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# Tag Info

38

Let's look at the formula and equivalent circuit for a transmission line. (1) Impedance rather than reactance. Reactance refers to the opposition to the change in current (of an inductor) or voltage (for a capacitor) - single components. The transmission line has $R,L$ and $C$ components - impedance is the ratio of voltage phasor to current phasor. (...

32

A transmission line has distributed inductance and capacitance along its entire length. We can think of it as infinitely many little inductors and capacitors along the line: simulate this circuit – Schematic created using CircuitLab Each inductor serves to limit the rate at which the capacitor can charge. But, as we divide the line into increasingly ...

28

With electrical transmission lines, it all has to do with the speed of light being finite, thus so is the speed of EM propagation in a wire. You can think of a wire as a long series of infinitesimal capacitors (connected by infinitesimal inductors). If you start charging the capacitors at one end, you have to keep pumping charge into the wire to charge ...

21

Adding to what Phil said: Now imagine everything starts out at 0 Volts and Amps in this long chain of inductors and capacitors, then you put a voltage step in one end. The way the inductors slow down how the capacitors are charged, a steady current will flow, which will be proportional to the voltage you put in. Since you have a voltage and a current ...

21

The impedance of a transmission line, in ohms, is the ratio of voltage wave and current wave that travels down the line. For a 100 ohm line for instance, a 1 volt wave will always be accompanied by a 10mA wave. Intuitively, the current wave delivers charge to the parts of the line that have to 'charge up' to the voltage of the voltage wave. If the 100 ohm ...

16

USB cables do require some precision engineering. There are stringent requirements on value of differential impedance, quality of interconnects, and amount of losses per cable. The high-speed part of USB cable, even at USB 2.0 480 Mbps data rate, is made of a twisted pair of wires, all wrapped into a shield. This makes it a "bi-axial" cable. The cable is ...

13

The short answer is you're going to be fine, route it above VCC and make sure you have some VCC to GND decoupling caps near your chip. Plus your route is pretty short at 600mil, I've seen some people do terrible things to USB routes that still end up working :) I think the best way to understand this is to consider where your return current will flow. ...

12

Jim had a very good answer. To expand on a few, however: 2) 50 Ohms is 50 Ohms (kind of). The dielectric constant of a material IS slightly frequency dependent. Therefore, the trace height and width you choose for 1 GHz will be a slightly different impedance at 10 GHz (if you need to worry about the difference, you probably already know about the difference!...

12

Very few do. If your chip's datasheet does not say it has 50-ohm termination, then it almost certainly does not. Traditional CMOS and TTL logic does not provide matching termination, though a few specialized types (line drivers?) might. Typically drivers are low impedance and receivers are high impedance (with some capacitance). Traditional ECL (emitter-...

11

When to consider a digital signal fast enough to treat the interconnect (PCB trace) as a transmission line depends on two things: Electrical length of the signal Rise/fall-time (which is another way to say frequency content) As a rule-of-thumb(*) whenever the electrical length of the trace is longer than some fraction of the rise/fall time, you will see ...

11

No, the 50-Ohms is not a convention for PCB tracks to carry signals. The 50 Ohms is a standard for coaxial cables and corresponding interconnects - dozens and dozens various SMA/SMB, BNC, type-N, etc. connectors. In fact, typical (thin) PCB traces have 65 - 80 - 100 Ohm characteristic impedance on a typical stack-up (7 mils or 12 mils of FR4 between ground ...

10

You might consider adding an attenuator probe to your PCB consisting of a single series resistor. The 50-ohm coax cable to your oscilloscope can be any length - the oscilloscope input at the far end of this cable must be 50-ohm-terminated, not left to its default 1MEG simulate this circuit – Schematic created using CircuitLab The series resistor Rs ...

9

I've been working on a Gigabit Ethernet project for months, and so far I checked datasheets from Realtek, TI, Microchip, and a few reference designs, here is what I've found out. 0. High-speed routing guidelines are sometimes ignored for RGMII, but big manufacturers recommend them. In many low-cost products, the RGMII signals are routed with no regards of ...

8

Scheme #1 is terminating only the differential mode signal, not the common mode. Scheme #2 is terminating both differential and common mode. Even with a perfectly symmetrical differential output signal you will have what we call "differential to common mode conversion" in the cable. So at the receiver you will have both common mode and differential mode. ...

8

Probe cable is lossy coax. Achieving a matched condition with an oscilloscope probe is virtually impossible because the source impedance of the circuit under test is unknown and generally different from the scope's 1MΩ or 50Ω input impedance. On top of that the input impedance of the oscilloscope has a reactive component as do most circuits under test, ...

