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Companion stars of Type Ia supernovae and hypervelocity stars

# Companion stars of Type Ia supernovae and hypervelocity stars

Bo Wang National Astronomical Observatories/Yunnan Observatory, the Chinese Academy of Sciences, Kunming 650011, China
Zhanwen Han National Astronomical Observatories/Yunnan Observatory, the Chinese Academy of Sciences, Kunming 650011, China
###### Key Words.:
stars: binaries: close – stars: supernovae: general – stars: white dwarfs – stars: individual: (US 708)
###### Abstract

Context:Recent investigations of the white dwarf (WD) + He star channel of Type Ia supernovae (SNe Ia) imply that this channel can produce SNe Ia with short delay times. The companion stars in this channel would survive and potentially be identifiable.

Aims:In this Letter, we study the properties of the companion stars of this channel at the moment of SN explosion, which can be verified by future observations.

Methods:According to SN Ia production regions of the WD + He star channel and three formation channels of WD + He star systems, we performed a detailed binary population synthesis study to obtain the properties of the surviving companions.

Results:We obtained the distributions of many properties of the companion stars of this channel at the moment of SN explosion. We find that the surviving companion stars have a high spatial velocity (400 km/s) after SN explosion, which could be an alternative origin for hypervelocity stars (HVSs), especially for HVSs such as US 708.

Conclusions:

## 1 Introduction

Type Ia supernovae (SNe Ia) play an important role in the study of cosmic evolution, especially in cosmology. They have been applied successfully in determining cosmological parameters (e.g., and ; Riess et al. 1998; Perlmutter et al. 1999). It is generally believed that SNe Ia are thermonuclear explosions of carbon–oxygen white dwarfs (CO WDs) in binaries (for the review see Nomoto et al. 1997). However, there is still no agreement on the nature of their progenitors (Hillebrandt & Niemeyer 2000; Podsiadlowski et al. 2008; Wang et al. 2008), and no SN Ia progenitor system has been conclusively identified from before the explosion.

Over the past few decades, two families of SN Ia progenitor models have been proposed, i.e., the double-degenerate (DD) and single-degenerate (SD) models. Of these two models, the SD model is widely accepted at present (Nomoto et al. 1984). It is suggested that the DD model, which involves the merger of two CO WDs (Iben & Tutukov 1984; Webbink 1984; Han 1998), likely leads to an accretion-induced collapse rather than to an SN Ia (Nomoto & Iben 1985). For the SD model, the companion is probably a main-sequence (MS) star, a slightly evolved subgiant star (WD + MS channel), or a red-giant star (WD + RG channel) (e.g., Hachisu et al. 1996, 1999a,b; Li van den Heuvel 1997; Langer et al. 2000; Han Podsiadlowski 2004, 2006; Chen Li 2007, 2009; Meng et al. 2009; Lü et al. 2009; Wang, Li & Han 2009). An explosion following the merger of two WDs would leave no remnant, while the companion star in the SD model would survive and be potentially identifiable (Podsiadlowski 2003). There has been no conclusive proof yet that any individual object is the surviving companion star of an SN Ia. It will be a promising method to test SN Ia progenitor models by identifying their surviving companion stars.

Yoon Langer (2003) followed the evolution of a CO WD + He star system with a CO WD and a He star in a 0.124 d orbit. In this binary, the WD accretes He from the He star and grows in mass to the Chandrasekhar (Ch) mass. SNe Ia from this binary channel can neatly avoid H lines. Recently, Wang et al. (2009a) systematically studied the WD + He star channel of SNe Ia. In the study, they carried out binary evolution calculations of this channel for about 2600 close WD binaries, in which a CO WD accretes material from an He MS star or an He subgiant to increase its mass to the Ch mass. The study shows the parameter spaces for the progenitors of SNe Ia. By using a detailed binary population synthesis (BPS) approach, Wang et al. (2009b) find that the Galactic SN Ia birthrate from this channel is and that this channel can produce SNe Ia with short delay times (45140 Myr) from the star formation to SN explosion. The companion star in this channel would survive and show distinguishing properties.

In recent years hypervelocity stars (HVSs) have been observed in the halo of the Galaxy. HVSs are stars with a velocity so great that they are able to escape the gravitational pull of the Galaxy. However, the formation of HVSs is still unclear (for a recent review see Tutukov & Fedorova 2009). It has been suggested that such HVSs can be formed by the tidal disruption of a binary through interaction with the super-massive black hole (SMBH) at the Galactic center (GC) (Hills 1988; Yu & Tremaine 2003). The first three HVSs have only recently been discovered serendipitously (e.g., Brown et al. 2005; Hirsch et al. 2005; Edelmann et al. 2005). Up to now, about 17 HVSs have been discovered in the Galaxy (Brown et al. 2009; Tillich et al. 2009), most of which are B-type stars, probably with masses ranging from 3 to 5  (Brown et al. 2005, 2009; Edelmann et al. 2005). One HVS, HE 0437-5439, is known to be an apparently normal early B-type star. Edelmann et al. (2005) suggests that the star could have originated in the Large Magellanic Cloud, because it is much closer to this galaxy (18 kpc) than to the GC (see also Przybilla et al. 2008).

At present, only one HVS, US 708, is a subdwarf O (sdO) star, and Hirsch et al. (2005) speculatS that US 708 is formed by the merger of two He WDs in a close binary induced by the interaction with the SMBH in the GC and then escaped. Recently, Perets (2009) has suggested that US 708 may have been ejected as a binary from a triple disruption by the SMBH, which later on evolved and merged to form an sdO star. Because of the existence of the short orbital periods (1 h) for the WD + He star systems, Justham et al. (2009) argues that the WD + He star channel of SNe Ia may provide a natural explanation for stars like US 708.

Han (2008a) obtained the distributions of many properties of the surviving companions from the WD + MS channel of SNe Ia. The properties can be verified by future observations. The purpose of this Letter is to investigate the properties of the surviving companions of the WD + He star channel and to explore whether HVSs such as US 708 could have been released from the binaries that produced SNe Ia. In Section 2, we describe the BPS approach and the simulation results for the properties of the surviving companions. Finally, a discussion is given in Section 3.

## 2 The distributions of properties of the companion stars

In the WD + He star channel, the progenitor of an SN Ia is a close WD binary system, which has most likely emerged from the CE evolution (Paczyński 1976) of a giant binary system. The CE ejection is still an open problem. Here, we use the standard energy equations (Webbink 1984) to calculate the output of the CE phase. The CE is ejected if

 αce(GMfdonMacc2af−GMidonMacc2ai)=GMidonMenvλRdon, (1)

where is a structure parameter that depends on the evolutionary stage of the donor, is the mass of the donor, the mass of the accretor, the orbital separation, the mass of the donor’s envelope, the radius of the donor, and the indices and denote the initial and final values, respectively. The right side of the equation represents the binding energy of the CE, the left side shows the difference between the final and initial orbital energy, and is the CE ejection efficiency, i.e., the fraction of the released orbital energy used to eject the CE. For this prescription of the CE ejection, there are two highly uncertain parameters (i.e., and ). As in previous studies, we combine and into one free parameter , and set it to be 0.5 (e.g., Wang et al. 2009b).

To obtain the distributions of properties of the surviving companions, we performed a Monte Carlo simulation in the BPS study. In the simulation, by using the Hurley’s rapid binary evolution code (Hurley et al. 2000, 2002), we followed the evolution of sample binaries from the star formation to the formation of the WD + He star systems according to three evolutionary channels (i.e., the He star channel, the EAGB channel, and the TPAGB channel; for details see Wang et al. 2009b). If a binary system evolves to a WD + He star system, and if the system, at the beginning of the Roche lobe overflow (RLOF) phase, is located in the SN Ia production regions in the plane of (, ) for its , where , and are, respectively, the orbital period, the secondary’s mass, and the WD’s mass of the WD + He star system at the beginning of the RLOF (see Fig. 8 of Wang et al. 2009a), we assume that an SN Ia is produced, and the properties of the WD binary at the moment of SN explosion are obtained by interpolation in the three-dimensional grid (, , ) of the 2600 close WD binaries calculated in Wang et al. (2009a).

In the BPS study, the primordial binary samples are generated in the Monte Carlo way. We adopted the following input for the simulation (e.g., Han et al. 1995a, 2002, 2003, 2007; Han 2008b; Wang et al. 2009b). (1) The star-formation rate (SFR) is taken to be constant over the past 15 Gyr. For the constant SFR, we assume that a binary with its primary more massive than is formed annually (e.g., Han et al. 1995b). (2) The initial mass function (IMF) of Miller & Scalo (1979) is adopted. (3) The mass-ratio distribution is taken to be constant. (4) The distribution of separations is taken to be constant in for wide binaries, where is the orbital separation. (5) The orbits are assumed to be circular.

