text:
[ Research Article ]
Asian Journal of Atmospheric Environment - Vol. 4, No. 3, pp.141-149
ISSN: 1976-6912 (Print) 2287-1160 (Online)
Print publication date 31 Dec 2010
Received 26 May 2010 Accepted 23 Jul 2010

# Vertical Change in Extinction and Atmospheric Particle Size in the Boundary Layers over Beijing: Balloon-borne Measurement

Bin Chen1), 2) ; Guang-Yu Shi1) ; Maromu Yamada3) ; Dai-Zhou Zhang4) ; Masahiko Hayashi5) ; Yasunobu Iwasaka6), *
1)State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
3)Center for Innovation, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan
4)Faculty of Environmental and Symbiotic Sciences, Prefectural University of Kumamoto, 3-1-100 Tsukide, Kumamoto 862-8502, Japan
5)Department of Earth System Science, Graduate School of Science, Fukuoka University, 8-19-1 Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan
6)Frontier Science Organization, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan

Correspondence to: *Tel: +81-76-234-4645, E-mail: kosa@staff.kanazawa-u.ac.jp

## Abstract

Aerosol size and number concentration were observed in the atmospheric boundary layer over Beijing (from near the ground to 1,200 m) on March 15 (a clear day) and 16 (a dusty day), 2005. The results were further compared with lidar measurements in order to understand the dependency of extinction on the particle size distribution and their vertical changes. The boundary layer atmosphere was composed of several sub-layers, and a dry air layer appeared between 400 and 1,000 m under the influence of dust event. In this dry air layer, the concentration of the fine-mode particles (diameter smaller than 1.0 μm) was slightly lower than the value on the clear day, while the concentration of coarse-mode particles (diameter larger than 1.0 μm) was remarkably higher than that on the clear day. This situation was attributed to the inflow of an air mass containing large amounts of Asian dust particles and a smaller amount of fine-mode particles.

The results strongly suggest that the fine-mode particles affect light extinction even in the dusty atmosphere. However, quantitatively the relation between extinction and particle concentration is not satisfied under the dusty atmospheric conditions since laser beam attenuates in the atmosphere with high concentration of particles. Laser beam attenuation effect becomes larger in the relation between extinction and coarse particle content comparing the relation between extinction and fine particle content. To clarify this problem technically, future in situ measurements such as balloon-borne lidar are suggested. Here extinction was measured at 532 nm wavelength. Measurements of extinction at other wavelengths are desired in the future.

## Keywords:

Extinction, Vertical distribution of aerosol, Particle size, Boundary layer, Visibility

## 1. INTRODUCTION

Atmospheric aerosols play an important role in climate change by affecting global cloud albedo, radiative transfer, ozone chemistry, acid rain, and visibility (Jacobson et al., 2001; Seinfeld et al., 1998; Charlson et al., 1992). They also have an important and direct effect on the atmospheric environment and human health (Chan et al., 1997). Increases in the number concentration of aerosol particles, especially the fine particulate matter (particles with diameter smaller than 2.5 μm: PM2.5), may also have significant health effects (Donaldson et al., 1998; Dockery et al., 1994), reduce visibility, and concentrate large amounts of poisonous compounds in the air (An et al., 2000; Prospero et al., 1999). Such harmful effects have understandably caused widespread public concern (Zhang et al., 2003). Indeed, particulate matter is one of the key pollutant types in most of the big cities in China (Wang et al., 2000).

Reduced visibility has become a severe issue in many large cities of the world and can cause inconveniences in daily life disturbing traffic and transportation including delay of flights. Many studies have investigated issues of reduced visibility and particulate matter(Wang et al., 2006; Song et al., 2003; Yan et al., 2003; Chan et al., 1999; Baik et al., 1996; Larson et al., 1989).

These studies have mostly been based on ground observations and few measurements have investigated the particulate characteristics and visibility in the free atmosphere or boundary mixing layer over urban regions of China. Knowledge of free atmospheric particulate content and its effect on slant visibility is essential to control safely aviation system considering that activities of big cities including Beijing city are strongly depending on air transportation. Beijing, which has approximately 14.4 million inhabitants, has been experiencing rapid economic growth and urbanization for decades, and consequently faces many problems due to pollution. In addition, previous studies have focused on monitoring PM10 and PM2.5 particles, without clarifying size dependences.

The results described here is the first case study discussing the vertical change in aerosol size/number concentration and extinction (at wavelength=532 nm) in the boundary layer over Beijing, China, on the basis of the measurement with balloon-borne optical particle counters and lidar.

## 2. MEASUREMENTS AND METHODS

### 2. 1 Observation Sites

Tethered-balloon measurements were performed at the Atmospheric Observing Experimental Base of Institute of Atmospheric Physics (39.82°N, 116.47°E), close to the south 5th Ring Road in Beijing, south-east of the central part of the city (Matsuki et al., 2005; Fig. 1). Lidar measurements were performed at the China-Japan Friendship Environmental Protection Centre (39.98°N, 116.42°E) which is near the northcentral part of the 4th Ring Road and locates 18km north of the tethered-balloon site. The distance between the two observation sites was about 10 km. This difference, as described later, can reasonably be neglected considering the atmospheric structures of the planetary boundary layer observed here.

Observation sites. (A) Base of the atmospheric observation; (B) China-Japan Friendship Environmental Protection Centre.

### 2. 2 Instrumentation and Measurements

The tethered-balloon (TB) observation system is similar to that described by Matsuki et al. (2005), so we describe the specifications of the system only briefly here. Fig. 2 shows the balloon train used. A barometer was mounted on the balloon to measure the atmospheric pressure, which we used to calculate altitude. The balloon system could reach an altitude of 1,200 m above the ground within 30-40 minutes. It was therefore possible to monitor the changes in aerosols at a time resolution of about 1 hour. The size and number concentration of the atmospheric aerosols were observed in 20-second intervals with an optical particle counter (OPC), which sized particles at diameters of 0.3, 0.5, 1.0, 2.0, and 5.0 μm(Yamashita et al., 2003). A detailed description of the counter was given by Hayashi et al. (1998). Vertical profiles of the number concentration of particles, temperature, relative humidity, and air pressure in the same air mass were obtained by balloon-borne measurements at high vertical resolution (about 0.5 m, with resolution dependent on the sounding speed). Table 1 lists the observation time and meteorological conditions near the surface at the balloon site.

Balloon train.

Surface meteorological records and observing time.

Detailed specifications of the polarization lidar, which was installed at the China-Japan Friendship Environmental Protection Centre, can be found in Shimizu et al. (2004). The lidar measurements were made using backscattering light from atmospheric particles at 532 nm wavelength (second harmonic wavelength of Nd: YAG laser). Scattering light received by a telescope with a diameter of 20 cm, and detected with photomultiplier tubes. A detailed description of the lidar was given by Sugimoto et al. (2003).

