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# How do LIGO and Virgo know that a gravitational wave has its origin in a neutron star or a black hole?

It is being said that gravitational-wave detectors are now able to distinguish neutron star waves from those originating from black holes.

Two Questions:

1. How do LIGO and Virgo know that a gravitational wave has its origin in a neutron star or a black hole, if their gravitational fields, except for their intensity, are identical in that space beyond the radius that defines them?

2. Is this identification accurate and reliable?

The most obvious — though possibly least convincing — way is by noting the "mass gap": the heaviest neutron stars we know of (by other means) are lighter than 3 solar masses, while the lightest black holes we know of (by other means) are heavier than 5 solar masses. So if the constituents of a binary that LIGO detects have masses in one group or the other, LIGO/Virgo folks sort of expect that the objects are really in that group. If you look at the current confirmed detections (shown in the image below), you'll notice that there is indeed a significant gap between the masses of the neutron stars and the masses of the black holes. But part of LIGO/Virgo's job is to look for things that we can't find by other means, which might show us that there are lighter black holes (BHs) or heavier neutron stars (NSs) than we expect otherwise. So they don't stop there.

It's also possible to look for "tidal effects". Before two NSs (or one NS and one BH) actually touch, the matter in the neutron star will get distorted in ways that a black hole can't. The build up of this distortion takes energy, which comes out of the orbital energy of the binary, and that loss of energy imprints itself on the orbital motion — most prominently, on the "phase" of the binary which is the most accurately detected aspect of the inspiral. So when the OP says the BH and NS "gravitational fields, except for their intensity, are identical in that space beyond the radius that defines them", that's not quite true. It's true for isolated nonspinning objects (thanks to Birkhoff's theorem, which I guess is what the OP was thinking of), but it's not true for objects in binaries, and not once you get below the radius of the NS.

That brings up another important difference: NSs merge (basically) when their matter comes into contact with the other member of the binary, which is significantly earlier than BHs come into contact with each other. The BH radius is much smaller than the NS radius, so essentially a pair of BHs get to keep going for a while, going faster and faster than they would if a NS were present. This talk of distances is a bit imprecise, so it's better to talk about the GW signal observed at large distances from the binary (e.g., on Earth). You could — in principle — see this effect in the GW signal where the BH signal would just keep getting faster and stronger after the NS signal "shuts off". Of course, it's not really shut off; complicated stuff happens after NSs merge.

After the objects merge, they continue to exhibit huge differences. For example, if there's a NS involved, some matter can get flung out in a "tail" or into a disc around the central remnant. This extra motion of the matter (that wouldn't happen if there were only BHs) can generate its own gravitational waves, which could possibly be detected directly. More likely, the NS will "smear out" and just not be as good at emitting gravitational waves, so the peak amplitude will be smaller. However, after BHs merge, we know that they "ringdown" exponentially quickly. Basically, BHs have a very fast, simple, and well understood ringdown phase, whereas NSs have a messy and non-exponential aftermath. For example, we frequently talk about "mountains" on NSs afterward, which will continue to spin and give off sort of mildly damped but mostly continuous waves. Of course, it is possible that a merger with one or two NSs will end up with a single BH at the end, which will also ringdown, but before or in addition to that, we expect a lot of other complicated stuff to happen. [Note that the binary NS merger shown in the figure below ends up in a question mark, meaning that we're not entirely sure whether the remnant is a huge NS or a tiny BH.]

I should explain that these merger and post-merger effects happen at pretty high frequencies (because NSs are relatively low-mass objects), whereas LIGO and Virgo start to become much less sensitive as you go to higher frequencies (because at high frequencies there just aren't enough photons arriving at the interferometer's output; the number of photons per period, say, becomes quite random and therefore noisy). So it's not entirely clear whether or not we'll be able to see the "shutoff" or "mountains" with current detectors. A lot depends on unknown physics, and our ability to create good models for the signals given off by merging NSs. But it is true that we have not yet seen any direct evidence for them as of early 2019. So the last two items I described have not yet featured in claims about whether any source involves NSs or BHs.

But one thing that will tell us for sure if there was much matter involved — and was the reason we were so sure about the binary NS LIGO/Virgo announced in 2017 — is the presence of electromagnetic signals. Obviously, a pair of BHs on their own won't give off any obvious electromagnetic signal, whereas those huge amounts of matter when a NS is involved should give off some signal. If we detect an electromagnetic "counterpart", we can be much more confident that there was a lot of matter involved; if we don't detect any, it's unlikely that there was much matter in the system.

So there's no one piece of evidence that proves beyond doubt that there were only NSs or only BHs involved, but a collection of evidence that points in that direction. And really, how sure we are of the conclusion depends on a lot of factors. If the signal is very "loud" and clear, and the masses are very far from the mass gap, we can be particularly sure about our conclusions. But if the signal is from a source that's very far away, or is otherwise hard to measure, and if the masses are close to that mass gap, then we wouldn't be too sure about our conclusions. For all the systems confirmed so far, I think it's fair to say that most GW astronomers are extremely confident in the conclusions, but there are certainly more detections on the way that will be much more uncertain.

• @RobJeffries Notice that I didn't actually refer to "separation", because it really depends on precisely what you mean by "separation", but as a numerical relativist / GW astronomer, I can tell you that the meaning of "merger" that we actually use will be much later for BHs than for NSs. To put it another way: the signal will continue to evolve regularly to higher frequencies and amplitudes for a BBH than it would if a NS were involved. – Mike May 1 '19 at 18:36
• The maximum frequency for the first GW source was about 150 Hz. What was it for the neutron star merger? You mean at a fixed mass? – ProfRob May 1 '19 at 18:43
• Now you're comparing apples to oranges, and I'm not sure where you get 150 Hz, but typically the peak frequency of a BNS system will be 50% to 75% of the peak frequency of a BBH system with the same masses — depending on the EOS assumed for the NSs. (Of course, I never suggested that we could get ~30$M_\odot$ NSs to match the first GW source, so this only applies for masses where NSs could reasonably exist.) – Mike May 1 '19 at 19:18
• A bit surprisingly there aren't too many good plots of this, so this is an old one with very short signals, but Fig. 6 of arxiv:1306.4065 has pretty clear comparisons in the frequency domain. Fig. 3 of 1603.01286 has more recent simulations, but cuts off the higher frequencies of the BBH signal. And Figs. 5 and 6 of 1812.06988 have some decent time-domain comparisons. – Mike May 2 '19 at 2:33
• But it is true that I wasn't clear about that, so I'll edit my answer. – Mike May 3 '19 at 16:13

A measurement of the frequency of the gravitational waves from a binary system tells you the orbital period of the binary. The rate of change of that frequency tells you how fast that orbital period is changing. The combination of the two uniquely determines the "chirp mass". Basically, a lower mass binary system exhibits a much slower rate of increase in frequency during the chirp.

To first order, the following equation can be used: $$\frac{df}{dt} = \left(\frac{96}{5}\right)\left(\frac{GM_c}{c^3}\right)^{5/3}\pi^{8/3} f^{11/3}\, ,$$ where $$f$$ is the observed GW frequency and $$M_c$$ is the chirp mass, defined by $$M_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}\, ,$$ where $$m_1$$ and $$m_2$$ are the component masses.

Whilst the individual component masses are not determined, the chirp mass does give a lower limit to the total mass $$M_T \geq 2.3 M_c$$ and a lower limit to the mass of the most massive component $$m_1 \geq 1.15M_c$$.

So for example, the first detected gravitational wave source had a chirp mass of about 28 solar masses suggesting a minimum primary mass of about 32 solar masses.

Such large masses are totally incompatible with neutron star sources. Neutron stars have a maximum mass of about 3 solar masses (and the most massive yet observed are about 3 solar masses).

In addition we can observe what the frequency at the maximum gravitational wave amplitude. This frequency basically increases until the merger of the two begins. For the first GW source that is at about 150 Hz. You can then use Kepler's third law (with some small relativistic corrections) to estimate that the separation of the two objects In the case of the first GW source this yields a separation of only $$\sim 300$$ km.

The combination of these two analyses (cramming $$60+$$ solar masses of material into what amounts to about twice the Schwarzschild radius of a black hole of that mass) strongly suggests merging black holes.

On the contrary, the merger of lower mass neutron stars yields much lower chirp masses and a much higher peak frequency and separation. However, to discriminate between the merger of a neutron star and a black hole of the same total mass requires a careful analysis of the signal just prior to and after the merger and comparison with the predictions of General Relativity. Neutron stars have a surface and are affected by tidal forces which feeds into subtle differences in the GW signal.

Finally, a clinching piece of evidence is that the merger of neutron stars is expected to result in a significant "electromagnetic counterpart" - a kilonnova - that can be observed by other telescopes at Gamma ray through to optical and infrared wavelengths. Such a counterpart has been seen in the only confirmed neutron star merger signal, but is not expected to be present when black holes merge.