The simulation gives current-epoch distributions of many properties of companions at the moment of SN explosion, e.g., the masses, the orbital periods, the orbital separations, the orbital velocities, the effective temperatures, the luminosities, the surface gravities, the surface abundances, the mass-transfer rates, the mass-loss rates of the optically thick stellar winds, etc. The simulation also shows the initial parameters of the primordial binaries and the WD binaries that lead to SNe Ia. Figures 1-5 are selected distributions that may be helpful for identifying the surviving companion stars.

## 3 Discussion

Figures 1 and 2 show the distributions of the masses, the orbital velocities, the effective temperatures, and the surface gravities of companion stars at the moment of SN explosion. In Figure 1, the companion star has an orbital velocity of 300500  for a corresponding mass of 0.61.7  at the moment of SN explosion. In the three formation channels of the WD + He star systems, SNe Ia mainly come from the He star channel. For this formation channel, the recorded properties at each step show that a primordial binary system with a primary mass , a secondary mass , and an orbital period would evolve to a close WD + He star system with a WD mass , a He star mass , and an orbital period . Finally, the WD + He star system results in an SN Ia explosion and survives a companion star.

However, Figures 1 and 2 are for the moment of SN explosion, which could be modified by the explosion. The SN ejecta will interact with its companion. The companion stars will be stripped of some mass and receive a kick velocity that is perpendicular to the orbital velocity. Adopting a similar method to Meng, Chen & Han (2007), we estimated the stripped mass for the companion stars of the WD + He star channel and find that the stripped mass is very low, e.g., a 1.243 companion star with 0.274 only loses a mass of 0.015 in our simulation. We also roughly obtain the kick velocity based on the momentum conservation equation.111 However, in reality, the collision between the SN ejecta and its companion cannot be elastic. If we adopt the inelastic collision in our calculations, the kick velocity will only decrease by 3%5%. Thus, the inelastic collision is no significant influence on the kick velocity. The kick velocity mainly depends on the ratio of separation to the radius of companions at the moment of SN explosion, , and the leading head velocity of SN ejecta, which is assumed to be in the range of 11000 to 13500 km/s. The velocity of 11000 km/s is from the SN ejecta kinetic energy  erg corresponding to the lower limit of normal SN Ia kinetic energy, while the velocity of 13500 km/s is from the SN ejecta kinetic energy  erg corresponding to the upper limit of the kinetic energy (Gamezo et al. 2003). We can obtain the spatial velocity by the formula , where and are the kick velocity and the orbital velocity of the companion star at the moment of SN explosion, respectively. In Figure 3, we show the distribution of the spatial velocity for the surviving companion stars of the WD + He star channel. We see that the surviving companion stars have high spatial velocities (400 km/s) that almost entirely exceed the gravitational pull of the Galaxy nearby the sun. Thus, the surviving companion stars from the WD + He star channel could be an alternative origin for HVSs.

US 708 is an extremely He-rich sdO star in the Galaxy halo, with a heliocentric radial velocity of + (Hirsch et al. 2005). We note that the local velocity relative to the Galatic center may lead to a higher observation velocity for the surviving companion stars, but this may also lead to a lower observation velocity. Considering the local velocity near the sun (220 km/s), we find that 30% of the surviving companion stars may be observed to have velocity for a given SN ejecta velocity 13500 km/s. In addition, the asymmetric explosion of SNe Ia may also enhance the velocity of the surviving companions. Thus, a surviving companion star in the WD + He star channel may have a high velocity like US 708 (see also Justham et al. 2008).

The companion stars are out of thermal equilibrium at the moment of SN explosion. For He stars, the equilibrium radii are lower than at the moment of SN explosion. Thus, the surface gravity at equilibrium should be greater than the one in Figure 2; e.g., a 1.243 He star with 0.274 and (, )=(4.70, 5.65) will be a 0.267 He star and with (, )=(4.75, 5.67) after the He star is back to the thermal equilibrium. It is well known that a shock will develop after impact by the ejecta. A large part of the material in the companion’s envelope is heated by the shock, and some of the material is vaporized from the surface of the companion stars. Following a similar method to Chen & Li (2007), we also estimated the vaporized mass, and find that the mass loss from the companion stars is not significant (5%). Such a surviving companion star may be significantly overluminous or underluminous depending on the amount of heating (e.g., Podsiadlowski 2003). Figure 2 could be a starting point for further studies of this kind.

Figure 4 shows the distributions of orbital periods and secondary masses of the WD + He star systems at the moment of SN explosion. The orbital periods and secondary masses of the WD + He star systems at this moment are basic input parameters when one simulates the interaction between SN ejecta and its companion. It is suggested that, for hot stars with radiative envelopes (such as He stars), tidal forces may be inefficient for synchronization (e.g., Zahn 1977). However, the recent study by Charpinet et al. (2008) supports efficient tidal synchronization for hot subdwarf stars. The work by Toledano et al. (2007) also indicates, on the other hand, that even stars with radiative envelopes may have efficient tidal interaction on a time scale comparable to convective envelopes. Thus, we make an assumption that the companion stars co-rotate with their orbits. In Figure 5, we show the distributions of equatorial rotational velocities of the companions stars. We see that the surviving companion stars are fast rotators, so their spectral lines should be broadened noticeably.

The simulation in this Letter was made with . If we adopt a higher value for , e.g., 1.5, the birthrate of SNe Ia would be a little bit higher and the delay time from the star formation to SN explosion longer. This is because binaries emerging from CE ejections tend to have longer orbital periods for a large and are more likely to be located in the SN Ia production region (Fig. 8 of Wang et al. 2009a). Due to the lack of WD binaries with short orbital periods () for a large , the companions with orbital velocity would be noticeably absent in Figure 1.

The distributions are results of the current epoch for a constant SFR. For a single starburst, most of the SN explosions occur between 45 Myr and 140 Myr after the starburst; i.e., SNe Ia from the WD + He star channel will be absent in old galaxies. The Galactic SN Ia birthrate from this channel is (Wang et al. 2009b). By multiplying the birthrate with a typical MS lifetime of He stars,  yr, we estimated the current number of this type of HVSs in the Galaxy to be . In future investigations, we will employ the Large sky Area Multi-Object fiber Spectral Telescope (LAMOST) to search the HVSs originating from the surviving companion stars of SNe Ia.

###### Acknowledgements.
We thank an anonymous referee for his/her valuable comments that helped us to improve the paper. B.W. thanks Drs S. Justham, X.-C. Meng and W.-C. Chen for their helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant No. 10821061), the National Basic Research Program of China (Grant No. 2007CB815406), and the Yunnan Natural Science Foundation (Grant No. 08YJ041001).

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# 4.6: Hydrostatic Weighing

#### Reinforcement Exercises

Calculate the density of an that object that weighs 12 N, but has an apparent weight of only 8.5 N.

The previous equation is very similar to the body density equation used for hydrostatic weighing, but you will notice a slight difference. The previous equation determines the average density of the object including any hollow parts containing trapped air, but the body density equation is designed to determine the average density of body tissues only, not including trapped air. Therefore, the body density equation is modified to account for a volume of air trapped inside the body, known as the residual volume (RV). Also different standard symbols are used to designate body density, apparent weight, and water density.

Formulas used in calculating residual lung volume, body density, and body fat percentage. Image Credit: Measure Body Fat Via Under Water Weighing by via Instructables

[1]

We arrived at equation (1) by starting with the definition of density as mass divided by its volume:

$\rho=\frac{m_{O}}{V_{O}}$

We can find the mass of an object if we divide its weight by g:

$m=\frac{W_{O}}{g}$

Inserting that result for mass into the density equation we have:

$\rho=\frac{W_{O}}{g V_{O}}$

For a completely submerged object the volume of water displaced is equal to the volume of the object, so we can replace with .

$\rho=\frac{W_{O}}{g V_{D}}$

Using the definition of density again, we can replace the volume of water displaced with the displaced water mass () divided by water density ().

$\rho_W)} = \frac{W_O}{g m_D}\rho_W \end{equation*}$

Archimedes' Principle which tells us that the buoyant forcepushing upward on objects in a fluid is equal to the weight displaced fluid. Therefore we can replace with .

$\rho=\frac{W_{O}}{F_{B}} \rho_{W}$

Making that replacement in our density equation we have:

$\rho=\frac{W_{O}}{W_{O}-F_{A}} \rho_{W}$

We now have an equation that allows us to calculate the density of an object by measuring only its weight and apparent weight, as long as we know the density of the fluid we are using.

