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Basic properties of Fermi blazars and the “blazar sequence”

# Basic properties of Fermi blazars and the “blazar sequence”

Dingrong Xiong, Xiong Zhang, Jinming Bai and Haojing Zhang
National Astronomical Observatories/Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China
Department of Physics, Yunnan Normal University, Kunming 650500, China
E-mail: ynzx@yeah.net
###### Abstract

By statistically analyzing a large sample which includes blazars of Fermi detection (FBs) and non-Fermi detection (NFBs), we find that there are significant differences between FBs and NFBs for redshift, black hole mass, jet kinetic power from “cavity” power, broad-line luminosity, and ratio of core luminosity to absolute V-band magnitude (), but not for ratio of radio core to extended flux () and Eddington ratio. Compared with NFBs, FBs have larger mean jet power, and while smaller mean redshift, black hole mass, broad-line luminosity. These results support that the beaming effect is main reason for differences between FBs and NFBs, and that FBs are likely to have a more powerful jet. For both Fermi and non-Fermi blazars, there are significant correlations between jet power and the accretion rate (traced by the broad-emission-lines luminosity), between jet power and black hole mass; for Fermi blazars, the black hole mass does not have significant influence on jet power while for non-Fermi blazars, both accretion rate and black hole mass have contributions to the jet power. Our results support the “blazar sequence” and show that synchrotron peak frequency () is associated with accretion rate but not with black hole mass.

###### keywords:
radiation mechanisms: nonthermal – galaxies: active – BL Lacertae objects: general – quasars: general – gamma-rays: theory – X-rays: general
pagerange: Basic properties of Fermi blazars and the “blazar sequence”LABEL:lastpagepubyear: 2002

## 1 Introduction

Blazars are often divided into two subclasses of BL Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs). FSRQs have strong emission lines, while BL Lacs have only very weak or non-existent emission lines. The classic division between FSRQs and BL Lacs is mainly based on the equivalent width (EW) of the emission lines. Objects with rest frame EW Å are classified as FSRQs (e.g. Scarpa & Falomo 1997; Urry & Padovani 1995). Many authors have proposed that EW alone is not a good indicator of the distinction between the two classes of blazars (Scarpa & Falomo 1997; Ghisellini et al. 2011; Sbarrato et al. 2012, 2014; Giommi et al. 2012, 2013; XZ14). Ghisellini et al. (2011) introduced a physical distinction between the two classes of blazars, based on the luminosity of the broad line region measured in Eddington units. The dividing line is of the order of . The result also was confirmed by Sbarrato et al. (2012) and XZ14. Giommi et al. (2012, 2013) suggested that blazars should be divided in high and low ionization sources.

In this paper, we constructed a large sample of blazars, including Fermi blazars from XZ14 and non-Fermi blazars, and studied the properties of Fermi blazars and the blazar sequence. The paper is structured as follows: in Sect. 2, we present the samples; the results are presented in Sect. 3; discussions and conclusions are presented in Sect. 4. The cosmological parameters , and have been adopted in this work. The energy spectral index is defined such that .

## 2 The samples

The selection criteria for the sample were that we tried to select the largest group of blazars included in BZCAT (Massaro et al. 2009: the Roma BZCAT) with reliable broad line luminosity (used as a proxy for disk luminosity), redshift, black hole mass and jet kinetic power. The sample of Fermi blazars was directly from XZ14. Due to be classed into non-clean 2LAC, the four fermi blazars (2FGL J0204.0+3045; 2FGL J0656.2-0320; 2FGL J1830.1+0617; 2FGL J2356.3+0432) in XZ14 were not included in our sample. In order to have reliable sample, we did not consider the candidate blazars of unknown type (BZU called in BZCAT) in our sample. Also for the same reason, non-Fermi blazars, which were detected by EGRET or recorded in 1LAC but missed in 2LAC, were not included in our sample. In addition, to reduce the uncertainty, we tried to select the data from a same paper and/or a uniform method. The detailed information and calculating methods for broad line luminosity, black hole mass are seen in XZ14. Cavagnolo et al. (2010) searched for X-ray cavities in different systems including giant elliptical galaxies and cD galaxies and estimated the jet power required to inflate these cavities or bubbles, obtaining a tight correlation between the “cavity” power and the radio luminosity at 200-400 MHz

which is continuous over decades in and and . Making use of the correlation between and from Cavagnolo et al. (2010), Meyer et al. (2011) chose the low-frequency extended luminosity at 300 MHz as an estimator of the jet power for blazars. Their 300 MHz extended luminosity was extrapolated from 1.4 GHz extended radio emission or obtained from spectral decomposition. Following Meyer et al. (2011), Nemmen et al. (2012) estimated the jet kinetic power for a large sample of Fermi blazars. We also used Equation (1) to get jet kinetic power from “cavity” power (this is what we mean when we refer to the jet kinetic power in the rest of the paper). In our sample, in order to reduce uncertainty, the jet kinetic power only is gained from extended 1.4 GHz radio data but not from spectral decomposition. The extended 1.4 GHz radio data can not be obtained for all blazars. The proportion of blazars of jet kinetic power estimated from extended radio luminosity in all sources was 28%. From Wang et al. (2004), the uncertainty in the relation was dex; the uncertainty on the zero point of the line width-luminosity-mass relation was approximately 0.5 dex; the correlation for quasar host galaxies had an uncertainty of 0.6 dex. So given the intrinsic uncertainty of the different black hole mass estimators and the heterogeneity of sample, we estimated that the individual BH masses may have an uncertainty as large as 1 dex. The uncertainty in was dominated by the scatter in the correlation of Cavagnolo et al. (2010) and corresponded to 0.7 dex. The uncertainties on broad line luminosity were based on the standard deviation of Mg II/Ly, H/Ly and C IV/Ly in the composite quasar spectrum of Francis et al. (1991) (Wang et al. 2004). We assumed that the uncertainties on broad line luminosity were to be 0.5 dex for all sources.

For the sake of exploring the blazar sequence, we also collected and/or calculated the synchrotron peak frequency and the peak luminosity of the synchrotron component . The and of our Fermi blazars were collected from Finke (2013) and Meyer et al. (2011), and the and of non-Fermi blazar from Nieppola et al. (2006, 2008), Meyer et al. (2011), Wu et al. (2009), Aatrokoski et al. (2011). Generally, the SED was fitted by using a simple third-degree polynomial function. However, many blazars were lack of observed SED. Abdo et al. (2010c) have conducted a detailed investigation of the broadband spectral properties of the -ray selected blazars of the Fermi LAT Bright AGN Sample (LBAS). They assembled high-quality and quasi-simultaneous SED for 48 LBAS blazars, and their results have been used to derive empirical relationships that estimate the position of the two peaks from the broadband colors (i.e. the radio to optical, , and optical to X-ray, , spectral slopes) and from the -ray spectral index. Ackermann et al. (2011) used the empirical relationships for finding the peak frequency of synchrotron component of 2LAC clean blazars from the slopes between the 5 GHz and 5500 Å flux, and between the 5500 Å and 1 KeV flux. Finke (2013) used their results for finding . The rest of authors fitted SED to obtain . When blazars were missed in the above literatures, we used the empirical relationships of Abdo et al. (2010c) to find . Firstly, we collected the fluxes of the blazars at 5 GHz, 5500 Å and 1 KeV from BZCAT, NASA/IPAC Extragalactic Database: NED, Veron-Cetty & Veron (2010) and Ackermann et al. (2011). When more than one flux or magnitude was found, we took the most recent one. Apparent magnitude of optical V band can be converted into flux as with flux KJy and in units of KJy (Mead et al. 1990). All flux densities were K-corrected according to , where was the spectral index and , , and for BL Lacs, and for FSRQs (Cheng et al. 2000). The luminosity was calculated from the relation , and was the luminosity distance. The 1 KeV flux density was transferred from the 0.1-2.4 KeV flux density given in the BZCAT or NED using . After then we used the Equation (1) of Abdo et al. (2010c) to get and . Finally, we used the empirical relationships of Abdo et al. (2010c) for finding the peak frequency of synchrotron component. For blazars without X-ray flux, we adopted to estimate ( from linear regression analysis: ). The proportion of blazars (that the is estimated by the linear regression equation) in all sources is 20%. When the peak luminosity of synchrotron component was not got from above literatures, we used radio luminosity at 5 GHz to estimate the synchrotron peak luminosity ( from linear regression analysis: ). The proportion of blazars (that the is estimated by the linear regression equation) in all sources is 60%.

The ratio of the beamed radio core flux density to the unbeamed extended radio flux density ( with , ) has routinely been used as a statistical indicator of Doppler beaming and thereby orientation (Orr & Browne 1982; Urry & Padovani 1995; Kharb et al. 2010). In order to compare Fermi blazars with non-Fermi blazars, we also estimated at 1.4 GHz. When more than one was obtained, we took the most recent one.

The relevant data for Fermi blazars can be seen in Table 1 of XZ14. The relevant data for non-Fermi blazars were listed in Table 1 with the following headings: column (1), name of the Roma BZCAT catalog; column (2), other name; column (3) is right ascension (the first entry) and declination (the second entry); column (4), redshift from NED; column (5), logarithm of the synchrotron peak frequency (the first entry) and logarithm of the peak luminosity of the synchrotron component in units of (the second entry); column (6), logarithm of jet kinetic power in units of ; column (7), logarithm of black hole mass in units of and references; column (8), logarithm of broad-line luminosity in units of and references; column (9), logarithm of the ratio of the beamed radio core flux density to the unbeamed extended radio flux density and references. For black hole mass or broad-line luminosity, when more than one value was obtained, we took the mean value.

In total, we have a sample containing 244 clean Fermi blazars (187 FSRQs and 57 BL Lacs) and 469 non-Fermi blazars (370 FSRQs and 99 BL Lacs).

