text:
# American Institute of Mathematical Sciences

March  2012, 11(2): 809-828. doi: 10.3934/cpaa.2012.11.809

## On the structure of the global attractor for non-autonomous dynamical systems with weak convergence

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  January 2011 Revised  January 2011 Published  October 2011

The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so--called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert's question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations.
Citation: Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
##### References:
 [1] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Lecture Notes in Mathematics, (1970). Google Scholar [2] I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979). Google Scholar [3] B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity,", Moscow, (1966). Google Scholar [4] T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations,, J. Differ. Eqns., 251 (2011). Google Scholar [5] D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267. Google Scholar [6] D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915. Google Scholar [7] D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations,, Dynamical systems and boundary value problems, (1987), 143. Google Scholar [8] D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511. Google Scholar [9] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004). Google Scholar [10] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669. Google Scholar [11] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009). Google Scholar [12] D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010). Google Scholar [13] D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of non-autonomous difference equations,, Nonlinear Analysis TMA, 56 (2004), 465. Google Scholar [14] D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations,, J. Math. Anal. Appl., 340 (2008), 374. Google Scholar [15] C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978). Google Scholar [16] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19. Google Scholar [17] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3. Google Scholar [18] B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967). Google Scholar [19] A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280. Google Scholar [20] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [21] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems,, J. Dyn. Diff. Eqns, 13 (2001), 107. Google Scholar [22] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982). Google Scholar [23] A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich,, Systems and Control Letters, 52 (2007), 257. Google Scholar [24] V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964). Google Scholar [25] V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977). Google Scholar [26] G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971). Google Scholar [27] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence,, Differential Equations, 11 (1975), 1246. Google Scholar [28] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985). Google Scholar [29] R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of non-linear differential equations,, Mat. Sb. N.S., 41 (1957), 431. Google Scholar [30] T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975). Google Scholar [31] V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167. Google Scholar [32] V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214. Google Scholar [33] V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964). Google Scholar [34] V. I. Zubov, "Theory of Oscillations,", Nauka, (1979). Google Scholar

show all references

##### References:
 [1] N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Lecture Notes in Mathematics, (1970). Google Scholar [2] I. U. Bronsteyn, "Extensions of Minimal Transformation Group,", Noordhoff, (1979). Google Scholar [3] B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity,", Moscow, (1966). Google Scholar [4] T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of second-order monotone differential equations,, J. Differ. Eqns., 251 (2011). Google Scholar [5] D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients,, Differential Equations, 22 (1986), 267. Google Scholar [6] D. N. Cheban, $\mathbb C$-analytic dissipative dynamical systems,, Differential Equations, 22 (1986), 1915. Google Scholar [7] D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations,, Dynamical systems and boundary value problems, (1987), 143. Google Scholar [8] D. N. Cheban, Global pullback atttactors of C-analytic nonautonomous dynamical systems,, Stochastics and Dynamics, 1 (2001), 511. Google Scholar [9] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1., River Edge, (2004). Google Scholar [10] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations,, J. Dynamics and Differential Equations, 20 (2008), 669. Google Scholar [11] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations,", Hindawi Publishing Corporation, (2009). Google Scholar [12] D. N. Cheban, "Global Attractors of Set-Valued Dynamical and Control Systems,", Nova Science Publishers, (2010). Google Scholar [13] D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of non-autonomous difference equations,, Nonlinear Analysis TMA, 56 (2004), 465. Google Scholar [14] D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations,, J. Math. Anal. Appl., 340 (2008), 374. Google Scholar [15] C. Conley, "Isolated Invariant Sets and the Morse Index,", Region. Conf. Ser. Math., (1978). Google Scholar [16] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I,, Vestnik MGU, 6 (1961), 19. Google Scholar [17] B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II,, Vestnik MGU, 1 (1962), 3. Google Scholar [18] B. P. Demidovich, "Lectures on Mathematical Theory of Stability,", Moscow, (1967). Google Scholar [19] A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity,, Journal of Differential Equations, 9 (1971), 280. Google Scholar [20] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems,", Amer. Math. Soc., (1988). Google Scholar [21] M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems,, J. Dyn. Diff. Eqns, 13 (2001), 107. Google Scholar [22] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge Univ. Press, (1982). Google Scholar [23] A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich,, Systems and Control Letters, 52 (2007), 257. Google Scholar [24] V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations,", Nauka, (1964). Google Scholar [25] V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations,", Nauka, (1977). Google Scholar [26] G. R. Sell, "Topological Dynamics and Ordinary Differential Equations,", Van Nostrand-Reinhold, (1971). Google Scholar [27] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence,, Differential Equations, 11 (1975), 1246. Google Scholar [28] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations,", \cStiin\cta, (1985). Google Scholar [29] R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of non-linear differential equations,, Mat. Sb. N.S., 41 (1957), 431. Google Scholar [30] T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14,, Springer-Verlag, (1975). Google Scholar [31] V. V. Zhikov, On stability and unstability of Levinson's centre,, Differentsial'nye Uravneniya, 8 (1972), 2167. Google Scholar [32] V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of non-linear operator equations,, Mat. Sbornik, 90 (1973), 214. Google Scholar [33] V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application,", Noordhoof, (1964). Google Scholar [34] V. I. Zubov, "Theory of Oscillations,", Nauka, (1979). Google Scholar
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# Causes “libGL error: failed to load driver: swrast” an Huge Overhead in Parallel under Version 10.3?

Under Linux (SUSE Linux Enterprise 11) for Mathematica Version 10.2/10.3 I observe the following error in parallel

libGL error: Try again with LIBGL_DEBUG=verbose for more details.
libGL error: failed to load driver: swrast

for the sub-kernels. This error does not occur for the master-kernel just for the sub-kernels. I also do not observe this kind of failure for Mathematica Version 10.0.

It seems to me that this failure causes an huge overhead in parallel when I execute the following function

LaunchKernels[12]

exportFrames[baseName_, a_, ext_:"jpg"] := Module[{l = Length[a], d},
d = Length@IntegerDigits[l];
ParallelTable[Export[baseName <> IntegerString[i, 10, d] <> "." <> ext,   Style[a[[i]],Magnification->1.5]], {i, l}]
]

exportFrames["Exp11krmovie",a1]

This function let me render and export several hundreds or thousands graphics in parallel. For instance, under Mathematica Version 10.0 I can render and export a test sample of several hundreds graphics in less than a quarter of an hour. In contrast, under Mathematica 10.2/10.3 I can only generate 1/3 of the sample in an hour.

Inspection by the network tool iftop reveals an one-sided network traffic. Moreover, the tool htop shows me that only the master-kernel is working and the sub-kernels are only showing up sporadical. As a consequence, there is no continuous flow of rendering and exporting the graphic files as under Mathematica 10.0. In average, each 4 minutes it starts to generate a set of 12 graphic files, whereas the whole cycle ends at least after 2 minutes.

Any idea how I can get rid of this error for the sub-kernels? How can I reset the graphic rendering from a software rendering back to a hardware rendering under Mathematica 10.3?

• I would suggest getting in touch with support with your SystemInformation[]. Are you using an NVIDIA video card? Is there any difference if you do ParallelEvaluate[SetOptions[DeveloperInstallFrontEnd, DeveloperLaunchFlags -> "-mesa"]]; before exporting? – ilian Oct 30 '15 at 21:07
• @ilian Thanks a lot for this great pointer! It works now!! The network traffic is reduced and more balanced now. The whole set of graphics is now generated in a few minutes with an AMD/ATI video card. – Holger I. Meinhardt Oct 31 '15 at 11:32

I noticed this problem recently (Feb 2016) following an update to Raspian's Jessie on the Raspberry Pi. My solution is probably specific to the RPi, and I apologize in advance if I am considered to have hijacked your question.