## Specific Gravity

The ratio of the density of a substance to that of water is known as the specific gravity. Specific gravity can be determined by hydrostatic weighing. If we simply divide both sides of our density equation by the density of water we will have a formula for the specific gravity:

$S G=\frac{\rho}{\rho_{W}}=\frac{W_{O}}{W_{O}-F_{A}} \label{2}$

#### Reinforcement Exercises

Calculate the specific gravity of an that object that weighs 12 N, but has an apparent weight of only 8.5 N.

4.6: Hydrostatic Weighing is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Lawrence Davis.

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## When our two-state PDA constructed from CFG is non-deterministic PDA?

We can always convert our GNF-CFG/CNF-CFG to a two-state PDA but i’m wondering when our PDA is non-deterministic? i’m sure we can not make DPDA for non-Deterministic-CFL , and i suspect that same rule which applied for differing between DCFG and non-Deterministic-CFG also applied here. i mean when we have non-Deterministic δ (delta) implied non-Deterministic edge in our PDA. if my suspicion is right , then for every DCFL exist at least one DPDA. Am i right?

``R ≝ Production Rules of CFG (x,y,"LBL") is a labeled-edge between x and y with “LBL” as a label  ∀r∊R: r= (A,aⱰ) ( A∊V ⋀ a∊T ∧ Ɒ∊V*) add (q,q,"a,A/Ɒ") to E Add (q,q,"ε,z/Sz′") to E Add (q,f,"ε,z′/z′") to E ``

## If anything can be verified efficiently, must it be solvable efficiently on a Non-Deterministic machine?

Suppose, I wanted to verify the solution to $$2$$^$$3$$. Which is $$8$$.

The $$powers~of~2$$ have only one 1-bit at the start of the binary-string.

## Verify Solution Efficently

``n = 8 N = 3 IF only ONE 1-bit at start of binary-string:   IF total_0-bits == N:    if n is a power_of_2:      OUTPUT solution verified, 2^3 == 8 ``

A solution will always be approximately $$2$$^$$N$$ digits. Its not possible for even a non-deterministic machine to arrive to a solution with $$2$$^$$N$$ digits faster than $$2$$^$$N$$ time.

Question

Can this problem be solved efficently in non-deterministic poly-time? Why not if the solutions can be verified efficently?

## Does this imply Hamiltonian path cannot be decided in nondeterministic logspace?

Suppose I nondeterministically walk around in a graph with n vertices.

When looking for a Hamiltonian path, at some point I’ve walked n/2 vertices.

There are (n choose n/2) different combinations of vertices I could have walked (meaning the unordered set, not the ordered walk).

Each of those states must be distinct from one another.

If not, then, depending on the remaining n/2 vertices, I would decide the wrong answer.

Therefore, midway through my processing, at n/2, I need (n choose n/2) different states. That is too big for logspace.

Therefore you cannot decide a Hamiltonian path by nondeterministically walking around.

Does this imply Hamiltonian path cannot be decided in nondeterministic logspace – at least by “walking around”?

## Why aren’t distributed computing and/or GPU considered non-deterministic Turing machines if they can run multiple jobs at once?

So we know a nondeterministic Turing machine (NTM) is just a theoretical model of computation. They are used in thought experiments to examine the abilities and limitations of computers. Commonly used to dicuss P vs NP, and how NP problems cannot be solved in polynomial time UNLESS the computation was done on the hypothetical NTM. We also know an NTM would use a set of rules to prescribe more than one action to be performed for any given situation. In other words, attempt many different options simultaneously.

Isn’t this what distributed computing does across commodity hardware? Run many different possible calculations in parallel? And the GPU, does this within a single machine. Why isn’t this considered an NTM?

## Statement TRUE or FALSE: there exist more non-deterministic TMs than deterministic TMs

Statement TRUE or FALSE:

There exist more non-deterministic TMs than deterministic TMs

For my point of view (I ask if it’s correct):

Under the assumption $$P \neq NP$$, the Statement is TRUE because (1) P is a subset on NP and (2) P is a special case of non-determinism.

Under the assumption $$P = NP$$, the Statement is FALSE, because for each deterministic TM we have a non-deterministic TM.

## Writing a non-deterministic Turing machine

I got an exercise in my college in which ive been asked to write a Non deterministic Turing Machine that accepts words of this form:

Ive manage to find some sort of solution but in a Deterministic way

## Can a Non-Deterministic Pushdown Automata recognize \$ \# a^nb^{2^n} \# \$ which a TM can?

$$\# a^nb^{2^n} \#$$

such that

• The alphabet of the machine is {, a, b, x}.

• The symbol x will never appear on the input a.

• The contents of the tape at completion may be anything.

• The head begins on the lefthand #.

• n ≥ 0.

I know that a Turing machine could recognize this language. But can a NPDA recognize this language too? I am thinking it can but I do not know how to start proving how/why?

## Converting non-deterministic algorithm to deterministic

I was thinking about an non-deterministic algorithm to generate all the subsets of the $$\{1..n\}$$ set.

``Subsets(n)     S' = null     for i = 1..n             u = choice(1..n)         if u in S'            fail         S' = S' + u     for_each u in S'          print u ``

How would I convert this algorithm into a deterministic form? I know that it is a lot more complicated to convert the `choice` statement in a deterministic manner but I want to get a general idea about maybe representing it in a deterministic manner.

## motivation and idea of defining non-deterministic Turing machine

This is a very basic question but I spent some time reading and find no answer. I am not computer science majored but have read some basic algorithm stuff, for example, some basic sorting algorithms and I also have some basic knowledge of how computers operate. However, I am really interested in the idea of a Turing machine, especially the non-deterministic one.

I have read Wiki about the definitions of a Turing machine (and watched some youtube videos) and I sort of accept that, although I really feel that this is a huge jump from an algorithm to an abstract machine. From my understanding (you are more than welcome to correct me):

1. A Turing machine is a machine performing works specified on a cookery book (algorithm).
2. The pages of the cookery book represent the “states” of your machine and each page contains a table saying that which state and which cell your machine will move to given the alphabet the machine read and your current state. (NB. This is not a function but a partial function because it is possible that the machine stops.)
3. So, to guess the idea and motivation of defining an abstract Turing machine, I imagine that the algorithm corresponds to the partial map, the memory of the computer corresponds to the (infinitely long) tape and what’s finally on the tape is the answer to the question you wanna solve.

So, Turing machine looks like a machine to realize any algorithm to solve problems. One just “translates” any algorithm to a set of mysterious simple rules (i.e. the partial function) and let the machine do the laboring job and then we get the solution.

In this respect, Turing machine is always deterministic, because algorithms are deterministic. It tells you what to do next precisely. This is no uncertainty. Turing machine is just a machine to realize any algorithm.

OK, This is very abstract and I sort of accept it. However, then I read something called non-deterministic Turing machine (NTM) and then I was knocked down. A NTM is pretty much similar to a Turing machine except that the partial function is now replaced by a “relation”. That is, it is a one-to-multiple map and it is no longer a (partial) function.

Could someone explain to me why we need such multiple options? I would never expect to encounter something uncertainty in the implementation of an algorithm. It is like telling the machine: you first do A, then if you find yourself in a state B and find the data is now B’ then you choose for yourself one of the 10 allowed next steps?

Are NTM’s corresponding to a set of algorithms that need uncertainty? for example the generation of random numbers? If no, why do we need to allow multiple choices for a Turing machine?

Any help will be appreciated!

## What publication first introduced the concept of a non-deterministic Turing machine?

What publication first introduced the concept of a non-deterministic Turing machine?

Turing did not define the concept in his 1936 paper.


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 P: 117 The easiest bases for the space of nxn matrices is just the matrices with a one as one entry and zeros everywhere else. If you can show that there is a way to make all of these matrice as linear combinations of invertible nxn matrices (just do it explicitly) you are done. Also remember that the general linear group is not a group of matrices but of isomorphisms of vector spaces. depending on the bases one isomorphism can have different matrices.


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# Need help solving second order DE.

1. Jan 2, 2010

### Joans

Hello, I hope I am writing to right part of a forum...

I made a differential equation when I was solving my problem, but unfortunately I am not capable of solving such equation since I am only 12th grader.
Or maybe it is not possible to solve it at all??

$$\frac{5\sqrt{3}}{18}\frac{d^{2}x}{dt^{2}} = 5 - \frac{3\sqrt{3}}{R}\frac{dx}{dt}$$

R is unknown.

$$\frac{d^{2}x}{dt^{2}} = x(t)$$

$$\frac{dx}{dt} = v(t)$$

v(0) = 0
v(5) = 15

I need to find equation describing x(t). Jap, v is velocity, and nope it is not my homework.