7

Yes, a smooth bend would be better for signal integrity than the square corner. All other things being equal, that corner sticking out represents an unnecessary impedance "bump" in the transmission line. Although in this case, the dimensions are so short that you'd probably never be able to measure the difference. The solder fillets on the nearby components ...

6

It appears that I should solve for odd characteristic impedances of 50 Ohms each. Is this true? Yes, the differential-mode impedance is equal to twice the odd-mode impedance (at least for symmetric geometries). My board house can do 5/5 mil trace/space. Is there any reason not to use these values? If you can fit a design with larger gaps and wider ...

6

This seems the simplest mathematical way to derive characteristic impedance. Consider a "lump" of transmission line connected to the continuation of that transmission line ($Z_0$): - R is series resistance of cable for a given length L is series inductance of cable for a given length G is parallel conductance of cable for a given length C is parallel ...

6

What you're asking about is called a transmission line taper. In general, there's no analytical solution to describe the reflections. The link in Chris L's answer (if you follow through to Klopfenstein's paper) gives some examples of specific taper shapes where something close to an analytical answer has been found. The basic way to study it is to imagine ...

6

What is the difference between characteristic impedance and input impedance in transmission lines? Characteristic impedance ($Z_0$) depends on the transmission line and its physical properties. Mathematically it can be shown that if you know the inductance (L), capacitance (C), resistance (R) and conductance (G) per unit length, $Z_0$ is: - $\sqrt{\... 6 I am by no means an expert in this field (there is a lot going to consider), but... Are USB/DVI/Ethernet cables a trivially easier solution for tens of cm? USB, DVI, and Ethernet are all very different, so it's difficult to generalize, but I would say "yes, somewhat". Maybe not for 10-30 cm, but 50 cm+ is definitely getting up there. DVI is definitely ... 6 A rule of thumb is that characteristic impedance becomes seriously important when your trace length gets above a tenth of a wavelength. Below that you can mostly get away with treating your trace like a capacitor. 50 ohm single ended is a standard mostly used by the radio world. Digital signals that are fast enough to need impedance matching usually use ... 6 I like to use this sort of BNC to scope probe adapter, which frequently is included in the accessory kit you get with the probe. You put it on the tip of your oscilloscope probe, then plug it into a perfectly ordinary BNC connector. Since the probe has a high resistance (9 MΩ for a 10x probe) positioned right at the tip, there's minimal loading from this ... 5 There is a simple hand-waving explanation why the effective impedance of a (ideal) transmission line is a constant. Other explanations leave some confusion on how do we "select" Li and Ci in the model of transmission line. What are these Li and Ci exactly? First, once we say "transmission line", we are talking about long wires. How long? Longer than the ... 5 Continuous varying impedances are used all the time for impedance matching. If you have a very capacitive part of a trace (for example, where a large component pad might be), you can have a relatively inductive transition before or after it to "balance" it out. What will end up happening is that the reflections will "stack up" but, instead of being at one ... 5 When we talk about the "impedance" of a pcb trace, we are talking about the characteristic impedance of a uniform transmission line. The characteristic impedance depends mainly on H, W, and$\epsilon_r$in the figure above. To get a 50 Ohm characteristic impedance, you just have to define your trace width in the proper proportion to the H of your ... 5 The velocity that electricity propagates at is related to the dielectric in the cable. In free-space, an EM wave travels at the speed of light but there is only the permittivity of free space ($\epsilon_0$)to hinder it: - So, as the permittivity ($\epsilon_r\\$) rises, velocity decreases. Magnetic permeability is the other factor but this barely changes ...

5

At a risk of collecting few negative points, I will try to answer this question as follows: The "characteristic impedance" has no direct physical meaning. It is just a constant in amplitude coefficients in the solution to the "Telegrapher's Equation", which describes propagation of a sinusoidal electromagnetic wave(s) along a special geometry of uniform ...

5

It seems there's a fairly simple way to get a good approximation of the impedance of a random piece of cable. It requires a fast oscilloscope, a pulse generator with a fast rising edge, and a potentiometer with a range that includes the cable's impedance (cable impedance is usually less than 500 ohms.) The cable you want to test must also be long enough ...

4

This fascinating Bell Labs video beautifully demonstrates wave motion, SWR, and impedance matching sections on a completely mechanical tabletop device without needing mathematics. It's presented in a way that even a layperson can understand these concepts. Reflection of waves from free and clamped ends Superposition Standing waves and resonance Energy loss ...