There are a couple things about the gravitational wave signal from a kilonova that distinguish it from the gravitational wave signal from merging binary black holes, some of which are discussed in the discovery paper for GW170817:

• The signal provides information on the chirp mass of the system, a quantity that can be used to infer the actual masses of the components. In the case of merging neutron stars, these masses (a few solar masses in total) should of course be substantially lower than the masses of merging stellar-mass black holes (a few tens of solar masses).
• The internal structure of the neutron stars should lead to tidal deformations that are visible in reconstructions of the signal. Black holes do not exhibit the same deformations.
• There should be electromagnetic counterparts to the gravitational wave that would not be expected in a binary black hole merger, and these transients were observed across the electromagnetic spectrum after GW170817 was detected.

Shortly after the first detection, the LIGO/Virgo collaborations published a paper in Annalen der Physik about The basic physics of the binary black hole merger GW150914. This is aimed at the general public, and shows with some back-of-the-envelope calculations why they think that the signal was due to two colliding black holes. It discusses the chirp mass and shows that the objects must have been extremely compact, since the final orbit was extremely small. This is already well summarized in Rob's great answer, but I suggest to read the paper itself, which is a pretty easy read.

• Indeed Bas, that is highly recommended reading (and I have of course read it!) Do you perhaps know of any publication that has model chirps directly compared between BHBs and NSBs with the same total mass? – ProfRob May 2 '19 at 9:55
• @RobJeffries not one paper in particular. I am not an expert on waveform modeling, but there are tons about tidal effects in BNS. As mentioned in Mike's answer, these show up as a fingerprint in especially the phase of the inspiral signal. – Bas Swinckels May 2 '19 at 11:58


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# Convert to Interval Notation (x-4)(x+1)<0

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Add to both sides of the equation.
Set the next factor equal to and solve.
Set the next factor equal to .
Subtract from both sides of the equation.
Consolidate the solutions.
Use each root to create test intervals.
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is not less than the right side , which means that the given statement is false.
False
False
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is less than the right side , which means that the given statement is always true.
True
True
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is not less than the right side , which means that the given statement is false.
False
False
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
The solution consists of all of the true intervals.
Convert the inequality to interval notation.
Convert to Interval Notation (x-4)(x+1)<0

format: plain


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Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

networkx.algorithms.isomorphism.GraphMatcher.initialize¶

GraphMatcher.initialize()[source]

Reinitializes the state of the algorithm.

This method should be redefined if using something other than GMState. If only subclassing GraphMatcher, a redefinition is not necessary.


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# NAG Library Routine Document

## 1Purpose

g05kgf initializes the selected base generator to generate a non-repeatable sequence of variates. The base generator can then be used by the group of pseudorandom number routines (see g05khfg05kjf, g05ncf, g05ndf, g05pdfg05pjf, g05pxfg05pzf, g05rcf, g05rdf, g05ryf, g05rzf and g05safg05tlf) and the quasi-random scrambled sequence initialization routine, g05ynf.

## 2Specification

Fortran Interface
 Subroutine g05kgf (
 Integer, Intent (In) :: genid, subid Integer, Intent (Inout) :: state(lstate), lstate, ifail
#include <nagmk26.h>
 void g05kgf_ (const Integer *genid, const Integer *subid, Integer state[], Integer *lstate, Integer *ifail)

## 3Description

g05kgf selects a base generator through the input value of the arguments genid and subid, and then initializes it based on the values taken from the real-time clock, resulting in the same base generator yielding different sequences of random numbers each time the calling program is run. It should be noted that there is no guarantee of statistical properties between sequences, only within sequences.
A definition of some of the terms used in this description, along with details of the various base generators can be found in the G05 Chapter Introduction.

## 4References

L'Ecuyer P and Simard R (2002) TestU01: a software library in ANSI C for empirical testing of random number generators Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal http://www.iro.umontreal.ca/~lecuyer
Maclaren N M (1989) The generation of multiple independent sequences of pseudorandom numbers Appl. Statist. 38 351–359
Matsumoto M and Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator ACM Transactions on Modelling and Computer Simulations
Wichmann B A and Hill I D (2006) Generating good pseudo-random numbers Computational Statistics and Data Analysis 51 1614–1622
Wikramaratna R S (1989) ACORN - a new method for generating sequences of uniformly distributed pseudo-random numbers Journal of Computational Physics 83 16–31

## 5Arguments

1:     $\mathbf{genid}$ – IntegerInput
On entry: must contain the type of base generator to use.
${\mathbf{genid}}=1$
NAG basic generator.
${\mathbf{genid}}=2$
Wichmann Hill I generator.
${\mathbf{genid}}=3$
Mersenne Twister.
${\mathbf{genid}}=4$
Wichmann Hill II generator.
${\mathbf{genid}}=5$
ACORN generator.
${\mathbf{genid}}=6$
L'Ecuyer MRG32k3a generator.
See the G05 Chapter Introduction for details of each of the base generators.
Constraint: ${\mathbf{genid}}=1$, $2$, $3$, $4$, $5$ or $6$.
2:     $\mathbf{subid}$ – IntegerInput
On entry: if ${\mathbf{genid}}=2$, subid indicates which of the $273$ sub-generators to use. In this case, the  sub-generator is used.
If ${\mathbf{genid}}=5$, subid indicates the values of $k$ and $p$ to use, where $k$ is the order of the generator, and $p$ controls the size of the modulus, $M$, with $M={2}^{\left(p×30\right)}$. If ${\mathbf{subid}}<1$, the default values of $k=10$ and $p=2$ are used, otherwise values for $k$ and $p$ are calculated from the formula, ${\mathbf{subid}}=k+1000\left(p-1\right)$.
If ${\mathbf{genid}}=6$ and  the range of the generator is set to $\left(0,1\right]$, otherwise the range is set to $\left(0,1\right)$; in this case the sequence is identical to the implementation of MRG32k3a in TestU01 (see L'Ecuyer and Simard (2002)) for identical seeds.
For all other values of genid, subid is not referenced.
3:     $\mathbf{state}\left({\mathbf{lstate}}\right)$ – Integer arrayCommunication Array
On exit: contains information on the selected base generator and its current state.
4:     $\mathbf{lstate}$ – IntegerInput/Output
On entry: the dimension of the state array, or a value $<1$. If the Mersenne Twister (${\mathbf{genid}}=3$) is being used and the skip ahead routine g05kjf or g05kkf will be called subsequently, then you must ensure that ${\mathbf{lstate}}\ge 1260$.
On exit: if ${\mathbf{lstate}}<1$ on entry, then the required length of the state array for the chosen base generator, otherwise lstate is unchanged. When ${\mathbf{genid}}=3$ (Mersenne Twister) a value of $1260$ is returned, allowing for the skip ahead routine to be subsequently called. In all other cases the minimum length, as documented in the constraints below, is returned.
Constraints:
• if ${\mathbf{genid}}=1$, ${\mathbf{lstate}}\ge 17$;
• if ${\mathbf{genid}}=2$, ${\mathbf{lstate}}\ge 21$;
• if ${\mathbf{genid}}=3$, ${\mathbf{lstate}}\ge 633$;
• if ${\mathbf{genid}}=4$, ${\mathbf{lstate}}\ge 29$;
• if ${\mathbf{genid}}=5$, ${\mathbf{lstate}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left(k+1\right)×p+9,14\right)+3$, where $k$ and $p$ are defined by subid;
• if ${\mathbf{genid}}=6$, ${\mathbf{lstate}}\ge 61$;
• otherwise ${\mathbf{lstate}}<1$.
5:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ or $-1$ unless the routine detects an error or a warning has been flagged (see Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{genid}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{genid}}=1$, $2$, $3$, $4$, $5$ or $6$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{lstate}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lstate}}\le 0$ or ${\mathbf{lstate}}\ge 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-1$
Required length of state array returned in lstate but state array not initialized.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

g05kgf is not threaded in any implementation.

None.

## 10Example

In order to preserve the statistical properties of the base generators, g05kgf should only be called once. If multiple streams of values are required then one of the methods described in Section 2.1.1 in the G05 Chapter Introduction should be used.
However, for illustrative purposes only, this example calls g05kgf twice. At each call a sample of $500$ values from a discrete uniform distribution are generated and then the two samples are compared.

### 10.1Program Text

Program Text (g05kgfe.f90)

None.

### 10.3Program Results

Program Results (g05kgfe.r)


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Research

# Tripled coincidence point results for generalized contractions in ordered generalized metric spaces

Hassen Aydi1, Erdal Karapınar2* and Wasfi Shatanawi3

Author Affiliations

1 Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011 Hammam Sousse, Tunisia

2 Department of Mathematics, Atılım University, Incek, 06836 Ankara, Turkey

3 Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

For all author emails, please log on.

Fixed Point Theory and Applications 2012, 2012:101  doi:10.1186/1687-1812-2012-101

 Received: 4 August 2011 Accepted: 21 June 2012 Published: 21 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we establish some tripled coincidence point results for a mixed g-monotone mapping F : X3 X satisfying (ψ, ϕ)-contractions in ordered generalized metric spaces. Also, an application and some examples are given to support our results.

### Introduction and preliminaries

Banach contraction principle is one of the core subject that has been studied. It has so many different generalizations with different approaches. One of the remarkable generalizations, known as Φ-contraction, was given by Boyd and Wong [1] in 1969. In 1997, Alber and Guerre-Delabriere [2], introduced the notion of a weak ϕ-contraction which generalizes Boyd and Wong results, so Banach's result. Very recently, inspired from the notion of weak ϕ-contractions, a new concept of ( ψ, ϕ)-contractions was introduced (see e.g. [3-7]).