I noticed upon starting Mathematica that I would get a similar LibGL swrast driver error. Following the suggestions mentioned here (which is alluded to in the error message you receive) I set the LIBGL_DEBUG environment variable via export LIBGL_DEBUG=verbose and then re-ran Mathematica, receiving the following verbose message list:

libGL: OpenDriver: trying /usr/lib/arm-linux-gnueabihf/dri/tls/swrast_dri.so
libGL: OpenDriver: trying /usr/lib/arm-linux-gnueabihf/dri/swrast_dri.so
libGL: dlopen /usr/lib/arm-linux-gnueabihf/dri/swrast_dri.so failed (/usr/lib/arm-linux-gnueabihf/dri/swrast_dri.so: cannot open shared object file: No su
ch file or directory)
libGL: OpenDriver: trying ${ORIGIN}/dri/tls/swrast_dri.so libGL: OpenDriver: trying${ORIGIN}/dri/swrast_dri.so
libGL: dlopen ${ORIGIN}/dri/swrast_dri.so failed (${ORIGIN}/dri/swrast_dri.so: cannot open shared object file: No such file or directory)
libGL: OpenDriver: trying /usr/lib/dri/tls/swrast_dri.so
libGL: OpenDriver: trying /usr/lib/dri/swrast_dri.so
libGL: dlopen /usr/lib/dri/swrast_dri.so failed (/usr/lib/dri/swrast_dri.so: cannot open shared object file: No such file or directory)
libGL error: unable to load driver: swrast_dri.so
libGL error: failed to load driver: swrast

For the RPi, the issue was not a missing symbolic link as assumed in the solution linked to above. I therefore suspected a missing package. Searching for swrast_dri.so on https://packages.debian.org/ reveals that this file is in the package libgl1-mesa-dri which I installed (sudo apt-get install libgl1-mesa-dri). Now, running Mathematica (after unsetting the verbose debug variable with unset LIBGL_DEBUG resulted in a warning-free start to Mathematica.

As I mentioned earlier, my situation is likely different; however it might be worthwhile to check your packages dpkg --get-selections | grep libgl1-mesa-dri and see if installing it also solves your problem.

• I checked the situation on my computer system, and find out that it is neither related to a broken symlink nor to a missing installation of libgl1-mesa-dri. The libgl1-mesa-dri are installed on my computer, but not used anymore after I have changed to proprietary drivers. The proposed workaround works fine on my system, even though the drawback is that I have not to forget to include these lines in my Mathematica code. – Holger I. Meinhardt Feb 9 '16 at 12:41
• I figured as much - I suspect my answer is more appropriate for those trying to solve the corresponding problem on the RPi. – bobthechemist Feb 9 '16 at 12:44
• Hi Bob, if you have time, would you mind testing this package on a Raspberry Pi (2)? I would like to make sure that it works on someone else's machine before making a full release, and this is the first time I compiled it on a Raspberry Pi. Running the test suite on an RPi 1 is slow (half an hour). A quicker way to test is to just load the package and look at the version number: Get["IGraphM"]; IGVersion[]. If there are no error messages shown then it worked! – Szabolcs Feb 27 '16 at 19:51
• @Szabolcs I get this, which suggests that some of it is working. I kept getting a network error when downloading the zip file but the source link worked. – bobthechemist Feb 28 '16 at 1:04
• @bobthechemist Thanks for trying it! The source archive is perfect, but I think Directory[] should be /home /pi/projects and not /home/pi/projects/IGraphM when you issue <<IGraphM  (in the case when the package is not installed in ~/.WolframEngine/Applications, just just loaded from the current directory). – Szabolcs Feb 28 '16 at 17:15

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# Isoperimetric sets in Carnot groups

created on 18 Sep 2001
modified on 07 Jun 2002

[BibTeX]

Accepted Paper

Inserted: 18 sep 2001
Last Updated: 7 jun 2002

Journal: Houston J. Math.
Year: 2002

Abstract:

We prove the existence of isoperimetric sets in any Carnot group, that is, sets minimizing the intrinsic perimeter among all measurable sets with prescribed Lebesgue measure. We also give some properties of these isoperimetric sets. Namely we show that, up to a null set, they are open, bounded, their boundary is an Ahlfors-regular set of dimension $Q-1$, where $Q$ denotes the homogeneous dimension of the group, and they satisfy the condition B. In the particular case of the Heisenberg group we also prove that any reduced isoperimetric set is a domain of isoperimetry. Moreover all these properties are satisfied with implicit constants that depend only on the dimension of the group and on the prescribed Lebesgue measure.

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Ocular filters in the eyes of many vertebrates, including humans, absorb wavelengths shorter than approximately 400nm. These filters prevent the β-band of a visual pigment from being exposed to ultraviolet radiation, essentially narrowing the spectral sensitivity of the different photoreceptor classes. A comparison of different hypothetical visual systems is used to show that von Kries colour constancy is improved by ocular filtration of ultraviolet radiation, whilst there is no reduction in colour discrimination. Furthermore, it is shown that the asymmetric spectral spacing of different photoreceptor classes present in the human visual system may benefit colour constancy. The results are interpreted in relation to predictions of von Kries colour constancy for a standard human observer.

The eyes of many vertebrates contain ocular filters that absorb ultraviolet radiation (300–400nm) (Fig.1). For example, in primates protein pigmentation in the lens of the eye prevents radiations with wavelengths shorter than approximately 400nm from reaching the retina (Zigman et al., 1976). The possible function of this type of optical filtering has previously been explained as improving optical performance through reduced chromatic aberration, haze and glare, or to prevent damaging ultraviolet radiation from reaching the retina (for reviews see Muntz, 1972; Douglas and Marshall, 1999).

Human colour vision is based on three classes of photoreceptor with spectral absorbance maxima at approximately 420, 534 and 564nm (Bowmaker and Dartnall, 1980). These receptor peaks can be taken as reasonably representative of the asymmetrically spaced receptors in many Old-World primates (Bowmaker, 1991). Each class of photoreceptor also has the potential to absorb ultraviolet radiation as a result of the cis-band of the chromophore, which produces a common secondary absorbance peak at approximately 340nm (Stavenga et al., 1993). The primary receptor peak is termed the α-band and the secondary peak the β-band. Interestingly, the perception of ultraviolet radiation by humans has been directly observed in aphakic patients who have had their lenses surgically removed (Wald, 1945; Davenport and Foley, 1979).

Illumination quality for most diurnal animals is highly variable because of continuously changing atmospheric conditions (Lythgoe, 1979), and the ability of a visual system to correctly identify a coloured stimulus in spectrally variable illumination is termed colour constancy (Hurvich, 1981; MacAdam, 1985; Kaiser and Boynton, 1996). Colour constancy is usually considered to be complete only if the colour of a stimulus is identical in spectrally different illumination sources, and an incomplete correction is described as approximate colour constancy. It is known that in many circumstances colour constancy is imperfect in humans (Hurvich, 1981; MacAdam, 1985; Kaiser and Boynton, 1996). Chromatic adaptation is likely to be an important mechanism of colour constancy and can be explained by the von Kries coefficient law (von Kries, 1902 in MacAdam, 1970; Hurvich, 1981; MacAdam, 1985; Kaiser and Boynton, 1996). von Kries colour constancy assumes that the relative spectral sensitivity of different photoreceptor classes does not change, but that the relative sensitivity of photoreceptors vary to achieve constancy.

Theoretical analyses show that von Kries colour constancy is limited both by an increase in the spectral breadth and an overlap of different photoreceptor classes (Worthey and Brill, 1986; Dyer, 1999; Kevan et al., 2001), and recent studies have suggested that one purpose of ocular filters may be to improve von Kries colour constancy (Vorobyev et al., 1998; Dyer, 1999). For example, Vorobyev et al. (Vorobyev et al., 1998) modelled the tetrachromatic colour visual systems of two birds, the pigeons and the Pekin robins, and showed that coloured oil droplets above the bird’s cones would narrow the spectral sensitivities of the photoreceptors and improve von Kries colour constancy. Dyer (Dyer, 1999) modelled the trichromatic colour vision of the honeybee (Apis mellifera), which has photoreceptor peaks at approximately 350, 440 and 540nm (Peitsch et al., 1992), and showed that increases in the spectral breadth of the photoreceptors caused by the β-band peaks limit von Kries colour constancy. It was therefore suggested that screening pigments in the eyes of some insects (e.g. Peitsch et al., 1992) may serve to reduce the influence of the β-band peak. For animals with visual systems that do not have an α-band photoreceptor absorbing in the ultraviolet and do not need to have optics that transmit these wavelengths, it may thus be beneficial for colour constancy to have ocular filters that absorb ultraviolet radiation before it reaches the retina.