It would be great if someone help me a bit, in school do not teach how to solve differential equations, nor second order. :)

2. Jan 2, 2010

### rock.freak667

If R is constant, then you have a 2nd order DE with constant coefficients. Which can be solved by writing down the roots of the auxiliary equation.
http://www.efunda.com/math/ode/linearode_consthomo.cfm" [Broken]

Last edited by a moderator: May 4, 2017
3. Jan 3, 2010

### Joans

Yes R is a constant.
Okay I guess I understood a bit:

$$x(t) = c_{1}e^{\frac{18(\sqrt{\frac{27}{R^{2}}+\frac{50\sqrt{3}}{9}}-\frac{3\sqrt{3}}{R})}{5\sqrt{3}}t}+c_{2}e^{-\frac{18(\sqrt{\frac{27}{R^{2}}+\frac{50\sqrt{3}}{9}}+\frac{3\sqrt{3}}{R})}{5\sqrt{3}}t}$$

anyway to me it gets too crazy.

x(0)'=0
x(5)'=15

is it possible to solve this equation normally that I would know R & c1 & c2 ?

Last edited: Jan 3, 2010
4. Jan 3, 2010

### rock.freak667

Since you have three unknowns you'd need at least one more condition to find R.

5. Jan 3, 2010

### Joans

I understood, thanks for help!


text:
# 5r-5s=35

## Simple and best practice solution for 5r-5s=35 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.

If it's not what You are looking for type in the equation solver your own equation and let us solve it.

## Solution for 5r-5s=35 equation:

Simplifying
5r + -5s = 35
Solving
5r + -5s = 35
Solving for variable 'r'.
Move all terms containing r to the left, all other terms to the right.
Add '5s' to each side of the equation.
5r + -5s + 5s = 35 + 5s
Combine like terms: -5s + 5s = 0
5r + 0 = 35 + 5s
5r = 35 + 5s
Divide each side by '5'.
r = 7 + s
Simplifying
r = 7 + s`

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text:
journal_list | How to participate | E-utilities
Yoon, Wi-Ho, and Jin: An Off-site Screening Process for the Public in Radiation Emergencies and Disasters

### Background

A contamination screening process for the local population in radiation emergencies is discussed.

### Materials and Methods

We present an overview of the relevant Korean governmental regulations that underpin the development of an effective response system. Moreover, case studies of foreign countries responding to mass casualties are presented, and indicate that responses should be able to handle a large demand for contamination screening of the local public as well as screening of the immediate victims of the incident.

### Results and Discussion

We propose operating procedures for an off-site contamination screening post operated by the local government for members of the public who have not been directly harmed in the accident. In order to devise screening categories, sorting strategies assessing contamination and exposure are discussed, as well as a psychological response system.

### Conclusion

This study will lead to the effective operation of contamination screening clinics if an accident occurs. Furthermore, the role of contamination screening clinics in the overall context of the radiation emergency treatment system should be clearly established.

### Introduction

The main role of the NREMC is to provide medical treatment for the immediate victims of the accident, such as the provision of consultation and technical support regarding medical relief in radiation emergencies and disasters and the collection of materials to evaluate the radiation exposure dose [1]. The Korea Institute of Nuclear Safety (KINS) has the mission of establishing and operating the technical support headquarters for radiation protection, while nuclear plant operators are responsible for establishing and operating emergency headquarters in case of nuclear plant accidents. The law states that the local radiation safety response centers in the emergency planning zones of each city and province are to be controlled by the respective local governments in order to facilitate prompt gathering and evacuation of the residents. To that end, peacetime relief stations are to be designated and maintained.
However, due to the unique nature of radiation emergencies and disasters, radiation contamination may occur in areas neighboring the accident site in a short period of time. Moreover, the public may exhibit a heightened fear of radiation. Thus, proper contamination screening and protective measures are essential for more than the immediate victims. To that end, the NREMC is in charge of providing contamination screening and treatment not only for immediate victims of the accident (the injured), but also inhabitants (possible casualties) of the region affected by the accident. In addition, the NREMC runs off-site radiation emergency clinics near the accident site [2].
However, the main object of treatment in the initial stage of the accident remains immediate victims and off-site responders. Moreover, in order to ensure the prompt transport of victims, the center should be located close to the accident site. Therefore, it may not be reasonable to have the center provide contamination screening for local inhabitants at the same time. Furthermore, despite being at some distance from the accident site, if the fear of radiation intensifies in the public, the need to provide contamination screening effectively for those who live farther from the accident site should be also considered. Therefore, the roles of the center should be functionally divided.
This study discusses contamination screening steps for local inhabitants in case of a large-scale radiation emergency or disaster. To that end, we reviewed the current system of radiation-protective measures in Korea as well as instances of radiation emergencies and disasters that have occurred outside the country, with a focus on the characteristics of the cases and the responses of the respective countries. Moreover, we review the functions of off-site contamination screening in the U.S., with the aim of establishing a system for off-site contamination screening clinics that fits circumstances in Korea. Finally, we discuss how contamination screening should be evaluated in order to establish effective processes.

### 1. Relevant law and roles of local governments

According to the Act on Measures for the Protection of Nuclear Facilities, etc. and Prevention of Radiation Disasters, if a radiation accident resulting in a radiological emergency occurs, a radiation disaster prevention response body should be promptly mobilized. In particular, if an accident occurs in large-scale nuclear facilities, such as nuclear power plants, organized government-level responses should be planned and regular education and training should be offered to maintain a response capability sufficient for actual accidents.
At the time of the occurrence of the accident, the Central Headquarters, whose head is the head of the NSSC, is to be launched. The roles of expert organizations in different areas in the local command system in emergency are stipulated in the law, and the regional radioactive disaster prevention and response headquarters are to be established and operated by local government. These centers are to play a crucial role in protecting the inhabitants of the affected areas. Moreover, the centers must plan the evacuation of many residents from the area as speedily as possible.
In particular, the off-site command centers for the prevention of radioactive disasters (“off-site command centers”) are to take urgent measures for the protection of residents, such as evacuation, implementing restrictions on food intake, and distributing thyroid-protective agents. In carrying out protective measures for individual residents, the centers use concrete criteria for acceptable environmental radiation doses and space doses (Figure 1).
In case of an emergency, the respective local governments where nuclear power plants are located should announce preparedness guidelines to gather inhabitants to the designated place and evacuate them to the designated relief stations via planned transport vehicles. Nevertheless, in general, local governments formulate plans only for the designation and operation of assembly places for evacuation and relief stations: No specific standards exist for contamination screening for ordinary residents, beyond the immediate victims of an accident or those who may have suffered serious internal or external contamination [2].

The KIRAMS of the NREMC has a radiation emergency medical response system for providing medical care for radiation victims or possible casualties in case of an emergency. It is stipulated that if an accident occurs, the head of the KIRAMS should organize a medical support team to be dispatched to the site with the cooperation of primary and secondary radiation emergency medical institutions in nearby regions and promptly operate off-site clinics.
The treatment system of the off-site clinics is classified into a triage team, a decontamination team, a treatment team, and a transport team. The triage team divides victims depending on whether they have sustained medical injuries or radiological injuries. The victims categorized into the latter group are then graded from 1 to 3: victims with an injury level of 2 or 3, who do not exhibit signs or symptoms of exposure and contamination with radiation or radioactive materials, are considered to require long-term follow-up observation.
Based on these criteria, many residents who have not suffered direct injuries from the accident may fall into this category. A primer on radiation emergency treatment states that lower-priority casualties also warrant contamination measurements and recordings thereof, as well as long-term follow-up observation [1, 3].
However, the most crucial function of the off-site clinic is to provide radiological medical treatment for victims and possible casualties transported from the accident site, which is facilitated by being located as close to the safety perimeter within the security perimeter as possible. Therefore, it is important to establish the clinics in a timely manner in order to offer prompt treatment to victims who have suffered an injury or who are highly likely to have been injured. This suggests that the evaluation of the residents of neighboring areas for contamination is not the primary duty of these clinics (Figure 2).