## 3. RESULTS AND DISCUSSION

### 3. 1 The Relationship between Visibility and Extinction

The relationship between visibility and atmospheric extinction is expressed as (Koschmieder et al., 1924)

 (1)

where Lv is the visibility in kilometres and b is the atmospheric extinction per kilometre. This formulation is widely used in the routine measurement of the meteorological observatory in the world and regarded as one of the global standards.

The atmospheric extinction can be broken into three parts: b=br+ba+bp, where br is the molecular Rayleigh scattering, ba is the gaseous absorption, and bp is the extinction due to aerosols. Under standard conditions (sea-surface temperature 15°C, air pressure 1013.25 hPa, air density 1.2250 kg/m3), the value of Rayleigh scattering is 0.013 km-1 (Peundorf et al., 1957). Some atmospheric gas species, especially NO2, may have absorption properties in visible light wavelength ranges. Previous research has verified that the contribution of particulate aerosols to the extinction occupies more than 90% of the total b over Beijing (Liu et al., 2004). Hence it is easily confirmed that the empirically determined equation (1) is roughly satisfied in the atmosphere over Beijing, and the visibility has approximately inverse correlation with the particle extinction. According to previous investigations, the size distributions and chemical compositions of atmospheric aerosols have been strongly affected by dust particles and particles of anthropogenic origin in the Beijing region (Wu et al., 2009; Li et al., 2008; Xu et al., 2008; Yuan et al., 2007; Wang et al., 2000; Liu et al., 1999). It is therefore interesting to analyse the relationship between extinction and the size-segregated number concentration of particles, in a case study of a large city in a developing country. However, this relationship cannot reveal which particle size controls the extinction.

### 3. 2 Vertical Structures of Particle Number Concentrations and Extinctions

As described above, the distance between the lidar observation and balloon sounding sites is about 10 km. Balloon-borne observations took about 1 hour to collect one dataset covering from near the ground to 1,200 m. Horizontal scale of observation field (duration time of observation×typical wind speed) is considered to be about 7-10 km taking into horizontal wind speed of 2-3 m/sec during the observation. Hence the difference in the observation locations in the present analysis is negligible. As shown in Table 1, the weather conditions of the two observing days (050315 and 050316 represent March 15, 2005 and March 16, 2005, respectively) were quite different: the former was a fine day, while the latter was more polluted, as shown by the weather report of the China Meteorological Administration. Figs. 3 and 4 present the vertical profiles of atmospheric relative humidity, potential temperature, size-segregated particle number concentration, and light extinction for the respective days.

Potential temperature (K) and relative humidity (R.H., %) plotted separately for the two observation days (Left: March 15, right: March 16). The black lines represent the potential temperature and the gray lines represent the relative humidity.

Profiles of particle number concentrations (L-1) in different size ranges and extinction ratio (km-1) on March 15 (up) and 16 (down).

Comparing the particle concentration of March 15 with that of March 16 (Fig. 4), there is large difference in concentrations of particles with diameter >2.0 μm, and values of March 16 (dusty day) are about 10 times higher than those of March 15 (clear day) from near the ground to about 1,200m height. From the profiles of coarse mode particle contents, it is suggested that the dust storm disturbed aerosols not only in the boundary mixing layer (potential temperature shows clear decline slope from the ground to about 80 m and narrow decline region at about 400 m. Here we define the region from near the ground to 400 m as the boundary mixing layer of March 16) but also in the free atmosphere (Figs. 3 and 4). In the free atmosphere of March 16, relatively dry air was observed suggesting inclusion of air mass having different origins and/or histories (L4′ and L5′ in Fig. 3). Interestingly, extremely enhanced content of coarse mode particles was found in the dry air mass (Fig. 4).

Concerning with the concentration of fine particles, very interesting feature was observed in the dusty day (March 16) (Fig. 4). The fine particle concentration decreased in the free atmosphere (corresponding to the layers of L4′ and L5′), which were apparently lower than the level of the clear day (March 15). The decrease of the fine particle concentration has been, on the basis of ground-based particle sampling, frequently reported during dust episode by many investigators (e.g., Zhang et al., 2008). When the air mass such as the case in the free troposphere of 16 March descends to near the ground, high concentration of dust particles is observed without effect of pollutants including fine particles formed secondly from polluted gases.

The profile of potential temperature on March 15 suggests active vertical mixing compared to that of 16 March, especially from near the ground to about 800 m height. It can be suggested that the stable thermodynamic structure of 16 March, especially from near the ground to about 80 m, caused polluted air because the pollutants could not diffuse upwards and their concentrations became higher in the low atmosphere. Here, considering vertical profiles of potential temperature and relative humidity, we divided the boundary layer into six sub-layers (L1-L6) for 050315 and seven sub-layers (L1′-L7′) for 050316 measurements. If we decide the boundary layer according to the position of the first clear inversion in the profiles of potential temperature, the boundary layer height will be about 150 m on March 15 and about 200 m on March 16 (Fig. 3). However, it is reasonable to determine the height of the boundary layer top considering humidity profiles in addition to the potential temperature profiles since the ground is large source of water vapour, and heights of about 950 m or 1,150 m of March 15 and about 430 m of March 16 would be top of the boundary layer (Fig. 3).

Fig. 4 shows the relation of the vertical profiles between the particle number concentrations in different size ranges and light extinction at the wavelength of 532 nm. The extinction (αsl) and particle number concentration (Csl) in each sub-layer are given by

 ${\alpha }_{sl}=\frac{{\int }_{{\delta }_{z}}\alpha \left(z,r\right)dz}{{L}_{sl}}$ (2)

and

 ${C}_{sl}=\frac{{\int }_{{\delta }_{z}}C\left(z,r\right)dz}{{L}_{sl}}$ (3)

where α(z, r) and C(z, r) are extinction coefficient and particle number concentration as a function of altitudes (z) in each sub-layer, respectively. Lsl represents the layer thickness in each sub-layer (=layer top height-layer bottom height), respectively.

### 3. 3 Correlation between the Extinction and Particle Number Concentration

Figs. 5 and 6 show the correlation between the extinction coefficients and the particle number concentrations in different size ranges in the case of 050315 (the clear day) and 050316 (the dusty day), respectively. According to (2) and (3), six data points of particle concentration and extinction were obtained for the clear day (Fig. 5) and seven data points were obtained for the dusty day (Fig. 6). Close positive correlations existed between the extinction coefficients and the particle number concentrations for particle diameters smaller than 1.0 μm in both cases.

Correlation between the extinction coefficients (km-1) and the particle number concentrations (L-1) in different size ranges in the case of 050315 (clear day).

Correlation between extinction coefficients (km-1) and particle number concentration (L-1) in each size range in the case of 050316 (dusty day). The dotted lines (L1′ to L4′) show the correlation between the extinction and number concentration, neglecting data points L5′, L6′ and L7′.