An intriguing question in the evolution of trichromatic colour visual systems for Old-World primates and humans is that the three receptor classes are asymmetrically spaced (Mollon, 1989; Goldsmith, 1990). In humans, the short-wavelength-sensitive (SWS) and long-wavelength-sensitive (LWS) cones absorb radiation maximally at approximately 420nm and 564nm, respectively, so the medium-wavelength-sensitive (MWS) cones would need to be maximally sensitive at approximately 492nm for the three receptor classes to be evenly spaced, assuming the same region of the spectrum is to be sampled. Theoretical analyses of colour visual systems suggest that symmetrically spaced receptors across the spectrum should be optimal for the discrimination of differently coloured stimuli (Barlow, 1982; Vorobyev and Menzel, 1999). Several authors have suggested that the asymmetric trichromatic visual systems in Old-World primates may be an adaptation that is well suited to detecting yellow- and orange-coloured fruits from green foliage (Mollon, 1989; Regan et al., 1998; Sumner and Mollon, 2000), and Barlow (Barlow, 1982) demonstrated that the small spectral separation of the MWS and LWS cones may benefit spatial resolution in human vision. Another possible explanation for asymmetric receptor spacing lies in the genetic evolution of human trichromatic vision: the MWS receptor may lie at approximately 534nm because of how far and how fast the spectral position of this receptor is able to evolve from the long-wavelength subfamily of pigments in mammals (Goldsmith, 1990). However, the green opsin genes of rabbits (Oryctolagus cuniculus) and rats (Rattus norvegicus) have been sequenced (Radlwimmer and Yokoyama, 1998), and an absorbance maximum at about 509nm was demonstrated. Sun et al. (Sun et al., 1997) sequenced the mouse green cone, which has a maximum absorbance of radiation at approximately 508nm, and suggested that there are genetic tuning mechanisms that could allow the long-wavelength subfamily of pigments in mammals to have absorption maxima at wavelengths as short as 487nm. Studies of the evolution of visual pigments indicate that genetic tuning may occur over relatively short periods; for example, divergent fish species possess photoreceptors that appear to have adapted to their present photopic environment over the last 20000 years (Bridges and Yoshikami, 1970; Bridges, 1974). This suggests that the asymmetric spacing of receptors in Old-World primates may be an adaptation to their visual environment, rather than a result of genetic invariability. Osorio (Osorio, 1997) suggested an alternative possibility for the asymmetric spectral spacing of receptors by demonstrating that a reduction in the spectral separation of the MWS and LWS receptors benefits von Kries colour constancy for coloured fruits. This idea, together with the optimised ability to detect stimuli against a mature leaf background (Sumner and Mollon, 2000), provides an interesting and plausible explanation for the curious spectral spacing of photoreceptors in Old-World primates.

In the present study, hypothetical visual systems (HVSs) are used to approach two questions about colour vision. (i) Are ocular filters that absorb ultraviolet radiation likely to benefit colour constancy? (ii) Is the asymmetric spectral spacing of photoreceptor classes in Old-World primates and humans likely to benefit colour constancy? The efficiency of von Kries colour constancy is also determined for a standard human observer to allow a calculation of perceptual colour difference in spectrally variable illumination (MacAdam, 1985).

### Measuring the reflectance spectra of stimuli

The reflectance spectra of 24 differently coloured stimuli (Table1) were measured using a Varian DMS100 spectrophotometer fitted with a diffuse reflectance attachment. The spectral range examined included 300–700nm, to allow an analysis of how a visual system including the β-band peak might view the stimuli. The spectrophotometer was calibrated against a Varian pressed polytetrafluoroethylene powder standard. The 24 stimuli include 12 man-made and 12 naturally occurring samples, and represent a range of colours for normal human vision. These included fruit items, coloured card (Optix Australia), white paper (Reflex Australia), coloured fabric material and a piece of sawn mountain ash wood (all purchased from local stores). In addition, leaf and bark samples were collected from the gardens at Monash University. To measure the reflectance spectra of the fruit, leaf and bark samples, a cut 3cm×2cm section was used. Measurements were made immediately, and no visible discolouration of the fruit occurred in this time (Fig.2).

### Illumination sources

From the data of Judd et al. (Judd et al., 1964) two radiation sources were used to represent spectrally different distributions of daylight. These included direct daylight (a mixture of direct sunlight and diffuse skylight) with a correlated colour temperature (CCT)=6500K, and shade (diffuse skylight only) with a CCT=10000K (Fig.3). The two radiation sources represent the problem for the animal of identifying objects illuminated alternatively by sunlight and shade, as finding food might be in a natural foraging environment (Dyer, 1998).

### CIE 1931 standard observer

To model von Kries colour constancy for normal human colour vision the (λ) (red), (λ) (green) and (λ) (blue) colour-matching functions for a standard observer as specified by the Commission Internationale de l’Eclairage (CIE) 1931 (see Wyszecki and Stiles, 1967 for details) were used. These colour-matching functions are an internationally agreed standard for quantitatively specifying colour for an average or typical human observer (Wyszecki and Stiles, 1967; MacAdam, 1985). To estimate the efficiency of von Kries colour constancy for a CIE 1931 standard observer it is possible to use MacAdam’s ellipses (MacAdam, 1942; MacAdam, 1985) to calculate a just-noticeable distance in colour space. MacAdam’s ellipses are an experimentally determined set of ellipses for different regions of CIE 1931 colour space that allow the calculation of a perceivable difference in stimuli colour for a CIE 1931 standard observer (Wyszecki and Stiles, 1967; MacAdam, 1985).

For a given stimulus I(λ) viewed under an illumination D(λ), the tristimulus values (X, Y and Z) are given by.

1

for i=1–3, where λ is wavelength, i(λ)=(λ), (λ) and (λ), respectively, and Ti=X, Y and Z, respectively. The variable K is the von Kries coefficient (see equation 2), and the visual system is assumed to be adapted to a stimulus reflecting radiation equally at all wavelengths. As the goal of the calculation is to determine the colour shift of a given stimulus in spectrally different illumination, the absolute brightness of the stimulus is considered to remain constant and the only change is to the relative spectral signal. As colour matching data does not exist for ultraviolet wavelengths (since we do not normally see these wavelengths), the integration was calculated over the range 400–700nm.

2

for i=1–3.

Chromaticity coordinates (x, y, z) are computed from the tristimulus values (X, Y and Z) (equation 3). The value of z is defined since z=1−(x+y) (Wyszecki and Stiles, 1967):

$\mathit{x}/\mathit{X}{=}\mathit{y}/\mathit{Y}{=}\mathit{z}/\mathit{Z}{=}1/(\mathit{X}{+}\mathit{Y}{+}\mathit{Z}).$
3

The chromaticity coordinates were plotted in CIE 1931 colour space (see Fig.4) and constant metric coefficients for MacAdam’s ellipses were used to calculate the perceptible change in the colour of a stimulus I(λ) considering spectrally variable illumination and von Kries colour constancy (Wyszecki and Stiles, 1967; MacAdam, 1985). Using equation 4, a calculated distance of 2.0 units is equivalent to one just-noticeable distance, or the smallest perceivable colour step for a standard observer (Wyszecki and Stiles, 1967; MacAdam, 1985), and Table1 shows the predicted colour shift in just-noticeable distances. The choice of one just-noticeable distance as a criterion to designate approximate colour constancy (i.e. the perceived colour of a stimulus is not exactly the same) is probably very strict. Colour constancy in the biological sense is concerned with correctly identifying a stimulus by its reflectance properties independently of illumination colour (i.e. how similar is stimulus colour when illumination colour changes), whilst the formulation for MacAdam’s ellipses are based on the perception of small colour differences (i.e. can two stimuli be perceived as different). However, it is useful to have a benchmark to quantify the change in the appearance of a stimulus in spectrally variable illumination, and colour difference Δc is more tangible than colour similarity:

4

where Δx and Δy are the differences in the coordinates of the stimulus in colour space for spectrally different illumination conditions (assuming von Kries colour constancy). The constant metric coefficients g11, g12 and g22 are based on MacAdam’s ellipses and are given elsewhere (Wyszecki and Stiles, 1967; MacAdam, 1985).

