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# 20.2: Design and Optimization of Separators

Once oil and gas are brought to the surface, our main goal becomes that of transportion of the oil and gas from the wellhead to the refinery (for final processing) in the best possible form. All equipment and processes required to accomplish this are found at the surface production facility. Hence, all surface production starts right at the wellhead. Starting at the wellhead, the complex mixture of produced fluids makes its way from the production tubing into the flow line. Normally, many wells are drilled to effectively produce the hydrocarbons contained in the field. From each of these wells emerge one or more flow lines depending on how many layers are being produced simultaneously. Depending on the physical terrain of the area and several other environmental factors, each of the flow lines may be allowed to continue from the wellhead to a central processing facility commonly referred as a production platform or a flow station. If not, all the flow lines or several of them empty their contents into a bigger pipeline called a bulk header, which then carries the fluids to the production platform. The combination of the wellhead, the flow lines, bulk headers, valves and fittings needed to collect and transport the raw produced fluid to the production platform is referred to as the gathering system.

The gathered fluids must be processed to enhance their value. First of all, fluids must be separated into their main phasial components—namely, oil, water, and natural gas. The separation system performs this function. For this, the system is usually made up of a free water knock-out (FWKO), flow line heater, and oil-gas (two-phase) separators. We will be looking at the design of this last component.

The physical separation of these three phases is carried out in several steps. Water is separated first from the hydrocarbon mixture (by means of the FWKO), and then the hydrocarbon mixture is separated into two hydrocarbon phases (gas and oil/condensate). A successful hydrocarbon separation maximizes production of condensate or oil, and enhances its properties. In field applications, this is accomplished by means of stage separation. Stage separation of oil and gas is carried out with a series of separators operating at consecutively reduced pressures. Liquid is discharged from a higher-pressure separator into the next-lower-pressure separator. The purpose of stage separation is to obtain maximum recovery of liquid hydrocarbons from the fluids coming from the wellheads and to provide maximum stabilization of both the liquid and gas effluents.

Surface Production Facility: The physical installation where fluids coming from the wellhead are separated into three main constituents: water, oil, and natural gas.

Figure 20.1: Purpose of a Surface Production Facility

Usually it is most economical to use three to four stages of separation for the hydrocarbon mixture. Five or six may payout under favorable conditions, when — for example — the incoming wellhead fluid is found at very high pressure. However, the increase in liquid yield with the addition of new stages is not linear. For instance, the increase in liquids gained by adding one stage to a single-stage system is likely to be substantial. However, adding one stage to a three or four stage system is not as likely to produce any major significant gain. In general, it has been found that a three stage separating system is the most cost effective. Figure 20.2 shows this typical configuration.

Figure 20.2: Three-stage Surface Separation Facility

Under the assumption of equilibrium conditions, and knowing the composition of the fluid stream coming into the separator and the working pressure and temperature conditions, we could apply our current knowledge of VLE equilibrium (flash calculations) and calculate the vapor and liquid fractions at each stage. However, if we are looking at designing and optimizing the separation facility, we would like to know the optimal conditions of pressure and temperature under which we would get the most economical profit from the operation. In this context, we have to keep in mind that stage separation aims at reducing the pressure of the produced fluid in sequential steps so that better and more stock-tank oil/condensate recovery will result.

Separator calculations are basically performed to determine:

• Optimum separation conditions: separator pressure and temperature
• Compositions of the separated gas and oil phases
• Oil formation volume factor
• Producing Gas-Oil ratio
• API gravity of the stock tank oil

Let us look at the case of three-stage separation. In general, temperature conditions in the surface separation facility are very much determined by the atmospheric condition and incoming stream temperatures. As for pressures, the very first separator pressure is controlled by the gathering lines coming from well heads, thus there is not much room for playing with pressure in the first separator. The same arguments are valid for the last stage of separation (stock tank), which usually operates at atmospheric conditions. Therefore, we are only left with the middle separator for optimization.

As it turns out, the key to designing a three stage separation system is finding the optimum pressure at which to operate the second separator. The question that we would answer is “what is the pressure that will result in the best quality liquid going out of the stock tank for sales?” We do not want to do this empirically. This is, we do not want to play with the second stage separator pressure in the field, until we ultimately find the optimum condition. What we can do, using our phase behavior knowledge, is to find this optimum middle stage pressure applying our understanding of VLE equilibrium.

Figure 20.3 shows the typical effect of playing with the middle separator pressure on the quality and quantity of produced oil/condensate at the stock tank. Quality and quantity are measured in terms of properties, such as API and Bo, and the overall GOR at the separation facility.

Figure 20.3: Selection of Optimum Middle Separator Pressure

The optimum value of pressure for the middle stage is the one that produces the maximum liquid yield (by minimizing GOR and Bo) of a maximum quality (by maximizing stock-tank API gravity). The smaller the value of GOR and Bo, the larger the liquid yield. The higher the API gravity of the stock-tank fluid, the more profitable its commercialization. From Figure 20.2, we see that this condition is found at neither extreme (low/high) values of middle stage pressure. There is, in fact, an optimal value for middle stage pressure. This is the value we are looking for.

The Phase Behavior model that we have described throughout these series of lectures provides the basic framework for the type of calculations required here. Additionally, we discussed how API and Bo can be calculated using the output of the phase behavior model. While doing the calculations for a 3-stage separating system, keep in mind that we have minimal control over feed pressure, as we do not want to inhibit the well (high-pressure separator). We do not control the sales line pressure (stock-tank pressure) either. The control that we do have is the operating pressure of the middle separator.

Recall that finding the optimum pressure calls for, in part, finding the minimum gas to oil ratio (GOR, SCF/STB). We are dealing, in this case, with total GOR. The total GOR is the cumulative amount of gas from all three separators divided by the amount of liquid/condensate leaving the stock tank. During our discussion on Bo-calculations, we called “nst” the moles of liquid leaving the stock tank per mole of feed entering the separation facility. This number can be obtained by sequentially flashing 1 lbmol of feed through each of the separation stages. Recalling the definition of GOR,

$G O R=\frac{\text { Total Volume of Gas produced (in standard cubic feet) }}{\text { Total Volume of Liquid produced (in stock-tank barrels) }}=\frac{\left(V_{g}\right)_{S C}}{\left(V_{o}\right)_{s r} 15.615} \label{20.1}$

where:

$\left(V_{o}\right)_{s r}=\frac{n_{s t}\left(Z_{o}\right)_{st} R T_{s t}}{P_{s t}}$

$\left(V_{g}\right)_{s c}=379.4 \frac{S C F}{l b m o l} \cdot\left(n_{g}\right) T O T A L=379.4 \cdot\left(1-n_{s t}\right)$

[basis: 1 lbmol of feed]

Therefore,

$G O R=\frac{5.615}{379.4} R\left(\frac{n_{s t}}{1-n_{s t}}\right)\left(\frac{T_{s t}}{P_{s t}}\right)\left(Z_{o}\right)_{s c}[S C F / S T B] \label{20.2}$

Usually, the stock tank is considered to operate at standard conditions (psc, Tsc). Now you are ready to make your own surface separation design!


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# Negri bodies are viral factories with properties of liquid organelles

## Abstract

Replication of Mononegavirales occurs in viral factories which form inclusions in the host-cell cytoplasm. For rabies virus, those inclusions are called Negri bodies (NBs). We report that NBs have characteristics similar to those of liquid organelles: they are spherical, they fuse to form larger structures, and they disappear upon hypotonic shock. Their liquid phase is confirmed by FRAP experiments. Live-cell imaging indicates that viral nucleocapsids are ejected from NBs and transported along microtubules to form either new virions or secondary viral factories. Coexpression of rabies virus N and P proteins results in cytoplasmic inclusions recapitulating NBs properties. This minimal system reveals that an intrinsically disordered domain and the dimerization domain of P are essential for Negri bodies-like structures formation. We suggest that formation of liquid viral factories by phase separation is common among Mononegavirales and allows specific recruitment and concentration of viral proteins but also the escape to cellular antiviral response.

## Introduction

Replication and assembly of many viruses occur in specialized intracellular compartments known as viral factories, viral inclusions or viroplasms1, 2. These neo-organelles formed during viral infection concentrate viral proteins, cellular factors and nucleic acids to build a platform facilitating viral replication. They might also prevent the activation of host innate immunity and restrain the access of viral machineries to cellular antiviral proteins. Such factories are widespread in the viral word and have been identified for a variety of non-related viruses.

The location and the nature of viral factories are very heterogeneous. They depend on the genome composition (DNA or RNA) and on the viral replication strategy. The first viral factories which were characterized were those formed by large DNA viruses such as the Poxviridae, the Iridoviridae and the Asfaviridae3,4,5,6. Those factories are devoid of membrane, located in close proximity to the microtubule organizing center. They recruit mitochondria, contain molecular chaperones such as HSP proteins and are surrounded by a vimentin cage. In the case of positive strand RNA viruses, viral factories are associated with rearrangements of membranes from diverse organelles (Mitochondria, ER, and so on) leading to the formation of double-membrane vesicles7,8,9. These vesicles seem to remain connected to the cytoplasm by channels which allow ribonucleotide import and product RNA export.

Several negative strand RNA viruses also induced the formation of membrane-less cytoplasmic inclusions which, in the case of rhabdoviruses10, 11 and filoviruses12, have been demonstrated to harbor several viral replication stages. In the case of rabies virus (RABV), those inclusions are called Negri bodies (NBs) and can reach several microns in diameter11, 13, 14.