For particles of diameter larger than 2.0 μm(050316 case), assessing their contribution is difficult using only lidar data (Fig. 6) because the intensity of the laser beam possibly decreased as the laser beam penetrated the layer of large optical depth. Therefore the extinction deduced from the lidar measurements will be underestimated above about 800 m, comparing the profiles of lidar returns of 050315 and 050316 in Fig. 4. The open symbols in Fig. 6 represent values that were considered to be disturbed by the large optical depth layer (L5′, L6′, and L7′). The correlation between fine-mode particle concentration and extinction is clearly positive even if the values higher than 800m are neglected. However, correlation between coarse-mode particle concentration and extinction is still negative even though the values of L5′, L6′ and L7′ are neglected, and effect of laser beam attenuation is not completely avoided.

Our results suggest that the contribution of fine-mode particles (diameter smaller than 1.0 μm) to light extinction or the visibility is important not only near the ground, but also over the planetary boundary layer (or the free troposphere) in Beijing. The attenuation of lidar beam is caused in dusty atmosphere, and consequently large underestimation of extinction is made. This is important problem to be solved when we want to estimate slant visibility from lidar return. To clarify this problem technically, future in situ measurements such as balloon-born OPCs are suggested. Here, we estimated extinction only at wavelength of 532 nm. Observations at other wavelengths would also be useful to discuss the relation between extinction and aerosol concentration.

## 4. SUMMARY

We performed tethered-balloon observations twice in spring (on dusty and clear days) in Beijing, China. Vertical profiles of size-segregated particle concentration were obtained with the OPC. To understand the relationship between the visibility and the particulate matter suspended in the urban atmosphere, the vertical profiles of the OPC datasets and the corresponding lidar data were compared. The present observations reveal that the concentration of fine-mode particles (diameter <1.0 μm) was positively correlated with the extinction. Consequently, fine-mode particles strongly contribute to light extinction and visibility in the boundary layer over Beijing, even under different weather conditions. However, the extinction deduced from the lidar return is certainly underestimated under the dusty atmospheric condition and correspondence between particle concentration and extinction is strongly disturbed, especially in size ranges of coarse particles due to strong lidar return attenuation.

The balloon-borne measurements showed that the aerosol layer was composed of some sub-layers. The sub-layers sometimes contained apparently higher content of particles than the value near the ground in dusty day, and therefore it will be hard to estimate slant visibility only from the lidar return data since intensity of laser pulse rapidly decreases during penetration in such high concentration particle layer. In situ measurements will be effective to correct underestimation in lidar return intensity, and the results presented here are only case study and much more observations are desired in future.

## Acknowledgments

The work was supported by China’s 973 Plan 2006-CB403705 and The Ministry of Science and Technology of China (MOST) project 2009DFA2265. The authors are thankful to Dr. A. Shimizu and Dr. N. Sugimoto for preparing and processing the lidar data.

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### Fig. 1.

Observation sites. (A) Base of the atmospheric observation; (B) China-Japan Friendship Environmental Protection Centre.

Balloon train.

### Fig. 3.

Potential temperature (K) and relative humidity (R.H., %) plotted separately for the two observation days (Left: March 15, right: March 16). The black lines represent the potential temperature and the gray lines represent the relative humidity.

### Fig. 4.

Profiles of particle number concentrations (L-1) in different size ranges and extinction ratio (km-1) on March 15 (up) and 16 (down).

### Fig. 5.

Correlation between the extinction coefficients (km-1) and the particle number concentrations (L-1) in different size ranges in the case of 050315 (clear day).

### Fig. 6.

Correlation between extinction coefficients (km-1) and particle number concentration (L-1) in each size range in the case of 050316 (dusty day). The dotted lines (L1′ to L4′) show the correlation between the extinction and number concentration, neglecting data points L5′, L6′ and L7′.

### Table 1.

Surface meteorological records and observing time.

Time Items
Temperature (°C) Relative humidity (%) Wind speed (m/s) Wind direction (deg.) Cloud (0-10) (Total/Low) Visibility (km) Weather records
2005/03/15
16:30-18:30
14.1 18 2.5 SSW 0/0 20.0 Clear
2005/03/16
16:30-18:30
14.0 27 1.6 SSW 10/0 9.0 Smoke screen


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# 9.31 GroebnerFactorizationPackage¶

Solving systems of polynomial equations with the Gröbner basis algorithm can often be very time consuming because, in general, the algorithm has exponential run-time. These systems, which often come from concrete applications, frequently have symmetries which are not taken advantage of by the algorithm. However, it often happens in this case that the polynomials which occur during the Gröbner calculations are reducible. Since FriCAS has an excellent polynomial factorization algorithm, it is very natural to combine the Gröbner and factorization algorithms.

GroebnerFactorizationPackage exports the groebnerFactorizegroebnerFactorizeGroebnerFactorizationPackage operation which implements a modified Gröbner basis algorithm. In this algorithm, each polynomial that is to be put into the partial list of the basis is first factored. The remaining calculation is split into as many parts as there are irreducible factors. Call these factors p1,…,pn. In the branches corresponding to p2,…,pn, the factor p1 can be divided out, and so on. This package also contains operations that allow you to specify the polynomials that are not zero on the common roots of the final Gröbner basis.

Here is an example from chemistry. In a theoretical model of cyclohexane, C6H12, the six carbon atoms each sit in the center of gravity of a tetrahedron that has two hydrogen atoms and two carbon atoms at its corners. We first normalize and set the length of each edge to 1. Hence, the distances of one fixed carbon atom to each of its immediate neighbours is 1. We will denote the distances to the other three carbon atoms by x, y and z.

A. Dress developed a theory to decide whether a set of points and distances between them can be realized in an n-dimensional space. Here, of course, we have n=3.

mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [ [0,1,1,1,1,1],

[1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0] ]

 [01111110183x83110183y183101831x83101183y8310]

Type: SquareMatrix(6,DistributedMultivariatePolynomial([x,y,z],Fraction Integer))

For cyclohexane the distances have to satisfy this equation.

eq := determinant mfzn

 -x2y2+223x2y-259x2+223xy2-3889xy-25027x-259y2-25027y+1457581

Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)

They also must satisfy the equations given by cyclic shifts of the indeterminates.

groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z],

[z,x,y])]

 [[xy+xz-223x+yz-223y-223z+1213,xz2-223xz+259x+yz2-223yz+259y-223z2+3889z+25027,y2z2-223y2z+259y2-223yz2+3889yz+25027y+259z2+25027z-1457581],[x+y-219945625,y2-219945625y+4427675,z-46387],[x2-12xz-112x-56z+26518,y-z,z2-383z+2659],[x-259,y-113,z-113],[x-113,y-113,z-113],[x+53,y+53,z+53],[x-193,y+53,z+53]]

Type: List List DistributedMultivariatePolynomial([x,y,z],Fraction Integer)

The union of the solutions of this list is the solution of our original problem. If we impose positivity conditions, we get two relevant ideals. One ideal is zero-dimensional, namely x=y=z=11/3, and this determines the boat form of cyclohexane. The other ideal is one-dimensional, which means that we have a solution space given by one parameter. This gives the chair form of cyclohexane. The parameter describes the angle of the back of the chair.

groebnerFactorizegroebnerFactorizeGroebnerFactorizationPackage has an optional Boolean-valued second argument. When it is true partial results are displayed, since it may happen that the calculation does not terminate in a reasonable time. See the source code for GroebnerFactorizationPackage in groebf.input for more details about the algorithms used.

format: plain


text:
# Layman's doubts about Gen Relativity

I already did. Go back and re-read a little more closely. Don't hesitate to ask questions about the parts you either don't understand or feel are incomplete. I glossed over some parts, like the derivation of the equation of the null geodesics.
You seem to still suffer not understanding even the fact that there is no such thing as
$$\frac{\zeta^{\nu}}{d\lambda}$$

in any part of mathematics; let alone physics.

