and Pablo Picasso an intellectual of cubist construction and meta-layered perspective placing subjective psychologic perception into the empirical world?

The work of a the passion of Christ when the word means perserverence through suffering may be the best example of artistic devotion to an aethetic cause if we allow real values to intrude into the aesthete. Passion with sobriety is the best kind.

I stipulate that being in some way compelled to attend an Opera instead of reading philosophy or cosmology would be a terrible fate. So is a succession of Presidents of the United States that would be guided by emotion or red flags stimulating deficit spending instead of rational, analytical informed opinion about ecological economics and the way to move socially in a Thomas Moore direction or at least with a George Luis Borges sort of amusement aesthetically with intellectual genius.

Some prefer irrational motivation for living, yet it requires a good healthy ecosphere to run amuck in decadence instead of one with a prospect for mass extinction. I believe one might sublimate the aesthetic of romanticism within an intellectual appreciation of beauty and avoid the excesses of Nietzsche-

Edited by garycgibson on Jan 5, 2012 - 9:44 PM. Reason: improvement
This inactive thread has been archived. To continue the topic, start a new one.


text:
# BellInequalityMax

 Other toolboxes required BellInequalityMax Computes the maximum value of a Bell inequality CVX NPAHierarchyParallelRepetitionXORGameValue Nonlocality and Bell inequalities no

BellInequalityMax is a function that computes the maximum possible value of a given Bell inequality or nonlocal game under either classical mechanics, quantum mechanics, or general no-signalling theories. In the classical and non-signalling cases, an exact value is computed, whereas the value computed in the quantum case is only an upper bound (found using the NPA hierarchy).

## Syntax

• BMAX = BellInequalityMax(COEFFICIENTS,DESC,NOTATION)
• BMAX = BellInequalityMax(COEFFICIENTS,DESC,NOTATION,MTYPE)
• BMAX = BellInequalityMax(COEFFICIENTS,DESC,NOTATION,MTYPE,K)

## Argument descriptions

• COEFFICIENTS: A matrix or tensor that specifies the coefficients of the Bell inequality in either full probability, full correlator, or Collins-Gisin notation.
• DESC: A vector of 4 elements describing the number of outputs of Alice, number of outputs of Bob, number of inputs of Alice, and number of inputs of Bob, in this order.
• NOTATION: A string indicating in which notation are the coefficients of the Bell inequality given. Must be one of 'fp', 'fc', or 'cg', for full probability, full correlator, or Collins-Gisin, respectively.
• MTYPE (optional, default 'classical'): A string indicating which type of theory should be used when computing the maximum value of the Bell inequality. Must be one of 'classical', 'quantum', or 'nosignal'. If MTYPE = 'quantum' then only an upper bound on the Bell inequality is computed, not necessarily is best upper bound (see the argument K below).
• K (optional, default 1): If MTYPE = 'quantum' then this is a positive integer indicating what level of the NPA hierarchy should be used when bounding the Bell inequality. Higher values of K give better bounds, but require more memory and time. Alternatively, K can be a string of a form like '1+ab+aab', which indicates that an intermediate level of the hierarchy should be used, where this example uses all products of 1 measurement, all products of one Alice and one Bob measurement, and all products of two Alice and one Bob measurement. Use plus signs to separate the different categories of products, as above. The first character of this string should always be a number, indicating the base level to use. If MTYPE is anything other than 'quantum' then K has no effect.

## Examples

### The CHSH inequality

One formulation of the CHSH inequality[1] says that if $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are $\{-1,+1\}$-valued measurement settings, then in classical physics the following inequality holds: $\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle + \langle A_2 B_1 \rangle - \langle A_2 B_2 \rangle \leq 2.$ Similarly, the best bound on this quantity is $2\sqrt{2}$ in a quantum mechanical setting (this is Tsirelson's bound)[2], and it is $4$ in no-signalling theories. All three of these bounds can be found as follows:

>> M = [0 0 0;0 1 1;0 1 -1];
>> desc = [2 2 2 2];
>> BellInequalityMax(M,desc,'fc','classical')

ans =

2

>> BellInequalityMax(M,desc,'fc','quantum')

ans =

2.8284

>> BellInequalityMax(M,desc,'fc','nosignal')

ans =

4

### The CHSH inequality again

Another (equivalent) formulation of the CHSH inequality is in Collins-Gisin notation. We write in terms of only the probabilities of obtaining one of the outcomes, as the other probabilities are redundant given the normalization and no-signalling conditions. In this formulation, the following inequality holds in classical theories: $p(11|11) + p(11|12) + p(11|21) - p(11|22) - p_A(1|1) - p_B(1|1) \leq 0.$ Similarly, the best bound on this quantity is $1/\sqrt{2} - 1/2$ in a quantum mechanical setting, and it is $1/2$ in no-signalling settings. These three bounds can be verified as follows:

>> M = [0 -1 0; -1 1 1; 0 1 -1];
>> desc = [2 2 2 2];
>> BellInequalityMax(M,desc,'cg','classical')

ans =

0

>> BellInequalityMax(M,desc,'cg','quantum')

ans =

0.2071

>> BellInequalityMax(M,desc,'cg','nosignal')

ans =

0.5000

### The CHSH game

The CHSH inequality can be reformulated as a nonlocal game, as is favoured in computer science[3]. The rules of the game are as follows:

• A referee asks Alice a question $x \in \{0,1\}$ and Bob a question $y \in \{0,1\}$ with uniform probability.
• Alice and Bob then reply with answers $a \in \{0,1\}$ and $b \in \{0,1\}$ respectively.
• Alice and Bob win if and only if $a \oplus b = xy$.