### 3. Case studies of responses in other countries

After a nuclear power plant disaster occurred, Fukushima Prefecture in Japan, conducted emergency radiation exposure testing for screening the residents neighboring the nuclear plant for contamination. Simple external contamination tests were provided at some community health centers and gymnasiums for nearly a year from March 13, 2011, starting from two days after the occurrence of the accident [4].2) From March 13, when the emergency radiation exposure testing was initiated, 114,488 people were tested over the course of 18 days, of whom 102 were determined to have been contaminated based on the criterion for total decontamination (100,000 cpm) set by the Japanese government at the time.
Later, over the course of approximately a year, a total of 243,992 people received testing. Additionally, Fukushima Prefecture conducted thyroid testing for children beginning on March 24, 2011, as well as whole-body counting to measure 137Cs and 134Cs in the body. Judging from the irregularity in the test periods and the number of tests performed, we conjecture that the testing was done sporadically according to public sentiment, due to the unique nature of the disaster in the country at the time.
A similar case occurred in Goiania, Brazil, where 112,000 residents, nearly 10% of the whole population of the city, were screened for contamination in a large stadium, which led to extreme social chaos [5]. Such cases in different countries show that the ripple effects from radiation contamination and exposure can be explosive, due to the characteristics of large-scale radiation disasters. Moreover, regarding public protective measures, neighboring residents’ concerns about the accident emphasize the importance of having sufficient capacity for contamination screening of a large number of people.

### 1. Operations in other countries

The Centers for Disease Control and Prevention (CDC) in the U.S. have established a systematic process for contamination and exposure screening of residents in case of a large-scale radiation emergency or disaster, and provide education and training for responders on a regular basis. To that end, the organization recommends the operation of community reception centers (CRCs) in accident sites in case of the occurrence of an accident. This is part of the early response plan for such accidents, and the goal is to exhibit the same level of preparedness as in the response to care for the seriously injured. The CDC defines such activities as part of radiation protection plans for neighboring residents.
The CDC classifies the screening procedures of CRCs into initial triage, medical treatment, contamination screening, decontamination, registration, and radiation dose evaluation. Moreover, they provide radiation contamination screening for examinees, decontamination for those who have been contaminated, and establish immediate response processes for those who need long-term follow-up and additional measures. Furthermore, CRCs minimize the consumption of medical resources and facilitate the effective operation of relief stations after accidents by allowing off-site radiation emergency clinics to focus on the immediate victims of the accident [6].
In particular, the CDC regularly recommends in its guidelines that the emergency response bodies of local governments, which are in charge of the residents’ contamination screening, and the health authorities of the respective regions should evacuate and rescue residents at the same time. Through such measures, the local governments responsible for accident sites can carry out the processes involved in contamination screening, prepare response plans for searching for potential survivors of radiation exposure, and help the public to feel more secure after an accident.
The services provided by CRCs are targeted at residents who have not suffered a direct injury from the accident. To that end, the U.S. government recommends that CRCs be established in community health centers, relief stations, and emergency shelters (alternate care sites, points of dispensing, and neighborhood emergency help centers) to enable the centers to carry out their mission outside the security perimeter and to improve residents’ access to the centers. Since they are located close to the relief facilities, the centers appear to be capable of promptly implementing follow-up measures, such as transferring those seriously contaminated to a hospital and sending the rest home or to relief stations, in case of a large-scale radiation emergency or disaster (Figure 3).

### 2. Programs in Korea

In Korea, the NREMC of the KIRAMS conducts regular radiation emergency treatment trainings simulating an accident in a nuclear facility. The focus of the training used to be the operation of off-site radiation emergency treatment clinics. However, in an effort to examine response systems targeting residents, on November 20, 2014, the center facilitated a joint training with Busan City regarding the Kori Nuclear Power Plant and ran a support center for victims and a radiation effects counseling center, in which more than 500 residents participated.
Under conditions similar to those found in actual accidents, a relatively large number of residents were trained to evacuate and to take protective measures in relief stations. Simple contamination screening and experimental training on group psychological consultations were provided in the radiation effects counseling center. By further developing such a program (providing objective screening systems in addition to psychological relief strategies), we may be able to more effectively manage residents in case of a radiation disaster. Moreover, screening and following up those who may have been exposed to radiation may prove helpful in identifying the long-term effects of accidents.

### 3. Establishing off-site contamination screening clinics

In the prior literature on radiation emergency treatment, the installation types of off-site radiation emergency clinics have been classified according to the accident site [3]. It is recommended that the clinics should be operated based on the perimeters that determine the extent to which the effects of the accident can be minimized: the safety perimeter, security perimeter, and clean area. Moreover, it is recommended to establish off-site radiation emergency clinics within the security perimeter, in order to allow efficient triage of victims arriving from the accident site and easy provision of medical treatment for them (Figure 2).
In the past, contamination monitoring for residents who did not suffer direct harm from the accident was carried out in off-site radiation emergency clinics. In general, in response to nuclear power plant accidents, the relief stations run by local governments are located at a certain distance from the plant: This may correspond to the clean area according to the existing recommendations.
However, despite the presence of a significant distance between contamination screening sites for neighboring residents and the accident site, due to psychological factors, the clinics may need to attend to residents voluntarily coming from remote areas. Therefore, it may be necessary to establish additional clinics. Moreover, based on previous cases, off-site contamination screening clinics need to be operated for a long period of time on a large scale. In this regard, increasing residents’ accessibility to the clinics is very important, and this may be accomplished by connecting the clinics to relief stations. The operation of off-site radiation emergency clinics should be on a short-term basis in the early stages of the accident, as they focus on the immediate victims of the accident. In particular, the functions of such clinics should not include contamination screening of residents.
In general, local governments designate public buildings, including schools, in nearby areas as relief stations in order to serve local inhabitants effectively.3) As accidents are highly likely to require large-scale responses, it is necessary to operate multiple relief stations. When establishing and operating off-site contamination screening clinics, local governments can dispatch staff in small groups to disparate stations.
The advantage of such a method is that unnecessary travel by residents can be minimized. However, sufficient measuring equipment, such as portal monitors and survey meters, must be provided. Additionally, transport vehicles for residents who exhibit severe contamination should be prepared in each station. Other considerations include the possibility that many relief stations are established in certain regions, in which case a large number of support personnel and volunteers may be used in conducting off-site contamination screening in nearby large indoor stadiums.
In such cases, off-site internal and external contamination screening can be performed systematically. Moreover, the inclusion of professional psychologists means that effective radiation emergency and disaster response activities incorporating group psychological counseling can be provided. Furthermore, inhabitants from remote areas can be screened at a certain level. However, appropriate transport vehicles for the residents seen at each station must be provided. Additionally, systematic screening processes should be established and practiced via training under ordinary circumstances to enable prompt screening. Therefore, planning of the establishment and operation of off-site contamination screening should be performed according to the conditions of each local government.

### 4. Off-site contamination screening system

We designed a resident contamination screening process strategy effective in areas with a dense distribution of stations. If transport vehicles for residents are appropriately provided at each station, facilities that can accommodate large populations, such as an indoor stadium, may be suitable. Moreover, a transport system for patients who are contaminated or who warrant urgent medical attention is necessary, as such cases may occur without notice throughout the overall screening process. It would be effective to triage examinees who need urgent care by positioning medical staff at the entrance of the screening site (Figure 4).
Ambulatory examinees pass through a portal monitor for external whole-body contamination screening. Based on the 137Cs action level, the appropriate external contamination screening alarm setpoint is estimated within 8 σ of the background count, according to Equation 1. Under conservative assumptions, the background rate is 7,000 cps, the measurement efficiency is 1%, and the measuring time is 2 s (walk-through mode). Ta refers to the alarm trip point of the portal monitor (cps), Br is the background rate (cps), SSTD is the alarm set point, and t signifies the measuring time.4,5)
##### (1)
$Ta=Br+(SSTD×Brt)$
The examinees who exhibit values above the alarm setpoint are sent for partial contamination screening and receive additional testing. It would be effective to identify the site of contamination by screening intensively where external contamination is suspected based on the reports of the examinee, using a hand-held survey meter. If the measurement does not exceed 1 μSv·h−1 at the 10 cm point from the surface of the screened area, the examinee is to be sent for registration; if the measurement exceeds 1 μSv·h−1, the examinee is to be sent for decontamination [7].
If, despite appropriate undressing and decontamination, the measurement exceeds the screening criteria, the examinee is then to be sent to the additional testing stage for internal contamination testing and extra measures. Incorporating in vivo and in vitro bioassays, which are used in response to radiological emergencies and disasters, into this stage with support from expert institutions would enable effective off-site responses.
The examinees who pass through the portal monitor without triggering an alarm can move to the registration stage. In this stage, the examinees provide general demographic information and are sent to the survey stage, where they record personal information related to the accident and the results of their contamination evaluation. In the final stage, group counseling sessions for alleviating the mental distress of the examinees would be appropriate. Subsequently, the entity operating the screening clinics will issue verifiable records, such as a certificate of completion of the screening process, and send examinees home or to relief stations.