Mustafa and Sims [8] introduced the notion of generalized metric spaces or simply G-metric spaces as a generalization of the concept of a metric space. Based on the concept of G-metric space, Mustafa et al. [9-11] proved several fixed point theorems for mapping satisfying different contractive condition. The study of common fixed point was initiated by Abbas and Rhoades [12]. The first result for contractive mappings in ordered G-metric spaces was obtained by Saadati et al. [13]. Bhashkar and Lakshmikantham [14] introduced the concept of a coupled fixed point of a mapping F : X × X X and proved some coupled fixed point theorems in ordered metric spaces. Some authors obtained some interesting coupled fixed point theorems in G-metric spaces (see e.g. [15-18]). For more results on G-metric spaces, one can refer to the papers [9-13,15-28].

Throughout the paper, ℕ* is the set of positive integers. In 2004, Mustafa and Sims [8] introduced the concept of G-metric spaces as follows:

Definition 1. (see [8]). Let X be a non-empty set, G : X × X × X → ℝ+ be a function satisfying the following properties

(G1) G(x, y, z) = 0 if x = y = z,

(G2) 0 < G(x, x, y) for all x, y X with x y,

(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z X with y z,

(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = ⋯ (symmetry in all three variables),

(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a X (rectangle inequality).

Then the function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X, G) is called a G-metric space.

Every G-metric on X defines a metric dG on X by

d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) , for all x , y X . (1)

Example 2. Let (X, d) be a metric space. The function G : X × X × X → [0, +∞), defined by

G ( x , y , z ) = max { d ( x , y ) , d ( y , z ) , d ( z , x ) } ,

or

G ( x , y , z ) = d ( x , y ) + d ( y , z ) + d ( z , x ) ,

for all x, y, z X, is a G-metric on X.

Definition 3. (see [8]). Let (X, G) be a G-metric space, and let {xn} be a sequence of points of X, therefore, we say that {xn} is G-convergent to x X if lim n , m + G ( x , x n , x m ) = 0 , that is, for any ε > 0, there exists N ∈ ℕ such that G(x, xn, xm) < ε, for all n, m N. We call x the limit of the sequence and write xn x or lim n + x n = x .

Proposition 4. (see [8]). Let (X,G) be a G-metric space. The following are equivalent:

(1) {xn} is G-convergent to x,

(2) G(xn, xn, x) → 0 as n → +∞,

(3) G(xn, x, x) → 0 as n → +∞,

(4) G(xn, xm, x) → 0 as n, m → +∞.

Definition 5. (see [8]). Let (X, G) be a G-metric space. A sequence {xn} is is called a G-Cauchy sequence if, for any ε > 0, there is N ∈ ℕ such that G(xn, xm, xl) < ε for all m, n, l N, i.e., G(xn, xm, xl) → 0 as n, m, l → ∞.

Proposition 6. (see [8]). Let (X, G) be a G-metric space. Then the following are equivalent:

(1) the sequence {xn} is G-Cauchy,

(2) for any ε > 0, there exists N ∈ ℕ such that G(xn, xm, xm) < ε, for all m, n N.

Definition 7. (see [8]). A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X, G).

Definition 8. Let (X, G) be a G-metric space. A mapping F : X × X × X X is said to be continuous if for any three G-convergent sequences {xn}, {yn} and {zn} converging to x, y and z respectively, {F(xn, yn, zn)} is G-convergent to F(x, y, z).

Following the paper of Berinde and Borcut [29], Aydi, Karapınar and Postolache [30] introduced the following definitions:

Definition 9. (see [30]). Let (X, ≤) be a partially ordered set and F : X × X × X X, g : X X. The mapping F is said to has the mixed g-monotone property if for any x, y, z X

x 1 , x 2 X , g x 1 g x 2 F ( x 1 , y , z ) F ( x 2 , y , z ) , y 1 , y 2 X , g y 1 g y 2 F ( x , y 1 , z ) F ( x , y 2 , z ) , z 1 , z 2 X , g z 1 g z 2 F ( x , y , z 1 ) F ( x , y , z 2 ) . (2)

Definition 10. (see [30]). Let F : X × X × X X and g : X X. An element (x, y, z) is called a tripled coincidence point of F and g if

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

The point (gx, gy, gz) is called a point of coincidence of F and g.

Definition 11. (see [30]). Let F : X × X × X X and g : X X. An element (x, y, z) is called a tripled common fixed point of F and g if

F ( x , y , z ) = g x = x , F ( y , x , y ) = g y = y , F ( z , y , x ) = g z = z .

Definition 12. (see [30]). Let X be a non-empty set. Let F: X × X × X X and g: X X are such that

g ( F ( x , y , z ) ) = F ( g x , g y , g z )

whenever x, y, z X, then F and g are said to be commutative.

Khan et al. [31] introduced the concept of altering distance function as follows:

Definition 13. (altering distance function, [31]) The function ψ : [0, + ∞) → [0, + ∞) is called an altering distance function if the following properties are satisfied:

(1) ψ is continuous and non-decreasing,

(2) ψ(t) = 0 if and only if t = 0.

Let Ψ be the set of altering distances. Again, we denote by Φ the set of functions ϕ : [0, +∞) → [0, +∞) such that

(i) ϕ is lower-continuous and non-decreasing,

(ii) ϕ(t) = 0 if and only if t = 0.

The notion of a fixed point of N-order was first introduced by Samet and Vetro [32]. Later, Berinde and Borcut [29] proved some tripled fixed point results (N = 3) in partially ordered metric spaces (see also [33-36]). In this paper, we establish tripled coincidence point results for mappings F : X3 X and g : X X involving nonlinear contractions in the setting of ordered G-metric spaces. Also, we present an application and some examples in support of our results.

### Main results

Before stating our results, we give the following useful lemma.

Lemma 14. Consider three non-negative real sequences {an}, {bn} and {cn}. Suppose there exists α ≥ 0 such that

lim n + max { a n , b n } = 0 a n d lim n + max { a n , b n , c n } = α .

Then, lim sup n + c n = α .

Proof. First, we have cn ≤ max{an, bn, cn}, then

lim sup n + c n α . (3)

For all n ∈ ℕ, we have

0 max { a n , b n , c n } - c n max { a n , b n } + c n - c n = max { a n , b n } ,

which implies that lim sup n + ( max { a n , b n , c n } - c n ) 0 . Having in mind that

lim sup n + ( max { a n , b n , c n } - c n ) lim sup n + max { a n , b n , c n } - lim sup n + c n = α - lim sup n + c n ,

so it follows that

lim sup n + c n α . (4)

By (3) and (4), we get that lim sup n + c n = α . □

The aim of this paper is to prove the following theorem.

Theorem 15. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 X and g : X X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc gw, we have

ψ ( G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) ) ψ ( m a x { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } ) - ϕ ( m a x { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } ) . (5)

Assume that F and g satisfy the following conditions:

(1) F(X3) ⊆ g(X),

(2) F has the mixed g-monotone property,

(3) F is continuous,

(4) g is continuous and commutes with F.

Suppose there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Proof. Suppose x0, y0, z0 X are such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0), and gz0 F(z0, y0, x0). Since F(X3) ⊆ g(X), we can choose gx1 = F(x0, y0, z0), gy1 = F(y0, x0, y0) and gz1 = F(z0, y0, x0). Then gx0 gx1, gy0 gy1 and gz0 gz1. Similarly, define gx2 = F(x1, y1, z1), gy2 = F(y1, x1, y1) and gz2 = F(z1, y1, x1). Since F has the mixed g-monotone property, we have gx0 gx1 gx2, gy2 gy1 gy0 and gz0 gz1 gz2. Continuing this process, we can construct three sequences {xn}, {yn} and {zn} in X such that

g x n = F ( x n - 1 , y n - 1 , z n - 1 ) g x n + 1 = F ( x n , y n , z n ) , g y n + 1 = F ( y n , x n , y n ) g y n = F ( y n - 1 , x n - 1 , y n - 1 ) ,

and

g z n = F ( z n - 1 , y n - 1 , x n - 1 ) g z n + 1 = F ( z n , y n , x n ) .