RABV (Mononegavirales order, Rhabdoviridae family, Lyssavirus genus) is a neurotropic virus, which remains a substantial health concern as it causes fatal encephalitis in humans and animals and still kills > 55,000 people worldwide every year mainly in Asia and Africa. It has a negative stranded RNA genome (about 12 kb) encoding five proteins. The genome is encapsidated by the nucleoprotein (N) to form a helical nucleocapsid in which each N protomer binds to nine nucleotides15. The nucleocapsid is associated with the RNA dependent RNA polymerase (L) and its cofactor the phosphoprotein (P) to form the ribonucleoprotein (RNP) which is enwrapped by a lipid bilayer derived from a host cell membrane during the budding process. The matrix protein M and the glycoprotein G are membrane-associated proteins. M protein is located beneath the viral membrane and bridges the condensed RNP and the lipid bilayer. G protein is an integral transmembrane protein that is involved in viral entry16. The virus enters the host cell through the endocytic pathway via a low-pH-induced membrane fusion process catalyzed by G. The RNP is then released into the cytoplasm and serves as a template for transcription and replication processes that are catalyzed by the L–P polymerase complex. During transcription, a positive-stranded leader RNA and five capped and polyadenylated mRNAs are synthesized. The replication process yields nucleocapsids containing full-length antigenome-sense RNA, which in turn serve as templates for the synthesis of genome-sense RNA. Replication strictly depends upon ongoing protein synthesis to provide the N protein necessary to encapsidate nascent antigenomes and genomes. Neo-synthesized genomes either serve as templates for secondary transcription or are condensed and assembled with M proteins to allow budding of neo-synthesized virions at a cellular membrane16.

We have previously demonstrated that viral transcription and replication take place within NBs11. NBs contain all the replication machinery (L, N and P)11 together with M17 and several cellular proteins including HSP7018 and the focal adhesion kinase (FAK)19. It has been recently shown that NBs are in close proximity to stress granules (SGs)20, which are membrane-less liquid cellular organelles consisting of mRNA and protein aggregates21, 22 that form rapidly in response to a wide range of environmental cellular stresses and viral infections23.

In this report, using a recombinant RABV expressing a fluorescent P protein, we have investigated the physical nature of NBs. Live imaging and FRAP analysis demonstrated that they have liquid organelles properties. We have also characterized the role of the cytoskeleton on the dynamics of NBs and transport of RNPs. This revealed that RNPs are ejected from NBs by a cytoskeleton-independent mechanism and are further transported along microtubules. Finally, we developed a minimal system which recapitulates NBs properties and allows us to identify the P domains required for NBs formation.

## Results

### Characterization of inclusions formed during RABV infection

RABV-infected BSR cells (MOI = 0.5) were fixed and permeabilized at different times post infection (p.i.), and the structures formed in the cytosol were analyzed by immunofluorescence with an anti-N antibody (Fig. 1a). At each time, the structures were counted and classified based on their size (i.e., the surface of their projection) (Figs. 1b–d). We distinguished small dots (surface < 0.26 µm2) (Fig. 1b), inclusions of intermediate size (0.26 µm2 < surface < 3.7 µm2) (Fig. 1c) and large inclusions (surface > 3.7 µm2) (Fig. 1d). At early time p.i. (Fig. 1a,c, 8 h), there is a limited number (up to 2 per cell) of spherical intracytoplasmic inclusions of intermediate size, which correspond to the first NBs formed during the infection11, 20. These structures grow with time (Fig. 1a,e) and sometimes lose their spherical aspect (Fig. 1a, 24 h). At later stages of infection (16 h and 24 h), large inclusions (1–3 per cell in most cases) are observed together with an increased number of intermediate inclusions and an explosion of the number of punctate structures (Fig. 1a–d).

The cellular location of inclusions and punctate structures was analyzed by confocal microscopy. At 24 h p.i., punctate structures are mainly detected in the basal section, whereas inclusions (intermediate and big ones) are located in median sections of the cell (Fig. 1f).

This was confirmed by electron microscopy performed on fixed sample (Fig. 1g,h). Cytoplasmic inclusions are rather observed in median section (Fig. 1h). They are initially highly spherical and devoid of membranes (Fig. 1h, 8 h). At later stage (Fig. 1h, 24 h), they are often associated with membranes—most probably derived from the ER11 – and their shape appears to be altered. At that time, basal section of the cells reveals the presence of typical condensed RNPs (Fig. 1g). Therefore, the small punctate structures observed by immunofluorescence most probably correspond to condensed RNPs. The inclusions, either large or intermediate, will be referred as NBs thereafter.

### NBs are liquid structures

To gain insight into the dynamic of NBs inside the cytoplasm, live cell imaging was performed. For this, BSR cells were infected with the previously described recombinant virus rCVSN2C-P-mCherry20, which expresses the P protein C-terminally fused to the mCherry fluorescent protein. This recombinant virus behaves as wild type and, particularly, exhibits similar kinetics of NBs formation (Supplementary Fig. 1).

Up to 16 h p.i., NBs appear to be highly spherical structures (Figs. 1a,h) as judged by the distribution of their axial ratio which is < 1.2 in 70% of cases (Fig. 2a). This suggests that they consist of viscous liquids, similarly to the liquid-like nature of P granules24, 25, SGs21, 22 and nuclear bodies26. Consistent with this view, we frequently observed that when two or more NBs contact one another, they readily fuse and round up into a single-larger sphere (Fig. 2b, Supplementary Movies 1 and 2). Furthermore, we also sometimes observed spherical bubbles crossing the NBs (Fig. 2c, Supplementary Movie 3). The directionality of their movement suggests that they are vesicles being trafficked through the NBs. This reinforces the idea that NBs are made of a fluid phase, which can reversibly deform when encountering a physical barrier.

It has been reported that liquid organelles respond rapidly to cellular osmotic shock27. To test whether NBs also behave in this way, BSR cells that have been infected with rCVSN2C-P-mCherry for 16 h, were incubated with DMEM medium diluted 5 times in water. This hypotonic shock resulted in a fast and complete disappearance of NBs followed by the reformation of inclusions of intermediate size in 10–15 min (Fig. 2d, Supplementary Movie 4).

To definitively demonstrate the fluid nature of NBs, we performed fluorescence recovery after photobleaching (FRAP) experiments. For safety reason, we used a G gene deleted recombinant RABV (rCVSN2CΔG-P-mCherry), which was pseudotyped with the RABV G glycoprotein. This recombinant RABV also induces the formation of NBs, which incidentally demonstrates that G synthesis is not required for NB formation.

FRAP experiments were first performed on BSR cells transfected with a plasmid expressing P-mCherry (Fig. 2e and Supplementary Fig. 2). P-mCherry expressed alone was diffuse in the cytosol (Supplementary Fig. 2) and fluorescence recovery was fast although biphasic (Supplementary Table 1). The half time of the first phase for a photobleached region of 2.7 µm diameter took about 0.4 s corresponding to an approximate diffusion coefficient of 5 × 10−12 m2 s−1 for P-mCherry in the cytosol (Supplementary Table 1), a value which is consistent with that expected for a protein of such a molecular weight. Full fluorescence recovery was achieved after ~ 30 s. Similar data were obtained in cells infected by rCVSN2CΔG-P-mCherry, when cytosolic P (i.e., outside the NBs) was photobleached (Fig. 2f and Supplementary Fig. 3).

When P located inside NBs was photobleached (Fig. 2g and Supplementary Fig. 4), the recovery of the fluorescence signal was again biphasic but much slower (Supplementary Table 1). The half time of the first phase for a photobleached region of 2.7 µm diameter is ~5.2 s (Fig. 2g), corresponding to a diffusion coefficient of 3.9 × 10−13 m2 s−1 (Supplementary Table 1). Such a self-diffusion rate is consistent with those of other membrane-less organelles, such as nuclear speckles and nucleoli21, 28. After 2 min, the fluorescence recovery is not complete, but only reaches 78 ± 8% ( ± SD, n = 12). The recovery profile along a diameter revealed that fluorescence came back first to the periphery a few seconds after photobleaching and then displays a homogeneous distribution within the structure after ~1 min (Fig. 2h). Therefore, the FRAP data confirm that NBs have liquid properties and that P protein can shuttle between the cytosol and the inner of the NBs.

We have previously shown that SGs are formed in the cytoplasm of RABV infected cells20. As SGs have also been demonstrated to be liquid droplets21, 22, we investigated if SGs and NBs can fuse. Cells were transfected by a plasmid encoding G3BP-GFP, which is a marker of SGs. Then, 1 h post transfection, they were infected with rCVSN2C-P-mCherry. As previously described, we observed SGs in close proximity to NBs (Fig. 3, Supplementary Movie 5). However, we never observed mixing content of the two structures: the P protein remains in NBs whereas G3BP remains in SGs. Furthermore, in some circumstances, we identified small SGs located inside NBs confirming the non-miscibility of the two liquid phases (Fig. 3).

### Role of the cytoskeleton on the NBs fate and RNPs transport

To investigate the role of the cytoskeleton in NB morphogenesis and dynamics, we used several drugs which interfere with the organization of the cytoskeleton. Treatment of RABV infected cells with Nocodazole (NCZ), a drug which depolymerizes the microtubules, results in the disappearance of NBs of intermediate size (Fig. 4a,c) and a significant decrease in the number of small dots corresponding to condensed RNPs (Fig. 4a,b). In general, in the presence of NCZ, a single-large NB is observed (Fig. 4a,d) which is bigger than those observed in absence of the drug. This is evidenced by the significant increase of the average surface of the largest NB present in an infected cell (Fig. 4e). Taxol, which stabilizes the microtubule network, has no effect on the structures.

Cell treatment by Cytochalasin D (Cyto D), which inhibits actin polymerization, results in an increased fragmentation of NBs, which is apparent when considering the average surface of the largest NB present in an infected cell (Fig. 4e).

Live cell imaging was then performed in RABV infected cells. We observed ejections of punctate structures out of NBs (Fig. 5a, Supplementary Movies 6 and 7). These structures, which probably correspond to condensed RNPs, are rapidly transported further away across the cell (Supplementary Movie 8). Ejections are also observed when cells are treated by Nocodazole (Fig. 5b, Supplementary Movie 9). However, the ejected structures are not transported and remain in the vicinity of the NBs (Fig. 5b, Supplementary Movie 9).

We further characterized the transport of RNPs inside the cytoplasm (Fig. 5b). The histograms in Fig. 5c show that their velocity is the same in non-treated cells or in cells treated with Taxol or Cytochalasin D. However, in presence of Nocodazole, RNPs remain immobile in the vicinity of the NBs (Fig. 5b,c). Altogether with the fact that the measured velocities are in agreement with those of other cargoes for which transport is microtubule dependent, this indicated that RNPs transport requires the integrity of the microtubule network.