### 5. Survey and certificate of completion of screening

A large number of people can swamp the off-site contamination screening site in a short period of time. In order to overcome the challenge of carrying out contamination screening for many residents with limited equipment and manpower, we developed a self-report survey form for prompt screening, in which examinees fill in basic personal information and describe any symptoms they might experience.
At the entrance, examinees who did not trigger an alarm at the portal monitor during whole-body screening record their demographic information, their location at the time of the accident, and clinical symptoms, and participate in a group counselling session to complete the screening process. By doing so, the process can be performed speedily. Moreover, the records filled out by the examinees can be used as evidence for follow-up observations. However, for examinees whose measurements were higher than the alarm setpoint, information should be recorded describing whether partial screening, decontamination, or additional testing were performed, along with the measurement values and the name of the examiner. These records will play an important role in identifying the long-term effects of radiation (Figure 5).
If the entities operating screening sites or local governments issue and use documents to prove the completion of contamination screening or decontamination for examinees who are not found to be injured or contaminated based on the screening results, such an action can be used as an effective response measure to address psychological concerns among members of the public that may occur in disasters. In fact, at the time of the Fukushima Daiichi nuclear disaster, victims who were not accommodated within Fukushima Prefecture due to regional circumstances were moved to nearby areas. However, misinformation about radiation blocked effective relief activities.6) Furthermore, such documents may be used to encourage voluntary screening among people who are potentially contaminated or who have been exposed to radiation.

### Conclusion

This study established a contamination screening process for a large number of residents in case of a large-scale radiation accident, which would include nuclear power plant accidents. The current radiation emergency treatment system is focused on the establishment and operation of an emergency treatment system as well as the provision of medical support for immediate victims according to radiation disaster prevention laws and manuals. Nonetheless, it appears that the demand for contamination screening is high, considering cases outside Korea, due to the unique nature of radiation and psychological factors that may come into play. Moreover, the screening process should be able to be extended to neighboring residents who are not directly affected by the accident.
To that end, we reviewed off-site contamination screening systems in different countries and the content of radiation emergency treatment training materials in Korea. Moreover, we established a local government-initiated off-site contamination screening clinic operation system capable of providing effective contamination screening to a large number of residents in case of a radiation emergency or disaster. Nonetheless, due to limitations in the human resources and equipment of local governments, education and training should be provided under ordinary circumstances for the staff members of local agencies, with the support of expert organizations. This will lead to the effective operation of contamination screening clinics if an accident occurs. Furthermore, the role of contamination screening clinics in the overall context of the radiation emergency treatment system should be clearly established.

### Acknowledgements

This study was supported by a grant of the Korea Institute of Radiological and Medical Sciences (KIRAMS), funded by Ministry of Science, ICT and Future Planning, Republic of Korea (1711031804/50445-2016).

### Notes

1) Act on Measures for the Protection of Nuclear Facilities, etc. and Prevention of Radiation Disasters (Law number 12665).

2) Activities of emergency radiation exposure testing, Disaster countermeasures headquarters of Fukushima Prefecture, 2014.

3) Radioactive disaster prevention plan of Busan Metropolitan City, Busan Metropolitan City, 2014.

4) Canberra Inc. MiniSentry user’s manual. Rev.00. 2006;34–36.

5) Norwegian Radiation Protection Authority. Triage, monitoring and treatment (TMT) of people exposed to ionizing radiation following a malevolent act. 2009;80–94.

6) Japanese reaction to Fukushima Daiichi nuclear disaster, Wikipedia, 2015.

### References

1. National Radiation Emergency Medical Center (KIRAMS). Introduction to radiation emergency medical treatment (in Korean). NREMC;3rd edition. 2014. p. 89–93.
2. National Radiation Emergency Medical Center (KIRAMS). Operating manual of radiation emergency medical post (in Korean). 2014. p. 27–36.
3. International Atomic Energy Agency. Generic procedures for medical response during a nuclear or radiological emergency. EPR Medical manual. 2005; 37–42.
4. Korea Institute of Nuclear Safety. Radiation effect report of Fukushima NPP accident (In Korean). KINS/ER-219. 2012; 25–36.
5. International Atomic Energy Agency. The radiological accident in Goiania. STI/PUB/815. 1988; 41–59.
6. U.S. Center for Disease Control and Prevention. Population monitoring in radiation emergencies. 2nd edition. 2014. p. 11–17.
7. International Atomic Energy Agency. Generic procedure for assessment and response during a radiological emergency. IAEA-TECDOC-1162. 2000; 55–67.
##### Fig. 1
Flow diagram of the radiation emergency response system in Korea.
##### Fig. 2
Field triage during a radiation emergency.
##### Fig. 3
Network diagram of community reception centers (CRCs).
##### Fig. 4
Off-site screening process.


text:
# Math Help - Can someone check my work?

1. ## Can someone check my work?

Nevermind

2. Originally Posted by EdenM
If anyone can help me understand out to do this problem:

Find a formula for the sum x^{k}+x^{k+1}+x^{k+2}+⋯+x^264

It would be a huge help as I cant figure it out and its been days.

Thanks
It's equal to $(1 + x + x^2 + .... + x^{264}) - (1 + x + x^2 + .... + x^{k-1})$ and you have a geometric series in each bracket ....


text:
Chapter 18

Death after traumatic injury has been described in terms of a trimodal distribution. Immediate and acute (<24 hours) deaths usually result from uncontrolled hemorrhage, but infections and multiple organ dysfunction syndrome, which often arise from infection, are responsible for a significant proportion of late deaths. Indeed, infection is responsible for most deaths in patients who survive longer than 48 hours after trauma.1 Trauma-related infections are generally divided into those that result directly from the injury (e.g., due to contamination that occurs in conjunction with the traumatic injury) and nosocomial infections that arise in the health care setting, secondary to treatment of the injury. The pathogens involved can be exogenous or endogenous bacteria, depending on the mechanism of injury and/or the iatrogenic cause. Most post-traumatic infections are polymicrobial and involve a mixture of aerobic and anaerobic organisms.2 In one series, 37–45% of all trauma patients experienced infectious complications during their initial hospitalization. Furthermore, in the same study, 80% of trauma patients staying at least 7 days in the intensive care unit met systemic inflammatory response syndrome (SIRS) criteria.3 Therefore, it is important that all caregivers understand the principles of surgical infections in the context of trauma patients. This chapter discusses the following: factors that normally prevent infection, how trauma disrupts or overwhelms normal host defenses, how to recognize and treat the most common infectious complications after traumatic injury, principles that can be employed to prevent infection, and how those principles can be applied chronologically during the treatment of trauma patients.

Humans have evolved mechanisms to avoid infection despite the ubiquitous presence of bacteria in our environment and throughout our bodies. Under normal circumstances there is a balance between bacteria, intact environmental barriers, and host defenses (see Fig. 18-1). With surgery in general, and trauma in particular, there is a disruption in this balance that significantly increases the probability of developing an infection (Fig. 18-2). Bacteria are abundant on the surface of the skin, within the oral cavity, and present in increasing numbers down the length of the gastrointestinal tract. Bacterial numbers differ at various locations, and the pathogenic species and their respective numbers at different anatomic sites are summarized in Table 18-1. Trauma disrupts the environmental barriers that prevent bacteria from gaining access to normally sterile regions of the body. When inoculation of bacteria into normally sterile sites occurs, infection will ensue if bacteria can proliferate faster than the host defense mechanisms can eradicate them. Furthermore, there is potential for much greater disruption of normal barriers with trauma than occurs with elective surgery as there is often concomitant hypoperfusion (shock), devitalized tissue, and retained foreign bodies.

Table 18-1 Pathogenic Microorganisms Present at Various Anatomic Sites

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# zbMATH — the first resource for mathematics

Homology and closed geodesics in a compact negatively curved surface. (English) Zbl 0728.53031
Let S be a compact smooth surface of genus $$g\geq 2$$ and with strictly negative curvature. A well known result of G. Margulis [Funct. Anal. Appl. 3(1969), 335-336 (1970); translation from Funkts. Anal. Prilozh. 3, No.4, 89-90 (1969; Zbl 0207.203)] gives an asymptotic formula for the number of closed geodesics on S with length less than t. For a fixed homology class $$\alpha \in H_ 1(S,{\mathbb{Z}})$$, the author gives an asymptotic formula for geodesics within the specified homology class. Let $$\gamma$$ be a closed geodesic, let $$\lambda$$ ($$\gamma$$) be its length and let $$[\gamma]\in H_ 2(S,{\mathbb{Z}})$$ be the associated homology class. If $$\pi$$ (t,$$\alpha$$) is the number of closed geodesics $$\gamma$$ for which $$\ell (\gamma)\leq t$$ and $$[\gamma]=\alpha$$, then there exist constants C, $$h>0$$ such that $$\pi (t,\alpha)/(Ce^{ht}/t^{g+1})\to 1$$ as $$t\to \infty$$. This generalizes a result due to A. Katsuda and T. Sunada for the special case of constant negative curvature [Am. J. Math. 110, No.1, 145-155 (1988; Zbl 0647.53036)]. The author extends their proof which depends on properties of L-functions. He does this by using a different approach to these L-functions which is based on a thermodynamic analysis of generalized zeta functions.
Reviewer: D.Hurley (Cork)

##### MSC:
 53C22 Geodesics in global differential geometry 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry
Full Text:


text:
# Elongation in bar with unequal applied forces

How is elongation in a uniform rod with unequal forces acting on opposite sides calculated? If applied forces are equal and opposite, the elongation is defined by the formula ($\delta = \frac{FL}{AE}$). How does the solution change for the case when forces are unequal (as shown)?