### Hypothetical visual systems

Three HVSs were used in the study to allow a comparative evaluation of how visual systems with different features might perform in spectrally variable illumination. The relative spectral sensitivity of different photoreceptor classes for each HVS were calculated using the vitamin A1 template (Stavenga et al., 1993). The region of the spectrum considered for these visual systems included 300–700nm.

#### HVS1

For HVS1, the SWS, MWS and LWS receptor peaks were taken to be 420, 534 and 564nm, respectively, on the basis of data for cones in human colour vision (Bowmaker and Dartnall, 1980). These receptor peaks are very similar to those reported in a number of different Old-World primate species (Bowmaker, 1991). Radiation with a wavelength shorter than 400nm was assumed to be totally absorbed by ocular filters (Fig.5A). This HVS approximately represents human colour vision.

#### HVS2

The HVS2 α-band receptor peaks are identical to HVS1; however, it is assumed that there is no filtration of radiation by ocular filters, and ultraviolet radiation is absorbed by a common β-band peak maximally sensitive at 340nm (Stavenga et al., 1993) (Fig.5B). There is some evidence that the spectral position of the β-band peak may vary with the spectral position of the α-band receptor (Palacios et al., 1998); however, an invariable position is used here to simplify the model and because of the paucity of other data. Minor shifts of the β-band peak towards longer wavelengths should not greatly change the spectral breadth and overlap of the three receptor classes, nor the overall predictions of the model.

#### HVS3

For HVS3, the SWS and LWS receptors are identical to HVS1, but the MWS receptor is assumed to absorb radiation maximally at 492nm. All radiation at wavelengths shorter than 400nm is assumed to be absorbed by ocular filters. This visual system represents approximately symmetrically spaced receptors without a β-band peak (Fig.5C). The placement of the MWS receptor at 492nm seems plausible because of the findings (Sun et al., 1997) suggesting that mammals may have the genetic tuning mechanisms to have ‘green’ cone pigments with absorption maxima as short as 487nm.

### Colour distance for HVS

For the three HVSs the amount of radiation absorbed by each receptor P (SWS, MWS and LWS) was calculated using equations 5 and 6. Chromaticity coordinates (sws, mws and lws) were calculated using equation 7, and these were subsequently plotted onto Cartesian coordinates to represent colour space for each HVS (see Fig.6A–C). A Euclidean colour distance was calculated using Pythagoras’ theorem (equation 8) (Wyszecki and Stiles, 1967).

For a given stimulus I(λ) viewed under an illumination D(λ) by receptor type S(λ), the relative amount of radiation absorbed by each receptor is given by:

5

where the von Kries coefficient (K) is given by:

6

Chromaticity coordinates were calculated as described by Wyszecki and Stiles (Wyszecki and Stiles, 1967).

$\mathit{sws}/\mathit{P}(SWS)\ {=}\ \mathit{mws}/\mathit{P}(MWS)\ {=}\ \mathit{lws}/\mathit{P}(LWS){=}1/{[}\mathit{P}(SWS){+}\mathit{P}(MWS){+}\mathit{P}(LWS){]}.$
7

The relative colour shift of a stimulus considering spectrally variable illumination is given by:

8

where Δlws and Δmws are the differences in the coordinates of the stimulus in colour space for different illumination conditions (assuming von Kries colour constancy).

To determine the relative performance of von Kries colour constancy for HVS2 (visual system with no ocular filtering) compared with HVS1 (visual system with ocular filtering), the percentage difference in relative colour shift was calculated. An increase in colour shift (e.g. colour shift for HVS2 is greater than colour shift for HVS1) is represented by a positive number, and a decrease in predicted colour shift is represented by a negative number. This method was repeated for each of the 24 stimuli individually, and the data are presented as a percentage increase (or decrease) in colour shift (Fig.7). This procedure was also repeated for each of the 24 stimuli to compare HVS3 (symmetrically spaced photoreceptors) with HVS1 (asymmetrically spaced photoreceptors), and again the data are presented as a percentage increase (or decrease) in colour shift (Fig.7).

### Relative colour discrimination of the stimuli for HVS

To evaluate the relative ability of the three HVSs to discriminate the 24 stimuli in Table1, the standard deviation of the distribution of the loci in each of the respective colour spaces (Fig.6) was calculated. This was done by calculating the mean of the 24 lws and the 24 mws chromaticity coordinates (i.e. the centre of gravity of the 24 stimuli in colour space), and the mean distance (standard deviation) of loci from this point in colour space. A larger standard deviation represents a larger spread of loci and indicates a relatively better ability to discriminate between stimuli.

The reflectance spectra of the 24 stimuli are shown in Fig.2. For clarity of display, the curves are presented in separate graphs (Fig.2A–C), but no special significance applies to these divisions. All the naturally occurring samples (fruit, leaves, bark) reflect less than 10% ultraviolet radiation. The results for leaf foliage are consistent with larger studies of foliage reflectance spectra (Chittka et al., 1994). Since fruits and foliage have a similar ultraviolet reflectance, it may be of little value for a visual system trying to discriminate these stimuli to evaluate this region of the spectrum. However, many of the man-made stimuli reflect significant proportions of ultraviolet radiation (Fig.2).

Fig.4 shows the distribution of the 24 stimuli in colour space for a CCT=6500K and the colour matching functions of a CIE 1931 standard observer. The perceptual change in colour of these stimuli, considering spectrally variable illumination and von Kries colour constancy, are given in Table1. For 23 of the 24 stimuli only, approximate colour constancy is achieved by chromatic adaptation as the predicted colour distance exceeds the threshold for one just-noticeable distance. This result is consistent with other findings that von Kries-type colour constancy achieves approximate colour constancy for normal human vision (MacAdam, 1985). It should be noted that the values of just-noticeable differences for a CIE standard observer cannot be directly compared with relative colour shift values for the HVSs. The calculations of just-noticeable differences are based on data from colour-matching experiments with CIE 1931 specified standard human viewers (MacAdam, 1985), whist the calculation of relative colour shift is simply the Euclidean distance for a HVS that does not actually exist in nature. The just noticeable differences are used here to demonstrate that von Kries colour constancy is not complete (except in one case) for a standard human viewer. The purpose of the colour-shift values is that they allow a comparison of performance for HVSs that natural selection has not favoured.

To evaluate how an increase in the spectral breadth of the photoreceptors due to the inclusion of the β-band might affect von Kries colour constancy, it is possible to compare the relative performance of HVS2 with that of HVS1. The calculated values do not represent perceptual colour distance, but the relative performance indicates which visual system is better able to correct for spectrally variable illumination. Fig.7 shows that for most of the 24 stimuli, there is predicted to be a larger colour shift for HVS2, and for a few of the stimuli (e.g. 9, 15 and 20) the predicted colour shift is over twice as large as for HVS1. A comparison of the mean colour distances of all 24 stimuli for HVS1 and HVS2 suggests that HVS1 is approximately 33% better at correcting for spectrally variable illumination. Since von Kries colour constancy is predicted to be approximate for a CIE 1931 standard observer, a decrease in the ability to identify coloured stimuli in spectrally variable illumination is likely to reduce the effectiveness of colour vision. It appears, therefore, that ocular filters that prevent ultraviolet radiation reaching the β-band improve the effectiveness of colour constancy, which agrees with the theoretical considerations suggested by Worthey and Brill (Worthey and Brill, 1986). However, it is interesting to note that the relative performance of colour constancy for five of the stimuli is predicted to be slightly better for HVS2 (Fig.7). The explanation for this may be that, for some long-wavelength-rich stimuli that also weakly reflect ultraviolet radiation, the degree of adaptation for a visual system with β-peaks is enhanced compared with that of a visual system without β-peaks. This appears to be because whilst a weak ultraviolet reflectance has little influence on the overall colour signal of a stimulus, the state of adaptation of the receptors is benefited by the input of the β-band. Similar anomalies of chromatic adaptation are discussed elsewhere (Jameson and Hurvich, 1989).