If, for some integer n0, we have ( g x n 0 + 1 , g y n 0 + 1 , g z n 0 + 1 ) = ( g x n 0 , g y n 0 , g z n 0 ) , then F ( x n 0 , y n 0 , z n 0 ) = g x n 0 , F ( y n 0 , x n 0 , y n 0 ) = g y n 0 , and F ( z n 0 , y n 0 , x n 0 ) = g z n 0 ; i.e., ( x n 0 , y n 0 , z n 0 ) is a tripled coincidence point of F and g. Thus we shall assume that (gxn+1, gyn+1, gzn+1) ≠ (gxn, gyn, gzn) for all n ∈ ℕ; i.e., we assume that either gxn+1 gxn or gyn+1 gyn or gzn+1 gzn. For any n ∈ ℕ*, we have from (5)

ψ ( G ( g x n + 1 , g x n , g x n ) ) : = ψ ( G ( F ( x n , y n , z n ) , F ( x n 1 , y n 1 , z n 1 ) , F ( x n 1 , y n 1 , z n 1 ) ) ) ψ ( max { G ( g x n , g x n 1 , g x n 1 ) , G ( g y n , g y n 1 , g y n 1 ) , G ( g z n 1 , g z n 1 ) } ) ϕ ( max { G ( g x n , g x n 1 , g x n 1 ) , G ( g y n , g y n 1 , g y n 1 ) , G ( g z n , g z n 1 , g z n 1 ) } ) ψ ( max { G ( g x n , g x n 1 , g x n 1 ) , G ( g y n , g y n 1 ) g y n 1 ) , G ( g z n , g z n 1 , g z n 1 ) } ) , (6)

ψ ( G ( g y n , g y n , g y n + 1 ) ) : = ψ ( G ( F ( y n 1 , x n 1 , y n 1 ) , F ( y n 1 , x n 1 , y n 1 ) , F ( y n , x n , y n ) ) ) ψ ( max { G ( g y n 1 , g y n 1 , g y n ) , G ( g x n 1 , g x n 1 , g x n ) } ϕ ( max { G ( g y n 1 , g y n 1 , g y n ) , G ( g x n 1 , g x n 1 , g x n ) } ) ψ ( max { G ( g y n , g y n 1 , g y n 1 ) , G ( g x n , g x n 1 , g x n 1 ) , G ( g z n , g z n 1 , g z n 1 ) } ) , (7)

and

ψ ( G ( g z n + 1 , g z n , g z n ) ) : = ψ ( G ( F ( z n , y n , x n ) , F ( z n - 1 , y n - 1 , x n - 1 ) , F ( z n - 1 , y n - 1 , x n - 1 ) ) ) ψ ( max { G ( g z n , g z n - 1 , g z n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g x n , g x n - 1 , g x n - 1 ) } ) - ϕ ( max { G ( F ( z n , y n , x n ) , F ( z n - 1 , y n - 1 , x n - 1 ) , F ( z n - 1 , y n - 1 , x n - 1 ) ) } ) ψ ( max { G ( g z n , g z n - 1 , g z n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g x n , g x n - 1 , g x n - 1 ) } ) . (8)

Since ψ:[0,+∞) → [0, +∞) is a non-decreasing function, for a, b, c ∈ [0,+∞), we have ψ(max{a, b, c}) = max{ψ(a), ψ(b), ψ(c)}. Then, from (6), (7), and (8), it follows that

ψ ( max { G ( g x n + 1 , g x n , g x n ) , G ( g y n , g y n , g y n + 1 ) , G ( g z n + 1 , g z n , g z n ) } ) = max ( { ψ ( G ( g x n + 1 , g x n , g x n ) ) , ψ ( G ( g y n , g y n , g y n + 1 ) ) , ψ ( G ( g z n + 1 , g z n , g z n ) ) } ) ψ ( max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g z n , g z n - 1 , g z n - 1 ) } ) .

The fact that ψ is non-decreasing yields that

max { G ( g x n + 1 , g x n , g x n ) , G ( g y n , g y n , g y n + 1 ) , G ( g z n + 1 , g z n , g z n ) } max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g z n , g z n - 1 , g z n - 1 ) } . (9)

Thus, {max{G(gxn+1, gxn, gxn),G(gyn, gyn, gyn+1),G(gzn+1, gzn, gzn)}} is a positive non-increasing sequence. Hence there exists r ≥ 0 such that

lim n + max { G ( g x n + 1 , g x n , g x n ) , G ( g y n , g y n , g y n + 1 ) , G ( g z n + 1 , g z n , g z n ) } = r . (10)

Having in mind that ϕ is non-decreasing, then

ϕ ( max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g z n , g z n - 1 , g z n - 1 ) } ) ϕ ( max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) } ) , (11)

so from (6)-(8), we get that

ψ ( max { G ( g x n + 1 , g x n , g x n ) , G ( g y n , g y n , g y n + 1 ) , G ( g z n + 1 , g z n , g z n ) } ) = max ( { ψ ( G ( g x n + 1 , g x n , g x n ) ) , ψ ( G ( g y n , g y n , g y n + 1 ) ) , ψ ( G ( g z n + 1 , g z n , g z n ) ) } ) ψ ( max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) , G ( g z n , g z n - 1 , g z n - 1 ) } ) - ϕ ( max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n , g y n - 1 , g y n - 1 ) } ) . (12)

On the other hand,

0 max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n - 1 , g y n - 1 , g y n ) } max { G ( g x n , g x n - 1 , g x n - 1 ) , G ( g y n - 1 , g y n - 1 , g y n ) , G ( g z n , g z n - 1 , g z n - 1 ) } , (13)

so by (10), the real sequence {max{G(gxn, gxn-1, gxn-1),G(gyn-1, gyn-1, gyn)}} is bounded. Thus, there exists a real number r1 with 0 ≤ r1 r and subsequences {xn(k)} of {xn} and {yn(k)} of {yn} such that

lim k + max { G ( g x n ( k ) + 1 , g x n ( k ) , g x n ( k ) ) , G ( g y n ( k ) , g y n ( k ) , g y n ( k ) + 1 ) } = r 1 . (14)

We rewrite (12)

ψ ( max { G ( g x n ( k ) + 1 , g x n ( k ) , g x n ( k ) ) , G ( g y n ( k ) , g y n ( k ) , g y n ( k ) + 1 ) , G ( g z n ( k ) + 1 , g z n ( k ) , g z n ( k ) } ) ψ ( max { G ( g x n ( k ) , g x n ( k ) 1 , g x n ( k ) 1 ) , G ( g y n ( k ) , g y n ( k ) 1 , g y n ( k ) 1 ) , G ( g z n ( k ) , g z n ( k ) 1 , g z n ( k ) 1 ) } ) ϕ ( max { G ( g x n ( k ) , g x n ( k ) 1 , g x n ( k ) 1 ) , G ( g y n ( k ) , g y n ( k ) 1 , g y n ( k ) 1 ) } ) . (15)

Letting k → +∞ in (15), having in mind (10), (14), the continuity of ψ and the lower semi-continuity of ϕ, we obtain

ψ ( r ) = lim sup k + ψ ( max { G ( g x n ( k ) + 1 , g x n ( k ) , g x n ( k ) ) , G ( g y n ( k ) , g y n ( k ) , g y n ( k ) + 1 ) , G ( g z n ( k ) + 1 , g z n ( k ) , g z n ( k ) ) } ) lim sup k + ψ ( max { G ( g x n ( k ) , g x n ( k ) - 1 , g x n ( k ) - 1 ) , G ( g y n ( k ) , g y n ( k ) - 1 , g y n ( k ) - 1 ) , G ( g z n ( k ) , g z n ( k ) - 1 , g z n ( k ) - 1 ) } ) - lim inf k + ϕ ( max { G ( g x n ( k ) , g x n ( k ) - 1 , g x n ( k ) - 1 ) , G ( g y n ( k ) , g y n ( k ) - 1 , g y n ( k ) - 1 ) } ) ψ ( r ) - ϕ ( r 1 ) ,

which implies that ϕ(r1) = 0, and using a property of ϕ, we find r1 = 0. Thanks to Lemma 14 together with (10) and (14), it yields that

r = : lim k + max { G ( g x n ( k ) , g x n ( k ) - 1 , g x n ( k ) - 1 ) , G ( g y n ( k ) , g y n ( k ) - 1 , g y n ( k ) - 1 ) , G ( g z n ( k ) , g z n ( k ) - 1 , g z n ( k ) - 1 ) } = lim sup k + G ( g z n ( k ) , g z n ( k ) - 1 , g z n ( k ) - 1 ) . (16)

For any k ∈ ℕ, we rewrite (8) as

ψ ( G ( g z n ( k ) + 1 , g z n ( k ) , g z n ( k ) ) ) ψ ( max { G ( g z n ( k ) , g z n ( k ) - 1 , g z n ( k ) - 1 ) , G ( g y n ( k ) , g y n ( k ) - 1 , g y n ( k ) - 1 ) , G ( g x n ( k ) , g x n ( k ) - 1 , g x n ( k ) - 1 ) } ) - ϕ ( max { G ( g z n ( k ) , g z n ( k ) - 1 , g z n ( k ) - 1 ) , G ( g y n ( k ) , g y n ( k ) - 1 , g y n ( k ) - 1 ) , G ( g x n ( k ) , g x n ( k ) - 1 , g x n ( k ) - 1 ) } ) . (17)

Again, letting k → +∞ in (17), having in mind (10), (16) and by the properties of ψ, ϕ, we obtain

ψ ( r ) = lim sup k + ψ ( G ( g z n ( k ) + 1 , g z n ( k ) , g z n ( k ) ) ) lim sup k + ψ ( max { g z n ( k ) , g z n ( k ) 1 , g z n ( k ) 1 ) , G ( g y n ( k ) , g y n ( k ) 1 , g y n ( k ) 1 ) , G ( g x n ( k ) , g x n ( k ) 1 , g x n ( k ) 1 ) } ) lim inf k + ϕ ( max { G ( g z n ( k ) , g z n ( k ) 1 , g z n ( k ) 1 ) , G ( g y n ( k ) , g y n ( k ) 1 , g y n ( k ) 1 ) , G ( g x n ( k ) , g x n ( k ) 1 , g x n ( k ) 1 ) } ) ψ ( r ) ϕ ( r ) ,

which gives that ϕ(r) = 0, so r = 0, i.e., by (10),

lim n + max { G ( g x n + 1 , g x n , g x n ) , G ( g y n , g y n , g y n + 1 ) , G ( g z n + 1 , g z n , g z n ) } = 0 . (18)

Our next step is to show that {gxn}, {gyn} and {gzn} are G-Cauchy sequences.