## 3 The results

### 3.1 The distributions

The redshift distributions of the various classes are shown in Fig. 1. From Fig. 2 of Roma-BZCAT, the redshift distributions of BL Lacs are much closer than that of FSRQs and their distribution peaks at , whereas FSRQs show a broad maximum between 0.6 and 1.5. There are only very few BL Lacs at redshift higher than 0.8. So the redshift distributions from our sample are consistent with the results of Roma-BZCAT. The redshift distributions for Fermi blazars are and mean value is ; for non-Fermi blazars, the redshift distributions are and mean value is . The mean values for Fermi BL Lacs, Fermi FSRQs, non-Fermi BL Lacs and non-Fermi FSRQs are , , and respectively. Through nonparametric Kolmogorov-Smirnov (KS) test, we get that the redshift distributions between all Fermi blazars and all non-Fermi blazars, between Fermi BL Lacs and non-Fermi BL Lacs, between Fermi FSRQs and non-Fermi FSRQs are significant difference (chance probability ). Based on above results, it is shown that compared with non-Fermi BL Lacs, Fermi BL Lacs have larger mean redshift while compared with non-Fermi FSRQs, Fermi FSRQs have smaller mean redshift.

The black hole mass distributions of the various classes are shown in Fig. 2. The black hole mass distributions between all Fermi blazars and all non-Fermi blazars, between Fermi BL Lacs and non-Fermi BL Lacs, between Fermi FSRQs and non-Fermi FSRQs are significant difference (). So we can get that compared with non-Fermi blazars, Fermi blazars have smaller mean black hole mass.

The jet kinetic power distributions of the various classes are shown in Fig. 3. The jet kinetic power distributions between all Fermi blazars and all non-Fermi blazars, between Fermi FSRQs and non-Fermi FSRQs are significant difference (). So Fermi blazars have larger mean jet kinetic power than non-Fermi blazars.

The broad-line luminosity distributions of the various classes are shown in Fig. 4. Because non-Fermi BL Lacs almost have not broad-line data, we only compare broad-line luminosity distributions between Fermi FSRQs and non-Fermi FSRQs. The mean values for Fermi FSRQs and non-Fermi FSRQs are and respectively. From KS test, the broad-line luminosity distributions between Fermi FSRQs and non-Fermi FSRQs are significant difference (). We also compare the Eddington ratio distributions between Fermi FSRQs and non-Fermi FSRQs (see Fig. 5; from Netzer (1990)). However, the result of KS test shows that they do not have significant difference (). The mean values for Fermi FSRQs and non-Fermi FSRQs are and respectively.

The distributions for ratio of the beamed radio core flux density to the unbeamed extended radio flux density (core prominence parameter ) are shown in Fig. 6. The distributions for Fermi blazars are and mean value is ; for non-Fermi blazars, the distributions are and mean value is . The mean values for Fermi BL Lacs, Fermi FSRQs, non-Fermi BL Lacs and non-Fermi FSRQs are , , and respectively. The distributions between all Fermi blazars and all non-Fermi blazars, between Fermi BL Lacs and non-Fermi BL Lacs, between Fermi FSRQs and non-Fermi FSRQs are not significant difference (). Kharb et al. (2010) found that the ratio of the radio core luminosity to the k-corrected optical luminosity () appears to be a better indicator of orientation than the traditionally used radio core prominence parameter (). Wills & Brotherton (1995) defined as the ratio of the radio core luminosity to the k-corrected absolute V-band magnitude (): , where , and the k-correction is, with the optical spectral index, . Making use of the above Equations, we obtain for our sample. The distributions for are shown in Fig. 7. The mean values of for Fermi blazars and non-Fermi blazars are and respectively. The distribution between all Fermi blazars and all non-Fermi blazars is significant difference (). Therefore, Fermi blazars have larger mean than non-Fermi blazars, which supports that compared with non-Fermi blazars, Fermi blazars are more beamed.

### 3.2 Black hole mass vs jet power

Figure 8 shows black hole mass as a function of jet power. Different symbols correspond to blazars of different classes. Pearson product-moment analysis (hereafter called Pearson analysis) is applied to analyze the correlations between black hole mass and jet power for all blazars. The results show that the correlations between black hole mass and jet power for both Fermi blazars and non-Fermi blazars are significant (Fermi blazars: number of points , significance level , coefficient of correlation ; non-Fermi blazars: , , ).

### 3.3 Broad line luminosity and disk luminosity vs jet power

Figure 9 shows broad line luminosity as a function of jet power. The results of Pearson analysis show that there are significant correlations between broad line luminosity and jet power for Fermi blazars and non-Fermi blazars (, , ; , , ). Linear regression is applied to analyze the correlation between broad line luminosity and jet power. And we obtain for Fermi blazars; for non-Fermi blazars (95% confidence level and ). The Analysis of Variance (ANOVA) is used to test the results of linear regression which shows that it is valid for the results of linear regression (value , probability . Figure 10 shows disk luminosity as a function of jet power. From Fig. 10, it is seen that compared with non-Fermi blazars, Fermi blazars are closer to the ; the jet powers of some BL Lacs are larger than the disk luminosity while some are opposite; the jet power is much smaller than the disk luminosity for most of FSRQs.

We use multiple linear regression analysis to get the relationships between the jet power and both the Eddington luminosity and the broad line region luminosity for Fermi and non-Fermi blazars with 95% confidence level and :

 logPjet=0.49(±0.06)logLBLR−0.01(±0.16)logLEdd+23.6(±7), (2)
 logPjet=0.7(±0.14)logLBLR−0.46(±0.17)logLEdd+34.71(±4.9). (3)

The ANOVA shows that it is valid for the results of multiple linear regression (. From Equations (2) and (3), we see that for Fermi blazars, the black hole mass does not have significant influence on jet power while for non-Fermi blazars, both accretion disk luminosity (referring to accretion rate; Sbarrato et al. 2014; Ghisellini et al. 2014) and black hole mass have contributions to the jet power.

### 3.4 The blazar sequence

The plot of our sample is shown in Fig. 11. The plot of BZCAT catalog is given in Fig. 27 of Abdo et al. (2010c). From Fig. 11 of our sample, it is shown that FSRQs are exclusively located along the top-left/bottom-right band; the of most of BL Lacs are located between 0 and 0.8, and 0.8–2 for . As a comparison, it is found that the space of in Fig. 27 of Abdo et al. (2010c) has some BL Lacs while do not in our Fig. 11; for both our Fig. 11 and Fig. 27 of Abdo et al. (2010c), the top-right space of plot is empty; for the rest of space, our Fig. 11 is consistent with Fig. 27 of Abdo et al. (2010c). Selection effects may cause the deficiency of BL Lacs located in in our Fig. 11 because these BL Lacs are more likely to have nothing data about redshift, black hole mass, jet power. Padovani & Giommi (1995) and Abdo et al. (2010c) presented that the of HBL sources are located between 0.2 and 0.4, and 0.9–1.3 for . In our sample, this region mainly includes non-Fermi BL Lacs. Compared with non-Fermi FSRQs, Fermi FSRQs are more located in top-left of Fig. 11.

Figure 12 shows synchrotron peak luminosity versus synchrotron peak frequency for blazars and Fermi-detected narrow-line Seyfert 1 galaxy. The synchrotron peak luminosity and synchrotron peak frequency for four Fermi-detected narrow-line Seyfert 1 galaxies are collected from Abdo et al. (2009b). Making use of Pearson analysis, we find that there are strong anti-correlations between and for all blazars, Fermi blazars and non-Fermi blazars respectively (; ; ). In Fig. 12, we also find that the four Fermi-detected narrow-line Seyfert 1 galaxies mix in the area of blazars populated, and are in low –low region, which support that -loud narrow-line Seyfert 1 galaxies have similar mechanisms with blazars, and that the low –low blazars are more likely to have low black hole mass (see Chen & Bai 2011).

Ghisellini & Tavecchio (2008) extended the shape of SED from a one-parameter (observed bolometric luminosity) to a two-parameter sequence (black hole mass and accretion rate). So in Fig. 13, we give synchrotron peak frequency () versus black hole mass and Eddington ratio. From results of Pearson analysis, we get that there is not significant correlation between and black hole mass for all blazars () while there is significant anti-correlation between and Eddington ratio for all blazars ().

## 4 Discussions and conclusions

### 4.1 Possible biases in the evaluations of sample and parameters

For our sample, the selection criteria are: (i) we tried to select the largest group of blazars included in BZCAT with reliable broad line luminosity, redshift, black hole mass and jet kinetic power; (ii) in order to have reliable sample, we did not consider the candidate blazars of unknown type; (iii) the non-Fermi blazars, which were detected by EGRET or recorded in 1LAC but missed in 2LAC, were not included in our sample; (iv) the Fermi blazars only focused on clean 2LAC sample. Based on the above criteria, the following objects can be missed: blazars without measured redshift (mainly including BL Lacs), blazars without measured black hole mass, extended 1.4 GHz radio data and broad line data, blazars of uncertain type (BZU), blazars which were detected by EGRET or recorded in 1LAC but missed in 2LAC and blazars classed into non-clean 2LAC. From our selection criteria of sample, it is shown that the number of BL Lacs is missed more in our sample than that of FSRQs because compared with FSRQs, BL Lacs have much less information about redshift, black hole mass and broad line data. Via comparing our redshift distribution to redshift distribution of complete BZCAT sample, we find that the whole distribution and mean value of redshift from our sample agree with the results of complete BZCAT sample. From complete 2LAC sample (Fig. 12 of Ackermann et al. (2011)), it is shown that the redshift distribution peaks around for FSRQs and extends to ; the redshift distribution for BL Lacs extends to and shows a broad maximum between and . So through comparing redshift distributions between our Fermi sample (Fig. 1) and complete 2LAC sample, we find that our redshift distribution of Fermi blazars is similar to redshift distribution of complete 2LAC sample. However, this does not necessarily indicate that the distributions of all other quantities are also similar.

### 4.2 The basic properties of Fermi blazars

The jet formation remains one of the unsolved fundamental problems in astrophysics (e.g. Meier, Koide & Uchida 2001). Many models have been proposed to explain the origin of jets. In current theoretical models of the formation of jet, power is generated via accreting material and extraction of rotational energy of disc/black hole (Blandford & Znajek 1977; Blandford & Payne 1982), and then converted into the kinetic power of the jet. Our results show that the correlations between black hole mass and jet power for both Fermi and non-Fermi blazars are significant. The results of Pearson analysis show that there are significant correlations between broad line luminosity and jet power for both Fermi blazars and non-Fermi blazars, which supports that the jet power has a close link with accretion rate (Sbarrato et al. 2014; Ghisellini et al. 2014). The result is consistent with other authors (e.g. Rawlings & Saunders 1991; Falcke & Biermann 1995; Serjeant, Rawlings & Maddox 1998; Cao & Jiang 1999; Wang, Luo & Ho 2004; Liu, Jiang & Gu 2006; Xie et al. 2007; Ghisellini, Tavecchio & Ghirlanda 2009a; Ghisellini et al. 2009b, 2010, 2011; Gu, Cao & Jiang 2009; Sbarrato et al. 2012). Linear regression is applied to analyze the correlation between broad line luminosity and jet power, and we obtain for Fermi blazars and for non-Fermi blazars. In addition, from Equations (2) and (3), we find that for Fermi blazars, the black hole mass does not have significant influence on jet power while for non-Fermi blazars, both accretion disk luminosity (or accretion rate) and black hole mass have contributions to the jet power. These also can explain our result that the distributions of jet power between Fermi and non-Fermi blazars are significant difference.