I already did. Go back and re-read a little more closely. Don't hesitate to ask questions about the parts you either don't understand or feel are incomplete. I glossed over some parts, like the derivation of the equation of the null geodesics.
Where? You're making no sense; sorry! On one hand you said you believe that Fermi coordinates have a specific observer like Rindler coordinates. On the other hand you're saying you did prove that a non-Rindler observer sees the same result as Rindler observers do! (while you didn't as can be seen from you setting a consatnt $$x$$ for your emitter and receiver!) How can you explain this contradiction? More interestingly unreasonable is your leaky understanding of what a Rindler observer is:

Obviously. I thought that was clear from the fact that the emitter is on the ceiling and the detector is on the floor of an accelerating elevator. Most elevator designers try to avoid situations where the floor accelerates without the ceiling accelerating also.
Three points must be made here:

1- Looking at the acceleration vector of a Rindler observer,

$$\nabla_{\arrow{e}_0} \arrow{e}_0=\frac{1}{x}\frac{\partial}{\partial x}$$

it is obvious that these are accelerating with constant magnitude in the direction of $$\frac{\partial}{\partial x},$$ suggesting $$x=const.$$

2- These observers have in Minkowski coordinates hyperbolic world lines with the same asymptotes $$T=+X$$ and $$T=-X$$; the first coincides with the world line of a photon moving along the $$X$$-axis in the same direction as the accelerated observer (particle). Such observers are indeed uniformly accelerated that altogether form a Rindler reference frame within this frame physics is exactly the same as when the spacetime is analized using Minkowski coordinates. If the observer is no longer uniformly accelerating, then it is not called Rindler to whom according to the equivalence principle, physics only looks the same in a very small region. This is because the physical laws in a local reference frame in a gravitational field are equivalent to the physical laws in a uniformly accelerated frame.

3- The "time translation symmetry" property of Rindler metric only holds true for Rindler observers (look at the coframe field of Rindler metric and its first component i.e. starting with$$\sigma^0=-xt$$ one would lead to $$d\sigma^0=-xdt$$ only iff $$x=const.$$) which itself suggests that "physics as a whole has not changed" holds true in this case if this property resembles a corresponding property in Minkowski spacetime (basically "boost symmetry") that is just satisfied for Rindler observers.

Now if you got my point that a Rindler observer appears to be only uniformly accelerating, then stop making jerry-built claims that your example applies to non-Rindler observers as well. If you do not have prior studies on a physical problem, you're not forced to launch yourself into the related discussions.

Altabeh, I notice that you are still unable to demonstrate any experimental measurement which depends on the choice of Rindler or Minkowski coordinates.
I never said such nonsense. Either you're 1) escaping from the first fallacy you made here concerning OP's question which Weinberg and Papaterou corrected it or 2) overshadowing the second fallacy that straight line in GR is given by the transfomed formula of the straight line in the Euclidean space or 3) digressing from the main problem by confusing the Kruskal example with Rindler and making some fallacious claim concerning the latter.

I precisely stated everything above and it all falls upon your mind to be able to get what is going on in physics in this case.

AB

Last edited:
Dale
Mentor
there is no such thing as
$$\frac{\zeta^{\nu}}{d\lambda}$$
Oops, you are correct, I did make a typo. It should of course read:
$$\frac{d\zeta^{\nu}}{d\lambda}$$
Unfortunately I can no longer go back and correct the original.

On one hand you said you believe that Fermi coordinates have a specific observer like Rindler coordinates.
Yes.

On the other hand you're saying you did prove that a non-Rindler observer sees the same result as Rindler observers do!
Yes. I worked the same problem in Rindler coordinates for a Rindler observer and in Minkowski coordinates for a Minkowski observer and they agreed on the result.

DaleSpam said:
you are still unable to demonstrate any experimental measurement which depends on the choice of Rindler or Minkowski coordinates
I never said such nonsense.
Are you now agreeing with me that the measured result of any physical experiment does not depend on the choice of coordinate system used to analyze the experiment?

Assuming that is correct then let me state the following and see if you agree:

1) The result of any physical measurement does not depend on the coordinate system used to analyze the experiment

2) Experimentally, an inertial object is one where an attached accelerometer measures 0

3) By combining 1 and 2 all coordinate systems must agree if an accelerometer reads 0 and therefore all coordinate systems agree if an object is inertial

Last edited:
atyy
I think once again there is just a slight difference in terminology.

Altabeh's point is that a Lorentz inertial frame and a Rindler frame can each be associated with an observer. Even if these two observers are colocated and comoving, some of their measurements will differ (eg. frequency, due to Doppler effects).

Of course the difference between the measurements of the Lorentz inertial observer and Rindler observer will be the same in any coordinate system, since both observers exist in all coordinate systems, which is DaleSpam's point.

And I think we have all agreed that a freely falling observer has zero proper acceleration (following eg. Rindler and Wald's definition) along its entire worldline. With this observer, we can associate Fermi normal coordinates in which the Christoffel symbols are zero along his entire worldline, and so can define the freely falling observer to be "inertial" in a sense.

We also all agree that Fermi normal coordinates are only locally inertial, and that the Christoffel symbols are not zero once they are off the free falling wordline, and in fact even the worldline the second derivatives of the Christoffel symbols are not zero, so there is no global inertial frame associated with a freely falling observer, which is the answer to the third question of the OP, which was Altabeh's point in post #6.

?

Last edited:
Dale
Mentor
there is no global inertial frame associated with a freely falling observer, which is the answer to the third question of the OP, which was Altabeh's point in post #6.
The OP later clarified that he was interested in the "inertialness" of the object itself, not the local coordinate system.

atyy
The OP later clarified that he was interested in the "inertialness" of the object itself, not the local coordinate system.
3)When looked upon from Earth does a falling stone makes non-inertial frame of reference?
Is it safe now to tell that a very small particle in free fall, undergoing translational acceleration (in the kinematic sense) over a very long distance is actually an inertial frame for people on a perfectly still Earth?
This means that an accelerometer at rest gives the value of g at that point and a falling accelerometer records zero acceleration. I think it is a better and intuitive answer to my question.
The accelerometer mechanism is actually functioning in the frame of reference of the object itself. So, its zero reading doesn't shows us that it was in inertial motion (for us).