Assume the contrary, i.e., {gxn}, {gyn} or {gzn} is not a G-Cauchy sequence, i.e.,

lim n , m + G ( g x m , g x n g x n ) 0 , or lim n , m + G ( g y m , g y n , g y n ) 0 ,

or lim n , m + G ( g z m , g z n , g z n ) 0 . This means that there exists ε > 0 for which we can find subsequences of integers {mk} and {nk} with nk > mk > k such that

max { G ( g x m k , g x n k , g x n k ) , G ( g y m k , g y n k , g y n k ) , G ( g z m k , g z n k , g z n k ) } ε . (19)

Further, corresponding to mk we can choose nk in such a way that it is the smallest integer with nk > mk and satisfying (19). Then

max { G ( g x m k , g x n k - 1 , g x n k - 1 ) , G ( g y m k , g y n k - 1 , g y n k - 1 ) , G ( g z m k , g z n k - 1 , g z n k - 1 ) } < ε . (20)

By (G5) and (20), we have

G ( g x m k , g x n k , g x n k ) G ( g x m k , g x n k - 1 , g x n k - 1 ) + G ( g x n k - 1 , g x n k , g x n k ) < ε + G ( g x n k - 1 , g x n k , g x n k ) .

Thus, by (18) we obtain

lim k + G ( g x m k , g x n k , g x n k ) lim k + G ( g x m k , g x n k - 1 , g x n k - 1 ) ε . (21)

Similarly, we have

lim k + G ( g y m k , g y n k , g y n k ) lim k + G ( g y m k , g y n k - 1 , g y n k - 1 ) ε . (22)

lim k + G ( g z m k , g z n k , g z n k ) lim k + G ( g z m k , g z n k - 1 , g z n k - 1 ) ε . (23)

Again by (G5) and (20), we have

G ( g x m k , g x n k , g x n k ) G ( g x m k , g x m k - 1 , g x m k - 1 ) + G ( g x m k - 1 , g x n k - 1 , g x n k - 1 ) + G ( g x n k - 1 , g x n k , g x n k ) G ( g x m k , g x m k - 1 , g x m k - 1 ) + G ( g x m k - 1 , g x m k , g x m k ) + G ( g x m k , g x n k - 1 , g x n k - 1 ) + G ( g x n k - 1 , g x n k , g x n k ) < G ( g x m k , g x m k - 1 , g x m k - 1 ) + G ( g x m k - 1 , g x m k , g x m k ) + ε + G ( g x n k - 1 , g x n k , g x n k ) .

Letting k → +∞ and using (18), we get

lim k + G ( g x m k , g x n k , g x n k ) lim k + G ( g x m k - 1 , g x n k - 1 , g x n k - 1 ) ε , (24)

lim k + G ( g y m k , g y n k , g y n k ) lim k + G ( g y m k - 1 , g y n k - 1 , g y n k - 1 ) ε , (25)

lim k + G ( g z m k , g z n k , g z n k ) lim k + G ( g z m k - 1 , g z n k - 1 , g z n k - 1 ) ε . (26)

Using (19) and (24)-(26), we have

lim k + max { G ( g x m k , g x n k , g x n k ) , G ( g y m k , g y n k , g y n k ) , G ( g z m k , g z n k , g z n k ) } = lim k + max { G ( g x m k 1 , g x n k 1 , g x n k 1 ) , G ( g y m k 1 , g y n k 1 , g y n k 1 ) , G ( g z m k 1 , g z n k 1 , g z n k 1 ) = ε . (27)

Now, using inequality (5) we obtain

ψ ( G ( g x m k , g x n k , g x n k ) ) = ψ ( G ( F ( x m k - 1 , y m k - 1 , z m k - 1 ) , F ( x n k - 1 , y n k - 1 , z n k - 1 ) , F ( x n k - 1 , y n k - 1 , z n k - 1 ) ) ) ψ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) , G ( z m k - 1 , z n k - 1 , z n k - 1 ) } ) - ϕ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) , G ( z m k - 1 , z n k - 1 , z n k - 1 ) } ) (28)

ψ ( G ( g y m k , g y n k , g y n k ) ) = ψ ( G ( F ( y m k - 1 , x m k - 1 , y m k - 1 ) , F ( y n k - 1 , x n k - 1 , y n k - 1 ) , F ( y n k - 1 , x n k - 1 , y n k - 1 ) ) ) ψ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) } ) - ϕ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) } ) (29)

and

ψ ( G ( g z m k , g z n k , g z n k ) ) = ψ ( G ( F ( z m k - 1 , y m k - 1 , x m k - 1 ) , F ( z n k - 1 , y n k - 1 , x n k - 1 ) , F ( z n k - 1 , y n k - 1 , x n k - 1 ) ) ) ψ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) , G ( z m k - 1 , z n k - 1 , z n k - 1 ) } ) - ϕ ( max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) , G ( z m k - 1 , z n k - 1 , z n k - 1 ) } ) (30)

We deduce from (28)-(30) that

ψ ( max { G ( g x m k , g x n k , g x n k ) , G ( g y m k , g y n k , g y n k ) , G ( g z m k , g z n k , g z n k ) } = max { ψ ( G ( g x m k , g x n k , g x n k ) ) , ψ ( G ( g y m k , g y n k , g y n k ) ) , ψ ( G ( g z m k , g z n k , g z n k ) ) } ψ ( max { G ( x m k 1 , x n k 1 , x n k 1 ) , G ( y m k 1 , y n k 1 , y n k 1 ) , G ( z m k 1 , z n k 1 , z n k 1 ) } ) ϕ ( max { G ( x m k 1 , x n k 1 , x n k 1 ) , G ( y m k 1 , y n k 1 , y n k 1 ) } ) . (31)

On the other hand, since

max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) } max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) , G ( z m k - 1 , z n k - 1 , z n k - 1 ) } , (32)

then from (27),

lim sup k + max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) } ε .

Therefore, the real sequence { max { G ( x m k 1 , x n k 1 , x n k 1 ) , G ( y m k 1 , y n k 1 , y n k 1 ) } } is bounded. Thus, up to a subsequence (still denoted the same), there exists ε1 with 0 ≤ ε1 ε such that

lim k + max { G ( x m k - 1 , x n k - 1 , x n k - 1 ) , G ( y m k - 1 , y n k - 1 , y n k - 1 ) } = ε 1 . (33)

Inserting this in (31) and using (27), (33) together with the properties of ψ, ϕ, we get that

ψ ( ε ) = lim sup k + ψ ( max { G ( g x m k , g x n k , g x n k ) , G ( g y m k , g y n k , g y n k ) , G ( g z m k , g z n k , g z n k ) } lim sup k + ψ ( max { G ( x m k 1 , x n k 1 , x n k 1 ) , G ( y m k 1 , y n k 1 , y n k 1 ) , G ( z m k 1 , z n k 1 , z n k 1 ) } ) lim inf k + ϕ ( max { G ( x m k 1 , x n k 1 , x n k 1 ) , G ( y m k 1 , y n k 1 , y n k 1 ) } ) ψ ( ε ) ϕ ( ε 1 ) ,

which leads to ϕ(ε1) = 0, so ε1 = 0. By this and (27), due to Lemma 14, we obtain

lim sup k + G ( z m k - 1 , z n k - 1 , z n k - 1 ) = ε .

Combining this to (19) and (26), we find

lim sup k + G ( z m k , z n k , z n k ) = ε .