### 4.3 The blazar sequence

From Fig. 12 and making use of Pearson analysis, we find that there are strong anti-correlations between and for all blazars, Fermi blazars and non-Fermi blazars respectively. The results support the “blazar sequence” and are not same with results of Meyer et al. (2011). In Fig. 4 of Meyer et al. (2011), an “L”-shape in the plot seems to have emerged which destroys the “blazar sequence”. However, the “L”-shape in Meyer et al. (2011) may be a selection effect of sample. A very possible reason is that in interval of plot of Meyer et al. (2011), there are few blazars. If the interval is full of some blazars, then the “L”-shape will be weaken or disappeared. In our Fig. 12, the interval has many blazars which fill in the interval of Fig. 4 of Meyer et al. (2011). And the result is consistent with others. From Fig. 2 of Nieppola et al. (2008) and Fig. 2, 5 of Finke (2013), it was seen that their the interval still has some blazars. Giommi et al. (2012) used extensive Monte Carlo simulations to get the relation plot of which shows that the interval still has many blazars. The Compton dominance, the ratio of the peak of the Compton to the synchrotron peak luminosities, is essentially a redshift-independent quantity. Finke (2013) used Compton dominance to study the “blazar sequence”. The results of Finke (2013) showed that a correlation exists between Compton dominance and the peak frequency of the synchrotron component for all blazars, including ones with unknown redshift. The sample of Finke (2013) is the 2LAC clean sample. The -ray flares in blazars are likely to occur when the sources are in the high state (Fan et al. 1998). If this is true, the blazars from Finke (2013) are likely to stand for active states of the sources, not their necessarily averaged status. Finke (2013) also presented that SSC emission has the same beaming pattern as synchrotron, and thus Compton dominance does not depend on the viewing angle; however, EC emission does not have the same beaming pattern as synchrotron and SSC, and so Compton dominance is dependent on the viewing angle. Piner & Edwards (2013) have found that beaming factors in the -ray and radio bands are different yet correlated for TeV blazars. An obvious explanation for the ‘bulk Lorentz factor crisis’ is that the radio and -ray emissions are produced in different parts of the jet with different bulk Lorentz factors (Henri & Sauge 2006; Piner, Pant & Edwards 2008, 2010; Piner & Edwards 2013). Therefore, it is possible that the correlation between Compton dominance and peak frequency of the synchrotron component can be resulted from beaming effect. For our Fig. 12, if high high blazars are included, the correlation of will be destroyed. The results of Giommi et al. (2012) also showed that the phenomenological “blazar sequence” is a selection effect. Nieppola et al. (2008) proposed that after being Doppler-corrected, the anti-correlation between and become positive correlation. However, due to limit of Doppler factor, we can not determine the effects of beaming on the blazar sequence in this paper. Ghisellini & Tavecchio (2008) revisited the blazar sequence and proposed that the power of the jet and SED of its emission are linked to the mass of black hole and the accretion rate. From results of Pearson analysis, we get that there is not significant correlation between synchrotron peak frequency and black hole mass for all blazars while significant anti-correlation between synchrotron peak frequency and Eddington ratio for all blazars. The scatter in plot can be resulted from uncertainties of and black hole mass.

## Acknowledgments

We sincerely thank anonymous referee for valuable comments and suggestions. DRX also thanks Minfeng Gu for helpful suggestions. This work is financially supported by the National Nature Science Foundation of China (11163007, U1231203, 11063004, 11133006 and 11361140347) and the Strategic Priority Research Program “The emergence of Cosmological Structures” of the Chinese Academy of Sciences (grant No. XDB09000000). This research has made use of the NASA/IPAC Extragalactic Database (NED), that is operated by Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

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# How is ideal gas law applicable to real gas?

I read one of the questions here on physics.stack exchange proving how the speed of sound increases with temperature using ideal gas law equation and adiabatic index. Here's the link:

But how is the ideal gas law applicable to real gas? I'm sorry if this is an incredibly dumb question. I'm still in high school.

## 3 Answers

Ideal gas law works best for monatomic gasses at low pressures and high temperatures.

It doesn't take into account molecular size and intermolucular interactions, so when the effects of those are significant, then the equation will be less accurate.

It generally works well for gasses like air in the temperatures and pressures we are used to (air has a lot of diatomic gasses which don't interact highly with each other and behave fairly ideally at room temperature).

Wikipedia has a fairly good entry on it here.

The speed of sound is defined as $c^2 = \frac{\partial p}{\partial \rho}$, which for an ideal gas becomes $c^2 = \gamma \frac{p}{\rho}$.

For a real gas, the relationship to an ideal gas can be found through the compressibility factor $z$. This is a measure of how much a real gas deviates from an ideal gas. It shows up in the equations as:

$$P = z \rho R T$$

You can work through the math on it (or follow along on this page), but essentially for a real gas, the speed of sound uses the compressibility factor $z$ and a factor $n$:

$$n = \gamma \frac{z + T \partial z/\partial T \rvert_p}{z + T \partial z/\partial T\rvert_\rho}$$

such that:

$$c^2 = zn \frac{p}{\rho}$$

which can be related to the ideal gas speed of sound as:

$$c^2_{real} = z c^2_{ideal} \frac{z + T \partial z/\partial T \rvert_p}{z + T \partial z/\partial T\rvert_\rho}$$

For gases and conditions where intermolecular forces are unimportant, $z = 1$ and $\partial z/\partial T \approx 0$ and the ideal speed of sound is correct. This is true for a majority of gasses at ambient conditions. At very high pressures and/or low temperatures, real gas effects become important.

• I'll add that the $P = z \rho RT$ equation is still a fairly simplified version of real gas laws, which generally have much messier equations and are all different depending on which model you use and how it's applied. – JMac Mar 13 '17 at 12:08
• @JMac Indeed. I'm familiar with the Peng-Robinson and Redlich-Kwong, but I know there are many more. But given the background in the question, I don't know if it was worth really digging into cubic EOS's and more complex models at this stage. – tpg2114 Mar 13 '17 at 12:10
• Well he was wondering how an ideal gas law relates to a real gas. This equation helps to show an ignored factor, but the real gas equations help to illustrate how much is really not considered when we use the ideal gas law. The compressibility factor glosses over all the complexity of what we ignore by lumping it in a single term. I just think it's probably worth mentioning if you're showing the compressibility equation anyways. – JMac Mar 13 '17 at 12:13

Speed of sound = sqaure root of pressure upon density now here let the gas be ideal or real the pressure is going to increase with rise in temperature so this law can be used here

• This is very unclear about answering the question. – JMac Mar 13 '17 at 12:38

## protected by Qmechanic♦Mar 13 '17 at 14:36

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

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# Transformed Cosine & Sine Curves - Wave Function

## (Working with Degrees)

Transformed cosine and sine curves, sometimes called wave functions, are cosine and sine curves on which we have carried-out a series of transformations.

In their most general form, wave functions are defined by the equations: $y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix}+d$

and
$y = a.sin\begin{pmatrix}b(x-c)\end{pmatrix}+d$ Where:
• $$a$$ is known as the amplitude
• $$b$$ is known as the wave number, also called the angular frequency
• $$c$$ is known as the phase shift
• $$d$$ is known as the vertical shift or rest position.
In this section we define and learn how to find each of these when given a cosine or sine curve.

By the end of this, we should be quite comfortable with any cosine or sine curve looking like the one shown here:

## The Amplitude: $$a$$

The amplitude of a cosine, or a sine, curve is the amount by which the curve oscillates (either above or below) its "rest", or mean, value.

The amplitude is illustrated here. The pink curve is: $y = 3.cos(x)$ The dotted curve is the "regular" $$y = cos(x)$$ curve.

We can see that:

• the pink curve $$y=3.cos(x)$$ oscillates $$3$$ units either side of the $$x$$-axis, it has an amplitude of $$3$$: $$a = 3$$.
• the dotted curve $$y=cos(x)$$ oscillates $$1$$ unit either side of the $$x$$-axis, it has an amplitude of $$1$$: $$a = 1$$.
At times it won't be as easy to "simply see" the amplitude and we'll need a formula to find it

## Formula for amplitude $$a$$

Given the graph of a cosine, or sine, curve we can find the value of its amplitude using the formula: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ Where:

• $$y_{\text{min}}$$ is the lowest point on the curve.
• $$y_{\text{max}}$$ is the highest point on the curve.

## The Wave Number: $$b$$

Given the graph of either a cosine or a sine function, the wave number $$b$$, also known as angular frequency, tells us:

how many fully cycles the curve does every $$360^{\circ}$$ interval

It is inversely proportional to the function's period $$T$$. This means that the greater $$b$$ is: the smaller the period becomes.

The wave number $$b$$ is illustrated here, using the sine function defined by: $y = 2.sin(3x)$

We can see from the equation $$y = 2.sin(3x)$$ that the wave number is: $b = 3$ On the graph, this can be seen from the fact that the curve completes $$3$$ full cycles every $$360^{\circ}$$.

We can also find the wave number if we know a cosine, or sine, curve's period $$T$$. In that case, the wave number $$b$$ can be found using the formula given below.

## Formula: wave number $$b$$

Given a cosine, or sine, curve with period $$T$$, its wave number $$b$$ can be calculated using the formula: $b = \frac{360}{T}$

## Example

Consider the cosine curve shown here:

Given this curve has equation $$y = 3.cos(bx)$$, find the value of $$b$$.