Indeed, when live imaging was performed on cells which have been transduced by a baculovirus expressing a tubulin fused with the GFP before infection by rCVSN2C-P-mCherry, we directly observed the transport of RNPs along the microtubules (Fig. 5d, Supplementary Movie 10).

### A minimal system recapitulating NB properties

It has been previously demonstrated that co-expression of N and P after cell transfection also leads to the formation of cytoplasmic inclusions29. Indeed, in BSR cells constitutively expressing the T7 RNA polymerase (BSR-T7/5) and co-transfected by plasmids pTit-P and pTit-N, cytoplasmic spherical inclusions are observed (Fig. 6a).

The dependence of inclusions formation on the stoichiometry of the transfected plasmids was investigated. Spherical inclusions were observed with ratios of pTit-N and pTit-P plasmids going from 3:1 to 1:3. In limiting concentrations of one of the plasmids (ratios 9:1 or 1:9), no inclusions were observed even when both proteins were detected in the cell (Supplementary Fig. 6).

We investigated the physical properties of the inclusions formed in this minimal system. For this, we co-transfected BSR-T7/5 cells with pTit-N and pTit-P-mCherry. We then performed FRAP experiments on pTit-P-mCherry located in the cytoplasmic inclusions that were formed. Recovery profiles (Fig. 6b and Supplementary Fig. 7), although less homogeneous, are however similar to those observed for P located in NBs in infected cells (Fig. 2g and Supplementary Table 1). This confirmed that the N-P inclusions formed in this minimal system have the same liquid characteristics as the NBs.

N-P inclusions also concentrate several cellular partners of P previously shown to be located in NBs such as HSP70 and FAK (Supplementary Fig. 8). However, unlike NBs, N-P inclusions do not eject material in the cytosol (Supplementary Movie 11, same images acquisition as Supplementary Movie 7). A possible role of RABV M protein in pinching off and ejection of RNPs was investigated. Co-tranfection of pTit-N and pTit-P-mCherry together with pTit-M-GFP (which allows expression of M-GFP) did not result in material ejection from N-P inclusions (Supplementary Movie 12, same images acquisition as Supplementary Movies 7 and 11).

We then took advantage of this minimal system to identify P domains that are essential for inclusions formation. P is a modular protein (Fig. 6c) containing an N-terminal domain (PNTD) which binds N0 (the soluble form of N protein devoid of RNA), a C-terminal domain (PCTD) which binds to the N associated with RNA, and two central intrinsically disordered domains (IDD1 and IDD2) flanking a dimerization domain (DD)30. Therefore, pTit plasmids allowing the expression of P deletion mutants were co-transfected with pTit-N and the presence of N-P inclusions was investigated. This revealed that the domains DD, IDD2 and PCTD are required for NB-like structures formation (Fig. 6d).

Domains IDD2 and PCTD are phosphorylated on residues S162, S210 and S27131 (Fig. 6c). An eventual role of phosphorylation on the phase transition was investigated. Residues S162, S210 and S271 were mutated either into alanine residues (to abolish phosphorylation) or into aspartic residues (to mimic the phosphorylated state of the serine). None of those mutations affected P ability to form NB-like structures (Supplementary Fig. 9).

Finally, we delineated more precisely the part of IDD2 domain which is required for phase separation. Deletion of residues 151–181 did not affect P ability to form NB-like structures (Fig. 6e). Therefore, only the amino-terminal part of IDD2 (residues 132–150) is required for this process. This was confirmed by the deletion of residues 139–151, which abolished spherical inclusions formation (Fig. 6e).

## Discussion

In this study, we provide evidence that NBs are liquid droplets formed by phase separation. First, they form spherical structures (Figs. 1a,h and 2b) with a distribution of their axial ratios (Fig. 2a) which is very similar to that observed for other liquid organelles32. Second, they fuse together to form larger spherical structures (Fig. 2b). Third, they are crossed by spherical bubbles which are most probably vesicles being trafficked through their interior, showing that NBs can reversibly deform when encountering a physical barrier (Fig. 2c). Fourth, osmotic shock, induced by a rapid change from isotonic to hypotonic conditions, causes the rapid dissolution of NBs followed by the reformation of smaller spherical inclusions (Fig. 2d), a behavior which is reminiscent of that of the liquid cellular Ddx4-organelles27. Finally, the internal order within the NBs was assessed using FRAP measurements which definitively demonstrate their liquid nature (Fig. 2e–h).

Besides those physical properties, NBs share several other features with cellular liquid organelles21, 25, 27, 33, 34. Particularly, they are composed by viral RNA, an RNA binding protein (N), and a protein containing intrinsically disordered domains (P). Indeed, P and N expressed alone were able to form structures which recapitulate the properties of NBs (Fig. 6a). It is probable that in such N-P inclusions, N is associated with cellular RNAs and forms N-RNA rings and short RNP-like structures35. This minimal system allowed us to investigate the domains of P involved in this process. P dimerization domain, its N-RNA binding domain and the amino-terminal part of its second intrinsically disordered domain (IDD2) were absolutely required for NB-like structures formation (Fig. 6d). This could suggest that P dimers bridge RNPs to ensure the liquid phase cohesion. However, the P dimer is stable36 and the affinity of P dimers for N-RNA rings (which mimick the structure of the viral RNPs) is quite high (Kd = 160 ± 20 nM)37. Such strong interactions are not compatible with a liquid behavior. Furthermore, in vitro experiments have indicated that P dimers are associated with N proteins of the same nucleocapsid37. Therefore, there must be other weak interactions either between P dimers or between nucleocapsids which remain to be characterized. We suggest that those weak interactions are mediated by the amino-terminal part of IDD2 as such intrinsically disordered domains have been involved in the formation of fuzzy complexes38, 39.

The sequence of the amino-terminal part of IDD2 (residues 132–150, Supplementary Fig. 10) is strongly biased toward polar and charged residues which is a common feature of IDDs. It is also enriched in proline residues which, however, are not conserved among the lyssavirus genus. In fact, the IDD2 region (made of 51 residues) is not conserved except the two residues Q and T in position 147 and 148. This has to be compared with a 32% conservation of P amino acid sequence in the genus (Supplementary Fig. 10). Therefore, the ability of IDD2 to induce phase separation does not rely in its amino acid sequence but rather in some global physico-chemical properties which remain to be identified.

Depending on their physicochemical properties, proteins partition preferentially either in the cytosolic phase or the NB liquid phase. Therefore, phase separation is an efficient process to enrich the viral factories in factors which are required for viral transcription or replication. FRAP experiments revealed that P, although more concentrated in NBs, is able to shuttle between the cytosol and the NBs. This may help to recruit cellular partners in the viral factories. Indeed, several identified partners of P are recruited inside NBs. This is the case for the focal adhesion kinase FAK19 and heat shock protein HSP7018. All these proteins have been demonstrated to have proviral activities18, 19 and are also associated with NB-like structures in the minimal system (Supplementary Fig. 8A).

On the other hand, phase separation may also exclude proteins with antiviral properties. Particularly, we show that SGs, which are liquid organelles containing intracellular pattern recognition receptor acting as sensors of RNA virus replication20, 40, 41, do not fuse with NBs (Fig. 3). This indicates that SGs and NBs are forming non miscible liquid phases. Immiscibility of cellular liquid phases has already been observed and underlies the formation of nucleolar subcompartments42. It is worth noting that Rhabdoviruses may have evolved different ways of dealing with SGs: VSV infection also induces the formation of SG-like structures which, however, colocalize with viral replication proteins and RNA43.

In the presence of Nocodazole, which depolymerizes microtubules, the transport of RNPs inside the cytosol is inhibited (Fig. 5b,c). In agreement with this result, live cell imaging revealed a tight association between mobile RNPs and microtubules (Fig. 5d). This explains the previously observed inhibitory effect of Nocodazole on viral production11. On the other hand, the ejection of RNPs from NBs is not affected by Nocodazole and, therefore, is not microtubule-dependent. An attractive hypothesis is that a conformational change of RNPs (e.g., their compaction) modifies their physicochemical properties and drastically decreases their solubility in the NB liquid phase, which results in their ejection. Remarkably, no material ejection from NB-like structure is observed in the minimal system. Therefore, the trigger for ejection is only present upon RABV infection.

As depicted in the model presented in Fig. 7, ejected RNPs are transported in the cytosol far away from the NBs. Then, they can either be incorporated into new virions or give rise to new viral factories. This explains the burst of NBs of intermediate size after 12 h of infection. In presence of Nocodazole (Fig. 7), ejected RNPs remain in the vicinity of the NB. They form new viral factories which fuse with the NB. This explains why, in the presence of the drug, a large single NB is generally observed (Fig. 4).

Cytochalasin D, which depolymerizes actin filaments, has little effect on the number of inclusions and free RNPs formed all along the viral cycle. However, our data indicate a decrease of the size of the largest NBs (Fig. 4). This suggests that the actin filament network limits the fragmentation of NBs.

An overview of the literature suggests that the liquid nature of viral factories might be generalized to several other negative RNA viruses (either segmented or not). This is exemplified by FRAP experiments performed on fluorescent Borna Virus phosphoprotein which is also found in spherical inclusions in the nucleus of infected cells44. This is also suggested by data obtained (i) on filoviruses which forms spherical perinuclear inclusions which are viral replication centers12, 45, (ii) on several paramyxoviruses and pneumoviruses of which N and P proteins form spherical inclusions during the infection46,47,48 and (iii) on bunyaviruses of which genomes segments are found ‘aggregated’ in spherical structures49. Therefore, it appears that the formation of such liquid viral factories constitutes a signature of negative RNA viral infection. It is thus highly probable that the cell has developed mechanisms sensing the presence of such structures and that viruses have in turn developed strategies to increase the furtiveness of their factories. Finally, the characterization of the physicochemical nature of those liquid viral factories will pave the way toward innovative antiviral strategies.

## Methods

### Cells

N2A cells (mouse neuroblastoma, ATCC reference: CCL 131), U373-MG cells (human gliobastoma astrocytoma, ATCC reference: HTB17) were purchased from the ATTC organization (http://www.lgcstandards-atcc.org). BSR cells, a clone from BHK 21 (Baby Hamster Kidney) were obtained from Dr Anne Flamand (from the former Laboratoire de Génétique des Virus, Gif, France). All these cells were grown in Dulbeco’s modified eagle medium (DMEM) supplemented with 10% fetal calf serum (FCS).