Only top voted, non community-wiki answers of a minimum length are eligible


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## Anisotropic $$L^2$$-estimates of weak solutions to the stationary Oseen-type equations in 3D-exterior domain for a rotating body.(English)Zbl 1186.35163

Summary: We study the Oseen problem with rotational effect in exterior three-dimensional domains. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces in the whole three-dimensional space. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type and the theory of Calderon-Zygmund kernels in weighted spaces. For the extension of results to the case of exterior domains we use a localization procedure.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35B45 A priori estimates in context of PDEs 76E07 Rotation in hydrodynamic stability 76M30 Variational methods applied to problems in fluid mechanics
Full Text:

### References:

 [1] R. Farwig, A variational approach in weighted Sobolev spaces to the operator $$-\triangle+\partial/\partial x_{1}$$ in exterior domains of $$\mbi{R}^{3}$$, Math. Z., 210 (1992), 449-464. · Zbl 0727.35041 [2] R. Farwig, The stationary exterior 3-D problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447. · Zbl 0727.35106 [3] R. Farwig, An $$L^{q}$$-analysis of viscous fluid flow past a rotating obstacle, Tôhoku Math. J. (2), 58 (2006), 129-147. · Zbl 1136.76340 [4] R. Farwig, Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle, Banach Center Publ., 70 (2005), 73-84. · Zbl 1101.35348 [5] R. Farwig, T. Hishida and D. Müller, $$L^{q}$$-Theory of a singular “winding” integral operator arising from fluid dynamics, Pacific J. Math., 215 (2004), 297-312. · Zbl 1057.35028 [6] R. Farwig and T. Hishida, Stationary Navier-Stokes flow around a rotating obstacles, Funkcial. Ekvac., 50 (2007), 371-403. · Zbl 1180.35408 [7] R. Farwig, M. Krbec and Š. Nečasová, A weighted $$L^{q}$$-approach to Stokes flow around a rotating body, Ann. Univ. Ferrara Sez. VII Sci. Mat., 54 (2008), 61-84. · Zbl 1248.35158 [8] R. Farwig, M. Krbec and Š. Nečasová, A weighted $$L^{q}$$-approach to Oseen flow around a rotating body, accepted in Math. Methods Appl. Sci. · Zbl 1224.76034 [9] R. Finn, Estimates at infinity for stationary solution of Navier-Stokes equations, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.), 3, 51 (1959), 387-418. · Zbl 0106.39402 [10] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. · Zbl 0149.44606 [11] G. P. Galdi, On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications, Handbook of Mathematical Fluid Dynamics, 1 , (eds. S. Friedlander and D. Serre), Elsevier, North-Holland, Amsterdam, 2002, pp. 653-791. · Zbl 1230.76016 [12] G. P. Galdi, Steady flow of a Navier-Stokes fluid around a rotating obstacle, Essays and papers dedicated to the memory of Clifford Ambrose Truesdell III, II , J. Elasticity, 71 (2003), 1-31. · Zbl 1156.76367 [13] G. P. Galdi and A. L. Silvestre, On the steady motion of a Navier-Stokes liquid around a rigid body, Arch. Ration. Mech. Anal., 184 (2007), 371-400. · Zbl 1111.76010 [14] G. P. Galdi and A. L. Silvestre, Further results on steady-state flow of a Navier-Stokes liquid around a rigid body, Existence of the wake, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci., Kyoto, 2007, pp. 127-143. · Zbl 1119.76011 [15] M. Geissert, H. Heck and M. Hieber, $$L^{p}$$ theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle, J. Reine Angew. Math., 596 (2006), 45-62. · Zbl 1102.76015 [16] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, 749 , Springer-Verlag, Berlin, Heidelberg, New York, 1979. · Zbl 0413.65081 [17] T. Hishida, The Stokes operator with rotation effect in exterior domains, Analysis (Munich), 19 (1999), 51-67. · Zbl 0938.35114 [18] T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle, Arch. Ration. Mech. Anal., 150 (1999), 307-348. · Zbl 0949.35106 [19] T. Hishida, $$L^{q}$$-estimates of weak solutions to the stationary Stokes equations around a rotating body, J. Math. Soc. Japan, 58 (2006), 743-767. · Zbl 1184.35241 [20] J.-L. Impagliazzo, Resolution des equations de Navier-Stokes compressibles a l’aide de la methode de decomposition, These de doctorat de l’Universite de Toulon et du Var, 1997. [21] T. Kobayashi and Y. Shibata, On the Oseen equation in the three-dimensional exterior domains, Math. Ann., 310 (1998), 1-45. · Zbl 0891.35114 [22] H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J., 40 (1991), 1-27. · Zbl 0732.35068 [23] S. Kračmar, Š. Nečasová and P. Penel, Anisotropic $$L^{2}$$-estimates of weak solutions to the stationary Oseen type equations in $$\mbi{R}^{3}$$ for a rotating body, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci., Kyoto, 2007, pp. 219-235. · Zbl 1153.35060 [24] S. Kračmar, Š. Nečasová and P. Penel, Remarks on the non-homogeneous Oseen problem arising from modeling of the fluid around a rotating body, Proceedings of the International Conference “Hyperbolic equations”, held in Lyon, France, 2006, [25] S. Kračmar, A. Novotný and M. Pokorný, Estimates of three dimensional Oseen kernels in weighted $$L^{p}$$ spaces, In: Applied Nonlinear Analysis, London-New York, Kluwer Academic, 1999, pp. 281-316. · Zbl 0961.35105 [26] S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $$L^{p}$$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111. · Zbl 0988.76021 [27] S. Kračmar and P. Penel, Variational properties of a generic model equation in exterior 3D domains, Funkcial. Ekvac., 47 (2004), 499-523. · Zbl 1114.35053 [28] S. Kračmar and P. Penel, New regularity results for a generic model equation in exterior 3D domains, Banach Center Publ., 70 (2005), 139-155. · Zbl 1101.35350 [29] Š. Nečasová, Asymptotic properties of the steady fall of a body in a viscous fluid, Math. Methods Appl. Sci., 27 (2004), 1969-1995. · Zbl 1174.76306 [30] C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodynamik, Leipzig, Akad. Verlagsges. M.B.H., 1927. · JFM 53.0773.02 [31] H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Berlin, Birkhäuser, 2001. · Zbl 0983.35004 [32] E. A. Thomann and R. B. Guenther, The fundamental solution of the linearized Navier-Stokes equations for spinning bodies in three spatial dimensions,–,time dependent case, J. Math. Fluid Mech., 8 (2006), 77-98. · Zbl 1125.35076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