### Solution

We can see that:

• the curve completes $$2$$ cycles every $$360^{\circ}$$
• it has a period $$T = 180^{\circ}$$.
The first bullet point, above, tells us that this curve's wave number is equal to $$2$$, in other words: $b = 2$ Alternatively, we can use the fact that this curve's period is $$T = 180^{\circ}$$ and use our formula: \begin{aligned} b & = \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned}

## Finding the Phase Shift: $$c$$

The phase shift correponds to a horizontal translation of the cosine, or sine, curve.

Graphically, it can be seen by looking at how far to the right (or to the left) the curve without $$c$$ has been moved.

For example, the following graph shows the curve: $y = 3.sin\begin{pmatrix}x - 30 \end{pmatrix}$ as well as the curve $$y = 3.sin(x)$$ in dotted gray lines.

We can see that the phase shift of $$30^{\circ}$$ moves the entire curve $$y=3.sin(x)$$ $$30^{\circ}$$ to the right.

## Method

For a cosine curve, $$y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix} + d$$, or a sine curve, $$y = a.sin\begin{pmatrix}b(x-c)\end{pmatrix} + d$$, we can find the phase shift $$c$$ directly from its curve as follows:

• For a cosine curve $$y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix} + d$$

## Example

The following curve has equation $$y = a.cos\begin{pmatrix}b(x - c) \end{pmatrix}$$.

1. Find:
1. The value of the amplitude $$a$$.
2. The value of the wave number $$b$$.
3. The value of the phase shift $$c$$.
2. Given that this curve's equation can be written in the form: $y = a.cos\begin{pmatrix}bx+p\end{pmatrix}$ state the value of $$p$$.

### Solution

1. To find the amplitude we can use the formula we saw further-up: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ with:
• $$y_{\text{max}} = 4$$
• $$y_{\text{min}} = -4$$
So the amplitude is: \begin{aligned} a & = \frac{y_{\text{max}} - y_{\text{min}}}{2}\\ & = \frac{4 - (-4)}{2}\\ & = \frac{4+4}{2}\\ & = \frac{8}{2}\\ a & = 4 \end{aligned} At this stage we can state that this curve has an equation of the form: $y = 4.cos\begin{pmatrix}b(x - c) \end{pmatrix}$
2. To find the wave number $$b$$, we use our formula: $b = \frac{360}{T}$ Where the period $$T$$ can be read off the graph.
Indeed we can see that: $T = 180^{\circ}$ So we find: \begin{aligned} b &= \frac{360}{T}\\ & = \frac{360}{180} \\ b& = 2 \end{aligned} At this stage we can state that this curve has an equation of the form: $y = 4.cos\begin{pmatrix}2(x - c) \end{pmatrix}$
3. To find the phase shift $$c$$, we can compare the curve we're given to the curve without any phase shift, that's: $y = 4.cos\begin{pmatrix}2x \end{pmatrix}$ The phase shift will then equal to the amount by which the curve without the phase shift (shown in dotted gray below) would have to be horizontally translated to reach the curve we were initially given.

## The Vertical Shift: $$d$$

### (the "rest" position)

Given the graph of either a cosine or a sine function, to find the rest position $$d$$, which is the amount the curve has been translated vertically, we can use the following formula.

This is illustrated here:

• the pink curve shows $$y = 3.sin(x)+2$$
• the dotted curve shows $$y = 3.sin(x)$$

We can see that the pink curve, $$y=3.sin(x)+2$$ is the same as the dotted curve but it has been moved $$2$$ units upwards.

The pink curve $$y=3.sin(x)+2$$ oscillated above and below the "new" rest poistion $$y=2$$.

If we can't see/read the value of the vertical shift directly off the graph, we can use the following formula to find the value of $$d$$.

## Formula: vertical translation $$d$$

$d = \frac{y_{\text{max}} + y_{\text{min}}}{2}$ Where:

• $$y_{\text{min}}$$ is the lowest point on the curve.
• $$y_{\text{max}}$$ is the highest point on the curve.

## Example

The curve shown here has equation: $y = a.sin(bx)+d$

1. Find the value of $$a$$.
2. Find the value of $$b$$.
3. Find the value of $$d$$.

### Solution

1. The amplitude $$a$$ can be found using our formula: $a = \frac{y_{\text{max}} - y_{\text{min}}}{2}$ Looking at the curve:
We can see that:
• $$y_{\text{max}} = 1$$
• $$y_{\text{min}} = -7$$
So we find: \begin{aligned} a &= \frac{y_{\text{max}} - y_{\text{min}}}{2}\\ & = \frac{1 - (-7)}{2} \\ & = \frac{1+7}{2} \\ & = \frac{8}{2}\\ a & = 4 \end{aligned} $y = 4.sin(bx)+d$
2. The wave number $$b$$ can be found using the curve's period $$T$$ and the formula: $b = \frac{360}{T}$ Looking at the curve we can find its period:
We can see that its period is $$T = 180^{\circ}$$, so the formula leads to: \begin{aligned} b &= \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned} At this stage we can state that the curve has an equation of the form: $y = 4.sin(2x)+d$
3. To find the vertical shift $$d$$ we can use our formula: $d = \frac{y_{\text{max}} + y_{\text{min}}}{2}$ Looking at the curve we can see the maximum and minimum $$y$$-values:
Indeed we can see that:
• $$y_{\text{max}} = 1$$
• $$y_{\text{min}} = -7$$
So our formula becomes: \begin{aligned} d &= \frac{y_{\text{max}} + y_{\text{min}}}{2}\\ & = \frac{1+(-7)}{2} \\ & = \frac{1-7}{2}\\ & = \frac{-6}{2} \\ d & = -3 \end{aligned} Finally we can state this curve's equation: $y = 4.sin(2x)-3$

## Exercise 1

The curve, shown below, has equation: $y = a.sin(bx)$

1. Find the value of $$a$$.
2. Find the value of $$b$$.

1. The amplitude is $$a = 2$$.
2. The wave number is $$b=3$$.
This curve's equation is therefore: $y = 2.sin(3x)$

## Exercise 2

The curve, shown below, has equation: $y = a.cos\begin{pmatrix}x-c\end{pmatrix}$

1. Find the value of $$a$$.
2. Find the value of $$c$$.

1. The amplitude is $$a = 4$$.
2. The phase shift is $$c = 30^{\circ}$$.
This curve's equation is therefore: $y = 4.cos(x - 30)$

## Exercise 3

The curve, shown below, has equation: $y = a.cos\begin{pmatrix}b(x-c)\end{pmatrix}$

1. Find the value of $$a$$.
2. State this function's period.
3. Find the value of $$b$$.
4. Find the value of $$c$$.

1. The amplitude is $$a = 3$$.
2. The period is $$T = 180^{\circ}$$.
3. The wave number is: \begin{aligned} b &= \frac{360}{T} \\ & = \frac{360}{180} \\ b & = 2 \end{aligned}
4. The phase shift is $$c = 60^{\circ}$$.
The curve's equation is $$y = 3.cos\begin{pmatrix} 2(x-60)\end{pmatrix}$$


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# Homework Help: Particle Released From Narrow Potential - Fourier Transform

1. Mar 2, 2016

### BOAS

1. The problem statement, all variables and given/known data

A free particle moving in one dimension is initially bound by a very narrow potential well at the origin. At time $t = 0$ the potential is switched off and the particle is released; its wave function is:

$\psi (x,0) = N e^{-\frac{|x|}{\lambda}}$

where λ is a positive real constant giving the decay length of the initial wave function away from the origin.

(a) Calculate N.

(b) The wave function $\psi(x, 0)$ can be written as a superposition of stationary states, which for a free particle are planes waves,

$\psi (x,0) = \frac{1}{\sqrt{2 \pi}} \int^{\infty}_{\infty} dk c(k) e^{ikx}$

Calculate c(k) by applying the inverse Fourier transformation.

(c) How does $c(k)$ behave (i) if the initial spread of the wave function is very small (that is: if $\lambda$ is very small), and (ii) if $\lambda$ is very large? Bearing in mind that for a free particle $k$ is $\frac{p}{\hbar}$, can you see a connection between the dependence of $c(k)$ on $λ$, the width of the initial distribution, and the Heisenberg uncertainty principle?

2. Relevant equations

3. The attempt at a solution

I am stuck with part c of this question.

I find that $N = \frac{1}{\sqrt{\lambda}}$, so that $\psi (x,0) = \frac{1}{\sqrt{\lambda}} e^{-\frac{|x|}{\lambda}}$

$c(k) = \frac{1}{\sqrt{2 \pi} \sqrt{\lambda}} (\frac{2 \lambda}{1 + \lambda^2 k^2})$

I think that for very small $\lambda$, $(1 + \lambda^2 k^2) \approx 1$, so $c(k) \approx \frac{1}{\sqrt{2 \pi} \sqrt{\lambda}} 2 \lambda$. So for small $\lambda$, $c(k)$ is also small.

For large $\lambda$, $(1 + \lambda^2 k^2) \approx \lambda^2 k^2$ whick seems to suggest that $c(k)$ is also small for large $\lambda$.

I don't think i'm seeing the physics of what's going on here, and would really appreciate a conversation about the situation.

Thank you!

2. Mar 2, 2016

### blue_leaf77

I think this part of the problem asks you to investigate the behavior of the width of $|c(k)|^2$ for the two cases pertaining $\lambda$ mentioned above.

3. Mar 2, 2016

### BOAS

Ok, that makes sense.

Using the ideas of Full Width Half Maximum, I find that $c(k)_{max} = \sqrt{\frac{2}{\pi}} \sqrt{\lambda}$ and thus $\frac{1}{2} c(k)_{max} = \frac{\sqrt{\lambda}}{\sqrt{2 \pi}}$.

the values of k that give me the half maximum values are $k = \pm \frac{1}{\lambda}$, making the width of the function at FWHM $\frac{2}{\lambda}$ which corresponds to a broader peak for small values of $\lambda$ and a narrower peak for large values of $\lambda$.

Regarding the relation to Heisenberg's uncertainty principle, when the question says "the width of the initial distribution" is it talking about the wave function or the inverse Fourier transform?

4. Mar 2, 2016

### blue_leaf77

When mentioning the "width", the question refers to $\lambda$ which has a unit of length, therefore it must be talking about $\psi(x,0)$. By the way, when calculating the width, you should be using the probability distribution instead of the wavefunction.