BSR-T7/5 cells, a clone of BHK 21 constitutively expressing the T7 RNA polymerase50, were grown in DMEM supplemented with 10% FCS, supplemented with 2% Geneticin (Gibco-Life Technologies).

### Viruses

The challenge virus standard (CVS, French CVS 1151) strain of rabies virus was grown in BSR cells. rCVSN2C-P-mCherry virus encoding a P protein C-terminally fused to the fluorescent protein mCherry has been previously described20 and was grown in N2A cells.

### Antibodies and drugs

The rabbit polyclonal anti-P antibody was previously described11. Mouse monoclonal anti-N antibody (81C4), was produced in a mouse immunized with purified nucleocapsids. Both were used at a 1/1000 dilution. Anti-HSP70 MAb (SPA-810) (1/200 dilution) was obtained from Stressgen. Nocodazole (M1404), Taxol (Paclitaxel T1402), Cytochalasin, D (30385), were purchased from Sigma.

### Plasmids

The plasmid encoding G3BP-EGFP52 was kindly provided by Dr R Lloyd (Department of Molecular Virology and Microbiology, Baylor College of Medicine, Houston, TX 77584, USA). The plasmid pFAK-GFP encoding FAK in fusion with GFP has been described previously19.

pTit-P-mCherry plasmid encoding the P protein of CVS C-terminally fused to the mCherry fluorescent protein has been constructed using Gibson assembly kit (New England Biolabs). The P gene fused to the mCherry has been PCR amplified from the full-length genomic plasmid prN2C-P-mCherry20.

Plasmids pTit-PΔ, encoding truncated forms of P protein, have been constructed using Gibson assembly kit. PCR products encoding fragments of the P gene, with overlapping sequences, were assembled using Gibson assembly kit. The plasmids pTit-P S162A, pTit P S162D, pTit S210A, pTit P S210D and pTit P S271A, pTit P S271D were constructed by using QuikChange Site-Directed Mutagenesis Kit (Agilent Technologies).

Plasmids pTit-M, encoding the M protein of CVS and pTit-M-GFP, encoding M fused to the GFP fluorescent protein, have been constructed using Gibson assembly kit. In the latter construct, GFP was inserted between the N-terminal disordered domain and the globular C-terminal part of the M sequence (after residue 26).

In pTit plasmids53, the gene of interest is under the dependence of the T7 polymerase promoter and the corresponding transcripts contain an internal ribosomal entry site (IRES) located upstream the open reading frame.

### Construction of Recombinant Virus rCVSN2CΔG-P-mCherry

The full-length recombinant rCVSN2C-P-mCherry infectious clone was described previously20. The G coding sequence was removed from the genomic plasmid. The original full-length genomic plasmid was digested with SpeI and MluI restriction enzymes. Two overlapping fragments were amplified by PCR. The first one going from the SpeI site in the P gene to the beginning of the G coding sequence and the second one going from the end of the G coding sequence to the MluI site in the L gene. The PCR products and the digested plasmid were assembled using Gibson Assembly kit (New England Biolabs) to obtain the resulting plasmid, prN2C-P-mCherry-ΔG.

Recombinant virus (rCVSN2CΔG-P-mCherry) was recovered, as described previously with modifications19, 54. Briefly, N2A cells (106 cells) were transfected using Lipofectamine 2000 (Invitrogen) with 0.85 µg of full-length prN2C-P-mCherry-ΔG, in addition to 0.4 µg pTit-N, 0.2 µg pTit-P and 0.2 µg pTit-L53, which encode respectively the N, P and L proteins of rabies virus strain SAD-L16. These plasmids were cotransfected with 0.25 µg of a plasmid encoding the T7 RNA polymerase and the plasmid pCAGGS-G55 encoding the G protein of rabies virus strain Pasteur Virus (PV). Six days post-transfection, the supernatant was passaged on fresh N2A cells transfected with pCAGGs-G, and infectious recombinant viruses were detected 3 days later by the fluorescence of the P-mCherry protein.

### Treatments of cells with drugs

Cells were kept in Dulbecco modified Eagle medium containing 2 μM Nocodazole, 1.25 nM Taxol, 2.5 μM Cytochalasin D for 1 h before virus inoculation and during infection.

### Immunofluorescence staining and confocal microscopy

Cells were fixed for 15 min with 4% paraformaldehyde (PFA) and permeabilized for 5 min with 0.1% TX-100 in PBS. Cells were incubated with the indicated primary antibodies for 1 h at RT, washed and incubated for 1 h with Alexa fluor conjugated secondary antibodies (Thermo Fisher Scientific). Following washing, cells were mounted with Vectashield (Vector labs) containing DAPI. Images were captured using a Leica SP8 confocal microscope (63× oil-immersion objective).

### Quantification of rabies induced structures

BSR cells were infected with CVS virus at an MOI of 0.5. Cells were fixed at different time post-infection and immuno-stained with a mouse monoclonal anti-N antibody (81C4). For each cell, four planes were acquired by confocal microscopy and a max intensity Z-projection was done using ImageJ. Quantifications were performed using the image toolbox of MatLab software. Briefly, the grayscale image was converted to a binary image. The connected components (CC) of the output black and white image were then identified. For each CC, its area (i.e., the number of pixels) and its eccentricity e (i.e., that of the ellipse that has the same second-moments as the CC) were calculated using MatLab software. This allows the classification of the fluorescent structures formed in the cytoplasm based on their size, and the determination of the area of the largest NB in the cell. This also allows the calculation of the axial ratio a/b (where a and b are the long and short axes of the ellipse) of each inclusion using the formula: a/b = 1/(1−e 2)1/2.

### Live cell microscopy

For live-cell imaging, BSR cells were seeded onto 35-mm μdishes (Ibidi) 24 h before infection. Cells were infected with CVS-N2C-PmCherry rabies virus in DMEM fluorobrite medium (Invitrogen) supplemented with 5% FCS. Live-cell time-lapse experiments were recovered with a Zeiss AxioObserver epifluorescence microscope (63× oil-immersion objective). Cells were maintained at 37 °C and 5% CO2 during imaging.

For live imaging of stress granules, U373-MG cells were transfected using Lipofectamine 2000 (Invitrogen) with a plasmid encoding a G3BP-GFP fusion protein prior to cell infection as previously described20.

### Quantification of RNPs velocity

The speed and the trajectories of RNPs were determined manually using the “manual tracking” plugin of the ImageJ software. Only the RNPs that were tracked on four consecutives images were taken into account. The distance between two consecutive positions of a given RNP was measured, allowing the calculation of its instantaneous velocity. Instantaneous velocities along an RNP trajectory were averaged to give the RNP velocity.

### Labeling of tubulin in live cells

Cell-light Tubulin-GFP (Thermo Fisher Scientific) was used to label tubulin with green fluorescent protein (GFP) in live infected cells. Cell-light Tubulin-GFP is a modified baculovirus that encodes the human tubulin gene fused to the GFP. BSR cells were co-infected with CVS-PmCherry and Cell-light Tubulin-GFP. Live-cell time-lapse experiments were recovered with a Zeiss AxioObserver epifluorescence microscope (63× oil-immersion objective). Images were deconvoluated using the Huygens Imaging software (Scientific Volume Imaging). A blind deconvolution algorithm has been used.

### Electron microscopy

Infected BSR cells were fixed for 1 h at room temperature in 0.1 M cacodylate buffer (pH 7.2) containing 1% PFA, 2.5% glutaraldehyde and 1% tannic acid. Cells were then post-fixed for 1 h at 4 °C in 0.1 M cacodylate buffer (pH 7.2) containing 1% osmium and 0.8% potassium ferrycianide and stained for 1 h at 4 °C with 2% uranyl acetate in water. Cells were then dehydrated in increasing concentration of acetone and embedded by Epon with 2,4,6-tris(dimethylaminomethyl) phenol. Polymerization was carried out for 48 h at 60 °C. Ultrathin sections of Epon-embedded material were collected on copper palladium grids (200 mesh). During all the process, cells were kept as a monolayer. The cutting plane is parallel to the cell culture support. The first sections correspond to the basal plane of the cells. These sections were stained with lead citrate and uranyl acetate. Sections were examined with a MET Jeol 1400 electron microscope operated at 80 kV.

### FRAP

FRAP experiments were performed both on BSR cells infected with recombinant virus rCVSN2CΔG-P-mCherry at 16 h p.i. and on BSR-T7/5 cells co-transfected with pTit-N and pTit-P-mCherry at 24 h post-transfection. Data acquisitions were performed on an inverted Nikon Ti Eclipse E microscope coupled with a Spinning Disk (Yokogawa CSU-X1-A1) and cage incubator to control both temperature (37 °C) and CO2 concentration (5%). After excitation with a 561 nm laser (Cobolt Jive, 150 mW), fluorescence from mCherry was detected with a 100× oil immersion objective (Apochromat NA 1.49), a bandpass filter (607/36 Semrock), and a sCMOS camera (Hamamatsu, Orca-Flash4.0 LT). All the FRAP experiments were performed in identical conditions using iLas FRAP module (Roper Scientific): 5 s prebleach, 40 ms bleach, 120 s postbleach at a frame rate of 1 image every 500 ms. Bleaching was performed in a circular region (diameter 2.71 µm) located at the center of the field of view, with the 516 nm laser at 100%.

For the FRAP data analysis the mean intensity of every bleached region was measured and the recovery signal was normalized to the average of the prebleach signal and also corrected for the bleaching during post bleach imaging56. For this double normalization process, the background intensity was estimated by measuring a region outside the cell. To measure the bleaching of the sample during post bleach imaging, the whole cell fluorescence intensity was used for the FRAP experiments of the cytosol while a mean bleaching curve was used for the FRAP experiments of the NBs. To generate this mean bleaching curve, we realized experiments in the same conditions as for the FRAP but without the bleaching event and the mean bleach profile of 18 NBs gave a more robust correction than whole cell measurements due to the very weak numbers of NBs per cell and the strong impact of Z-axis drift of any surrounding NB in these conditions.