The performance of HVS3 compared to HVS1 shows that, for most of the 24 stimuli, von Kries colour constancy is predicted to be better for the asymmetrically spaced photoreceptors (Fig.7). A comparison of the mean colour distances of all 24 stimuli for HVS3 and HVS1 suggests that HVS1 is approximately 34% better at correcting for spectrally variable illumination. Using a different method from that of Osorio (Osorio, 1997), these results support his findings that the asymmetric spacings of colour receptor peaks present in Old-World primates improve von Kries colour constancy (compared with a symmetrically spaced visual system). Again, because colour constancy is predicted to be approximate for most stimuli considering a CIE 1931 standard observer (Table1), it is hypothesised that a decrease in the performance of colour constancy could reduce the reliability of colour vision. Vorobyev and Menzel (Vorobyev and Menzel, 1999) show that symmetrically spaced receptors should maximise the discriminability of different stimuli. The standard deviations of the distribution of 24 stimuli in separate colour spaces for HVS1, HVS2 and HVS3 were 0.092, 0.089 and 0.128 units, respectively. A larger standard deviation represents a larger mean distance between all of the stimuli and, hence, relatively better discriminability. For HVS1 and HVS2, discriminability is predicted to be similar, although the calculated value is slightly lower for HVS2 as a result of the desaturation of the receptor signals by the β-bands. The discrimination of stimuli by HVS3 is predicted to be better than that of HVS1. However, greater discriminability may result in a more difficult problem to solve in spectrally variable illumination and, hence, the possibility of poorer colour constancy.

Without any mechanism of colour constancy, reliable colour vision in spectrally variable illumination would be impossible. This is because, without colour constancy, a stimulus can occupy a very large number of positions in colour space (Dyer, 1998). When a visual system is able to partially correct for changes in illumination colour then colour constancy is described as approximate. The value of approximate colour constancy (i.e. the colour of a given stimulus does not appear identical in different illumination conditions) may be of great value to animals. For example, if the perceived colour of a stimulus changes under different illumination conditions, but to such a small extent that it can still be reliably identified from alternative stimuli, then approximate colour constancy is sufficient to meet the demands on that visual system.

Colour constancy, or approximate colour constancy, is likely to be achieved in the visual systems of animals through a variety of mechanisms. There is evidence that the problem of identifying stimuli by their reflectance properties in spectrally variable illumination may be partially solved at both the receptor and neural levels of visual processing. Zeki (Zeki, 1983) demonstrated that the V4 region of the cerebral cortex in monkeys appears to be important for constant colour recognition. However, colour constancy has also been demonstrated in goldfish (Carassius auratus) (Ingle, 1985; Dörr and Neumeyer, 1996; Dörr and Neumeyer, 2000), and these animals have virtually no cerebral cortex (Hubel, 1995). Evidence of chromatic adaptation contributing to colour constancy in humans has been demonstrated by Uchikawa et al. (Uchikawa et al., 1989). The predictions of von Kries colour constancy for a CIE 1931 standard observer suggest that approximate colour constancy is possible through chromatic adaptation (Table1), with the implication that any reduction in the performance of constancy could affect reliable colour vision. The comparative results for the three HVSs suggest that two features present in the visual systems of Old-World primates aid von Kries colour constancy (Fig.7). These include a narrowing of receptor spectral sensitivities by ocular filters that prevent ultraviolet radiation reaching the retina (compare Fig.5A,B), and an asymmetric spacing of the spectral sensitivities of the photoreceptor classes (compare Fig.5A,C). The finding that asymmetric receptor spacing is predicted to improve colour constancy agrees with the results of Osorio (Osorio, 1997). However, Sumner and Mollon (2000) have recently demonstrated that the spectral positions of the MWS and LWS receptors in Old-World primates appears to be ideally suited to detecting stimuli presented against a mature leaf background, and improved colour constancy due to asymmetric receptor spacing may be a fringe benefit of other demands on the visual system.

Fig. 1.

The spectral absorbance of ocular filters in the eyes of various vertebrates. (Redrafted from Muntz, 1972, with the permission of Springer-Verlag and W. R. A. Muntz.)

Fig. 1.

The spectral absorbance of ocular filters in the eyes of various vertebrates. (Redrafted from Muntz, 1972, with the permission of Springer-Verlag and W. R. A. Muntz.)

Fig. 2.

Reflectance spectra of the 24 stimuli (numbered 1–24) described in Table1. Division of the stimuli into A, B and C is purely to aid the reading of the reflectance curves.

Fig. 2.

Reflectance spectra of the 24 stimuli (numbered 1–24) described in Table1. Division of the stimuli into A, B and C is purely to aid the reading of the reflectance curves.

Fig. 3.

Spectral distribution of typical daylight with the relative quantity of radiation normalised to 1.0 at 560nm. Curves are based on the data of Judd et al. (Judd et al., 1964) and represent correlated colour temperatures of (A) 10 000K and (B) 6500K.

Fig. 3.

Spectral distribution of typical daylight with the relative quantity of radiation normalised to 1.0 at 560nm. Curves are based on the data of Judd et al. (Judd et al., 1964) and represent correlated colour temperatures of (A) 10 000K and (B) 6500K.

Fig. 4.

Plots of 24 coloured stimuli (numbered as in Table1) in CIE 1931 colour space. The chromaticity coordinates (x, y) of the stimuli are plotted considering an illumination of correlated colour temperature=6500K, and the spectral locus is a plot of pure spectral radiation.

Fig. 4.

Plots of 24 coloured stimuli (numbered as in Table1) in CIE 1931 colour space. The chromaticity coordinates (x, y) of the stimuli are plotted considering an illumination of correlated colour temperature=6500K, and the spectral locus is a plot of pure spectral radiation.

Fig. 5.

Spectral sensitivity of various hypothetical visual systems (HVSs) plotted using a vitamin A1 template described by Stavenga et al. (Stavenga et al., 1993) with the height of the photoreceptors adjusted to a maximum of 1.0. (A) HVS1 is based on human photoreceptor peaks at 420, 534 and 564nm (Bowmaker and Dartnall, 1980) and assumes that all radiations shorter than 400nm are absorbed by ocular filters. (B) HVS2 is based on the same photoreceptor peaks as HVS1, but assumes no ocular filtration and that ultraviolet radiations are absorbed by common β-band peaks maximally sensitive at 340nm. (C) HVS3 is based on three symmetrically spaced photoreceptors maximally sensitive at 420, 492 and 564nm and assumes that all radiations shorter than 400nm are absorbed by ocular filters.

Fig. 5.

Spectral sensitivity of various hypothetical visual systems (HVSs) plotted using a vitamin A1 template described by Stavenga et al. (Stavenga et al., 1993) with the height of the photoreceptors adjusted to a maximum of 1.0. (A) HVS1 is based on human photoreceptor peaks at 420, 534 and 564nm (Bowmaker and Dartnall, 1980) and assumes that all radiations shorter than 400nm are absorbed by ocular filters. (B) HVS2 is based on the same photoreceptor peaks as HVS1, but assumes no ocular filtration and that ultraviolet radiations are absorbed by common β-band peaks maximally sensitive at 340nm. (C) HVS3 is based on three symmetrically spaced photoreceptors maximally sensitive at 420, 492 and 564nm and assumes that all radiations shorter than 400nm are absorbed by ocular filters.

Fig. 6.