I've concluded that Altabeh's argument was right. The local flatness theorem holds only for infitesimal regions in space-time curvature and due to the long trajectory of the particle, it is non-inertial from outside. The continuous photon blueshifting/redshifting and apparent modification in c are some phenomenon that exhibit this thing.
The OP's question was a quantum superposition, which as we all know is not relativistically invariant, and hence observer dependent :tongue2:

George Jones
Staff Emeritus
Gold Member
We also all agree that Fermi normal coordinates are only locally inertial
Did you mean to write "We also all agree that for a freely falling observer, Fermi normal coordinates are only locally inertial"?

Rindler coordinates are Fermi normal coordinates.

atyy
Did you mean to write "We also all agree that for a freely falling observer, Fermi normal coordinates are only locally inertial"?

Rindler coordinates are Fermi normal coordinates.
Yes - thanks!

Yes. I worked the same problem in Rindler coordinates for a Rindler observer and in Minkowski coordinates for a Minkowski observer and they agreed on the result.
Thank goodness, you ultimately corrected yourself because I asked you

Could you now prove that a non-Rindler observer would also lead to the same result as in your example?

This shows you're now agreeing that Physics only in the eyes of the specific observer of Rindler coordinates doesn't change.

Are you now agreeing with me that the measured result of any physical experiment does not depend on the choice of coordinate system used to analyze the experiment?
I'm not "now" agreeing with you! This is all because you came to a "right" conclusion above as to what I was trying to say all along since the beginning of this thread. I gave away an example about Schwarzschild coordinates and said that is not "always" believable to say something like this and this had nothing to do with Rindler coordinates or any other experimentally proven result of the coordinate-independence property of any physical phenomena. As I already said, this could be probably true.

1) The result of any physical measurement does not depend on the coordinate system used to analyze the experiment
Do you remember I said there is a global transformation to bring Minkowski into Rindler form? From all observers using this transformation, only to those who are called Rindler the result of any physical measurement does not seem to depend on the coordinate system (Minkowski or Rindler.) That is to say, all observers uniformly accelerating with respect to each other agree on the result that Minkowski observers have measured in their own frame, as your example correctly shows this. The same thing is applicable in case of using Fermi coordinates.

2) Experimentally, an inertial object is one where an attached accelerometer measures 0
Yes but this is not globally possible in GR.

3) By combining 1 and 2 all coordinate systems must agree if an accelerometer reads 0 and therefore all coordinate systems agree if an object is inertial
I think atty responded to this in an awesome way:

Altabeh's point is that a Lorentz inertial frame and a Rindler frame can each be associated with an observer. Even if these two observers are colocated and comoving, some of their measurements will differ (eg. frequency, due to Doppler effects).

And I think we have all agreed that a freely falling observer has zero proper acceleration (following eg. Rindler and Wald's definition) along its entire worldline. With this observer, we can associate Fermi normal coordinates in which the Christoffel symbols are zero along his entire worldline, and so can define the freely falling observer to be "inertial" in a sense.
The OP later clarified that he was interested in the "inertialness" of the object itself, not the local coordinate system.
The OP clearly said in an early post he felt like my answer to his question was right. There was no talk of inertialness, I think.

I think we are now in a position to reach a middle ground on the issue. If no, please let me know.

AB


text:
# Numeric Input¶

Using the mouse for transformations is convenient, but if you require more precise control, you can also enter numeric values. After pressing the shortcut type a number to indicate the magnitude of the transformation. Then confirm or cancel. E.g. pressing S 2, Return will double the scale of an object.

Move G

By default and with no other key presses, the translation will occur along the X axis.

Rotation R

The rotation is in clockwise direction for positive values.

Scale S

Scaling works in almost identical fashion to translation. The primary difference is that by default, scaling applies equally to all three axes.

You can see the numbers you enter in the 3D Viewport footer.

Numeric input displayed in the footer.

Tipp

Numeric input can also be inputed in the Properties region.

## Simple Mode¶

Blender has two „modes“ a simple and an advanced one. Simple mode only accepts simple numbers. You can use basic text editing except selection.

Decimals Period

Decimals can be entered by pressing Period.

Negate Minus

Negate the whole value by pressing Minus.

Inverse Slash

Hitting Slash during number entry switches the number being entered to its reciprocal, e.g. 2 / results in 0.5 (1/2); 20 / results in 0.05 (1/20).

Reset Backspace

Hitting Backspace after having deleted all leading chars will first reset the edited value to initial state, and on second press, the whole number editing will be canceled, going back to usual transform with mouse.

Next/Previous Component Tab, Ctrl-Tab

To enter numeric values for multiple axes, use Tab or Ctrl-Tab. E.g. To move an object, one unit on all three axes press: G 1 and Tab 1 and Tab 1.

Non-number Inputs

You can also combine numeric input with Axis Locking to limit movement to a particular axis or tool specific shortcuts.

Use = or NumpadAsterisk to enable advanced mode, and Ctrl-= or Ctrl-NumpadAsterisk to switch back to simple mode.

It features:

• Units (cm, ", deg, etc.). See unit system.

• Basic operations from Python (+, *, /, **, etc.).

• Math constants and functions (pi, sin, sqrt, etc.). See Python’s math module.

You can still use the negate and inverse shortcuts (Minus, Slash), as well as non-number inputs, but you have to hold Ctrl to activate them.


text:
# Show that a certain set of positive real numbers must be finite or countable

Let $B$ be a set of positive real numbers with the property that adding together any finite subset of elements from $B$ always gives a sum of $2$ or less. Show that $B$ must be finite or at most countable.

$B$ = {$x \in R:x>0\}$, $x_1,x_2...x_n \in B$ such that $x_1+x_2+...+x_n \le 2$.

Question: for any $a,b$ $(a,b)$~$R$, but $B$ is $(0,+\infty)$ so why $B$ is not uncountable (taking as $a = 0$, and letting $b$->$\infty$)?

And why for $B$ being countable doesn't contradict: for any $a,b$ $(a,b)$~$R$?

P.S. I read Showing a set is finite or countable and understood it.

• It sounds like you are confused about what the question is asking. It is not saying that B is the set of positive real numbers; it is a set of positive real numbers. This means that B is a subset of the set of all positive real numbers. Aug 8 '18 at 17:23
• No. B is some unspecified subset of the positive real numbers such that if you add up a finite number of the elements of B the sum is always less than 2. For example, B could be {0.1, 0.2, 0.5, 1.1} or B could be the infinite set {1, 1/2, 1/4, 1/8, 1/16, ...}. Note that B cannot be {1, 1.4, 1.8} and B cannot be {1, 1/2, 1/3, 1/4, ...} Aug 8 '18 at 17:47
• B does not have to be an interval (0,a). In fact, it cannot be such an interval. Your job is to prove that B must be finite or countable. Intervals are uncountable. Aug 8 '18 at 17:49
• Well, I'm just trying to help clarify the question. I haven't given a solution. I think the hints and discussion below do that for you. But the point is you cannot assume anything about B except: 1. its elements are all positive real numbers and 2. if you take a finite number of elements from B and add them up then you get a sum less than two. Given any such B you have to prove that B is either finite or countable. Aug 8 '18 at 17:59
• Here is a proof that B cannot be the open interval (0,1). If it were then these numbers would be in B: 0.5, 0.51, 0.52, and 0.53. These numbers add up to more than 2. Do you see? You could do something similar to show that B cannot be (0,a) for any positive a. But you have to show more than that. You have to show B is finite or countable. Aug 8 '18 at 18:00

Hint 1: How many elements of $B$ can be in the set $[2,\infty)$?