Letting k → ∞ in (30) and using (27), we deduce

ψ ( ε ) ψ ( ε ) - ϕ ( ε ) ,

i.e., ε = 0, it is a contradiction. We conclude that {gxn}, {gyn} and {gzn} are G-Cauchy sequences in the G-metric space (X, G), which is G-complete. Then, there are x, y, z X such that {gxn}, {gyn} and {gzn} are respectively G-convergent to x, y and z, i.e., from Proposition 4, we have

lim n + G ( g x n , g x n , x ) = lim n + G ( g x n , x , x ) = 0 , (34)

lim n + G ( g y n , g y n , y ) = lim n + G ( g y n , y , y ) = 0 , (35)

lim n + G ( g z n , g z n , z ) = lim n + G ( g z n , z , z ) = 0 . (36)

From (34)-(36) and the continuity of g, we get thanks to Proposition 8

lim n + G ( g ( g x n ) , g ( g x n ) , g x ) = lim n + G ( g ( g x n ) , g x , g x ) = 0 , (37)

lim n + G ( g ( g y n ) , g ( g y n ) , g y ) = lim n + G ( g ( g y n ) , g y , g y ) = 0 , (38)

lim n + G ( g ( g z n ) , g ( g z n ) , g z ) = lim n + G ( g ( g z n ) , g z , g z ) = 0 . (39)

Since gxn+1 = F(xn, yn, zn), gyn+1 = F(yn, xn, yn) and gzn+1 = F(zn, yn, xn), so the commutativity of F and g yields that

g ( g x n + 1 ) = g ( F ( x n , y n , z n ) ) = F ( g x n , g y n , g z n ) , (40)

g ( g y n + 1 ) = g ( F ( y n , x n , y n ) ) = F ( g y n , g x n , g y n ) , (41)

g ( g z n + 1 ) = g ( F ( z n , y n , x n ) ) = F ( g z n , g y n , g x n ) . (42)

Now we show that F(x, y, z) = gx, F(y, x, y) = gy and F(z, y, x) = gz.

The mapping F is continuous, so since the sequences {gxn}, {gyn} and {gzn} are, respectively, G-convergent to x, y and z, hence using Definition 8, the sequence {F(gxn, gyn, gzn)} is G-convergent to F(x, y, z). Therefore, from (40), {g(gxn+1)} is G-convergent to F(x, y, z). By uniqueness of the limit, from (37) we have F(x, y, z) = gx.

Similarly, one finds F(y, x, y) = gy and F(z, y, x) = gz, and this makes end to the proof. □

Corollary 16. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 X and g : X X. Assume there exists k ∈ [0,1) such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc gw, we have

G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) k max { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } .

Assume that F and g satisfy the following conditions:

(1) F(X3) ⊂ g(X),

(2) F has the mixed g-monotone property,

(3) F is continuous,

(4) g is continuous and commutes with F.

Suppose there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Proof. If follows by taking ψ(t) = t and ϕ(t) = (1 - k)t for all t ≥ 0. □

Corollary 17. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 X and g : X X. Assume there exists k ∈ [0,1) such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc gw, we have

G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) k 3 ( G ( g x , g a , g u ) + G ( g y , g b , g v ) + G ( g z , g c , g w ) ) .

Assume that F and g satisfy the following conditions:

(1) F(X3) ⊆g(X),

(2) F has the mixed g-monotone property,

(3) F is continuous,

(4) g is continuous and commutes with F.

Suppose there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Proof. It suffices to remark that

k 3 ( G ( g x , g a , g u ) + G ( g y , g b , g v ) + G ( g z , g c , g w ) ) k max { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } . (43)

In the next theorem, we omit the continuity hypothesis of F. We need the following definition.

Definition 18. Let (X, ≼) be a partially ordered set and G be a G-metric on X. We say that (X, G, ≤) is regular if the following conditions hold:

(i) if a non-decreasing sequence {xn} is such that xn x, then xn x for all n,

(ii) if a non-increasing sequence {yn} is such that yn y, then y yn for all n.

Theorem 19. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Let F : X3 X and g: X X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc gw, we have

ψ ( G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) ) ψ ( max { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } ) - ϕ ( max { G ( g x , g a , g u ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } ) . (44)

Assume that (X, G, ≤) is regular. Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Proof. Proceeding exactly as in Theorem 15, we have that {gxn}, {gyn} and {gzn} are G-Cauchy sequences in the G-complete G-metric space (g(X), G). Then, there exist x, y, z X such that gxn gx, gyn gy and gzn gz. Since {gxn} and {gzn} are non-decreasing and {gyn} is non-increasing, using the regularity of (X, G, ≤), we have gxn gx, gzn gz and gy gyn for all n ≥ 0. Using (5), we get

ψ ( G ( F ( x , y , z ) , g x n + 1 , g x n + 1 ) = ψ ( G ( F ( x , y , z ) , F ( x n , y n , z n ) , F ( x n , y n , z n ) ) ) ψ ( max { G ( g x , g x n , g x n ) , G ( g y , g y n , g y n ) , G ( g z , g z n , g z n ) } ) ϕ ( max { G ( g x , g x n , g x n ) , G ( g y , g y n , g y n ) , G ( g z , g z n , g z n ) } ) . (45)

Letting n → +∞ in the above inequality, we obtain that

ψ ( G ( F ( x , y , z ) , g x , g x ) ) ψ ( 0 ) - ϕ ( 0 ) = 0 ,

which implies that G(F(x, y, z), gx, gx) = 0, i.e., gx = F(x, y, z).

Similarly, one can show that gy = F(y, x, y) and gz = F(z, y, x). Thus we proved that (x, y, z) is a tripled coincidence point of F and g. □

Similarly, we can state the following corollary.

Corollary 20. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Let F : X3 X and g: X X. Assume there exists k ∈ [0, 1) such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc gw, we have

G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) k max { G ( g x , g a , g y ) , G ( g y , g b , g v ) , G ( g z , g c , g w ) } .

Assume that (X, G, ≤) is regular. Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy 0 F(y0, x0, y0) and gz0 F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Corollary 21. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space. Assume that (X, G, ≤) is regular. Let F : X3 X and g : X X. Assume there exists k ∈ [0, 1) such that for x, y, z, a, b, c, u, v, w X, with gx ga gu, gy gb gv and gz gc > gw, we have

G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) k 3 ( G ( g x , g a , g u ) + G ( g y , g b , g v ) + G ( g z , g c , g w ) ) .

Suppose that (g(X),G) is G-complete, F has the mixed g-monotone property and F(X × X × X) ⊆ g(X). Also, assume there exist x0, y0, z0 X such that gx0 F(x0, y0, z0), gy0 F(y0, x0, y0) and gz0 < F(z0, y0, x0), then F and g have a tripled coincidence point in X, i.e., there exist x, y, z X such that

F ( x , y , z ) = g x , F ( y , x , y ) = g y , F ( z , y , x ) = g z .

Remark 22. Other corollaries could be derived from Theorems 15 and 19 by taking g = Idx.

Now, form previous obtained results, we will deduce some tripled coincidence point results for mappings satisfying a contraction of integral type in G-metric space. Let us introduce first some notations.

We denote by Γ the set of functions α: [0, +∞) → [0, +∞) satisfying the following conditions:

(i) α is a Lebesgue integrable mapping on each compact subset of [0, +∞),

(ii) for all ε > 0, we have

0 ε α ( s ) d s > 0 .

Let N ∈ ℕ* be fixed. Let {αi}1 ≤ i N be a family of N functions that belong to Γ. For all t ≥ 0, we denote (Ii)i = 1,...,N as follows:

I 1 ( t ) = 0 t α 1 ( s ) d s , I 2 ( t ) = 0 I 1 ( t ) α 2 ( s ) d s = 0 0 t α 1 ( s ) d s α 2 ( s ) d s , I N ( t ) = 0 I N - 1 ( t ) α N ( s ) d s .

We have the following result.

Theorem 23. Let (X, ≤) be a partially ordered set and (X, G) be a G-metric space such that (X, G) is G-complete. Let F : X3 X and g : X X. Assume there exist ψ ∈ Ψ and ϕ ∈ Φ such that for x, y, z, a, b, c, u, v, w X, wth gx ga gu, gy gb gv and gz gc gw, we have

I N ( ψ ( G ( F ( x , y , z ) , F ( a , b , c ) , F ( u , v , w ) ) ) ) I N ( ψ ( m a x { G ( g x , g a , g u ) , G ( g y , g b , g v


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In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 or, equivalently, if their greatest common divisor is 1. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

For example, 6 and 35 are coprime, but 6 and 27 are not coprime because they are both divisible by 3. In mathematics Six is the second smallest Composite number, its proper Divisors being 1, 2 and 3. 35 ( thirty-five) is the Natural number following 34 and preceding 36. In mathematics Six is the second smallest Composite number, its proper Divisors being 1, 2 and 3. 27 ( twenty-seven) is the Natural number following 26 and preceding 28. The number 1 is coprime to every integer. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD

Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n which are coprime to n. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to

## Properties

Figure 1. The numbers 4 and 9 are coprime because the diagonal does not intersect any other lattice points

There are a number of conditions which are equivalent to a and b being coprime:

• There exist integers x and y such that ax + by = 1 (see Bézout's identity). In Number theory, Bézout's identity or Bézout's lemma is a linear Diophantine equation.
• The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In other words, b is a unit in the ring Z/aZ of integers modulo a. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

As a consequence, if a and b are coprime and brbs (mod a), then rs (mod a) (because we may "divide by b" when working modulo a). In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Furthermore, if a and b1 are coprime, and a and b2 are coprime, then a and b1b2 are also coprime (because the product of units is a unit).

If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c. Euclid's lemma ( Greek) is a generalization of Proposition 30 of Book VII of Euclid's Elements. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane (See figure 1. )

The probability that two randomly chosen integers are coprime is 6/π2 (see pi), which is about 60%. Probability is the likelihood or chance that something is the case or will happen IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems See below.