After this normalization, individual FRAP curves were averaged to obtain a mean curve. As a single exponential function did not give a satisfactory fit, the mean curve was fitted by a double exponential function:

$$y\left( t \right) = {y_0} + {A_{\rm fast}}\left( {1 - {e^{ - {k_{\rm fast}}t}}} \right) + {A_{\rm slow}}\left( {1 - {e^{ - {k_{\rm slow}}t}}} \right)$$

The diffusion coefficient D associated with the fast phase was calculated using the following formula27:

$$D{\rm{ = }}\frac{1}{{{t_{\frac{1}{2}}}}}{\left( {\frac{x}{{{\rm 2Erfc}{^{ - 1}}\left( {0.5} \right)}}} \right)^2}$$

where Erfc is the complementary error function, $${\rm Erfc}{^{ - 1}}\left( {0.5} \right)\ {\sim}\ 0,4769$$, x is the radius of the beam and $${t_{\frac{{\rm{1}}}{2}}} = \frac{{{\rm ln}2}}{{{k_{\rm fast}}}}$$.

### Statistical analysis

All numerical data were calculated and plotted with mean±SD. Results were analyzed by unpaired two tailed Welch’s t-test or two tailed Mann Whitney U test. The statistical significances of the differences are indicated.

### Data availability

The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files, or are available from the corresponding authors upon request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

This work was supported by grants from the Fondation pour la Recherche Médicale (FRM DEQ20120323711) to Y.G. This work has benefited from the core facilities of Imagerie-Gif, supported by “France‐BioImaging” (ANR‐10‐INBS‐04‐01). We also thank Abbas Abou-Hamdan for careful reading of the manuscript. We thank Sean Whelan for bringing our attention to the sphericity of Negri bodies and Vincent Rincheval for helpful discussion.

## Author information

### Affiliations

1. #### Institute for Integrative Biology of the Cell (I2BC), CEA, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91198, Gif-sur-Yvette cedex, France

• Jovan Nikolic
• , Romain Le Bars
• , Zoé Lama
• , Nathalie Scrima
• , Cécile Lagaudrière-Gesbert
• , Yves Gaudin
•  & Danielle Blondel

### Contributions

D.B., J.N., R.L.B. and Y.G.: Conceived and designed the experiments, J.N., R.L.B., Z.L., N.S. carried out the experiments; C.L.-G., D.B., J.N., R.L.B. and Y.G. analyzed the data; D.B. and Y.G. supervised the research and wrote the manuscript. All authors contributed to editing the manuscript.

### Competing interests

The authors declare no competing financial interests.

### Corresponding authors

Correspondence to Yves Gaudin or Danielle Blondel.

## Electronic supplementary material

### DOI

https://doi.org/10.1038/s41467-017-00102-9

• ### Structural dissection of human metapneumovirus phosphoprotein using small angle x-ray scattering

• Max Renner
• , Guido C. Paesen
• , Claire M. Grison
• , Sébastien Granier
• , Jonathan M. Grimes
•  & Cédric Leyrat

Scientific Reports (2017)

format: plain


text:
Change size of section, subsection, subsubsection, paragraph and subparagraph title

I would like to increase the size of pretty much every headlines, so that it is more prominent than the normal text.

I read about the titlesec package, but can't seem to find good example how to use the package.

Using Gonzalo Medina does in fact change the font size, however, it changes the default font as well.

Before:

After:

Standard classes

Here's an exaple using titlesec:

\documentclass{article}
\usepackage{titlesec}

\titleformat*{\section}{\LARGE\bfseries}
\titleformat*{\subsection}{\Large\bfseries}
\titleformat*{\subsubsection}{\large\bfseries}
\titleformat*{\paragraph}{\large\bfseries}
\titleformat*{\subparagraph}{\large\bfseries}

\begin{document}

\section{Test section}
\subsection{Test section}
\subsubsection{Test section}
\paragraph{Test section}
\subparagraph{Test section}

\end{document}

I used the simplified version of \titleformat (i.e., \titleformat*) since the only desired change was in size, but you can use the extended version if "bigger" changes are required. These are the default definitions for the standard classes (taken from the documentation):

\titleformat{\section}
{\normalfont\Large\bfseries}{\thesection}{1em}{}
\titleformat{\subsection}
{\normalfont\large\bfseries}{\thesubsection}{1em}{}
\titleformat{\subsubsection}
{\normalfont\normalsize\bfseries}{\thesubsubsection}{1em}{}
\titleformat{\paragraph}[runin]
{\normalfont\normalsize\bfseries}{\theparagraph}{1em}{}
\titleformat{\subparagraph}[runin]
{\normalfont\normalsize\bfseries}{\thesubparagraph}{1em}{}

so you can make bigger changes. The following image shows both the standard sizes and the ones obtained with the above modifications:

And here's an exampe using the sectsty:

\documentclass{article}
\usepackage{sectsty}

\sectionfont{\LARGE}
\subsectionfont{\Large}
\subsubsectionfont{\large}
\paragraphfont{\large}

\begin{document}

\section{Test section}
\subsection{Test section}
\subsubsection{Test section}
\paragraph{Test section}
\subparagraph{Test section}

\end{document}

According to an edit to the original question, the OP wanted also a sans serif font:

\documentclass{article}
\usepackage{titlesec}

\titleformat*{\section}{\LARGE\bfseries\sffamily}
\titleformat*{\subsection}{\Large\bfseries\sffamily}
\titleformat*{\subsubsection}{\large\bfseries\sffamily}
\titleformat*{\paragraph}{\large\bfseries\sffamily}
\titleformat*{\subparagraph}{\large\bfseries\sffamily}

\begin{document}

\section{Test section}
\subsection{Test section}
\subsubsection{Test section}
\paragraph{Test section}
\subparagraph{Test section}

\end{document}

KOMA classes

The above approaches are to be used mostly with the default document classes (book, report, article); if the document class used is one of the classes of the KOMA-Script bundle, then it's not advisable to use titlesec (See Incompatibilities between KOMA-Script and titlesec) but to use the features provided by the KOMA classes:

\documentclass{scrartcl}

\setkomafont{section}{\LARGE}
\setkomafont{subsection}{\Large}
\setkomafont{subsubsection}{\large}
\setkomafont{paragraph}{\large}
\setkomafont{subparagraph}{\large}

\begin{document}

\section{Test section}
\subsection{Test section}
\subsubsection{Test section}
\paragraph{Test section}
\subparagraph{Test section}

\end{document}

memoir

For the memoir document class, the situation is analogous: is also not advisable to use titlesec (See About memoir and titlesec incompatibility) but to use the features provided by the class; in the case of the lower sectional units, the class provides the family of commands \setXheadstyle:

\documentclass[article]{memoir}

\begin{document}

\section{Test section}
\subsection{Test section}
\subsubsection{Test section}
\paragraph{Test section}
\subparagraph{Test section}

\end{document}
• Thanks. Works pretty good, except for the first modification. Because changing the size without further options, it changes to a different font. – cherrung Jun 13 '12 at 21:49
• @cherrung: How about just adding \sffamily to the list of corrections? – Werner Jun 13 '12 at 22:22
• @cherrung as Werner said, simply add \sffamily, as I did in the last code of my updated answer. – Gonzalo Medina Jun 13 '12 at 22:43
• @cherrung which document class are you using? The standard classes don't use a sans-serif font by default, so your last requirement makes me wonder if you are using some non-standard document class. – Gonzalo Medina Jun 13 '12 at 22:54
• @cherrung that is crucial information that you should have mentioned in your original question. KOMA-Script and titlesec are incompatible, so titlesec is not advisable here; I've provided an example on how to proceed with the KOMA classes in my updated answer. – Gonzalo Medina Jun 13 '12 at 23:35

You can just modify the sectional commands with the appropriate font size. Here's a small example in the article document class:

\documentclass{article}
\begin{document}
\section{A section}
\subsection{A subsection}
\subsubsection{A subsubsection}

\makeatletter
\renewcommand\section{\@startsection {section}{1}{\z@}%
{-3.5ex \@plus -1ex \@minus -.2ex}%
{2.3ex \@plus.2ex}%
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\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
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{\normalfont\large\bfseries}}% from \normalsize
\makeatother

\section{A section}
\subsection{A subsection}
\subsubsection{A subsubsection}

\end{document}

For a different font, replace \normalfont with the appropriate definition. For example, using \sffamily will yield:

titlesec package options

For simple cases, you can use \usepackage[<size>]{titlesec}, where <size> is big, medium, small, or tiny.

Example code

\documentclass{article}

\usepackage[medium]{titlesec}

\begin{document}
\section{A section}
\subsection{A subsection}
\subsubsection{A subsubsection}
\end{document}

For completeness I'll give a ConTeXt solution also.

You can change the heading style using:

The font switch \tf equals the body fontsize, \tfa is 1.2 times \tf; \tfb is 1.2 times \tfa and so on. If you want sans serif headings, then use


text:
Research website of Vyacheslav Gorchilin
2019-10-02
The conversion unit in the vector
The unusual name of this work is associated with non-traditional notions of space, and everything located in it, the world around us. In the first part we will show and prove that a scalar is a rolled vector, and discover how from the first "cook" the second, and Vice versa. This approach to things will allow a new look at the concept of measures and dimensions — length, mass, speed, acceleration, etc., which has long been well established in our minds. The work is interesting as classical physicists and netradiciniu, and some of its conclusions would relate to our esteemed and tireless seekers of free energy :)
1. Two points
Figure (1) shows the motion of two points relative to each other. Here, on the horizontal axis delayed values of the coordinates $$x$$ and the vertical coordinate value $$y$$. Under the "points" we mean the geometric points that have no dimensions [1], but in the figures for better clarity, we will depict it in small circles.
 Fig.1. The motion of one point relative to another in two-dimensional space
From a mathematical point of view, the movement of the red point relative to the blue (Fig. 1) describes a simple two-dimensional vector: $\mathbf{V} = \mathbf{i}\cdot 4\sin(x) + \mathbf{j}\cdot 1 \qquad (1.1)$ where: $$\mathbf{i}, \mathbf{j}$$ are unit vectors of the axes $$x$$ and $$y$$, respectively [2]. The basis formed by these axes in the theory of vector algebra are called orthonormal [3], as if to say more simple words, a unit vector is simply perpendicular to each other. The space formed by two vectors is two-dimensional [4]. The number $$1$$ in this formula, means the distance between the points along the coordinate $$y$$, and $$4$$ — amplitude of oscillations along the coordinate $$x$$. Indeed, if $$\sin(x)$$ tends to "-1", the red dot is moved on the coordinate of the left and reaches a value of "-4", and if $$\sin(x)$$ tends to "1" that moves to the right and reaches the value "4".