As you correctly note, the solution is different when the applied forces are not equal. The bar is not in static equilibrium: Both static and dynamic forces deform the bar in motion. These concepts are illustrated by superposition. $$\delta = \delta_\text{static} + \delta_\text{dynamic} \qquad = \frac {F_\text{less}L}{EA} + \frac{(F_\text{more} - F_\text{less})L}{2AE}$$

During changes in acceleration (when $\frac{\mathrm{d\;a(t)}}{\mathrm{dt}} \neq 0$), forces and accelerations within the damped solid body are transient, where $a(x,t)$, until they reach steady-state, where $\frac{\partial{\;a(x,t)}}{\partial{x}} = 0$.

Transient deformations in solid bodies are illustrated by a mass/spring system, where each mass element can be thought to represent differential mass element.

Newton's Second Law requires that the bar (of mass $M$) accelerate in the direction of $F_{net}$. $$\sum F \;\text{on bar:} \qquad F_\text{more}-F_\text{less} = Ma \qquad \Rightarrow \;\therefore a = \frac{F_\text{more} - F_\text{less}}{M}$$

The derivation of deformation is shown when a single force acts on the bar.

Dynamic Deformation:

A Free Body Diagram is taken at an arbitrary cross-section of the bar, where the mass of the split body is $m = (\frac{M}{L})x$. Summation of forces acting on $m$ is solved for $T(x)$.
$$\sum F \;\text{on split body:} \qquad F_{o} - T = ma$$ $$F_{o} - T = \overbrace{\left(\frac{M}{L}x\right)}^\text{m} \overbrace{\left(\frac{F_{o}}{M}\right)}^\text{a} = \frac{F_{o}}{L}x \qquad \Rightarrow \qquad T = F_{o} - \frac{F_{o}x}{L}$$ $$\therefore T = F_{o}\left(1-\frac{x}{L}\right)$$ The static axial deformation ($\delta = \frac{FL}{AE}$) written in differential form:
$$\mathrm{d \delta} = \frac{T \mathrm{dx}}{AE} = \frac{[F_{o}(1-\frac{x}{L})]\mathrm{dx}}{AE}$$ Integrate differential deformation over the length of the bar to determine total deformation: $$\delta = \int_0^L \mathrm{d \delta} \; = \frac{F_{o}}{AE} \int_0^L 1-\frac{x}{L} \mathrm{dx} \implies \; \delta = \frac{F_{o}L}{2AE}$$

The derivation can be generalized to include both forces, where integration of $T(x) = F_\text{more} - \dfrac{F_\text{more}-F_\text{less}}{L}x$ results in the same solution given by superposition.

$$\therefore \delta = \; \overbrace{\frac{F_\text{more}L}{2AE} + \frac{F_\text{less}L}{2AE}}^\text{Integration} \;=\; \overbrace{\frac{F_\text{less}L}{AE} + \frac{(F_\text{more}-F_\text{less})L}{2AE}}^\text{Superposition}$$

References:

$$T= F_\text{more} - \frac{(F_\text{more}-F_\text{less})x}{L}$$

where $T$ is tension in wire, $x$ is distance from $F_\text{more}$

Consider the same part,

Now,

Stress $$\frac{T}{A} = Y\frac{dn}{dx}\;,$$

where $n$ is elongation in wire.

$$\frac{T}{A}= F_\text{more} - \frac{(F_\text{more}-F_\text{less})}{LA}\;x= Y\frac{dn}{dx}\\\implies \frac{(F_\text{more} - F_\text{less})}{L}\; xdx= YdnA$$

On integrating both sides

$$F_\text{more}\;x - \frac{(F_\text{more}-F_\text{less})}{L}\;\frac{x^2}{2} = YNA$$

Putting the limits

$$F_\text{more}\;L - (F_\text{more}-F_\text{less})\;\frac{L}{2}= YNA\\ \implies \frac{F_\text{more}- \dfrac{(F_\text{more}- F_\text{less})}{2}}{A} = \frac{YN}{L}= \text{stress in wire}\;.$$

Hence elongation $$N =\frac{L\left(F_\text{more} - \dfrac{(F_\text{more}-F_\text{less})}{2}\right)}{AY}\;.$$

• This is the weirdest combination of text and math formatting combined. Instead of T=$F_{more} - (F_{more}-F_{less})x/L$ use $$T = F_{more} - (F_{less}-F_{more}) \frac{x}{L}$$ for example which renders as $$T = F_{more} - (F_{less}-F_{more}) \frac{x}{L}$$ – ja72 Dec 11 '15 at 19:35


text: I think the following is sufficiently elementary: a transposition in $S_n$ is an element of order 2 commuting with at least $2(n-2)!$ elements of the group. But $A_{n+1}$ does not have such an element if $n$ is large enough. Indeed, if $\sigma\in A_{n+1}$ is of order 2, then it is a product of $k$ independent transpositions where $k$ is even and $2\le k\le(n+1)/2$. The number of elements of $A(n+1)$ A_{n+1}$commuting with such$\sigma$equals$2^{k-1}k!(n+1-2k)!$, and this is smaller than$2(n-2)!$provided that$n\ge 6$. 1 I think the following is sufficiently elementary: a transposition in$S_n$is an element of order 2 commuting with at least$2(n-2)!$elements of the group. But$A_{n+1}$does not have such an element if$n$is large enough. Indeed, if$\sigma\in A_{n+1}$is of order 2, then it is a product of$k$independent transpositions where$k$is even and$2\le k\le(n+1)/2$. The number of elements of$A(n+1)$commuting with such$\sigma$equals$2^{k-1}k!(n+1-2k)!$, and this is smaller than$2(n-2)!$provided that$n\ge 6\$.


text:
# Question 66e6d

##### 1 Answer
Jun 27, 2016

The answer is (3) $27$

#### Explanation:

The idea here is that the temperature coefficient, which you'll sometimes see referred to as the ${Q}_{10}$ coefficient, relates the change in the rate of a reaction to a ${10}^{\circ} \text{C}$ increase in the temperature of the reaction.

Simply put, the ${Q}_{10}$ coefficient tells you how increasing the temperature of the reaction by ${10}^{\circ} \text{C}$ will affect its rate.

The equation that allows you to calculate the ${Q}_{10}$ coefficient looks like this

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {Q}_{10} = {\left({\text{rate"_2/"rate}}_{1}\right)}^{\frac{10}{{T}_{2} - {T}_{1}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Here

${T}_{1}$ - the initial absolute temperature of the reaction
${T}_{2}$ - the new absolute temperature of the reaction
${\text{rate}}_{1}$ - the rate at ${T}_{1}$
${\text{rate}}_{2}$ - the rate at ${T}_{2}$

Now, a ${Q}_{10}$ coefficient of $3$ means that with every ${10}^{\circ} \text{C}$ increase in temperature, the rate of the reaction increases by a factor of $3$.

It's important to notice that because you're dealing with change in temperature, you can skip the conversion to Kelvin.

That is the case because an increase by ${1}^{\circ} \text{C}$ is equal to an increase by $\text{1 K}$.