5. Mar 2, 2016

### BOAS

Earlier you said that I should calculate $|c(k)|^2$. Is that equivalent to both calculating the inverse Fourier transform of $|\psi (x,0)|^2$ and taking the modulus squared of the fourier transform of $|\psi (x,0)|$?

i.e, is it safe to take the modulus squared of what I calculated, or should I have started with the modulus squared of my wave function?

6. Mar 2, 2016

### blue_leaf77

It's not "safe", that way is indeed one of the correct ways to obtain $|c(k)|^2$ - calculate $c(k)$ first, which you have done, and then take its modulus square.

7. Mar 2, 2016

### BOAS

Ok, thank you.

I find the width of $|c(k)|^2 = \frac{2}{\lambda}$ at FWHM.

I find the width of $|\psi (x,0)|^2 = \frac{1}{\lambda}$ at FWHM.

I can't see the link between what I have done and the Heisenberg uncertainty principle. Has the fourier transform told me what the function looks like in momentum space?

8. Mar 2, 2016

### blue_leaf77

That cannot be true, the width of $\psi(x,0)$ should have a dimension of length. In the Heisenberg uncertainty principle, $\Delta x$ and $\Delta p$ are actually the standard deviations of the probability functions in position and momentum spaces. So, if you want to be exactly following the Heisenberg principle, you should calculate the standard deviations instead of the FWHM widths. But I think part c) just want you to verify that the widths, whatever formula used to calculate them, of the probability function in the two spaces are reciprocal of each other.


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## Marginal information for expectation parameters

### Abstract

The authors consider a special case of inference in the presence of nuisance parameters. They show that when the orthogonalized score function is a function of a statistic S, no Fisher information for the interest parameter is lost by using the marginal distribution of S rather than the full distribution of the observations. Therefore, no information for the interest parameter is recovered by conditioning on an ancillary statistic, and information will be lost by conditioning on an approximate ancillary statistic. This is the case for regular multivariate exponential families when the interest parameter is a subvector of the expectation parameter and the statistic is the maximum likelihood estimate of the interest parameter. Several examples are considered, including the 2 × 2 table.

View all

View all


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# Introduction

Making plots in oceanography (or anything, really) often requires creating some kind of “color map” – that is, having a color represent a field in a plot that is otherwise two-dimensional. Frequently this is done when making “image”-style plots (known in MatlabTM parlance as “pcolor” or pseudocolor plots), but could also be in coloring points on a 2D scatter plot based on a third variable (e.g. a TS plot with points colored for depth).

There are a whole bunch of different ways to make colormaps in R, including various approaches that are derived from the “tidyverse” and ggplot2 package for analyzing and plotting data. I don’t really use that approach for most of my work, so won’t touch on them here.

Instead, the purpose of this post (inspired by a question from a colleague who I know is a Matlab and Python user) is to show some of the ways various functions contained in the oce package cab be used to make colormaps and colorbars (or “palettes”). In particular, for the case where one wants a “discrete” (i.e. not continuous) colormap.

# The imagep() function

For making image-style plots, the function imagep() provided by the oce package is a handy function for quickly making nice-looking pseudocolor plots of matrices. The “p” in imagep() stands for “palette” or “pseudocolor”. Mostly, it is a wrapper around the base R function image(), to allow for increased control of the axes, colors, and palette specification.

The above example shows how to use imagep() with 3 different colormaps (the default, the classic “jet” scheme, and the cmocean package) to generate an image plot with a nice palette automatically placed on the side.

# The drawPalette() and colormap() functions

Under the hood, the imagep() function calls another function to actually draw the palette on the side of the plot – the drawPalette() function. That function can be called on it’s own, enabling plot building to be much more flexible. Additionally there is the colormap() function, which allows for detailed specification of the colormap properties to use, which can then be passed as an object to drawPalette().

For example, say we wanted to make a TS (temperature-salinity) plot of the Levitus surface data, but with each point colored by latitude in 10 degree increments. We do that first by making a colormap object:

(note that we use expand.grid() to make the number of lon/lat points match the matrices).

Looking inside the object, we can see some of the details that further plotting/palette functions can make use of:

Some of those fields are obvious (some probably aren’t to inexperienced users) but one field that is handy to know about is the zcol field. This encodes a color for every value in the original object based on the colormap specification. So, we can make a plot with the points colored based on the colormap using the argument col=cm\$zcol. We can also add the palette to the plot using the drawPalette() function, which has to be called before the main plot:

One problem that can happen when there are a lot of points is that overplotting obscures patterns in the colors. An easy way to fix this is to randomize the order of the plotted points with the sample() function:

## Custom color palettes with colormap()

In addition to the “known” color palettes that are included in R and oce (see also the cmocean package below, as well as RcolorBrewer), the colormap() function has arguments that allow for custom-built palettes. Specifically the x0, x1, col0 and col1 arguments, which are detailed in the help file as:

x0, x1, col0, col1: Vectors that specify a color map.  They must all be
the same length, with ‘x0’ and ‘x1’ being numerical values,
and ‘col0’ and ‘col1’ being colors.  The colors may be
strings (e.g. ‘"red"’) or colors as defined by rgb or
hsv.

The idea is that the x0 values define the numeric level of the bottom of the color ranges, the x1 values define the top of the color ranges, and the col0 and col1 the colors associated with the levels. An example:

(Note that to make the above plot, I had to fix a bug in oce that was making the zcol come out as “black” for all cases. Either build oce from source, or wait for the update to get pushed to CRAN in a month or two).


text:
Fraction Four Operations

## Difficult

Q1) $3\over10$ + $1\over3$ =
Q1) $4\over9$ x $2\over5$ =
Q1) 1$1\over2$ $\div$ 1$2\over3$ =
Q2) $5\over8$ - $3\over5$ =
Q2) $2\over7$ $\div$ $1\over2$ =
Q2) 1$1\over7$ $\div$ 2$1\over3$ =
Q3) $3\over7$ + $3\over10$ =
Q3) $1\over2$ x $2\over3$ =
Q3) 1$1\over8$ x 1$2\over5$ =
Q4) $5\over6$ - $2\over3$ =
Q4) $3\over5$ $\div$ $3\over7$ =
Q4) 1$1\over2$ x 1$3\over4$ =
Q5) $1\over4$ + $1\over3$ =
Q5) $6\over7$ x $3\over5$ =
Q5) 1$3\over5$ + 2$1\over3$ =
Q6) $1\over4$ + $1\over2$ =
Q6) $1\over2$ x $3\over4$ =
Q6) 1$2\over3$ x 2$1\over2$ =
Q7) $1\over3$ + $1\over4$ =
Q7) $1\over2$ x $3\over7$ =
Q7) 3$1\over2$ + 1$1\over3$ =
Q8) $1\over3$ + $1\over3$ =
Q8) $5\over7$ $\div$ $3\over5$ =
Q8) 2$1\over4$ + 3$1\over2$ =
Q9) $1\over2$ + $1\over5$ =
Q9) $3\over4$ $\div$ $3\over10$ =
Q9) 1$2\over5$ + 2$2\over3$ =
Q10) $6\over7$ - $2\over5$ =
Q10) $3\over7$ x $2\over3$ =
Q10) 1$1\over2$ + 1$1\over3$ =


text:
# Where exactly is the Morse code character ";" defined?

I've seen that quite often in Morse code tables (for example here), under punctuation, a semicolon ";"character is listed.

However, the ITU-R M.1677-1 doesn't specify that character as a part of the International Morse code.

So my question is: Where is the semicolon character actually defined?

• I think you might be on to something; –·–·–·, NNN being equivalent to ; might actually not be standardized. Jan 16, 2017 at 15:44


text:
Publication detail

# On iterated dualizations of topological structures

KOVÁR, M.

Original Title

On iterated dualizations of topological structures

English Title

On iterated dualizations of topological structures

Type

conference paper

Language

en

Original Abstract

A topology $\tau^d$ is said to be dual with respect to the topology $\tau$ on a set $X$ if $\tau^d$ has a closed base consisting of the compact saturated sets in the topological space $(X,\tau)$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated the problem no. 540 of J. D. Lawson and M. Mislove: {\it Does the process of iterating duals of a topology terminate by two topologies, dual to each other (1990, \cite{LM})?} As a matter of fact, for $T_1$ spaces, the problem was solved by G. E. Strecker, J. de Groot and E. Wattel (1966, \cite{GSW}) a long time before it was formulated by Lawson and Mislove, since in $T_1$ spaces, the dual operator studied by Lawson and Mislove coincides with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago. In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces, during the iterated dualization process there can arise at most four distinct topologies: the original topology $\tau$, then $\tau^d$, $\tau^{dd}$ and $\tau^{ddd}$. Finally, this result was generalized for all topological spaces by M. M. Kov\' ar (2001,\cite{Ko}). In this talk we will speak about the following rings of questions: \roster \item We will present some recent and hot results related to iterated dualizations of topological spaces. \item We will ask what happens with the dualizations if we leave the realm of spatiality. \item We will mention some (unsolved) problems related to dual topologies. \endroster

English abstract

A topology $\tau^d$ is said to be dual with respect to the topology $\tau$ on a set $X$ if $\tau^d$ has a closed base consisting of the compact saturated sets in the topological space $(X,\tau)$. In the well-known book{\it Open Problems in Topology}, edited by J. van Mill and G. M. Reed, there was stated the problem no. 540 of J. D. Lawson and M. Mislove: {\it Does the process of iterating duals of a topology terminate by two topologies, dual to each other (1990, \cite{LM})?} As a matter of fact, for $T_1$ spaces, the problem was solved by G. E. Strecker, J. de Groot and E. Wattel (1966, \cite{GSW}) a long time before it was formulated by Lawson and Mislove, since in $T_1$ spaces, the dual operator studied by Lawson and Mislove coincides with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago. In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces, during the iterated dualization process there can arise at most four distinct topologies: the original topology $\tau$, then $\tau^d$, $\tau^{dd}$ and $\tau^{ddd}$. Finally, this result was generalized for all topological spaces by M. M. Kov\' ar (2001,\cite{Ko}). In this talk we will speak about the following rings of questions: \roster \item We will present some recent and hot results related to iterated dualizations of topological spaces. \item We will ask what happens with the dualizations if we leave the realm of spatiality. \item We will mention some (unsolved) problems related to dual topologies. \endroster

RIV year

2002

Released

24.06.2002

Publisher

Shimane University in Matsue Osaka university

Location

Matsue, Japonsko

Pages from

42

Pages to

43

Pages count

2

BibTex

@inproceedings{BUT5185,
author="Martin {Kovár}",
title="On iterated dualizations of topological structures",
annote="A topology $\tau^d$ is said to be dual with respect to the topology $\tau$ on a set $X$ if $\tau^d$ has a closed base
consisting of the compact saturated sets in the topological space  $(X,\tau)$.
In the well-known book{\it Open Problems in Topology}, edited by J.  van Mill and  G. M. Reed, there was stated
the problem no. 540 of  J. D. Lawson and M. Mislove: {\it Does the process
of iterating duals of a topology terminate by two topologies, dual to each other (1990, \cite{LM})?}

As a matter of fact, for $T_1$ spaces, the problem  was solved by G. E. Strecker, J. de Groot and E. Wattel (1966, \cite{GSW})
a long time before it was formulated by Lawson and Mislove, since in $T_1$ spaces, the dual operator studied by
Lawson and Mislove coincides with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago.