Here and below, in bold we will denote vector quantities. Example: $$\mathbf{V},\, \mathbf{i},\, \mathbf{j},\, \mathbf{g}$$

What do you think will happen if we remove the ambient two-dimensional space, and leave our two points without coordinates $$x$$ and $$y$$? As we remember, our point has no dimensions, which means that the provisions of the "right" or "left".
 Fig.2. The motion of one point relative to another in one-dimensional space
What, then, is to observe one point relative to another? Obviously, this is an oscillatory motion relative to each other, and committed in one-dimensional space! If we denote the unit vector of this one-of coordinates $$\mathbf{g}$$, the equation of motion of the red point on the blue we can find from the Pythagorean theorem, and the vector will become even simpler: $\mathbf{V} = \mathbf{g}\cdot \sqrt{(4\sin(x))^2 + 1} \qquad (1.2)$ Obtained one-dimensional vector, and its dynamics depending on the parameter $$x$$, visualized in figure (2). Mentally it can be represented as a line connecting two points in Fig. 1, and if so, then it can be represented in the form of an ordinary scalar function: $f(x) = \sqrt{(4\sin(x))^2 + 1} \qquad (1.3)$
Now we need to resolve the emerging paradox that our points can simultaneously exist and move, as in the one-dimensional and two-dimensional space. How so? After all, we used to make sense of the world as see it and for us it definitely! Looking ahead we can say that the movement of a point is always one-dimensional, and large dimension will be defined by properties of its surrounding space and its measuring devices. But in order to realize this, we have to prove a small theorem that might turn your ideas about the world.

The materials used
1. Wikipedia. Point.
2. Wikipedia. Unit vector.
3. S. V. Karpov, V. G. Litovka, T. A. Manicheva, A. P. Filimonov. Elements of vector algebra.
4. Wikipedia. A two-dimensional space.


text:
# magnetic field outside a solenoid

I couldn't picture the idea behind the zero magnetic flux outside the solenoid properly.My teacher explained it as"Components of the magnetic field in other directions are cancelled by opposing fields from neighbouring coils. Outside the solenoid the field is also very weak due to this cancellation effect and for a solenoid which is long in comparison to its diameter, the field is very close to zero. "But I didn't understand anything at all.First he said about the components of magnetic field in other direction being cancelled by magnetic field, but didnt further explain and said nothing to support his arguement.If somebody has some simple,inituitive way of understanding this phenomenon then please explain it...

You first should understand that magnetic field are the circles (of magnetic force) around the wire with current. The direction of circles (counterwise or clockwise) depends on the direction of the currend (forth or back). Now, you can guess what happens to the magnetic field when you curve the wire into a circle. Then, you may also stack cricles of current next to each other to form a cylinder. Now, try to imagine what will be the magnetic field in each of these cases.

The magnetic fields from different currents add up (total field in every point is the sum of fields from different currents at that point).

I agree that symmetry, which tells that cancel is complete, is not obvious here. Because outside the loop, one wire is closer is to the outer region than the wire from the opposite side, which cancels out the first. But, this should give a qualitative picture, what is happening. The complexity of symmetry might be the same as zero gravity inside a massive sphere.

edit I am coming to conclusion that it is impossible to cancel out the outer field completely just because the field approaches infinity as we come closer to the wire from outside and there is nothing close to infinity from the opposite piece of the loop that can cancel this out. Solenoid just cancels out componets of the field perpendicular to its axis and parallel to the plain of the windings since windings are very close and, though current in them flows in one direction, say forward, upper wire induces left magnetic field between the wires whereas bottom wire produces the same right field. So, all the winds that blow in the planes between windings are eliminated. The axial fields inside the solenoid are added together. Axial outer are cancelled to some extent but not completely. But, this is not bad because, though they create the axial field, opposite to what solenodi should produce from its internal part, the irregularities at the edge of solinoid wrap it around and eliminate the negative effect.

See that though the major field is produced from left to right, some field blows from right from right to left outside. Yet, this does not harm the major flow because edges isolate the negative flow.

I tried to address this question in a short article that recently got published in the European Journal of Physics. https://arxiv.org/abs/1610.07876

You can also look at this article by Farley, Price, which uses a nice argument and physically motivates the problem: http://scitation.aip.org/content/aapt/journal/ajp/69/7/10.1119/1.1362694

Also, there is an exact calculation done in this old article: http://paginas.fe.up.pt/~ee08173/wp-content/uploads/2014/03/finite-solenoid.pdf

• This is the correct answer. It is really frustrating that nobody addresses the complete lack of rigor in all other answers of the type "the field on the outside is zero because the solenoid is infinite and magnetic lines reach zero density, as their flux has to be conserved". – Krastanov Jul 24 '17 at 21:57
• Could you consider writing a short explanation of all of this in your answer. Referencing own work is not against stackexchange rules, but the goal is to have somewhat of a self-contained answer before the references. – Krastanov Jul 24 '17 at 21:59
• @Krastanov: I'm not sure I can write down the arguments of either my paper or Richard Price's paper in a very concise way. I did try to write my argument on Quora once, and it became quite long. – Aritro Pathak Jul 25 '17 at 0:07
• The answers of the type you mention are really problematic. It leads to another fundamental thing ; that the correlation of density of field lines with field strength is due to the divergence of the field being 0. It still shocks me that so many people haven't been taught this/haven't thought about it very much/stubbornly refuse to accept it once told. There is this answer that addresses it on another thread: physics.stackexchange.com/a/267589/78842 , which is correct, and then my answer in the same thread. But many answers there and elsewhere are not very good. – – Aritro Pathak Jul 25 '17 at 1:20

What goes through the middle must go out one end and back in the other. The magnetic field on the outside is free to spread out and in doing so the Wb/m^2 goes low, but is not zero. You must distinguish between assumptions to prove a theory and real life examples. An infinitely long solenoid (for the theory) would have infinite cross-sectional area outside the solenoid (for the flux return path) so the Wb/m^2 would reduce to zero in math terms. Magnetic fields lines from each coil of solenoid will add, there's no cancelling going on here, or anywhere (just the math with infinity term as denominator means whole term becomes zero.)

Imagine a loop of current carrying wire and consider two small elements dl which are diametrically opposite.Try to find the magnetic field at a point outside this loop due to these two small elements which lies on a line which passes through both these small elements using Biot Savarts Law.When you try to find the direction of the magnetic field at this point using right hand rule you will find that they are in opposite directions and hevce cancel.Even after this cancellation there will be a small field present because the magnetic field due to the element which lies closer to the point will be greater than from the element which is further away.The distance between them is the diameter of the loop. Now imagine that this loop is part of solenoid which is infinitely long and try to find the contribution to the magnetic field to due the same two elements at a point outside the solenoid but located an infinite distance away along the length of the solenoid.Since it is an infinite distance away the angle between the elements and the position vector can be cosidered to be the same.Also the distances of the elements from the point can now be considered equal since the diameter is negligible in comparison and hence the field due to both can be considered be of equal magnitude and cancel completely giving rise to zero magnetic field. Hope this helps.

Magnetic field outside a long solenoid can never be zero. Because a solenoid of finite length will have edges and field will come out. However, if we imagine a straigth solenoid of infonite length or a toroid, it will not have edges from where lines of force can come out. therefore field would be zero outside a toroid or a strigth solenoid of infilite length.

First, I assume that the solenoid is infinitely long. If you choose a Polar coordinate with $z$ being along the solenoid's axis, based on symmetry you can argue that $\vec{B} = B(r) \hat{z}$. Also, at infinity the magnetic field should be zero.

Now we use Ampre's law for the following path: assume a rectangular path with one side parallel to the $z$ axis out side of the solenoid $C_1$, one side parallel to the axis again but at infinity $C_3$, and the other two sides of this rectangle connecting these two lines together $C_2 \& C_4$. Now if you write the Ampre's law, you see that the $\int \vec{B}.ds$ is zero along $C_3$, since $B$ at infinity is zero. $\int \vec{B}.ds$ is also zero along $C_2,C_4$ because the magnetic field is perpendicular to this path. So $\int \vec{B}.ds$ should also be zero along $C_1$, since there is no current passing through this closed path. So $\vec{B}(r) L = 0$, so $\vec{B}(r) = 0$.

The magnetic field outside an infinitely long AC solenoid is NOT zero, as shown by explicit calculation from Maxwell's equations by Abbot & Griffiths, American Journal of Physics 53, 1203 (1985), and also independently by Jacque Templin, American Journal of Physics 63, 916 (1995).

Every demonstration I've seen that shows the magnetic field outside an infinitely long DC solenoid is zero uses the approximation that the current of the solenoid is everywhere perpendicular to its axis, which, of course, is not true for any solenoid whose current is carried by a wire of non-vanishing dimensions. If the diameter of the wire is D and the circumference of the solenoid is C then there is a component of the current parallel to the axis of the solenoid, equal to (D/C)*I where I is the current in the wire. That component of the current makes a magnetic field outside the solenoid and though it may be a very weak field compared to that inside the solenoid, it doesn't vanish when the solenoid is infinitely long.

For an infinite solenoid, Ampère's theorem makes it possible to show very simply that the magnetic is uniform outside. If the magnetic field is zero to infinity, it is zero everywhere outside.