Plots of 24 coloured stimuli (numbered as in Table1) onto Cartesian coordinates to represent the colour space of various hypothetical visual systems (HVSs) considering an illumination of correlated colour temperature=6500K. Chromaticity coordinates (lws, mws) are the normalised stimulations of the long-wavelength-sensitive and medium-wavelength-sensitive photoreceptors. (A) Colour space for HVS1, which is a visual system based on human colour receptor peaks (420, 534 and 564nm; Bowmaker and Dartnall, 1980) and ocular filters absorbing radiations shorter than 400nm. (B) Colour space for HVS2, which is a visual system based on human colour receptor peaks and no ocular filters. Ultraviolet radiation is assumed to be absorbed by secondary β-band peaks that are maximally sensitive to 340nm radiation. (C) Colour space for HVS3, which is a visual system based on symmetrically spaced photoreceptors maximally sensitive at 420, 492 and 564nm and assumes that all radiations shorter than 400nm are absorbed by ocular filters (see Fig.5). The spectral locus is a plot of pure spectral radiation.

Fig. 6.

Plots of 24 coloured stimuli (numbered as in Table1) onto Cartesian coordinates to represent the colour space of various hypothetical visual systems (HVSs) considering an illumination of correlated colour temperature=6500K. Chromaticity coordinates (lws, mws) are the normalised stimulations of the long-wavelength-sensitive and medium-wavelength-sensitive photoreceptors. (A) Colour space for HVS1, which is a visual system based on human colour receptor peaks (420, 534 and 564nm; Bowmaker and Dartnall, 1980) and ocular filters absorbing radiations shorter than 400nm. (B) Colour space for HVS2, which is a visual system based on human colour receptor peaks and no ocular filters. Ultraviolet radiation is assumed to be absorbed by secondary β-band peaks that are maximally sensitive to 340nm radiation. (C) Colour space for HVS3, which is a visual system based on symmetrically spaced photoreceptors maximally sensitive at 420, 492 and 564nm and assumes that all radiations shorter than 400nm are absorbed by ocular filters (see Fig.5). The spectral locus is a plot of pure spectral radiation.

Fig. 7.

The relative performance of two hypothetical visual systems (HVS2 and HVS3) compared with HVS1 for a variety of differently coloured stimuli (numbered 1–24; see Table1) when considering spectrally variable illumination and von Kries colour constancy. The change in illumination colour is from a correlated colour temperature of 6500K to 10000K. A positive number indicates a relative increase in the predicted colour shift of a stimulus in colour space (poorer performance), and a negative number indicates a reduced colour shift (better performance). For example, an increase of 100% means the predicted colour shift is twice the magnitude of that calculated for HVS1. If the performance is identical to that of HVS1, then the relative increase is zero. HVS1 is based on three receptors with maximum absorption of radiation at 420, 534 and 564nm, plotted with a vitamin A1 template (Stavenga et al., 1993) and assuming all radiations shorter than 400nm are absorbed by ocular filters. HVS2 has the same main receptor peaks as HVS1, but does not have ocular filters that prevent the β-band absorbing short-wavelength radiations. HVS3 is based on symmetrically spaced receptors with maximum absorption of radiation at 420, 492 and 564nm and has ocular filters absorbing radiations shorter than 400nm (see Fig.5).

Fig. 7.

The relative performance of two hypothetical visual systems (HVS2 and HVS3) compared with HVS1 for a variety of differently coloured stimuli (numbered 1–24; see Table1) when considering spectrally variable illumination and von Kries colour constancy. The change in illumination colour is from a correlated colour temperature of 6500K to 10000K. A positive number indicates a relative increase in the predicted colour shift of a stimulus in colour space (poorer performance), and a negative number indicates a reduced colour shift (better performance). For example, an increase of 100% means the predicted colour shift is twice the magnitude of that calculated for HVS1. If the performance is identical to that of HVS1, then the relative increase is zero. HVS1 is based on three receptors with maximum absorption of radiation at 420, 534 and 564nm, plotted with a vitamin A1 template (Stavenga et al., 1993) and assuming all radiations shorter than 400nm are absorbed by ocular filters. HVS2 has the same main receptor peaks as HVS1, but does not have ocular filters that prevent the β-band absorbing short-wavelength radiations. HVS3 is based on symmetrically spaced receptors with maximum absorption of radiation at 420, 492 and 564nm and has ocular filters absorbing radiations shorter than 400nm (see Fig.5).

Table 1.

I am very grateful for information and advice that was provided by Professors J. K. Bowmaker, W. R. A. Muntz and D. G. Stavenga. I also wish to thank Dr L. L. Muir and Professor W. R. A. Muntz for their valuable comments on the manuscript. I am grateful for resources provided by Monash University and Royal Melbourne Institute of Technology University.

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text:
# pwm problem in mikroC -pic- [closed]

i am trying to do an simple PWM example in mikroC program using ISIS simulator for pic16f877a and i used the code in the program help and it didnt work

for(;;)
{
Pwm_Init(5000);
pwm_start();
delay_ms(500);
}

## closed as not a real question by Olin Lathrop, Leon Heller, stevenvh, Oli Glaser, W5VO♦Oct 7 '12 at 22:59

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question.

• There is too little to go on here - is this the whole of your code? How didn't it work? What is the error message? – Oli Glaser Oct 7 '12 at 21:04
• "Didn't work" is a useless description. – Olin Lathrop Oct 7 '12 at 21:12
• i think this the code used to start the pwm in mikroC i want to use pwm in a project but first i need to see how it works first and make sure that it works – alone Oct 8 '12 at 16:24
• it "didnt work" there is no errors in compiling but the problem that it do not work in the simulation ....i am sorry for being not clear in my question ..:) – alone Oct 8 '12 at 16:46

PWM needs at least 2 parameters. You only supply the 5000, which will probably be the frequency, but what about the duty cycle? If you don't initialize that and it defaults to 0 or 100 %, then you won't see anything.

edit
It appears you have to call the function Pwm_Change_Duty() with a value between 0 (minimum) and 255 (maximum) to set the duty cycle.
end of edit

Most microcontrollers with PWM will have more than 1 PWM channel, so you'll have to tell the program which channel you want to use as well.

Also, I don't know what Pwm_Init() and Pwm_start() do exactly, but normally they should be outside your main loop. Now you're executing them again and again. IMO should be

Pwm_Init(5000);
pwm_start();
for(;;)
{
delay_ms(500);
},

and then the delay() serves no function, so that you can omit it:

Pwm_Init(5000);
pwm_start();
for(;;)
{
}.

• ok i am just asking.... this is the code i saw in the help i used the code pwm_change_duty(value) and it didnt work so i think there is another thing to do so the pwm work on the ISIS – alone Oct 8 '12 at 16:16
• @alone - but have tried the Pwm_Change_Duty() I mention in my answer? And you also haven't answered Oli's question: what does "doesn't work" mean? What do you observe? If you can add details to the question it may be reopened. – stevenvh Oct 8 '12 at 16:22
• of course .... the code compile very will and there is no error in compiling but in the simulation it do not work thats my problem – alone Oct 8 '12 at 16:27


text:
# Another derivative problem

• Mar 25th 2008, 06:33 PM
lyzzi3
Another derivative problem
I'm so lost (Doh)
• Mar 25th 2008, 06:58 PM
mr fantastic
Quote:

Originally Posted by lyzzi3
I'm so lost (Doh)

You have nested chain rules.

Start off by letting $u = x^5$ and use the chain rule.

To get the derivative of $\cos (\cos u)$ with respect to u, let $w = \cos u$.

And you know how to differentiate $\cos w$ with respect to w, right?
• Mar 26th 2008, 04:29 AM
topsquark
Quote:

Originally Posted by lyzzi3
I'm so lost (Doh)

You have asked at least six questions of this kind. You would do well to talk to your instructor and see if they can help you learn this material better.

-Dan


text:
Domain.apply_masking(inplace=False)[source]

Apply masking as defined by the CF conventions.

Masking is applied to all metadata constructs with data.