In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces,
during the iterated dualization process there can arise at most four distinct topologies: the original topology
$\tau$,  then $\tau^d$, $\tau^{dd}$ and  $\tau^{ddd}$. Finally, this result was generalized for all
topological spaces by  M. M. Kov\' ar (2001,\cite{Ko}). In this talk we will speak about the following rings of  questions:

\roster
\item We will present some recent  and hot results related to iterated dualizations of topological spaces.
\item We will ask what happens with the dualizations if we leave the realm of spatiality.
\item We will mention some (unsolved) problems related to dual topologies.
\endroster

",
address="Shimane University in Matsue
Osaka university",
booktitle="Abstract of the International Conference on Topology and Its Applications - Topology in Matsue",
chapter="5185",
institution="Shimane University in Matsue
Osaka university",
year="2002",
month="june",
pages="42",
publisher="Shimane University in Matsue
Osaka university",
type="conference paper"
}


text:
# quotient singularities

Let $X$ be a relaively compact projective variety and has only quotient singularities then for any n-form $\Omega$ , $$\int_{X_{reg}}\Omega\wedge \bar \Omega$$ is bounded? what about the converse

• What do you mean by relatively compact? If $X$ is a normal projective variety which is $\mathbb Q$-Gorenstein, then for any local trivialization $\sigma$ of $mK_X$ defined on $U\subset X$, the integral $\int_{U_{\rm reg}}(\sigma\wedge \bar \sigma)^{1/m}$ is finite if and only if $U$ has klt singularities. In particular, as quotient singularities are klt, the integral you wrote is finite. But klt singularities are not quotient in general. – Henri Jan 12 '17 at 17:48
• relatively compact means its closure be compact. Can you give a reference? – pickasa Jan 12 '17 at 18:05
• If $X$ is projective, it is compact in the analytic topology. Maybe you meant quasi-projective? anyway it is not really important for the question here. – Henri Jan 12 '17 at 19:13

Let $X$ has log terminal singularities when $K_X$ is $\mathbb Q$-Cartier, see Proposition 1.17, of
The interesting part of your question is Kawamata's holomorphic extension theorem, say that if for a n-form $Ω$, $∫_XΩ∧\bar Ω<∞$ and $Ω|_{X∖D}$ be holomorphic where $D$ is the reduced normal crossing divisor on $X$, then $Ω_X$ is holomorphic also


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 Forthcoming seminars Seminar calendar List of seminars Archive by years Register a seminar Search RSS Forthcoming seminars

You may need the following programs to see the files

Iskovskikh Seminar
February 2, 2017 18:00, Moscow, Steklov Mathematical Institute, room 540

Geometry of moduli spaces of rank 2 vector bundles on curves and secant varieties

Eugene Cherkassky

National Research University "Higher School of Economics" (HSE), Moscow

Abstract: Let $L$ be a linear bundle on a smooth projective curve and consider the natural rational map from the extension space $\mathbb{P}_L = \mathbb{P}(Ext^1(L,\mathcal{O}))$ to the moduli space $M_{2,L}$ of vector bundles of rank $2$ with determinant $L$. Following Bertram's '92 paper I will construct a new variety $\widetilde{\mathbb{P}_L}$ with a sequence of blow-ups and a morphism to $M_{2,L}$, which allow to calculate dimensions of global sections of linear bundles on $M_{2,L}$ and answer some other questions on geometry of the moduli spaces.


text:
We don't bite newbies here... much PerlMonks

Newbie: uses/limits of perl in editing files

by wherethewild (Novice)
 on Nov 23, 2007 at 13:37 UTC Need Help??
wherethewild has asked for the wisdom of the Perl Monks concerning the following question:

Hello Monks, I'm a total newbie, not just to perl but to any form of scripting.

I'm trying to create a script which will take a text file that I have and run through it, changing bits here and there eg. on a line that starts with HEADER I want a number appended; I want an entire line inserted just before a line which starts with REMARK; and many other things.

So my question: is perl able to do this, or do I need to seek out another language/tool/something?

like I said...total newbie! Any words of wisdom greatly appreciated!

cheers,
wherethewild

• Comment on Newbie: uses/limits of perl in editing files

Replies are listed 'Best First'.
Re: Newbie: uses/limits of perl in editing files
by tirwhan (Abbot) on Nov 23, 2007 at 14:07 UTC

Welcome to the monastery. From the task description I'd say perl is very well suited. I'll give you an example for a program that roughly does what you describe

#!/usr/bin/perl

use warnings;
use strict;

my $filename = "whateveryourfileiscalled.txt"; my$newfile = "whateveryouwantthechangedfiletobecalled.txt";

open (my $rfh,"<",$filename) or die "Can't open file $filename :$!";
open (my $wfh,">",$newfile) or die "Can't open file $newfile :$!";

while (my $line = <$rfh>) {
if ($line =~ m/^HEADER/) { chomp$line;
my $number = 42; # change to whatever number you want to use$line .= $number."\n"; } if ($line =~ m/^REMARK/) {
print {$wfh} "Extra line\n" # Change to whatever extra line yo +u want } print {$wfh} $line; } close$rfh or die "Can't close $filename :$!";
close $wfh or die "Can't close$newfile : $!"; [download] Or you could do this in a perl oneliner (which will change the original file): perl -pi -e 'chomp;s/^(HEADER.*)$/${1}42/;s/^(REMARK.*)$/Extra line\n$+1/;$_.="\n"' whateveryourfileiscalled.txt
[download]
Caveat: Both of these are for systems where the line ending is "\n" (i.e. not Windows), adjust appropriately for other OSes. Update: the caveat is not actually correct, as pointed out by naikonta and wfsp, except possibly for the case outlined by Sixtease, also fixed in his update. Thanks to all of you.

All dogma is stupid.
Wow, I didn't know the print {$wfh}$line; construct. Does that disambiguate $wfh to be interpreted as a filehandle? Yes; because the "filehandle argument slot" (for lack of a better name) has to be a simple scalar value or a BLOCK. While it's superfluous in this particular case the block form is useful if you (for instance) have a hash of filehandles and want to use print {$handles->{$somekey} } "Yadda yadda yadda.\n"; directly rather than pulling it out into a tmp variable. The docs for print cover this. Yep, I picked that up from thedamians Perl Best Practices (a must-read for every Perl programmer IMO). As Fletch rightly points out, it's not necessary in this case, but I just use it wherever I print to a filehandle (easier to do than figure out what's wrong if I ever forget it :-) All dogma is stupid. Caveat: Both of these are for systems where the line ending is "\n" (i.e. not Windows), adjust appropriately for other OSes. I see no caveat in your example regarding \n. This character is just the Perl internal representative of a thing that constitutes line ending. So it will be whatever the underlying OSes (perl is run on) actually use to terminate lines. See how newlines are addressed in perlport. Open source softwares? Share and enjoy. Make profit from them if you can. Yet, share and enjoy! That's it! Wow, let's see how I go at adjusting it all for everything else I have to do to it. The weekend shall be fun. One problem though, and I guess it has something to do with the \n point you made at the end. Now I have <cr> appearing at the end of every line. I'm sitting on a Linux workstation running RedHat if that is at all helpful (I DID say I know nothing about this!) cheers wherethewild Here's my guess: Your file has Windows newlines (cr/lf), which your editor/viewer can deal with and shows it correctly. Then you add unix newlines (lf) on the lines you edit. Now there are mixed cr/lf and lf newlines, which confuses the editor and it shows the cr characters. If I'm correct, then I recommend either preprocess the file with the dos2unix tool or address this in the perl script itself update: The modified while loop could look like this: while (my$line = <$rfh>) { chomp$line;
if ($line =~ m/^HEADER/) { my$number = 42; # change to whatever number you want to use
$line .=$number;
}

if ($line =~ m/^REMARK/) { print {$wfh} "Extra line\n" # Change to whatever extra line yo
+u want
}

print {$wfh}$line, "\n";
}
[download]
Re: Newbie: uses/limits of perl in editing files
by Dominus (Parson) on Nov 23, 2007 at 15:06 UTC
Tie::File is nice for stuff like that.

It makes the file look like an array, with one line in each element. Then you modify the array. As you do, the changes appear in the file.

Re: Newbie: uses/limits of perl in editing files
by Sixtease (Friar) on Nov 23, 2007 at 14:08 UTC

You can do this with pretty much any scripting / programming language (if it has input/output capabilities and is turing-complete). And Perl may be the most comfortable one for this.

The code to do something like you said could look like

perl -pe '/^REMARK/ and print "the line you want to add\n"' < input_file > output_file
Re: Newbie: uses/limits of perl in editing files
by cdarke (Prior) on Nov 23, 2007 at 14:42 UTC
Exetending your requirements a little, there is a neat feature that is useful when replacing tokens, like your HEADER and REMARK. You can execute code from within a substitute statement, for example:
$line =~ s/(HEADER|REMARK)/mysub($1)/ge;
[download]
That will call user-written subroutine mysub every time HEADER and REMARK are found in the text. The argument passed is the text matched inside (). Whatever is returned by mysub will replace the token. It probably would not be worth it for the simple substitution you mentioned, but for more complex combinations it can be very powerful.
I was looking at those s/// but I wasn't sure I how I could get it to do some of the things I need as the text which has to be substitued is different from file to file and it also appears elswhere in the file, where it's not to be adjusted.