One can attempt a dimensional analysis to show that the field is zero at infinity. For a solenoid of radius $$R$$ , if the surface current density is $${{j}_{s}}$$, dimensional analysis shows that the magnetic field will be of the form $${{B}_{z}}={{\mu }_{0}}{{j}_{s}}f(r/R)$$

If, the radius of the solenoid is reduced to $$0$$ the currents cancels each over and the field must tend to $$0$$ : $$\underset{R\to 0}{\mathop{\lim }}\,f(r/R)=0$$ or $$\underset{u\to \infty }{\mathop{\lim }}\,f(u)=0$$

So $$\underset{r\to \infty }{\mathop{\lim }}\,f(r/R)=0$$ the field tends to 0 at infinity.

Rather formal proof ! Among my friends, some find it convincing, others do not!

• "Ampère's theorem makes it possible to show very simply that the magnetic is uniform outside." How can I do so? – Ma Joad Oct 29 '19 at 10:47


text:
# GlueX Physics Photoproduction

## Photoproduction

### Linear Polarization

In GlueX, the linearly polarized photons will be produced by coherent bremstrahlung. In this process, the photons in the coherent scattering peak from the crystal radiator are polarized. The direction of the polarization vector (in the lab) depends on the orientation of the crystal, but is fixed (in the lab). The degree of polarization of the coherent peak depends on the ratio of the photon energy to the end-point energy.

In the photo-production of mesons (via t-channel processes), the relevant

#### Reference Frames

The lab frame is typically taken with the z-axis along the incoming photon direction and the y-axis normal to the surface of the Earth. The x-axis is then taken ${\hat x}={\hat y}\times {\hat z}$. Related to the lab frame is the center of mass frame. The coordinate directions are the same, but the two frames are connected by a boost along the z-axis with $\beta =E_{{\gamma }}/(E_{{\gamma }}+m_{{p}})$ and the total energy in the center of mass is ${\sqrt {s}}={\sqrt {m_{{p}}^{{2}}+2m_{{p}}E_{{\gamma }}}}$.

In t-channel photo production, a particle X is produced recoiling against an out-going proton. The Mandelstam t is given as $t=(p_{{X}}^{{\mu }}-k^{{\mu }})(p_{{\mu X}}-k_{{\mu }})$ where k is the photon momentum and $p_{{X}}$ is the 4-momentum of the produced particle $X$.

### References

#### Experimental Papers

##### CLAS Publications
• First Measurement of the Polarization Observable E in the p⃗ (γ⃗ ,π+)n Reaction up to 2.25 GeV, S. Strauch et al. (CLAS Collaboration) arXiv.
• Data analysis techniques, differential cross sections, and spin density matrix elements for the reaction γp → φ p B. Dey et al., (CLAS Collaboration), Phys. Rev. C89, 055208 (2014), DOI
• Differential Photoproduction Cross Sections of the Σ0(1385), Λ(1405), and Λ(1520), K. Moriya et al. (CLAS Collaboration), Phys. Rev. C88 045201 (2013), DOI.
• Cross sections for the γp → K∗+Λ and γp → K∗+Σ0 reactions measured at CLAS, W. Tang et al. (CLAS Collaboration), Phys. Rev. C87 065204, (2013), DOI.
• Beam asymmetry Σ for π+ and π0 photoproduction on the proton for photon energies from 1.102 to 1.862 GeV, M. Dugger et al. (CLAS Collaboration), Phys. Rev. C88, 029901 (2013), DOI.
• Differential cross sections and recoil polarizations for the reaction γ p → K+ Σ0, B. Dey, et al., Phys. Rev. C82, 025202 (2010), DOI.
• Differential cross section and recoil polarization measurements for the γ p → K+ Λ reaction using CLAS at Jefferson Lab, M.E. McCracken et al (CLAS Collaboration), Phys. Rev. C85, 025201 (2010), DOI.
• Differential cross sections and spin density matrix elements for γp → ω p, M. Williams, et al., (CLAS Collaboration), Phys. Rev. C80, 065208 (2009), DOI, arXiv.
• Partial wave analysis of γ p → ω p and the search for nucleon resonances, M. Williams, et al., (CLAS Collaboration), Phys. Rev. C80, 065209 (2009), DOI, arXiv.
• Differential cross sections for the reactions γ p → η p and γ p → η' p, M. Williams, et al., (CLAS Collaboration), Phys. Rev. C<80>, 045213 (2009), DOI, arXiv.
• π+ photoproduction on the proton for photon energies from 0.725 to 2.875-GeV, M. Dugger et al. (CLAS Collaboration), Phys. Rev. C79, 065206 (2009),DOI.
• Search for the photo-excitation of exotic mesons in the π+ π+ π- system, M. Nozar et al., Phys. Rev. Lett. 102, 102002 (2009), DOI.
• Photoproduction of π+π- meson pairs on the protonM. Battaglieri et al., Phys. Rev. D80, 072005 (2009), DOI.
##### Other Publications
• First Measurement of the Σ Beam Asymmetry in η' Photoproduction off the Proton near Threshold, P. Levi Sandri et al. (GrAAL), submitted to Eur. Phys. J. C, (2015), arXiv.
• The polarization observables T, P, and H and their impact on γ p → p π 0 multipoles, J. Hartmann et al. (CBELSA/TAPS), Submitted for publication (2015), arXiv.
##### Older Publications
• On the apparent shift of the rho meson mass in photoproduction, P. Soeding, Phys. Lett. 19, 702, (1966), DOI.
• Interference Effects in Photoproduction of ρ0 Mesons, Allan S. Krass, Phys. Rev. 159, 1496 (1967), DOI
• On the Analysis of Vector-Meson Production by Polarized Photons, K. Schilling, P. Seyboth and G. Wolf, Nucl. Phys. B15 (1970), 397-412, DOI, pdf
• Bubble-Chamber Study of Photoproduction by 2.8 and 4.7 GeV Polarized Photons. I. Cross-Section Determination and Production of ρ0 and Δ++ in the Reaction γp → pπ+π- J. Ballam et al., Phys. Rev. D5 (1972), 545-589, DOI, pdf.
• Vector-Meson Production by Polarized Photons at 2.8, 4.7 and 9.3 GeV, J. Ballam et al., Phys. Rev. D7 (1973), 3150-3177, DOI, pdf.
• Spin-Parity Analysis of ρ' in the reaction γ p → π+ π+ π- π-p at 9 to 18 GeV, Joseph C.H. Park, Nucl. Phys. B58, (1973), 45-54, DOI, pdf.
• High-Energy Production and Decay of vector and Tensor Mesons, A.C. Irving and C. Michel, Nucl. Phys. B82, (1974), 282-325, DOI, pdf.
• The Reaction γ p → π+ π- π+ π- p at High-energy and dissociation into 4π, M. Davier, et al., Nucl. Phys. B58 (1973), 31-44, DOI, pdf.
• A Search for B and ρ'(1250) Production in the Reaction p π+π- + Neutrals at 2.8, 4.7 and 9.3 GeV, J. Ballam, et al., Nucl. Phys B76, 375-399 (1974), DOI, PDF.
• S-P Wave Interference in K+K- in Photoproduction Near K+K- Threshold, D.C. Fries et al., Nucl. Phys. B143, 408, (1978), DOI.
• The hadronic properties of the photon in high-energy interactions , T.H. Bauer, R.D. Spital, D.R. Yennie, and F.M. Pipkin Rev. Mod. Phys 50, No. 2, (1978) 261, DOI.

#### Theoretical Papers

• Polarization observables in the longitudinal basis for pseudo-scalar meson photoproduction using a density matrix approach B. Dey et al., Phys. Rev. C83, 055208 (2011), DOI.
• Vector meson photoproduction studied in its radiative decay channel, Qiang Zhao, J. S. Al-Khalili, and P. L. Cole, Phys. Rev. C71, 054004 (2005), DOI.
• Constraints on vector meson photoproduction spin observables W. M. Kloet and Frank Tabakin, Phys. Rev. C61, 015501 (1999), DOI.
• Spin information from vector-meson decay in photoproduction, W. M. Kloet, Wen-Tai Chiang and Frank Tabakin, Phys. Rev. C58, 1086 (1998), DOI.
• Completeness rules for spin observables in pseudoscalar meson photoproduction Wen-Tai Chiang and Frank Tabakin, Phys. Rev. C55, 2054 (1997), DOI.

#### Generalized Parton Distributions (GPD)

• Timelike Compton scattering: exclusive photoproduction of lepton pairs, E,.R. Berger et al., Eur.Phys.J.C23:675-689,2002 or hep-ph/0110062 pdf
• Photoproduction of a π ρ pair with a large invariant mass and transversity generalized parton distribution, M. El Beiyad et al., hep-ph/1001.4491 pdf

==Other==


text:
Derivative of Hyperbolic Cosine of a x

Theorem

$D_x \left({\cosh a x}\right) = a \sinh a x$

Proof

 $\displaystyle D_x \left({\cosh x}\right)$ $=$ $\displaystyle \sinh x$ Derivative of $\cosh x$ $\displaystyle \implies \ \$ $\displaystyle D_x \left({\cosh a x}\right)$ $=$ $\displaystyle a \sinh a x$ Derivative of Function of Constant Multiple

$\blacksquare$


text:
# What do eigenvalues/eigenvectors of the yield/forward rates covariance matrices mean?

I have 5 bonds (with maturities 1,2,3,4,5 years) which I calculated the yield curve for 10 days. I also calculated the forward rates from the yield rates. Now I've been told to calculate the covariance of the yield rates and the forward rates, as well as their eigenvalues/eigenvectors.

I'm assuming the covariance of the yield rates tell me how the bonds with different maturities move along with each other over time? But I'm not sure what the forward rates tell me.

-

The PCA analysis does not really tell you what the bonds do but it tells you how the rates move together. The variations of $n$ rates (i.e. 1 y, 2y, ...) are split up in (at first) abstract factors like $$\Delta R_i = \sum_{j=1}^n e_{i,j} f_j$$ where $\Delta R_i$ is the change in the rate $i$ and $f_j$ is factor $j$ and $e_{i,j}$ is the (factor loading=) influence of factor $j$ to the rate $i$. The factors coming from PCA are uncorrelated and ordered by the size of their variance (largest first). Then it turns out that usually all rates have $e_{i,1}$, the influence of the first factor, with the same sign. This means that a change in this factor is in the same direction for all rates. The $e_{i,2}$ have a different sign for short terms as opposed to longer terms. Thus the second factor influences short and long rates differently - this is interpreted as steepening/flattening factor. For the third one often sees a curvature pattern (same sign for short and long and another sign for the middle terms).