Hint 2: How many elements of $B$ can be in the set $[1,2)$?

Hint 3: How many elements of $B$ can be in the set $[0.5,1)$?

• At this rate you are going to need countably many hints :) Aug 8 '18 at 17:04
• @ArnaudMortier Can't he give a hint schema? Aug 8 '18 at 17:07
• 1) One $x = 2$, 2) set $x_1=1$ so one for sure, then I don't know how many $x_2,x_3,...x_n$ to choose so that they sum up to $1$. 3) The same, I don't know. Aug 8 '18 at 17:10
• Obviously $B\cap[1,\infty)$ has at most two elements, because if $x,y,z\ge1$ then $x+y+z>2$. Similarly $B\cap[2/n,\infty)$ has at most $n$ elements. (So $B$ is the union of countably many finite sets...) Aug 8 '18 at 17:17
• No, we certainly can't write $B$ as $(0,a)$. If $B=(0,a)$ then there are finite subsets of $B$ with sum larger than $2$. Aug 8 '18 at 18:56

Not only can we show that it's countable, it's not too difficult to construct an enumeration: given an element $b$, just count how many other elements of $B$ are larger. This has to be a finite integer, since if there were an infinite number of elements greater than $b$, then the sum of $n$ such elements would be greater than $nb$, so picking an $n>2/b$ would give a sum greater than 2.

For any $a\in B$, define $B_a\equiv\{b\in B:a<b\}$. The cardinality of $B_a$ must be less than $\lceil2/a\rceil$, otherwise the sum of any $\lceil2/a\rceil$ elements of $B_a$ would exceed 2. That is, $\sum_{i=1}^{\lceil2/a\rceil}x_i>\sum_{i=1}^{\lceil2/a\rceil}a=a\lceil2/a\rceil\ge2$ for any sequence $x_i$ in $B_a$, contradicting the requirements on $B$.

Define the function $f:B\rightarrow\mathbb{N}$ as $f(a)=\left|B_a\right|$.

Homework: If you can show that $f$ is one-to-one then you can conclude that $|B|\le|\mathbb{N}|$. That is, $B$ is either finite or countable.

Credit goes to Acccumulation for the outline of this proof.

• What if $a =$ sup$B$? Then $B_a$ is the empty set. If $B_a \equiv \{b \in B: a \leq b \}$, then the sum of any $\lceil r/a \rceil$ for $r > 2$ elements of $B_a$ would exceed 2. Jul 18 '19 at 23:08


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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 4.3.4.11. Let $X$ and $Y$ be simplicial sets. If either $X$ or $Y$ is finite, then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ of Construction 4.3.4.9 is a homeomorphism.

Proof. Without loss of generality, we may assume that $X$ is finite. Then the geometric realization $|X|$ is a compact Hausdorff space (Corollary 3.5.1.11). Using Remarks 4.3.4.6 and 4.3.3.29, we see that the functors

$\operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}_{|X|/} \quad \quad Y \mapsto |X \star Y|, Y \mapsto |X| \star |Y|$

preserve colimits. Consequently, if we regard $X$ as fixed, then the collection of simplicial sets $Y$ for which $T_{X,Y}$ is a homeomorphism is closed under colimits. Since every simplicial set can be realized as a colimit of standard simplices (Corollary 1.1.8.17), it will suffice to prove Proposition 4.3.4.11 in the special case where $Y = \Delta ^{q}$ is a standard simplex. In this case, $Y$ is also finite. Repeating the preceding argument (with the roles of $X$ and $Y$ reversed), we are reduced to proving that $T_{X,Y}$ is a homeomorphism in the case where $X = \Delta ^{p}$ is also a standard simplex. In this case, the desired result follows from Example 4.3.4.10. $\square$


text:
# $- string interpolation (C# reference) The $ special character identifies a string literal as an interpolated string. An interpolated string is a string literal that might contain interpolation expressions. When an interpolated string is resolved to a result string, items with interpolation expressions are replaced by the string representations of the expression results. This feature is available starting with C# 6.

String interpolation provides a more readable, convenient syntax to format strings. It's easier to read than string composite formatting. Compare the following example that uses both features to produce the same output:

string name = "Mark";
var date = DateTime.Now;

// Composite formatting:
Console.WriteLine("Hello, {0}! Today is {1}, it's {2:HH:mm} now.", name, date.DayOfWeek, date);
// String interpolation:
Console.WriteLine($"Hello, {name}! Today is {date.DayOfWeek}, it's {date:HH:mm} now."); // Both calls produce the same output that is similar to: // Hello, Mark! Today is Wednesday, it's 19:40 now.  ## Structure of an interpolated string To identify a string literal as an interpolated string, prepend it with the $ symbol. You can't have any white space between the  and the " that starts a string literal. The structure of an item with an interpolation expression is as follows: {<interpolationExpression>[,<alignment>][:<formatString>]}  Elements in square brackets are optional. The following table describes each element: Element Description interpolationExpression The expression that produces a result to be formatted. String representation of null is String.Empty. alignment The constant expression whose value defines the minimum number of characters in the string representation of the expression result. If positive, the string representation is right-aligned; if negative, it's left-aligned. For more information, see Alignment Component. formatString A format string that is supported by the type of the expression result. For more information, see Format String Component. The following example uses optional formatting components described above: Console.WriteLine("|{"Left",-7}|{"Right",7}|");

const int FieldWidthRightAligned = 20;
Console.WriteLine($"{Math.PI,FieldWidthRightAligned} - default formatting of the pi number"); Console.WriteLine($"{Math.PI,FieldWidthRightAligned:F3} - display only three decimal digits of the pi number");
// Expected output is:
// |Left   |  Right|
//     3.14159265358979 - default formatting of the pi number
//                3.142 - display only three decimal digits of the pi number

Beginning with C# 10, you can use string interpolation to initialize a constant string. All expressions used for placeholders must be constant strings. In other words, every interpolation expression must be a string, and it must be a compile time constant.