Was that clear?
This is a theoretical line is my file:
BOBBY X66666 A 345 674 A 123 488

The X66666 has to be changed to B22222. But the next file might have U33333 there instead of X66666, or worse still 666D3P, or even worse absolutely nothing at all. And I don't know what it might be unless I open up each of the text files and look what the previous program did to it (something I'm trying to avoid by learning this!). It SHOULD be that the spacing is constant across that line, but that's not guaranteed.

Anyway, that's some of what I'm trying to do. Thanks everyone for the speed and friendliness in helping me out!

BOBBY X66666 A 345 674 A 123 488

Assuming the piece you want to replace is the 2nd token in the line then you can do something like:

s/^(\S+)(\s+)(\S+)(.*)/${1}${2}B22222${4}/; [download] This reads like: (NOT WHITESPACE)(WHITESPACE)(NOT WHITESPACE)(EVERYTHING) That collects the first 3 pieces into variables$1-$3 then the remainder of the line into$4, then reassemblies the line with the pieces and the replacement.

Re: Newbie: uses/limits of perl in editing files
by dwm042 (Priest) on Nov 23, 2007 at 14:57 UTC
Perl is as close to a Swiss Army knife of a scripting language as exists. If you can't write the code yourself, you can, in most circumstance, find a solution written for you on CPAN.

Having said that, at this stage, you probably need a beginning text on writing Perl. Something like Learning Perl would be appropriate.

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Hamming Function

An Apodization Function chosen to minimize the height of the highest sidelobe. The Hamming function is given by

 (1)

Its Full Width at Half Maximum is . The corresponding Instrument Function is
 (2)

This Apodization Function is close to the one produced by the requirement that the Apparatus Function goes to 0 at . From Apodization Function, a general symmetric apodization function can be written as a Fourier Series
 (3)

where the Coefficients satisfy
 (4)

The corresponding apparatus function is

 (5)

To obtain an Apodization Function with zero at , use
 (6)

so
 (7)

 (8)

 (9)

 (10) (11)

The FWHM is 1.81522, the peak is 1.08, the peak Negative and Positive sidelobes (in units of the peak) are and 0.00734934, respectively.

Blackman, R. B. and Tukey, J. W. Particular Pairs of Windows.'' In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 98-99, 1959.


text:
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Jee Mains 2020

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Starting from the origin at ti...

# Starting from the origin at time t = 0 , with initial velocity 5hatj "ms"^(-1) , a particle moves in the x - y plane with a constant acceleration of (10hati+4hatj)ms^(-2) . At time t , its coordinates are (20 m,y_0 m) . The value of t and y_0 are , respectively :

Updated On: 27-06-2022

Text Solution

2 s and 18 m 5 s and 25 m 2 s and 24 m 4 s and 52 m

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

Transcript

hello Pawan starting from origin atticus 20 with velocity 5.8 metre per second particle moves in xy plane with constant acceleration of this at time t in it is best to find out the value of tea and why not if you see her initially you X is equals to zero and you why is given in the question 5 and if you see the acceleration so a x here is 10 and a why is 4G time the qualities 20 and it is this calculator will be this 20 the particle starts from rest with the displacement will be the best quality if you see the particle starts from origin at T = 20 to whatever is the displacement will be the X co-ordinate with its content that will cause 2

you XT plus half a t square so you XT here is zero and 10 by to you will write a letter to 30 square 2820 is equals to 5 square from here you will get the value of t square is equals to time will Bittu S very simple now if you write the one of them to apply the equation of motion so you like the second equation of motion to find out the time now we will again apply the second equation of motion in addition to get this y-coordinate it again if you see initially the y-coordinate he was zero so whatever will be the wide is present will be the coordinates to I will equals to from the second equation of motion which is already be applied in the acceleration to calculate the time now

find out the why do you want it is it into the time is 24 and two squares again 4 so this will be 10 + 16 by 28 is 8618 M Naik check the option have to say that is option is


text:
1
JEE Main 2022 (Online) 27th July Morning Shift
Numerical
+4
-1

Let $$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then $$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$ is equal to ______________.

2
JEE Main 2022 (Online) 29th June Morning Shift
Numerical
+4
-1

Let $$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at z1 $$\in$$ S and z2 $$\in$$ S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5$$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha$$ + $$\beta$$ is equal to ___________.

3
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1

Sum of squares of modulus of all the complex numbers z satisfying $$\overline z = i{z^2} + {z^2} - z$$ is equal to ___________.

4
JEE Main 2022 (Online) 28th June Morning Shift
Numerical
+4
-1

The number of elements in the set {z = a + ib $$\in$$ C : a, b $$\in$$ Z and 1 < | z $$-$$ 3 + 2i | < 4} is __________.

JEE Main Subjects
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Optics
Modern Physics
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Organic Chemistry
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Joint Entrance Examination


text:
# 12th International Symposium on Radiative Corrections (Radcor 2015) and LoopFest XIV (Radiative Corrections for the LHC and Future Colliders)

15-19 June 2015
UCLA Department of Physics & Astronomy Los Angeles, CA, USA
published June 29, 2017

This symposium combines the 12th International Symposium on Radiative Corrections (Radcor 2015) with LoopFest XIV (Radiative Corrections for the LHC and Future Colliders). This workshop provides a forum to coordinate activities focused on the theoretical challenges from current and future colliders.

We will focus on the potential for precision measurements, their role in searching for and disentangling physics beyond the Standard Model, progress in multi-loop and multi-leg calculations of Standard Model processes, new physics cross sections, new approaches and new theoretical developments. There will also be a few talks on more exotic theories and on the latest experimental progress.