Masking is applied according to any of the following criteria that are applicable:

• where data elements are equal to the value of the missing_value property;

• where data elements are equal to the value of the _FillValue property;

• where data elements are strictly less than the value of the valid_min property;

• where data elements are strictly greater than the value of the valid_max property;

• where data elements are within the inclusive range specified by the two values of valid_range property.

If any of the above properties have not been set the no masking is applied for that method.

Elements that are already masked remain so.

Note

If using the apply_masking method on a construct that has been read from a dataset with the mask=False parameter to the read function, then the mask defined in the dataset can only be recreated if the missing_value, _FillValue, valid_min, valid_max, and valid_range properties have not been updated.

New in version (cfdm): 1.8.9.0

Parameters
inplace: bool, optional

If True then do the operation in-place and return None.

Returns
Domain or None

A new domain construct with masked values, or None if the operation was in-place.

Examples:

>>> d = cfdm.example_field(0).domain
>>> x = d.construct('longitude')
>>> x.data[[0, -1]] = cfdm.masked
>>> print(x.data.array)
[-- 67.5 112.5 157.5 202.5 247.5 292.5 --]


text:
# zbMATH — the first resource for mathematics

On linearly Lindelöf and strongly discretely Lindelöf spaces. (English) Zbl 0930.54003
A topological space $$X$$ is said to be linearly Lindelöf if every open cover of $$X$$ which is a chain (under set-theoretic inclusion) admits a countable subcover. It has been known for many years [A. S. Mishchenko, Sov. Math., Dokl. 3, 1199-1202 (1962); translation from Dokl. Akad. Nauk SSSR 145, 1224-1227 (1962; Zbl 0121.17501)] that a linearly Lindelöf Tychonoff space need not be Lindelöf. (The present authors contribute another example, discovered independently by G. Gruenhage and R. Buzyakova, which “seems to be simpler than Mishchenko’s example”. This has the pleasing feature that it is a dense subgroup of $$\{0,1\}^{\aleph_\omega}$$, hence is homogeneous and of countable cellularity.)
In 1969 the first-listed co-author established a famous theorem of which the following may be viewed a the special case $$\kappa= \omega$$: Every first countable Lindelöf Tychonoff space $$X$$ satisfies $$|X |\leq 2^\omega$$. Perhaps the most striking result of the present paper is that the conclusion holds also if the hypothesis “Lindelöf” is weakened to “weakly Lindelöf”. A principal lemma is the following unexpected result: If $$Y$$ is a closed subset of a linearly Lindelöf Tychonoff space $$X$$ such that (a) $$|Y|\leq 2^\omega$$, (b) $$\chi(p,X)\leq\omega$$ for each $$p\in Y$$, and (c) every open cover of $$X$$ admits a subcover of cardinality $$\leq(2^\omega)^+$$, then $$Y$$ is the intersection of $$2^\omega$$-many open subsets of $$X$$.
The paper offers (for several properties $$P)$$ theorems of the form “linearly Lindelöf$$+ P\Rightarrow$$Lindelöf”. There are several unsolved problems, including: Does “linearly Lindelöf$$+ P\Rightarrow$$Lindelöf” when $$P$$ is one of these properties: Hewitt-complete (realcompact); locally compact; locally metrizable?

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54A35 Consistency and independence results in general topology
##### Keywords:
Lindelöf space; linearly Lindelöf space; free sequence
Full Text:


text:
# Schwinger effect

In the presence of a strong, constant electric field, electrons, e, and positrons, e+, will be spontaneously created.

The Schwinger effect is a predicted physical phenomenon whereby matter is created by a strong electric field. It is also referred to as the Sauter–Schwinger effect, Schwinger mechanism, or Schwinger pair production. It is a prediction of quantum electrodynamics (QED) in which electronpositron pairs are spontaneously created in the presence of an electric field, thereby causing the decay of the electric field. The effect was originally proposed by Fritz Sauter in 1931[1] and further important work was carried out by Werner Heisenberg and Hans Heinrich Euler in 1936,[2] though it was not until 1951 that Julian Schwinger gave a complete theoretical description.[3]

The Schwinger effect can be thought of as vacuum decay in the presence of an electric field. Although the notion of vacuum decay suggests that something is created out of nothing, physical conservation laws are nevertheless obeyed. To understand this, note that electrons and positrons are each other's antiparticles, with identical properties except opposite electric charge.

To conserve energy, the electric field loses energy when an electron–positron pair is created, by an amount equal to ${\displaystyle 2m_{\text{e}}c^{2}}$, where ${\displaystyle m_{\text{e}}}$ is the electron rest mass and ${\displaystyle c}$ is the speed of light. Electric charge is conserved because an electron–positron pair is charge neutral. Linear and angular momentum are conserved because, in each pair, the electron and positron are created with opposite velocities and spins. In fact, the electron and positron are expected to be created at (close to) rest, and then subsequently accelerated away from each other by the electric field.[4]

## Mathematical description

Schwinger pair production in a constant electric field takes place at a constant rate per unit volume, commonly referred to as ${\displaystyle \Gamma }$. The rate was first calculated by Schwinger[3] and at leading (one-loop) order is equal to

${\displaystyle \Gamma ={\frac {(eE)^{2}}{4\pi ^{3}c\hbar ^{2}}}\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}\mathrm {e} ^{-{\frac {\pi m^{2}c^{3}n}{eE\hbar }}}}$

where ${\displaystyle m}$ is the mass of an electron, ${\displaystyle e}$ is the elementary charge, and ${\displaystyle E}$ is the electric field strength. This formula cannot be expanded in a Taylor series in ${\displaystyle e^{2}}$, showing the nonperturbative nature of this effect. In terms of Feynman diagrams, one can derive the rate of Schwinger pair production by summing the infinite set of diagrams shown below, containing one electron loop and any number of external photon legs, each with zero energy.

The infinite set of Feynman diagrams relevant for Schwinger pair production.

## Experimental prospects

The Schwinger effect has never been observed due to the extremely strong electric-field strengths required. Pair production takes place exponentially slowly when the electric field strength is much below the Schwinger limit, corresponding to approximately 1018 V/m. With current and planned laser facilities, this is an unfeasibly strong electric-field strength, so various mechanisms have been proposed to speed up the process and thereby reduce the electric-field strength required for its observation.

The rate of pair production may be significantly increased in time-dependent electric fields,[5][6][7] and as such is being pursued by high-intensity laser experiments such as the Extreme Light Infrastructure.[8] Another possibility is to include a highly charged nucleus which itself produces a strong electric field.[9]

By electromagnetic duality, the same mechanism in a magnetic field should produce magnetic monopoles, if they exist. A search conducted using the Large Hadron Collider failed to detect monopoles, and analysis indicated a lower bound on monopole mass of 75 GeV/c2 at the 95% confidence level.[10]

In January 2022 researchers at the National Graphene Institute led by Andre Geim and a number of other collaborators reported the observation of an analog process between electron and holes at the Dirac point of a superlattice of graphene on hexagonal boron nitride (G/hBN) and another one of twisted bilayer graphene (TBG). An interpretation as Zener–Klein tunneling (a mix[11] between Zener tunneling and Klein tunneling) is also utilized.[12][13]