So in the case of a ${10}^{\circ} \text{C}$ increase, you would have

${T}_{2} - {T}_{1} = \text{10 K}$

and

Q_(10) = ("rate"_2/"rate"_1)^(10/10) = "rate"_2/"rate"_1

In your case,

3 = "rate"_2/"rate"_1 implies color(purple)(|bar(ul(color(white)(a/a)color(black)("rate"_2 = 3 xx "rate"_1)color(white)(a/a)|)))#

For your reaction, the temperature goes from ${35}^{\circ} \text{C}$ to ${65}^{\circ} \text{C}$. This is equivalent to

$\Delta T = {65}^{\circ} \text{C" - 35^@"C" = 30^@"C}$

So, if the rate of the reaction increases by factor of $3$ with every ${10}^{\circ} \text{C}$ increase in temperature, it follows that a

${30}^{\circ} \text{C" = 3 xx 10^@"C}$

increase will cause the rate to go up by

${\text{rate"_2 = overbrace(color(white)(aaa)3color(white)(aaa))^(color(darkgreen)("for a 10"^@"C increase")) xx overbrace(color(white)(aaa)3color(white)(aaa))^(color(darkgreen)("for a 10"^@"C increase")) xx overbrace(color(white)(aaa)3color(white)(aaa))^(color(darkgreen)("for a 10"^@"C increase")) xx "rate}}_{1}$

${\text{rate"_2 = 27 xx "rate}}_{1}$

Therefore, the rate of the reaction will increase by a factor of $27$.


text:
randRange( 2, 7 )

In the right triangle shown, AC = BC = AC. What is AB?

betterTriangle( 1, 1, "A", "B", "C", AC, AC, "?" );
AC * AC * 2

This is a 45-45-90 triangle because the two legs are congruent.

Thus, the hypotenuse is \sqrt{2} times as long as each of the legs.

AB = AC\sqrt{2}

2 * randRange( 2, 6 )

In the right triangle shown, AC = BC and AB = AB. How long are each of the legs?

betterTriangle( 1, 1, "A", "B", "C", "?", "?", AB );
AB * AB / 2

This is a 45-45-90 triangle because the two legs are congruent.

Thus, each leg is \frac{1}{\sqrt{2}} times as long as the hypotenuse.

AC = BC = \frac{1}{\sqrt{2}} AB = AB / 2 \sqrt{2}

2 * randRange( 2, 6 )

In the right triangle shown, AC = BC and AB = AB\sqrt{2}. How long are each of the legs?

betterTriangle( 1, 1, "A", "B", "C", "?", "?", AB + "\\sqrt{2}" );
AB * AB

This is a 45-45-90 triangle because the two legs are congruent.

Thus, each leg is \frac{1}{\sqrt{2}} times as long as the hypotenuse.

AC = BC = \frac{1}{\sqrt{2}} AB \sqrt{2} = AB

randRange( 2, 6 ) randFromArray([ [1, ""], [3, "\\sqrt{3}"] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and BC = BC + BCrs. How long is AB?

betterTriangle( 1, sqrt(3), "A", "B", "C", BC + BCrs, "", "?" );
4 * BC * BC * BCr

This is a 30-60-90 triangle.

Thus, the hypotenuse is twice as long as the shorter leg.

AB = 2 \cdot BC = 2 \cdot (BC + BCrs) = ( 2 * BC ) + BCrs

3 * randRange( 2, 6 ) randFromArray([ [1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3], [3, "\\sqrt{3}", (AC * 2), AC * AC * 4] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and AC = AC + ACrs. How long is AB?

betterTriangle( 1, sqrt(3), "A", "B", "C", "", AC + ACrs, "?" );
AB

This is a 30-60-90 triangle.

Thus, the hypotenuse is \frac{2}{\sqrt{3}} times as long as the longer leg.

AB = \frac{2}{\sqrt{3}} \cdot AC = \frac{2}{\sqrt{3}} \cdot (AC + ACrs) = ABs

randRange( 2, 6 ) randFromArray([ [1, ""], [3, "\\sqrt{3}"] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and AB = ( 2 * BC ) + BCrs. How long is BC?

betterTriangle( 1, sqrt(3), "A", "B", "C", "?", "", ( 2 * BC ) + BCrs );
BC * BC * BCr

This is a 30-60-90 triangle.

Thus, the shorter leg is half as long as the hypotenuse.

BC = \frac12 \cdot AB = \frac12 \cdot (( 2 * BC ) + BCrs) = ( BC ) + BCrs

3 * randRange( 2, 6 ) randFromArray([ [1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3], [3, "\\sqrt{3}", (AC * 2), AC * AC * 4] ]) randFromArray([ "\\angle A = 30^\\circ", "\\angle B = 60^\\circ" ])

In the right triangle shown, mAB and AB = ABs. How long is AC?

betterTriangle( 1, sqrt(3), "A", "B", "C", "", "?", ABs );
AC * AC * ACr

This is a 30-60-90 triangle.

Thus, the longer leg is \frac{\sqrt{3}}{2} times as long as the hypotenuse.

AC = \frac{\sqrt{3}}{2} \cdot AB = \frac{\sqrt{3}}{2} \cdot (ABs) = AC + ACrs


text:
# Even/odd composite functions

## Main Question or Discussion Point

Considering the composition of two functions ƒ · g

If g is even then does this mean that ƒ · g is even? why?

Or if g is odd and ƒ is even, then ƒ · g is even?

How can we show these statements?

Thanks.

Last edited:

Consider the fact that the identity function I(x) = x is odd and the absolute value function A(x)=|x| is even.

HallsofIvy
Homework Helper
Just like you would do any proof of this sort: use the definitions.

If g is even, then g(-x)= g(x). Now, what can you say of $f\cdot g(-x)= f(g(-x))$?

If g is odd then g(-x)= -g(x). If f is even then f(-x)= f(x). Now, what can you say of $f\cdot g(-x)= f(g(-x))$?

If g is even, then g(-x)= g(x). Now, what can you say of $f\cdot g(-x)= f(g(-x))$?
Could you explain a little bit more on this part please? Thanks.

f(-x)= f(x)
g(-x)= g(x)

f.g(-x) = f(-x(g(-x)))
f.g(-x) = f(g(x))

It is even? Because a function is even if:
f:(-a,a) -> R if for all $$x \in (-a,a)$$, f(x) = f(-x)

Regards,

Last edited:
tiny-tim
Homework Helper
Hi roam!

You need (for each part) a proof that starts "(ƒ · g)(-x) = … ", and finishes " … = (ƒ · g)(x)."

Hint: suppose g(3) = 7.

If g is even, what are (ƒ · g)(3) and (ƒ · g)(-3)?

If g is odd, what are (ƒ · g)(3) and (ƒ · g)(-3)?

If g is even, what are (ƒ · g)(3) and (ƒ · g)(-3)? even

If g is odd, what are (ƒ · g)(3) and (ƒ · g)(-3)? even

What if g is odd and f is even? would the (f · g) be even?

tiny-tim
Homework Helper
Hi roam!

No … you're missing the point!

Follow the hint … if g(3) = 7, what is (ƒ · g)(3) (how is (ƒ · g)(3) defined? and so what is it)? And what is (ƒ · g)(-3) (same procedure)?

HallsofIvy
Homework Helper
Could you explain a little bit more on this part please? Thanks.

f(-x)= f(x)
g(-x)= g(x)

f.g(-x) = f(-x(g(-x)))
NO! f.g(-x)= f(g(-x)) as you say below:
f.g(-x) = f(g(x))
and f(g(x))= f.g (x) doesn't it?

It is even? Because a function is even if:
f:(-a,a) -> R if for all $$x \in (-a,a)$$, f(x) = f(-x)
So you have just said, (f.g)(-x)= f.g(x), haven't you?

Regards,

and f(g(x))= f.g (x) doesn't it?
Sorry for the off-topic:

Is the "=" sign typically read as "is equal to" or as "equals"...?
I thought it would be the former but according to your question tag "doesn't it" you seem to use the latter.
Hm, probably both are possible

HallsofIvy


text:
# How would a drift diffusion model explain effects in reaction time but not accuracy?

How would a drift-diffusion model explain a case where a variable has an effect in reaction time but not in accuracy. I know that one explanation would be a speed-accuracy tradeoff, in which the decision boundaries are so far apart that there are unlikely to be any incorrect answers. But, is there another way that doesn't involve such a trade-off?

This could happen if the variable in question affected only the nondecision time parameter -- denoted $$T_{er}$$ in the diagram below (from this paper) -- which adds a constant "waiting time" before the diffusion process begins, thus shifting the reaction time distribution to the right.

• Ah! Thank you! So you are saying that if the variable forced participants to pause before beginning the random drift, that could lead to such an effect? – Dave May 21 at 0:00
• Technically, yes. But keep in mind the psychological interpretation of the nondecision time parameter. It's intended to represent time spent on extremely low-level biological & perceptual processes taking place before it's even registered in a subject's brain that a stimulus has been presented. The nondecision times are on the order of tens of milliseconds. So for a stimulus that makes subjects "pause" because, say, it's complicated, it's not psychologically plausible (or at least not consistent with the spirit of the model) to say that this is due to effects on the nondecision time parameter. – Jake Westfall May 21 at 0:26