Sessions
Plenary Session Monday June 15, 2015
Parallel Session I Monday afternoon June 15, 2015
Parallel Session II Monday afternoon June 15, 2015
Parallel Session III Monday afternoon June 15, 2015
Plenary Session Tuesday June 16, 2015
Parallel Session I Tuesday afternoon June 16, 2015
Parallel Session II Tuesday afternoon June 16, 2015
Parallel Session III Tuesday afternoon June 16, 2015
Plenary Session Wednesday June 17, 2015
Parallel Session I Wednesday morning June 17, 2015
Parallel Session II Wednesday morning June 17, 2015
Parallel Session III Wednesday morning June 17, 2015
Plenary Session Thursday June 18, 2015
Parallel Session I Thursday afternoon June 18, 2015
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Plenary Session Friday June 19, 2015
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 Plenary Session Monday June 15, 2015 The Standard Model at CMS PoS(RADCOR2015)001 M.A. Weber Dominant O(alpha*alpha_s) corrections to Drell–Yan processes in the resonance region PoS(RADCOR2015)002 S. Dittmaier Radiative corrections in bound states PoS(RADCOR2015)003 A. Czarnecki The NNLO subtraction scheme STRIPPER: implementation and application PoS(RADCOR2015)004 M. Czakon Precise predictions for supersymmetric Higgs boson masses and mixings PoS(RADCOR2015)005 W. Hollik Parallel Session I Monday afternoon June 15, 2015 Precise predictions for Higgs production in association with top quarks PoS(RADCOR2015)006 A. Denner, R. Feger and A. Scharf Implications of recent progress in heavy-quark hadroproduction PoS(RADCOR2015)007 M.V. Garzelli, S.O. Moch and G. Sigl Higher-order corrections to the Higgs-boson pT distribution PoS(RADCOR2015)008 J. Gao Higgs plus multi-jets in gluon fusion PoS(RADCOR2015)009 N. Greiner, S. Höche, G. Luisoni, M. Schonherr, J.C. Winter and V. Yundin The rare 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16, 2015 A coordinate description of partonic processes PoS(RADCOR2015)027 G. Sterman and O. Erdogan Rare exclusive decays of Higgs and Z bosons in QCD factorization PoS(RADCOR2015)028 M. Neubert A meta-analysis method for combination of PDF uncertainties PoS(RADCOR2015)029 P. Nadolsky Precision Higgs physics at hadron colliders PoS(RADCOR2015)030 C. Duhr An EFT Description of Forward Scattering and Factorization Violation PoS(RADCOR2015)031 I.W. Stewart Parallel Session I Tuesday afternoon June 16, 2015 Weak interactions in top-quark pair production at hadron colliders: An Update PoS(RADCOR2015)032 J. Kühn and P. Uwer N$^3$LO calculations for top-quark differential cross sections near partonic threshold PoS(RADCOR2015)033 N. Kidonakis Differential top pair production at NNLO in the $q \bar q$ channel PoS(RADCOR2015)034 G. Abelof, A. Gehrmann-De Ridder and I. Majer NNNLO determination of the bottom-quark mass from non-relativistic sum rules PoS(RADCOR2015)035 M. Beneke, A. Maier, 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Resummation in precision theory vs experiment PoS(RADCOR2015)045 B.F.L. Ward and A. Mukhopadhyay Parallel Session III Tuesday afternoon June 16, 2015 Radiative corrections for Pi_0 decays PoS(RADCOR2015)046 K. Kampf Jet production in the colorful NNLO framework PoS(RADCOR2015)047 A. Kardos Automated resummation for jet observables PoS(RADCOR2015)048 P.F. Monni The NNLO counterterm contributions to $\mathscr{B} (\bar B\rightarrow X_s \gamma)$ for an arbitrary charm quark mass PoS(RADCOR2015)049 A. Rehman, M. Steinhauser and M. Misiak NLO efforts in Herwig++ PoS(RADCOR2015)050 C. Reuschle, J. Bellm, S. Gieseke, D. Grellscheid, S. Plätzer, M. Rauch, P. Richardson, P. Schichtel, M. Seymour, A.K. Siodmok, A. Wilcock, N. Fischer, M. Harrendorf, G. Nail, A. Papaefstathiou and D. Rauch Colorful NNLO -- Completely local subtractions for fully differential predictions at NNLO PoS(RADCOR2015)051 G. Somogyi, F. Tramontano and Z. Trocsanyi Automated calculations of dijet soft functions in SCET PoS(RADCOR2015)052 G. Bell, R. Rahn and J. Talbert Plenary Session Wednesday June 17, 2015 Results from GENEVA PoS(RADCOR2015)053 C. Bauer NNLO phenomenology with jettiness subtraction PoS(RADCOR2015)054 R. Boughezal The Evolution of Jet Substructure PoS(RADCOR2015)055 J. Walsh Towards a four loop form factor PoS(RADCOR2015)110 R. Boels, B. Kniehl and G. Yang Parallel Session I Wednesday morning June 17, 2015 Higgs plus one jet production at NNLO PoS(RADCOR2015)056 X. Chen, T. Gehrmann, N. Glover and M. Jaquier Higgs Production: Calculating at N^3LO in QCD PoS(RADCOR2015)057 B. Mistlberger Higgs plus jet production from dimension-7 Higgs-gluon EFT PoS(RADCOR2015)058 M. Zeng Electroweak corrections of order alpha alpha_s to the decay witdh Gamma_(h\rightarrow bb) within SM PoS(RADCOR2015)059 L. Mihaila Parallel Session II Wednesday morning June 17, 2015 A toolbox to solve coupled systems of differential and difference equations PoS(RADCOR2015)060 C. Schneider, J. Ablinger, J. Bluemlein and A. De Freitas Advances in the method of differential equations for multi-Loop feynman integrals PoS(RADCOR2015)061 R. Schabinger Two-loop amplitudes from maximal unitarity PoS(RADCOR2015)062 K. Larsen Generalised unitarity for dimensionally regulated amplitudes within FDF PoS(RADCOR2015)063 W.J. Torres Bobadilla Parallel Session III Wednesday morning June 17, 2015 Present and future of the OpenLoops matrix element generator PoS(RADCOR2015)064 P. Maierhöfer Renormalization scheme dependence of the two-loop QCD corrections to the neutral Higgs-boson masses in the MSSM PoS(RADCOR2015)065 S. Borowka Complete NNNLO QCD prediction for the total top-anti-top S-wave pair production cross section near threshold in e+ e− annihilation PoS(RADCOR2015)066 P. Marquard Relation between the QCD resummation procedures for transverse-momentum distributions PoS(RADCOR2015)067 K. Tanaka Plenary Session Thursday June 18, 2015 QCD and electroweak results from ATLAS, and needs for new precision calculations PoS(RADCOR2015)068 J.W. Huston Higgs boson production through gluon fusion at N^3LO PoS(RADCOR2015)069 E. Furlan Fully differential VBF Higgs production at NNLO PoS(RADCOR2015)070 G. Zanderighi What sparsity and L1 optimization can do for you PoS(RADCOR2015)071 S. Osher Precision Forward Measurements at LHCb with Vector Bosons PoS(RADCOR2015)109 P. Ilten Parallel Session I Thursday afternoon June 18, 2015 Processes with Large Jet Multiplicity at the LHC PoS(RADCOR2015)072 F. Febres Cordero Jet-veto resummation for WW production PoS(RADCOR2015)073 P. Jaiswal Wgamma and Zgamma production at the LHC in NNLO QCD PoS(RADCOR2015)074 M. Grazzini, S. Kallweit and D. Rathlev NNLO QCD corrections for $Z$ boson plus jet production PoS(RADCOR2015)075 A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss and T. Morgan Status of diboson production in NNLO QCD PoS(RADCOR2015)076 D. Rathlev, S. Kallweit and M. Grazzini Threshold corrections at N^3LO in QCD for Drell-Yan and Higgs productions at the LHC PoS(RADCOR2015)077 V. Ravindran Comparisons of CMS data to NLO QCD in vector boson/photon processes PoS(RADCOR2015)078 E. Takasugi Parallel Session II Thursday afternoon June 18, 2015 Singularity structure of maximally supersymmetric scattering amplitudes PoS(RADCOR2015)079 J. Trnka Three- and Four-Loop Ultraviolet Finiteness in Supergravity Theories PoS(RADCOR2015)080 S. Davies Nonsupersymmetric Gravity at Two Loops PoS(RADCOR2015)081 J. Nohle Loop-tree duality and quantum field theory in four dimensions PoS(RADCOR2015)082 G. Sborlini FIRE5: a C++ implementation of Feynman Integral Reduction PoS(RADCOR2015)083 A. Smirnov Master integrals and generalized polylogarithms: focus on fast and efficient evaluation PoS(RADCOR2015)084 D. Tommasini, H.A. Frellesvig, C. Papadopoulos and C. Wever Integral reduction via algebraic curves PoS(RADCOR2015)085 Y. Zhang and A. Georgoudis Parallel Session III Thursday afternoon June 18, 2015 Jettiness as a subtraction scheme for NNLO PoS(RADCOR2015)087 X. Liu QCD NLO with Powheg matching and top threshold matching in WHIZARD PoS(RADCOR2015)088 J. Reuter, F. Bach, B. Chokoufe, W. Kilian, M. Stahlhofen and C. Weiss Hyperfine splitting in the bottomonium system on the lattice and in the continuum PoS(RADCOR2015)089 N. Zerf High-energy limit of Quantum Electrodynamics beyond Sudakov PoS(RADCOR2015)090 A. Penin Precision tests via radiative $μ$ and $τ$ leptonic decays PoS(RADCOR2015)091 M. Fael and M. Passera Muon decay in orbit and its role in New Physics searches PoS(RADCOR2015)092 R. Szafron Improved event generation of NLO and NNLO Monte Carlos PoS(RADCOR2015)086 W. Giele Plenary Session Friday June 19, 2015 Long-distance singularities in multi-leg scattering amplitudes PoS(RADCOR2015)093 E. Gardi Four-loop relation between the $\overline { \rm MS}$ and on-shell quark mass PoS(RADCOR2015)094 P. Marquard, A. Smirnov, V. Smirnov and M. Steinhauser Evaluating Feynman integrals by uniformly transcendental differential equations PoS(RADCOR2015)095 V. Smirnov A walk on sunset boulevard PoS(RADCOR2015)096 S. Weinzierl, L. Adams and C. Bogner TBA PoS(RADCOR2015)097 D.A. Kosower Conference summary PoS(RADCOR2015)098 K.R. Ellis Parallel Session I Friday afternoon June 19, 2015 High multiplicity processes with BlackHat and Sherpa PoS(RADCOR2015)099 D. Maitre, Z. Bern, F. Febres Cordero, K. Hamilton and D.A. Kosower NLO calculations matched with the Nagy-Soper parton shower PoS(RADCOR2015)100 M. Worek, M. Czakon, M. Kraus and H.B. Hartanto NLO predictions for effective field theory with MadGraph5 aMC@NLO PoS(RADCOR2015)101 C. Zhang NNLO Corrections to Dijet Production PoS(RADCOR2015)102 J. Currie NNLO matching and shower developments PoS(RADCOR2015)103 S. Prestel Parallel Session II Friday afternoon June 19, 2015 Simplified Differential Equations Approach for Master Integrals PoS(RADCOR2015)104 C. Papadopoulos, D. Tommasini and C. Wever A quasi-finite basis for multi-loop Feynman integrals PoS(RADCOR2015)105 A. von Manteuffel Calculation of multi-loop integrals with SecDec-3.0 PoS(RADCOR2015)106 J. Schlenk and T. Zirke Linear reducibility of Feynman integrals PoS(RADCOR2015)107 E. Panzer The calculation of a massive planar pentabox with a differential equation method PoS(RADCOR2015)108 C. Wever, C. Papadopoulos and D. Tommasini


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# Existence of nontrivial roots of a homogeneous polynomial over a finite field in extension fields

Consider a homogeneous polynomial, f, of total degree n in n variables, with coefficients in a prime order finite field, GF(p).

Are there any general rules regarding the existence of nontrivial roots of f in an extension field of GF(p)? I conjecture that there will exist nontrivial roots in GF(p)^k if k and n are not relatively prime, but I can't find any real proof.

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Take a linear form L in four variables whose coefficients form a basis of GF(p^4)/GF(p) and take the form of degree four which is the product of the conjugates of L. This won't have points in GF(p^2) and is a counterexample to your conjecture. – Felipe Voloch May 16 '13 at 20:25
My answer given yesteray is wrong. One can choose $p$ variables to have non-zero value, and set all others to be zero to get a non-trivial solution. Should I delete it? – P Vanchinathan May 19 '13 at 2:58
I don't understand how Felipe's suggestion would work. Those four coefficients can't be linearly independent over $GF(p^2)$, so won't any of the factors of the quartic form have zeros over $GF(p^2)$? – Jyrki Lahtonen May 24 '13 at 10:58

For a prime $p\ne2$, take the homogeneous polynomial $\sum_{j=1}^{p^2-1} X_j^{p^2-1}$ of degree $p^2-1$ in as many variables. For any non-zero element of $\mathbf{F}_{p^2}$ each term evaluates to 1 and so the given polynomial is the constant function $-1$ and has no zero. The number of variable $p^2-1$ is a multiple of 2, and yet has no solution in the quadratic extension of the prime field, providing another counterexample.
The above is wrong. One can choose $p$ variables to have non-zero value, and set all others to be zero to get a non-trivial solution. I am sorry to have rushed in without fully verifying. – P Vanchinathan May 18 '13 at 10:32


text:
Integrate the function$x\;logx$

$\begin{array}{1 1} \large \frac{\log x}{x^2}-\frac{x^2}{4}+c \\ \frac{\log x}{x^2}+\frac{x^2}{4}+c \\ \frac{\log x}{x}-\frac{x^2}{4}+c \\ \frac{\log x}{x}-\frac{x}{4}+c\end{array}$

Toolbox:
• (i)When there are two functions u and v and if they are of the form $\int u dv,$then we can solve it by the method of integration by parts$\int udv=uv-\int vdu$
• (ii)$\int\frac{1}{x}dx=log x+c.$
Given $I=\int xlog xdx.$

Clearly the given integral function is of the form $\int u dv$,so let us follow the method of integration by parts where $\int udv=uv-\int vdu$

Let u=log x.

On differentiating we get

$du=\frac{1}{x}dx.$

let dv=x dx.

On integrating we get

$v=\frac{x^2}{2}$.

On substituting for u,v,du and dv we get,

$\int xlog x=(log x.\frac{1}{x^2})-\int\frac{x^2}{2}.\frac{1}{x}dx.$

$\;\;\;\;=\frac{log x}{x^2}-\frac{1}{2}\int xdx.$

On integrating we get,

$\;\;\;\;=\frac{log x}{x^2}-\frac{1}{2}{x^2}{2}+c.$

$\;\;\;=\frac{log x}{x^2}-\frac{x^2}{4}+c.$

edited Feb 8, 2013