## References

1. ^ Sauter, Fritz (1931). "Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 69 (11–12): 742–764. doi:10.1007/bf01339461. ISSN 1434-6001.
2. ^ Heisenberg, W.; Euler, H. (1936). "Folgerungen aus der Diracschen Theorie des Positrons". Zeitschrift für Physik (in German). Springer Science and Business Media LLC. 98 (11–12): 714–732. arXiv:physics/0605038. doi:10.1007/bf01343663. ISSN 1434-6001.
3. ^ a b Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". Physical Review. American Physical Society (APS). 82 (5): 664–679. doi:10.1103/physrev.82.664. ISSN 0031-899X.
4. ^ A.I. Nikishov (1970). "Pair Production by a Constant External Field". Journal of Experimental and Theoretical Physics. 30: 660.
5. ^ Brezin, E.; Itzykson, C. (1970-10-01). "Pair Production in Vacuum by an Alternating Field". Physical Review D. American Physical Society (APS). 2 (7): 1191–1199. doi:10.1103/physrevd.2.1191. ISSN 0556-2821.
6. ^ Ringwald, A. (2001). "Pair production from vacuum at the focus of an X-ray free electron laser". Physics Letters B. Elsevier BV. 510 (1–4): 107–116. doi:10.1016/s0370-2693(01)00496-8. ISSN 0370-2693.
7. ^ Popov, V. S. (2001). "Schwinger mechanism of electron–positron pair production by the field of optical and X-ray lasers in vacuum". Journal of Experimental and Theoretical Physics Letters. Pleiades Publishing Ltd. 74 (3): 133–138. doi:10.1134/1.1410216. ISSN 0021-3640.
8. ^ I. C. E. Turcu; et al. (2016). "High field physics and QED experiments at ELI-NP" (PDF). Romanian Reports in Physics. 68: S145-S231.
9. ^ Müller, C.; Voitkiv, A. B.; Grün, N. (2003-06-24). "Differential rates for multiphoton pair production by an ultrarelativistic nucleus colliding with an intense laser beam". Physical Review A. American Physical Society (APS). 67 (6): 063407. Bibcode:2003PhRvA..67f3407M. doi:10.1103/physreva.67.063407. ISSN 1050-2947.
10. ^ Acharya, B.; Alexandre, J.; Benes, P.; Bergmann, B.; Bertolucci, S.; et al. (2022-02-02). "Search for magnetic monopoles produced via the Schwinger mechanism". Nature. Springer Science and Business Media LLC. 602 (7895): 63–67. doi:10.1038/s41586-021-04298-1. ISSN 0028-0836.
11. ^ Vandecasteele, Niels; Barreiro, Amelia; Lazzeri, Michele; Bachtold, Adrian; Mauri, Francesco (2010-07-20). "Current-voltage characteristics of graphene devices: Interplay between Zener–Klein tunneling and defects". Physical Review B. 82 (4): 045416. arXiv:1003.2072. Bibcode:2010PhRvB..82d5416V. doi:10.1103/PhysRevB.82.045416. ISSN 1098-0121. S2CID 38911270.
12. ^ Berdyugin, Alexey I.; Xin, Na; Gao, Haoyang; Slizovskiy, Sergey; Dong, Zhiyu; Bhattacharjee, Shubhadeep; Kumaravadivel, P.; Xu, Shuigang; Ponomarenko, L. A.; Holwill, Matthew; Bandurin, D. A. (2022-01-28). "Out-of-equilibrium criticalities in graphene superlattices". Science. 375 (6579): 430–433. arXiv:2106.12609. Bibcode:2022Sci...375..430B. doi:10.1126/science.abi8627. ISSN 0036-8075. PMID 35084955. S2CID 235623859.
13. ^ "Schwinger effect seen in graphene". Physics World. 2022-03-25. Retrieved 2022-03-28.


text:
# Set theory

1. Sep 18, 2009

### Fredrik

Staff Emeritus
I'm thinking about buying https://www.amazon.com/exec/obidos/tg/detail/-/0412606100/. Both books are getting excellent reviews at Amazon, especially Goldrei.

I would like to learn about the ZFC axioms, cardinals and ordinals, etc. I assume both books will cover those topics. I'm also very curious about something I heard for the first time today:
Am I reading this right? Every set in ZFC theory can be constructed from the empty set by iterating the power set operation?! That sounds weird. I know you can define the integers this way, and infinite ordinals I guess, but everything else?

0={}
1={0}
2={0,1}
...

If this one axiom leads to all the ZFC axioms, then I should have heard about this before, so I feel that I must have misunderstood something.

Do both of these books explain these things?

Last edited by a moderator: Apr 24, 2017
2. Sep 18, 2009

### Hurkyl

Staff Emeritus
Let V denote the class of all sets that are contained in some transfinitely-iterated power of the empty set. (As in the Wiki article)

Suppose the set S0 is not in V. Then S0 must contain an element, S1, not in V. Iterating this argument (and using the axiom of choice), we can construct an infinite chain $\cdots \in S_2 \in S_1 \in S_0$. This contradicts the axiom of foundation, therefore S0 cannot exist.

Now, this fact does not imply all the axioms of ZFC.

This fact cannot tell you "how many" ordinal numbers are available for use -- the class of ordinals is, roughly speaking, one of the "inputs" to this construction.

This fact cannot tell you much about what kinds of sets exist and how complex they can be. e.g. it cannot tell you whether or not there exist sets whose cardinality lies between those of Vw+1 and Vw.

This fact cannot even tell you that two sets are equal iff they have the same members!

3. Sep 18, 2009

### Fredrik

Staff Emeritus
If "this fact" refers to the idea that all sets are constructed by iterating the power set operation, then you have convinced me that it can't replace the ZFC axioms (probably not even a single one of them). You saved the best argument for last:

I should have realized that myself. I guess I didn't really think because I started the thread to ask what book I should get, and ended up throwing in a question that seems silly now.

I don't understand this. I'm not sure I understand the power set construction either, but it seems to me that the first steps are

V0={}
V1={{}}={0}
V2={{},{0}}={0,1}
V3={{},{0},{1},{0,1}}={0,1,{1},2}
V4={0,1,{1},{2},2,{0,{1}},{0,2},{1,{1}},{1,2},{{1},2},3}

Suppose your V is V4. You said that if S0 isn't in V, it must contain an element that isn't in V. But {0,{1}} isn't in V (it's a subset, but not a member) and it doesn't contain anything that isn't in V. So your proof seems to fail at the first step.

But I have no doubt that you'll explain what I have misunderstood and make me feel silly again.

4. Sep 19, 2009

### Hurkyl

Staff Emeritus
The (von Neumann) universe is constructed by iterating the power set operation. The theorem is that all sets are elements of the universe.

As for your example, the set you chose is a member of V4, so I don't understand what you were getting at.

Incidentally, the contrapositive of my claim is (hopefully) more obvious:
If every element of S is in V, then S is in V

This requires all of V -- if you stop iterating at some point, then it breaks down. For example, a set whose elements all lie in V4 is not necessarily a member of V4. (But it is necessarily a member of V5)

5. Sep 19, 2009

### Fredrik

Staff Emeritus
Sorry about the {0,{1}}. How about {0,{2}}? That's not in V and doesn't contain anything that isn't in V.

Also, my notation is inconsistent. I should have written 0,1,2,... on the left-hand sides, or V0,V1,V2,... on the right-hand sides. It doesn't make much sense to mix two notations the way I did.

6. Sep 19, 2009

### Hurkyl

Staff Emeritus
Equivocation fallacy. When I made that statement, I was using V to refer to the entirety of the Von Neumann universe. But you're using V to refer to the V4 subclass.

7. Sep 19, 2009

### Landau

Last edited by a moderator: Apr 24, 2017
8. Sep 19, 2009

### Fredrik

Staff Emeritus
OK, that wasn't at all clear, I think. But now that you've cleared that up, your proof is easy enough to understand. I needed some help from Wikipedia to understand why that sequence contradicts the axiom of foundation, but I think I got it now.

9. Sep 19, 2009

### Fredrik

Staff Emeritus
That doesn't seem like a bad thing. But the reviews at Amazon suggest that Potter is using some "new" approach to the subject. It doesn't seem to be about ZFC at all, but rather about some other set theory. If that's right, it's certainly not the best place to learn about ZFC, but I might still be interested in the things he is talking about. Have you read this book? Can you tell me what it's about?

10. Sep 20, 2009

### Fredrik

Staff Emeritus
I'm still about this. If we define V0={}, V1={V0}, V2={V0,{V0}}={V0,V1}, and so on, Vn is a set with 2n-1 members for n=1,2,3,... If these are the only sets we allow, there are no sets with 3 members. Is {1,2,3} not a set?!

11. Sep 20, 2009

### Hurkyl

Staff Emeritus
{1,2,3} is an element of the universe because it's an element of V5.

V is not the class whose elements are the Va: V is the class which is the union of all the Va.


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