Beginning with C# 11, the interpolated expressions can include newlines. The text between the { and } must be valid C#, therefore it can include newlines that improve readability. The following example shows how newlines can improve the readability of an expression involving pattern matching:

string message = $"The usage policy for {safetyScore} is { safetyScore switch { > 90 => "Unlimited usage", > 80 => "General usage, with daily safety check", > 70 => "Issues must be addressed within 1 week", > 50 => "Issues must be addressed within 1 day", _ => "Issues must be addressed before continued use", } }";  Also, beginning in C# 11, you can use a raw string literal for the format string: int X = 2; int Y = 3; var pointMessage =$"""The point "{X}, {Y}" is {Math.Sqrt(X * X + Y * Y)} from the origin""";

Console.WriteLine(pointMessage);
// output:  The point "2, 3" is 3.605551275463989 from the origin.

You can use multiple $ characters in an interpolated raw string literal to embed { and } characters in the output string without escaping them: int X = 2; int Y = 3; var pointMessage =$$"""The point {{{X}}, {{Y}}} is {{Math.Sqrt(X * X + Y * Y)}} from the origin"""; Console.WriteLine(pointMessage); // output: The point {2, 3} is 3.605551275463989 from the origin.  If your output string should contain repeated { or } characters, you can add more $ to designate the interpolated string. Any sequence of { or } shorter than the number or $ will be embedded in the output string. As shown in the preceding example, sequences longer than the sequence of $ characters embed the additional { or } characters in the output. The compiler issues an error if the sequence of brace characters is equal to or greater than double the length of the sequence of $ characters. You can try these features using the .NET 7 SDK. Or, if you have the .NET SDK 6.00.200 or later, you can set the <LangVersion> element in your csproj file to preview. ## Special characters To include a brace, "{" or "}", in the text produced by an interpolated string, use two braces, "{{" or "}}". For more information, see Escaping Braces. As the colon (":") has special meaning in an interpolation expression item, to use a conditional operator in an interpolation expression, enclose that expression in parentheses. The following example shows how to include a brace in a result string. It also shows how to use a conditional operator: string name = "Horace"; int age = 34; Console.WriteLine($"He asked, \"Is your name {name}?\", but didn't wait for a reply :-{{");
Console.WriteLine($"{name} is {age} year{(age == 1 ? "" : "s")} old."); // Expected output is: // He asked, "Is your name Horace?", but didn't wait for a reply :-{ // Horace is 34 years old.  An interpolated verbatim string starts with the $ character followed by the @ character. For more information about verbatim strings, see the string and verbatim identifier articles.

Note

Starting with C# 8.0, you can use the $ and @ tokens in any order: both $@"..." and @$"..." are valid interpolated verbatim strings. In earlier C# versions, the $ token must appear before the @ token.

## Implicit conversions and how to specify IFormatProvider implementation

There are three implicit conversions from an interpolated string:

1. Conversion of an interpolated string to a String instance. The string is the result of interpolated string resolution. All interpolation expression items are replaced with the properly formatted string representations of their results. This conversion uses the CurrentCulture to format expression results.

2. Conversion of an interpolated string to a FormattableString instance that represents a composite format string along with the expression results to be formatted. That allows you to create multiple result strings with culture-specific content from a single FormattableString instance. To do that, call one of the following methods:

The ToString(IFormatProvider) provides a user-defined implementation of the IFormatProvider interface that supports custom formatting. For more information, see the Custom formatting with ICustomFormatter section of the Formatting types in .NET article.

3. Conversion of an interpolated string to an IFormattable instance that also allows you to create multiple result strings with culture-specific content from a single IFormattable instance.

The following example uses implicit conversion to FormattableString to create culture-specific result strings:

double speedOfLight = 299792.458;
FormattableString message = $"The speed of light is {speedOfLight:N3} km/s."; System.Globalization.CultureInfo.CurrentCulture = System.Globalization.CultureInfo.GetCultureInfo("nl-NL"); string messageInCurrentCulture = message.ToString(); var specificCulture = System.Globalization.CultureInfo.GetCultureInfo("en-IN"); string messageInSpecificCulture = message.ToString(specificCulture); string messageInInvariantCulture = FormattableString.Invariant(message); Console.WriteLine($"{System.Globalization.CultureInfo.CurrentCulture,-10} {messageInCurrentCulture}");
Console.WriteLine($"{specificCulture,-10} {messageInSpecificCulture}"); Console.WriteLine($"{"Invariant",-10} {messageInInvariantCulture}");
// Expected output is:
// nl-NL      The speed of light is 299.792,458 km/s.
// en-IN      The speed of light is 2,99,792.458 km/s.
// Invariant  The speed of light is 299,792.458 km/s.

## Combining multiple interpolated strings

To combine multiple interpolated strings, use the interpolation expression for each:

var firstWord = "Hello";
var secondWord = "World";

var combinedInterpolatedString = $"{firstWord}, " +$"{secondWord}!";

Console.WriteLine(combinedInterpolatedString)
// output: "Hello, World!"

## Other resources

If you're new to string interpolation, see the String interpolation in C# interactive tutorial. You can also check another String interpolation in C# tutorial. That tutorial demonstrates how to use interpolated strings to produce formatted strings.

## Compilation of interpolated strings

If an interpolated string has the type string, it's typically transformed into a String.Format method call. The compiler may replace String.Format with String.Concat if the analyzed behavior would be equivalent to concatenation.

If an interpolated string has the type IFormattable or FormattableString, the compiler generates a call to the FormattableStringFactory.Create method.

Beginning with C# 10, when an interpolated string is used, the compiler checks if the interpolated string is assigned to a type that satisfies the interpolated string handler pattern. An interpolated string handler is a custom type that converts the interpolated string into a string. An interpolated string handler is an advanced scenario, typically used for performance reasons. You can learn about the requirements to build an interpolated string handler in the language specification for interpolated string improvements. You can build one following the interpolated string handler tutorial in the What's new in C# section. In .NET 6, when you use an interpolated string for an argument of type string, the interpolated string is processed by the System.Runtime.CompilerServices.DefaultInterpolatedStringHandler.

Note

One side effect of interpolated string handlers is that a custom handler, including System.Runtime.CompilerServices.DefaultInterpolatedStringHandler, may not evaluate all the expressions used as placeholders in the interpolated string under all conditions. That means side-effects in those expressions may not occur.

## C# language specification

For more information, see the Interpolated strings section of the C# language specification.


text:
# Bipyramid

Bipyramid
Set of bipyramids

(Example hexagonal form)
Faces 2n triangles
Edges 3n
Vertices n + 2
Face configuration V4.4.n
Symmetry group Dnh, [n,2], (*22n)
Dual polyhedron n-gonal prism
Properties convex, face-transitive
Net

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.

The face-transitive bipyramids are the dual polyhedra of the uniform prisms and will generally have isosceles triangle faces.

A bipyramid can be projected on a sphere or globe as n equally spaced lines of longitude going from pole to pole, and bisected by a line around the equator.

Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh.

## Volume

The volume of a bipyramid is $V = \frac{2}{3} Bh$ where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base.