Mathematical detail: the factor loadings are the eigenvectors of the covariance matrix of $\Delta R_i,i=1,\ldots,n$ and the variances of the factors are the squared eigenvalues.

Looking at the total variance explained by these it often turns out that $n$ rates can be described by the loadings to these 3 factors and the variances of these factors.

You can do this with spot rates and with forward rates. It would be interesting how a PCA of spot and forward rates together looks.

Note that such a reduction of dimensionality is an approximation and as always - take care doing it.

One of the first Google hits points to an article with more mathematical details:PRINCIPAL COMPONENT ANALYSIS by GRAEME WEST.

Problems of the interpretation are described here inPotential PCA interpretation problems for volatility smile dynamics by Reiswich and Tompkins.

-
Hi, for yield curve and forward rate analysis I recommend Overview of Forward Rates by Antti Ilmanen. Which forward rates to use? Why not all? $f(1,i), i = 2,\ldots,5$, $f(2,i), i = 3,\ldots,5$ and so on. From the construction of the forward rates the result can be guessed. E.g. $f(1,2) \approx 2 R_2 - R_1$ (approximately) so I guess the level factor will be present (due to the $2$) and the steepeining/flattening loading should be similar to $R1$ and $R_2$ but please do t he analysis and share the results. –  Richard Feb 4 '13 at 8:15


text:
# Beamer margins in bibliographies

In beamer I am using this code:

\begin{frame}[allowframebreaks]
\frametitle{REFERENCES}
\printbibliography
\end{frame}

to produce a bibliography that will span multiple slide pages (frames). All of this works well, except the left and right margins are way to the end of the frame. How can I bring those doggies back home?

• You should definitely reconsider using a biblography in a talk (especially when it's multiple pages long). It's a bunch of information thrown on the wall that nobody is interested in at that point. However, it makes sense for a version that is being published after the talk. – Marco Feb 29 '12 at 17:48
• @Marco: Did the OP mention anywhere that this is for a talk? It might as well be a 600 slides university course, for which a (multiple pages) bibliography would be more than appropriate. – Daniel Mar 15 '12 at 20:07

I had the same problem today, and here is my simple solution: I just define a single column using the "columns" environment. By default, Beamer will put that column in the center of the page, so this should take care of both margins.

\begin{frame}[allowframebreaks]{References}
\begin{columns}
\column{0.85\paperwidth}
\printbibliography
\end{columns}
\end{frame}

It's hard to tell what's going wrong without a MWE, but the narrow left margin might be due to the fact that your biblatex style doesn't account for beamer bibliography labels. This is the case for any standard style other than the numeric and alphabetic variants.

To fix this problem and increase the right margin, you can redefine the bibliography environment in your preamble.

\defbibenvironment{bibliography}
{\list{}
{\settowidth{\labelwidth}{\usebeamertemplate{bibliography item}}%
\setlength{\leftmargin}{\labelwidth}%
\setlength{\rightmargin}{\labelwidth}%
\setlength{\labelsep}{\biblabelsep}%
\setlength{\itemsep}{\bibitemsep}%
\setlength{\parsep}{\bibparsep}}}
{\endlist}
{\item}

With styles that already print labels in the bibliography, the text beamer template is preferable:

\setbeamertemplate{bibliography item}[text]

Under any document class, biblatex increases the left margin to accommodate the widest label among all entries represented in the bbl file. This rule holds regardless of whether or not the entry with the widest label is printed in the bibliography.

Other beamer templates make any numeric or alphabetic citation labels rather meaningless, unless you edit the style to print the labels with the bibliography items. In any case the label width needs to be adjusted. This can be done by adding the following patch to your preamble.

\makeatletter
\pretocmd{\blx@env@bibliography}
{\settowidth{\labelnumberwidth}{\usebeamertemplate{bibliography item}}%
\settowidth{\labelalphawidth}{\usebeamertemplate{bibliography item}}}{}{}
\makeatother


text:
English Információ A lap Pontverseny Cikkek Hírek Fórum

Rendelje meg a KöMaL-t!

VersenyVizsga portál

Kísérletek.hu

Matematika oktatási portál

P. 4672. A solid sphere of volume $\displaystyle 1~{\rm dm}^3$ made of Aluminium is connected with a thin thread to a wooden sphere made of pine tree, which has a density of $\displaystyle 0.5~\rm g/cm^3$. The spheres are fully immersed into water and they are at rest.

$\displaystyle a)$ What is the volume of the wooden sphere?

$\displaystyle b)$ What is the tension in the thin thread connecting the spheres?

(3 points)

Deadline expired on 10 December 2014.

Statistics on problem P. 4672.
 114 students sent a solution. 3 points: 78 students. 2 points: 25 students. 1 point: 3 students. 0 point: 7 students. Unfair, not evaluated: 1 solution.

• Problems in Physics of KöMaL, November 2014

•  Támogatóink: Morgan Stanley


text:
List of Fitness Bands and Artists : Vote for your favorites.

Fitness (often denoted ${\displaystyle w}$ or ω in population genetics models) is the quantitative representation of natural and sexual selection within evolutionary biology. It can be defined either with respect to a genotype or to a phenotype in a given environment. In either case, it describes individual reproductive success and is equal to the average contribution to the gene pool of the next generation that is made by individuals of the specified genotype or phenotype. The fitness of a genotype is manifested through its phenotype, which is also affected by the developmental environment. The fitness of a given phenotype can also be different in different selective environments.

With asexual reproduction, it is sufficient to assign fitnesses to genotypes. With sexual reproduction, genotypes are scrambled every generation. In this case, fitness values can be assigned to alleles by averaging over possible genetic backgrounds. Natural selection tends to make alleles with higher fitness more common over time, resulting in Darwinian evolution.

The term "Darwinian fitness" can be used to make clear the distinction with physical fitness. Fitness does not include a measure of survival or life-span; Herbert Spencer's well-known phrase "survival of the fittest" should be interpreted as: "Survival of the form (phenotypic or genotypic) that will leave the most copies of itself in successive generations."

Inclusive fitness differs from individual fitness by including the ability of an allele in one individual to promote the survival and/or reproduction of other individuals that share that allele, in preference to individuals with a different allele. One mechanism of inclusive fitness is kin selection.

Read more...


text:
"Geometry and Analysis on Groups" Research Seminar

Time: 2016.10.18, 15:00–17:00
Location: Seminarraum 9, Oskar-Morgenstern-Platz 1, 2.Stock
Title:"Amenability for étale groupoids, failures of Hulanicki's theorem and the weak containment story."
Speaker: Martin Finn-Sell (Universität Wien)
Abstract: In this talk, we will discuss variants of the classical theorem of Hulanicki from 1964 that shows that amenability of a discrete group can be characterised by the property that every representation is weakly contained in the left regular representation. A version of this result has been conjectured to hold for discrete groups acting on compact Hausdorff spaces, but has recently been shown to fail by Willett for certain pathological groupoids, far from actions of groups, originally constructed by Higson, Lafforgue and Skandalis in their work on the Baum-Connes conjecture - these groupoids are non-amenable, but they have the "weak containment" property. I will present a result (that is joint with Vadim Alekseev), in which we give a new example of groupoids that are not amenable, have weak containment and are much closer to group actions. The introductory part of this seminar will focus on weak containment of representations, amenability and the original theorem of Hulanicki, and the second half will focus on the broader question concerning groupoids - here we will understand what it means for a groupoid to be amenable, have the weak containment property and the overall scheme of the proof using representation theory of groups and homological techniques concerned with groupoid $$C^*$$–algebras.


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StudyWithGenius

Class 10 Maths MCQs for Arithmetic Progressions

Welcome to your Class 10 Maths MCQs for Arithmetic Progressions

1.
11th term of the A.P. -3, -1/2, ,2 …. Is
2.
If the 2nd term of an AP is 13 and the 5th term is 25, what is its 7th term?
3.
If the sum of three numbers in an A.P. is 9 and their product is 24, then numbers are
4.
If the 2nd term of an AP is 13 and the 5th term is 25, what is its 7th term?
5.
The 21st term of the AP whose first two terms are –3 and 4 is
6.
30th term of the A.P: 10,7, 4, …, is
7.
Class 10 Maths MCQs for Arithmetic Progressions
8.
If p, q, r and s are in A.P. then r - q is
9.
The famous mathematician associated with finding the sum of the first 100 natural numbers is
10.
The first term and common difference for the A.P. 3,1,-1,-3 is:
11.
If a=10 and d=10, then first four terms will be:
12.
The (n – 1)th term of an A.P. is given by 7,12,17, 22,… is
13.
The nth term of an A.P. is given by an = 3 + 4n. The common difference is
14.
The first four terms of an AP, whose first term is –2 and the common difference is –2, are
15.
In an Arithmetic Progression, if a=28, d=-4, n=7, then an is:


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# Doppler physics homework

1. Nov 28, 2005

### dustybray

I'm having trouble solving this one:

Two motorcycles are traveling in opposite directions at the same speed, when one of the cyclists blasts her horn, which has a frequency of 544 Hz. The other cyclist hears the frequency as 563 Hz. If the speed of sound in air is 344 m/s, what is the speed of the motorcycles? Ans. 5.88 m/s

If I use f[o] = f * ( ( v + v[o] ) / ( v – v ) )

Then I guess I could make v = -v[o]; so,
( f[o] / f ) = ( ( v + v[o] ) / ( v + v[o] ) )
or
( f[o] / f ) = ( ( v - v[o] ) / ( v - v[o] ) )

But how do I solve for v[o]??

Thanks,

dusty.......

2. Nov 28, 2005

### Staff: Mentor

Starting with f[o] = f * ( ( v + v[o] ) / ( v – v ) ),

then f[o]/f = ( ( v + v[o] ) / ( v – v ) ), and then

f[o]/f * ( v – v ) = ( ( v + v[o] ), and then let v = v[o].

But one may be confusing v's.

The equation should include the speed of sound, and in the initial equation, v would be the speed of sound, and on then solves for v,v[o], both being equal.