Two natural numbers a and b are coprime if and only if the numbers 2a − 1 and 2b − 1 are coprime. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

## Cross notation, group

If n≥1 is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)× or Zn*. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

## Generalizations

Two ideals A and B in the commutative ring R are called coprime if A + B = R. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime. In Number theory, Bézout's identity or Bézout's lemma is a linear Diophantine equation. In Ring theory, a branch of Abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single

If the ideals A and B of R are coprime, then AB = AB; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals. The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra.

The concept of being relatively prime can also be extended any finite set of integers S = {a1, a2, . In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. . . . an} to mean that the greatest common divisor of the elements of the set is 1. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime. In Mathematics, especially Number theory, a set of Integers is said to be pairwise coprime (or pairwise relatively prime, also known as mutually

Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor. )

## Probabilities

Given two randomly chosen integers A and B, it is reasonable to ask how likely it is that A and B are coprime. In this determination, it is convenient to use the characterization that A and B are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic). In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written

The probability that any number is divisible by a prime (or any integer), p is 1 / p. Hence the probability that two numbers are both divisible by this prime is 1 / p2, and the probability that at least one of them is not is 1 − 1 / p2. Thus the probability that two numbers are coprime is given by a product over all primes,

$\prod_p^{\infty} \left(1-\frac{1}{p^2}\right) = \left( \prod_p^{\infty} \frac{1}{1-p^{-2}} \right)^{-1} = \frac{1}{\zeta(2)} = \frac{6}{\pi^2}.$

Here ζ refers to the Riemann zeta function, the identity relating the product over primes to ζ(2) is an example of an Euler product, and the evaluation of ζ(2) as π2/6 is the Basel problem, solved by Leonhard Euler in 1735. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. The Basel problem is a famous problem in Number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735 In general, the probability of k randomly chosen integers being coprime is 1 / ζ(k).

There is often confusion about what a "randomly chosen integer" is. One way of understanding this is to assume that the integers are chosen randomly between 1 and an integer N. Then for each upper bound N, there is a probability PN that two randomly chosen numbers are coprime. This will never be exactly 6 / π2, but in the limit as $N \to \infty$, $P_N \to 6/\pi^2$.


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# Poisson noise for a squared contrast of density - expression of variance equal to $1/n^2$

1. I manage to compute the Poisson noise of a density field like this :

If I take $$N$$ the density of galaxies and compute the Shot noise with a Poisson distribution, I get :

$$\langle N^2\rangle - \langle N\rangle^2 = \langle N\rangle$$ so :

$$\langle N^2\rangle = \langle N\rangle + \langle N\rangle^2$$

Let's take the variable : $$X=\dfrac{N}{\langle N\rangle}-1$$

So, I get $$\langle X\rangle=0$$.

Then : $$\langle X^2\rangle = \left\langle\left (\dfrac{N}{\langle N\rangle}-1\right )^2\right\rangle=\dfrac{\langle N^2\rangle}{\langle N\rangle^2}-2+1 = \dfrac{\langle N^2\rangle}{\langle N\rangle^{2}}-1$$

Finally, I get : $$\sigma_x^2 = \langle X^2\rangle=\dfrac{1}{\langle N\rangle}+1-1$$

$$\Rightarrow\quad \sigma_x^2=\dfrac{1}{\langle N\rangle}$$

1. Now, I would like to do the same kind of calculation with variable $$Y=\left(\left(\dfrac{N}{\langle N\rangle}\right)-1\right)^2$$, and conclude normally that :

$$\sigma_y^2 = \dfrac{1}{\langle N\rangle^2}\quad(1)$$

But following the same reasoning with $$N$$ following the Poisson distribution, I can't manage to get this expression $$(1)$$.

UPDATE : i think that I have the proof :

$$\langle X^4\rangle = \Bigg\langle\bigg(\dfrac{N}{\langle N\rangle}-1\bigg)^2\,\bigg(\dfrac{N}{\langle N\rangle}-1\bigg)^2\Bigg\rangle$$

$$=\dfrac{\langle N^2\rangle^2}{\langle N\rangle^4}-2\dfrac{\langle N^2\rangle}{\langle N\rangle^2}+1$$

I) with $$\langle N^2\rangle = \langle N\rangle + \langle N\rangle^2$$

II) and squared :

$$\langle N^2\rangle^2 = \langle N\rangle^2 + 2\langle N\rangle^3 + \langle N\rangle^4$$

We conclude :

$$\langle X^4\rangle=\dfrac{1}{\langle N\rangle^2}+\dfrac{2}{\langle N\rangle} +1 - \dfrac{2}{\langle N\rangle} -2 + 1 = \dfrac{1}{\langle N\rangle^2}$$

• It's not clear to me what you mean by "Shot Noise", particularly as this is usually a synonym for noise that follows Poisson statistics. (e.g., en.wikipedia.org/wiki/Shot_noise) Jun 1 at 12:51
• @uhoh A Poisson distribution has (using the symbols from the Wiki page) an expectation value of $\lambda$ (not $k$) and a variance of $\lambda$. For the "density of galaxies" problem, $\lambda = \bar{n}_{\rm gal}$, so I don't understand why the OP says the "Shot Noise" should be $1/\bar{n}_{\rm gal}$. Jun 2 at 9:26
• @uhoh . Thanks for your answer. Unfortunately, I don't know yet how to justify that the expectation of Shot Noise $\bar{N}_{sp}$ should be equal to 0 like a white noise. I am also interested in the variance of Shot Noise : if I take a fixed value for the spectroscopic Shot Noise, then, the variance is equal to : $<(N_{sp}-\bar{N}_{sp})^{2}>=1/\bar{n}_{\rm gal}^{2}$. Could anyone confirm this result ? Regards Jun 2 at 10:34
• @youpilat13 I'm afraid I have no idea what you're trying to get at. Your last comment mentions "spectroscopic Shot Noise" -- whatever that's supposed to mean -- and then tries to relate that to the mean density of galaxies. That makes no sense. Jun 3 at 11:39
• @youpilat13 Have you tried asking about this on statistics.stackexchange? That seems like a much more appropriate place, since your question appears to be about estimating variances by calculations and manipulation of expectation values, and not really anything astronomical. Aug 11 at 13:03


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# Biodiversity and Ecosystem function

## Introduction

The recognition that species interactions may play important roles in ecosystems and the rapidly emerging interest in the biodiversity conservation have prompted ecologists to ask new questions on the relationships between 'biodiversity' and 'ecosystem function' (for example, Walker, 1992[1]; Schultze and Mooney, 1993[2]; Jones and Lawton, 1995[3]; Johnson et al., 1996[4]).

## Importance of ecosystem function

One reason for the interest in the functional role of biodiversity (rather than structural) in ecosystems is that society might be more likely to take action to preserve biodiversity if it could be shown that there was some direct economic gain by doing it (Bengtsson, 1998[5]). Over the last fifteen years, an increasing number of studies have focused on biodiversity. This is principally because the world’s flora and fauna are disappearing at rates greater than during historical mass extinction events (Chapin et al, 2001[6]). As suggested by Thomas et al. (2004[7]), there is an 18 to 35% risk of species-level extinction resulting from climate changes by the year 2050. Moreover, other processes, for example, agricultural expansion in response to an increasing demand for food, have a negative impact on biodiversity as a result of habitat destruction (Tilman et al., 2001[8]; Humbert and Dorigo, 2005[9]).

Biodiversity and Ecosystem function are central to both community and ecosystems ecology and need to be understood to predict, for example, how communities and ecosystems respond to environmental change (Bengtsson, 1998[5]) and on understanding how declining diversity influences ecosystem services on which humans depend (Duffy, 2003[10]).

## Research on Ecosystem Function

Research on Biodiversity - Ecosystem Function (the BEF agenda) has stimulated a new and highly productive intercourse between population, community, ecosystem, and conservation ecology (Kinzig et al. 2002[11]; Loreau et al. 2002[12]; Duffy, 2003[10]). Most experimental evidence for biodiversity effects on ecosystem function has come from terrestrial ecosystems, particularly grasslands (Naeem et al. 1994[13], Tilmann et al. 1997[14], Hector et al. 1999[15], Schmid et al. 2001[16]; Giller et al., 2004[17]). These studies have shown that changing biodiversity in natural ecosystems is likely to have far more complicated impacts on ecosystem function than predicted from changes in plant diversity alone (Duffy, 2003[10]). For example in trophic levels of plant communities, as diversity is lost from a system, impacts will also depend from the loss of predators which will evoke change in the structure of all trophic levels (Hairston et al. 1960[18]; Power 1990[19]; Estes et al. 1998[20]; Duffy, 2003[10]).

The mosaic of habitat patches in aquatic systems often is more spatially compact than in terrestrial environments, presenting more tractable experimental systems at the landscape scale (Schindler and Scheuerell 2002[21]). Because each aquatic ecosystem is composed of multiple habitat types, assessing the effects of biodiversity changes on the function of aquatic ecosystems requires experimental designs that allow a scaling up from individual homogenous patches to large scale, often highly heterogeneous areas (Giller et al. 2004[17]).