The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore:

$V = \frac{n}{6}hs^2 \cot\frac{\pi}{n}$

## Equilateral triangle bipyramids

Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are equilateral triangles, and thus the bipyramid is a deltahedron): the triangular, tetragonal, and pentagonal bipyramids. The tetragonal bipyramid with identical edges, or regular octahedron, counts among the Platonic solids, while the triangular and pentagonal bipyramids with identical edges count among the Johnson solids (J12 and J13).

 Triangular bipyramid Square bipyramid (Octahedron) Pentagonal bipyramid

## Forms

1. Triangular bipyramid - 6 faces - dual triangular prism
2. Square bipyramid (the regular octahedron is a special case) - 8 faces - dual cube
3. Pentagonal bipyramid - 10 faces - dual pentagonal prism
4. Hexagonal bipyramid - 12 faces - dual hexagonal prism
5. Heptagonal bipyramid - 14 faces - dual heptagonal prism
6. Octagonal bipyramid - 16 faces - dual octagonal prism
7. Enneagonal bipyramid - 18 faces - dual enneagonal prism
8. Decagonal bipyramid - 20 faces - dual decagonal prism
• ...n-gonal bipyramid - 2n faces - dual n-gonal prism
 3 4 5 6 8 10

## Symmetry groups

If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the n-agonal bipyramid has dihedral symmetry Dnh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group Oh of order 48, which has three versions of D4h as subgroups. The rotation group is Dn of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has three versions of D4 as subgroups.

## Star bipyramids

Self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points.

For example, a pentagrammic dipyramid is an isohedral star polyhedron composed of 10 intersecting isoceles triangles. It is the dual to the pentagrammic prism.

## Higher dimensions

In general, a bipyramid can be seen as an n-polytope constructed with a (n-1)-polytope in a hyperplane with two points in opposite directions, equal distance perpendicular from the hyperplane. If the (n-1)-polytope is a regular polytope, it will have identical pyramids facets.

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### Look at other dictionaries:

• bipyramid — (ˈ)bī+ noun Etymology: bi (I) + pyramid : dipyramid * * * bipyramidal /buy pi ram i dl/, adj. /buy pir euh mid/, n. Crystall. a form symmetrical about a plane dividing it into two pyramids. Also called dipyramid …   Useful english dictionary

• bipyramid — noun Date: 1897 a crystal consisting of two identical pyramids base to base • bipyramidal adjective …   New Collegiate Dictionary

• bipyramid — bipyramidal /buy pi ram i dl/, adj. /buy pir euh mid/, n. Crystall. a form symmetrical about a plane dividing it into two pyramids. Also called dipyramid. [BI 1 + PYRAMID] * * * …   Universalium

• bipyramid — noun A dipyramid …   Wiktionary

• bipyramid — bi·pyramid …   English syllables

• Hexagonal bipyramid — Type bipyramid Faces 12 triangles Edges 18 Vertices 8 Face configu …   Wikipedia

• Octagonal bipyramid — Type bipyramid Faces 16 triangles Edges 24 Vertices 10 Face configurat …   Wikipedia

• Decagonal bipyramid — Type bipyramid Faces 20 triangles Edges 30 Vertices 12 Face configurat …   Wikipedia

• Pentagonal bipyramid molecular geometry — In chemistry a pentagonal bipyramid (or dipyramid) is a molecular geometry with one atom at the centre with seven ligands at the corners of a pentagonal dipyramid. A perfect pentagonal bipyramid belongs to the molecular point group D5h . The… …   Wikipedia

• Johnson solid — The elongated square gyrobicupola (J37), a Johnson solid …   Wikipedia


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# Union of conjugacy classes of $O(n)$ is not a subgroup

Let $O(n)$ be the standard orthogonal group of real matrices. I am trying to prove the following:

$N = \bigcup_{g\in GL_n(\mathbb{R})}g\cdot O(n)\cdot g^{-1}$ is not a subgroup of $GL_n(\mathbb{R})$.

I know that if it was a subgroup then it was equal to the normal closure of $O(n)$ but I do not know what that is...

Motivation:

It is proved here that a linear automorphism $T:V \rightarrow V$ preserves some inner product on $V$ if and only if the matrix of $T$ w.r.t an arbitrary basis is similar to an orthogonal matrix. I want to prove a composition of two transformation of this type is not necessarily also of that type. (Which amounts to proving $N$ is not a subgroup, since closure under taking inverses clearly holds).

Note that $N$ is the set of real diagonalizable matrices that only have (complex) eigenvalues of magnitude $1$.
In the case of $n =2$: let $$A = \pmatrix{0&2\\-1/2&0}$$ Note that both $A$ and $A^T$ are elements of $N$. However, the matrix $AA^T$ has eigenvalues $4,1/4$, which are not of magnitude $1$.
You can generalize this counterexample by considering the block-matrix $$\pmatrix{A & 0\\0 & I_{n-2}}$$ Note that this counterexample still works if you restrict to the special orthogonal group.
• In fact: for $A \in N$, $AA^T$ (or $A^TA$) are elements of $N$ if and only if $A$ is normal (i.e. $AA^T = A^TA$). – Ben Grossmann Aug 12 '15 at 13:38


text:
# Difference between revisions of "0LAUK0 2015 01 Simulation"

 Home Content Plan Report Simulation Log Week1 Week2 Week3 Week4 Week5 Week6 Week7

The simulation test several different scheduling strategies and validate the outcome by evaluating the general happiness.

## Simulator

Discrete Event Simulation based on React.NET framework.

Source code is available on GitHub under GPL V3 license.

Compiled binary is available from dropbox

Hints: Use mouse wheel to switch between charts and click on bars will show detailed data.

## Simulator Setup

The simulator consists of 2 parts, logic and GUI.

#### Logic

The logic is implemented upon Discrete Event Simulation Frame by using React.NET and compiled into a dll for later integration.

The simulation logic has 3 process running.

Simulator
The simulator process is responsible for spawn all other processes and act as environment by adding passengers to bus stops.
Passenger
The passenger process is responsible for counting the on/off time.
Bus
The bus process is responsible to drive alone the destination list provided by the scheduler.

The scheduler is an interface which will be called by the Bus process each iteration for updating the destination list.

#### GUI

The GUI is written in C# using WPF for fast demonstration purpose. The data is displayed by using Modern UI Charts library.

#### Assumption and Approximations

There are three factors we used to calculate the general satisfaction.

Waiting time
The weight is 8.05.
Crowdeness
The weight is 7.97.
Travel time
The weight is 7.29.

These weight are taken from the survey result of people who take bus often.

The crowdness of each passenger is normalized to 0.0 - 1.0 by evaluating $\displaystyle{ \frac {Passenger Number}{Max Passenger Number} }$


text:
Contents

# Contents

## Idea

A judgement on the left of the “$\vdash$”-symbol of a sequent/hypothetical judgement.

antecedents $\vdash$ succedents

antecedents $\vdash$ consequent

Last revised on September 26, 2012 at 14:12:53. See the history of this page for a list of all contributions to it.