The optimal probability with which Alice and Bob can win this game in classical, quantum, and no-signalling theories are as follows:

>> M = chshd(2);
>> desc = [2 2 2 2];
>> BellInequalityMax(M,desc,'fp','classical')

ans =

0.7500

>> BellInequalityMax(M,desc,'fp','quantum')

ans =

0.8536

>> BellInequalityMax(M,desc,'fp','nosignal')

ans =

1

The auxiliary function chshd() above produces the predicate of the CHSH-d game, an intensively studied variant of the CHSH game that generalizes it to a larger alphabet. The rules are analogous, with the difference that each of $a,b,x,y$ are taken from the set $\{0,1,\ldots,d-1\}$ and sum is done modulo $d$.[4] The algorithm used to calculate the classical bound is parallelized and slightly smarter than naïve brute force[5]. It can handle up to $d=8$ quickly:

>> M = chshd(8);
>> desc = [8 8 8 8];
>> BellInequalityMax(M,desc,'fp','classical')
Starting parallel pool (parpool) using the 'local' profile ... connected to 2 workers.

ans =

0.4688

### The I3322 inequality

The I3322 inequality[6][7] is a Bell inequality where each of Alice and Bob has three possible inputs and 2 possible outputs. Its symmetric version in Collins-Gisin notation is given by: $-p(11|11) - p(11|12) - p(11|13) - p(11|21) - p(11|22) + p(11|23) - p(11|31) + p(11|32) + p_A(1|1) + p_B(1|1) \leq 1.$ Similarly, it is known that a value of (strictly) larger than $1+1/4$ is possible in quantum mechanics[8], and the best possible bound is $2$ in no-signalling theories. The following code computes the best possible classical and no-signalling bounds exactly, and computes several different upper bounds for the quantum mechanical setting:

>> format long
>> cvx_precision high
>> M = [0 1 0 0;1 -1 -1 -1;0 -1 -1 1;0 -1 1 0];
>> desc = [2 2 3 3];
>> BellInequalityMax(M,desc,'cg','classical')

ans =

1

>> BellInequalityMax(M,desc,'cg','quantum',1)

ans =

1.374999999999371

>> BellInequalityMax(M,desc,'cg','quantum','1+ab')

ans =

1.251470897748553

>> BellInequalityMax(M,desc,'cg','quantum',2)

ans =

1.250939721643942

>> BellInequalityMax(M,desc,'cg','quantum',3)

ans =

1.250875562046812

>> BellInequalityMax(M,desc,'cg','quantum',4)

ans =

1.250875384515790

>> BellInequalityMax(M,desc,'cg','nosignal')

ans =

2

We note that the values provided for the quantum upper bound above for K = '1+ab', K = 2, and K = 3 agree with the values found in [9] (subtracted 1 to account for the different form of the inequality). Furthermore, the value when K = 4 agrees with the value found in [8]. The exact value of the best upper bound in the quantum setting is an open problem, but is known to be approximately $1.250875384514$.

## Notes

In practice, K probably can't be any larger than 4 when MTYPE = 'quantum'.

## References

1. J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(15):880–884, 1969.
2. B.S. Tsirelson. Quantum generalizations of Bell's inequality. Lett. Math. Phys., 4:93, 1980. E-print
3. Richard Cleve, Peter Hoyer, Ben Toner, John Watrous. Consequences and Limits of Nonlocal Strategies. E-print: arXiv:quant-ph/0404076
4. M. Howard. Maximum nonlocality and minimum uncertainty using magic states. E-print: arXiv:arXiv:1501.05319 [quant-ph], 2015.
5. M. Araújo, F. Hirsch, and M. T. Quintino. Bell nonlocality with a single shot. E-print: arXiv:2005.13418
6. M. Froissart. Constructive generalization of Bell's inequalities. Nuov. Cim. B, 64:241, 1981
7. D. Collins and N. Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A: Math. Gen., 37(5):1175, 2004. E-print: arXiv:quant-ph/0306129
8. K.F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems. Phys. Rev. A, 82:022116, 2010. E-print: arXiv:1006.3032 [quant-ph]
9. M. Navascués, S. Pironio, and A. Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys., 10:073013, 2008. E-print: arXiv:0803.4290 [quant-ph]


text:
# Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$:

For each positive integer $k$, consider the sequence $x_n$ in $\mathbb{F}_3$

$x_i = 0$ for $i=1,2, \dots, k-1$, $x_k = 1$ and

$x_n = x_{n-1} + x_{n-k}^2$ for $n > k$.

Since we're working in a finite field, the sequence is eventually periodic. Let $N(k)$ denote the period. Can we write down a formula for $N(k)$? Can we give good bounds, etc? The values which have been computed are:

1,4,4,9,19,4,4,22,36,4,4,45,64,4,4,102,1082,231,4,188,272,4,412,225,202,4,4

Richard Pinch had an explanation for all the 4's as follows:

"The semi-ubiquitous 4's come from the fact that 2011 is a cycle when $k$ is $2 \bmod 4$ and 2201 when $k$ is $3 \bmod 4$. Of course the sequence does not have to lock on to that cycle: for example when $k$ is 18 or 23".

I can't find any more references to the problem than the ones that I give.

-
Did you try contacting Steve Finch directly? –  Wadim Zudilin Dec 29 '10 at 23:14
@Wadim: Yes, he told me that he knows nothing more than in his paper. –  Victor Miller Dec 30 '10 at 2:25