The most influential empirical research on biodiversity-ecosystem function linkages has been the series of experiments manipulating diversity in grasslands (reviewed by Tilman et al. 2002[22]) and in aquatic microbial microcosms (reviewed by Petchey et al. 2002[23]). Typically these have tested how ecosystem-wide biomass accumulation or metabolic rates change along gradients of species richness achieved by randomly assembling experimental communities from a pool of species. The grassland experiments have manipulated plant species richness, and sometimes also functional group richness. These studies have demonstrated significant positive correlations between species richness and plant biomass. Loreau et al. (2002[12]) provide a global overview of concepts and debates concerning the relationships between biodiversity and ecosystem function (Humbert and Dorigo, 2005[9]).

It has been clearly established that ecosystem function depends both on biotic factors and/or processes (such as the diversity and functions of the species, and interactions between species) and abiotic factors (such as climate or geology). However, what relative contribution these factors make is still a central question in the debate about diversity and ecosystem function (Huston and McBride, 2002[24]; Humbert and Dorigo, 2005[9]).

Species deletion stability can also be linked easily to removal experiments that address the consequences of species loss for ecosystem function (Thebault, et al. 2007[25]). With a few exceptions, theoretical work on the direct impact of species loss has focused on the study of secondary extinctions but has not considered associated changes in ecosystem properties (see King and Pimm 1983[26], Petchey et al. 2004[27]).

Many of the studies that dealt specifically with the mechanisms involved in the relationships between biodiversity and ecosystem function investigated the niche complementarity mechanism, stimulating both theoretical and experimental approaches (e.g., Naeem et al., 1994[13]; Loreau, 1998[28]). The sampling effect, difficult to distinguish from the niche complementarity, is defined as the greater likelihood of finding species with a strong impact on ecosystem function in highly diversified communities (e.g., Huston, 1997[29]; Hector et al., 1999[15]; Wardle, 1999[30]). These are not either-or mechanisms, but may be viewed as concomitant processes (Naeem, 2002[31]). Sampling effects are involved in community assembly, and thus in determining the number of phenotypic traits present in the community. Subsequently, this phenotypic diversity influences ecosystem processes through mechanisms that can be viewed as a continuum ranging from the selection of species with particular traits to complementarity among species with different traits (Loreau et al., 2001[32]).

Mathematical modelling has also been used recently, to investigate the relationships between biodiversity and ecosystem stability. For example, McCann et al. (1998[33]) have shown that weak to intermediate interaction strengths within food webs are important in promoting community persistence and stability (Humbert and Dorigo, 2005[9]).

## Theories and Hypothesis

In a review of the topic, Naeem et al. (2002[31]) proposed three hypotheses to account for linking biodiversity and ecosystem function:

• The first hypothesis is that species are primarily redundant, which means that one species can partially replace another. Many species have the same function, and the loss of one species can therefore be offset by some other species.
• The second hypothesis is that species are essentially singular, and make unique contributions to the ecosystem function. The loss or gain of species (generally referred to as Keystone or Key species) therefore has a measurable impact on the ecosystem function.
• The third hypothesis is that species impacts are context dependent such that the impact of the loss or gain of a species on ecosystem function is idiosyncratic and unpredictable.

What happens, will depend on the local conditions under which the species extinction or addition occurs (Humbert and Dorigo, 2005[9]) For further details on this topic, see the article Disturbances, biodiversity changes and ecosystem stability.

## How do we measure Ecosystem Function?

Describing or measuring ecosystem function is difficult, as it encompasses a number of phenomena (Hooper et al., 2005[34]). The overall function of an ecosystem is complex and involves many factors relating to the chemical, physical and biological components of the system. The way in which differences between species affect diversity-function relationships can be very complex (Lawton et al., 1998[35]; Ricotta, 2005[36]).

Functional diversity (FD), i.e. the diversity and range of functional traits possessed by the biota of an ecosystem (Wright et al., 2006[37]) or else defined by Tilman (2001[38]) as "those components of biodiversity that influence how an ecosystem operates or functions" (Ricotta, 2005[36]), is likely to be the component of biodiversity most relevant to the function of the ecosystems (Hooper et al., 2002[39]; 2005[40]; Heemsbergen et al., 2004[41]), even though, there is no clear relationship demonstrated between species diversity and ecosystem function (Somerfield et al., 2008[42]).

Whereas traditional diversity indices focus on species richness, rarity (Schmera 2003[43]) or the uncertainty of predicting species identity from abundance data (Magurran 1988[44]), functional diversity formulae are used to measure ‘‘those components of biodiversity that influence how an ecosystem operates or functions’’ (Tilman et al. 1997[14]; Schmera, Erös and Podani, 2009[45]). Functional Diversity relates the number, type and distribution of functions performed by organisms within an ecosystem (Diaz & Cabido, 2001[46]). It incorporates interactions between organisms and their environment into a concept that can portray ecosystem level structure in marine environments (Bremner et al., 2003[47]) and conjectures to be useful in predicting the consequences of changes in species richness and composition, or biodiversity in general, on ecosystem properties (Somerfield et al., 2008[48]). Many studies have focused on calculating Functional Diversity, in order to measure the Ecosystem Function. Methods and indices have been applied and tested on a long series of data concerning abiotic and biotic measures of fresh and sea water.

Recent methods for calculating functional diversity of a community, include Functional Attribute Diversity (FAD), as used in a study of Australian rangelands by Walker et al. (1999[49]), and Functional Diversity (FD) proposed more recently by Petchey and Gaston (2002[23]) which is computed as the total branch length of the functional dendrogram that results from clustering the species in trait space (Ricotta, 2005[36]). Trait variance, measured as the width of a trait distribution, has been proposed by Norberg (2004[50]). Beyond the simple measurement of diversity, Mason et al. (2005[51]) proposed also estimating functional richness, functional evenness and functional divergence, to enable descriptions of niche use and competitive interactions in communities. In order to take a functional approach and to use these new measures, however, we must have descriptors of the functional groups present in a community.

Most recently, based on the methodology proposed by the formers, Somerfield et al. (2008[42]) defined average functional distinctness (X+, from χαρακτηριστικό, meaning a trait) simply as the average resemblance among species in a sample. Incidentally, the same logic may be applied to Δ+ (Clarke and Warwick, 1998[52]). Once branch lengths are defined between taxonomic levels, a matrix of resemblances (Euclidean distances) between species becomes implicit, and the index is the average resemblance between species. In the same study, the authors concluded that the type of information we get from the functional level is complementary to the information we take from the taxonomic level.

## How do we calculate Ecosystem Function in practice?

The categorization of species into functional groups can be done by simply assigning each species found in the assemblage to a given a priori defined functional group (Hector et al., 1999[15]), or by standard multivariate clustering methods (Gitay & Noble, 1997[53]; Deckers, Verheyen, Hermy, & Muys, 2004[54]; Roscher et al., 2004[55]) - see also Biological Trait Analysis (BTA). To cluster species into functional groups, first, a set of functional traits thought to be of significance for ecosystem function is measured for each species obtaining an $S \times \tau$ matrix of $\tau$ functional traits measured on $S$ species (Petchey & Gaston, 2002[23]). Next, the trait matrix is converted into a distance matrix $\Delta$ of which the elements $\Delta_{i,j}$ embody the functional distances between the $i$-th and the $j$-th species such that $\Delta_{i,i}=0$ and $\Delta_{i,j}=\Delta_{j,i}$ for any $i \neq j$. Finally, the distance matrix is clustered with standard multivariate methods to separate species from different functional groups (Ricotta, 2005[36]). Generally, regardless of the proposed index, in most cases the information available for computing the FD of a given species assemblage is the set of pair wise species functional distances $\Delta_{i,j}$ (Ricotta, 2005[36]). For further details and explanations, see the article Measurements of biodiversity.

## Related articles

Ecosystem function
Measurements of biodiversity
Biological Trait Analysis
Biodiversity

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 The main author of this article is Vassiliki, MarkantonatouPlease note that others may also have edited the contents of this article. Citation: Vassiliki, Markantonatou (2020): Biodiversity and Ecosystem function. Available from http://www.coastalwiki.org/wiki/Biodiversity_and_Ecosystem_function [accessed on 18-01-2022] For other articles by this author see Category:Articles by Vassiliki, Markantonatou For an overview of contributions by this author see Special:Contributions/Markantonatou


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## 2018-12-01: not much math

I figured out exercise 10.4.3(b) in Analysis I only after seeing the trick in this answer (linked from this answer) of substituting in a quotient to get the general case; basically, since part (a) proved the case of $x \to 1$, in part (b) the trick is to see that $x/x_0 \to 1$ as $x \to x_0$. And then some algebra to get the thing to work. I had seen this sort of trick before, but didn’t think of it when I needed it, so I’ll have to keep it in mind more.

I read section 10.5 (l’Hopital’s rule) but didn’t do the exercises (yet).

Today was spent mostly relaxing and doing “first day of the month” bureaucracies (updating the front page of my website, updating Timelines Wiki pageviews table, submitting contract work hours to Vipul).


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# Econometrics-using log-log or continuously compounded annual rate of change

I am trying to run an econometric model, where I want to estimate the effect of credit growth on consumption growth, among other variables. Using data from Fred, I have the option of getting my variables as a continuously compounded annual rate of change. Are using these units and regressing normally the same as running a log-log regression?