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# Events Archive

## Event Archive

Christopher O'Neill - Department of Mathematics, University of California at Davis
Apr 26, 2018 - 1:50pm to 3:00pm
Department of Applied Mathematics - Seminar - Rettaliata Engineering Center, Room 119
An affine semigroup $$S$$ is a subset of $$\mathbb Z_{\geq 0}^k$$ that is closed under vector addition, and a factorization of $$a\in S$$ is an expression of $$a$$ as a sum of generators of $$S$$. The delta set of $$a$$ is a set of positive integers determined by the missing factorization lengths... read more
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4/26/18
Sonja Petrovic - Department of Applied Mathematics, Illinois Institute of Technology
Apr 25, 2018 - 12:50pm to 1:50pm
Department of Applied Mathematics - Seminar - Rettaliata Engineering Center, Room 122
This set of lectures covers some of the basics of how algebraic statistics is used for network modeling. Topics of this lecture include the following: Exponential random graph models. Log-linear models. Sufficient statistics. Example of a small network. Markov bases for conditional tests.
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4/25/18
Sonja Petrovic - Department of Applied Mathematics, Illinois Institute of Technology
Apr 24, 2018 - 12:45pm to 1:45pm
Center for Interdisciplinary Scientific Computation (CISC) - Seminar - Robert A. Pritzker Science Center, Room 121
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4/24/18


text:
# [Rd] problem with \eqn (PR#8322)

Kurt Hornik Kurt.Hornik at wu-wien.ac.at
Fri Nov 18 20:38:01 CET 2005

>>>>> Duncan Murdoch writes:

> On 11/18/2005 12:40 PM, Hin-Tak Leung wrote:
>> Martin Maechler wrote:
>>
>>>>>>>> "Hin-Tak" == Hin-Tak Leung <hin-tak.leung at cimr.cam.ac.uk>
>>>>>>>> on Fri, 18 Nov 2005 16:38:28 +0000 writes:
>>>
>>>
Hin-Tak> Your own fault. See below. It is basic LaTeX and any LaTeX person
Hin-Tak> can tell you the answer...(most probably haven't bothered...)
>>>
>>> No.  Whereas I partly agree that it's Ross fault'' trying to
>>> use too smart LaTex (and using outdated \bf instead of \mathbf),
>>> ;-)
>>>
>>> The bug is really there, since we are talking about the Rd "language",
>>> not LaTeX, an in Rd,  \eqn and \deqn are defined to have either
>>> one or two arguments -- where Ross used the 2-argument version
>>> correctly (in principle at least) --> See the manual "Writing R
>>> Extensions".
>>
>>
>> Forgive me for not reading R-ext carefully, but Ross's Rd code is
>> still "obviously" wrong in the lights of the two-argument \eqn:
>> (really doesn't differ from the 1-arg interpretaion of \eqn)
>>
>> \eqn{{\bf\beta}_j}{\bf\beta}_jnormal-bracket5bracket-normal{b(j)}
>>
>> In other words,
>> \eqn{...}{...}_...
>>
>> and the "_" is still outside of any maths environment, which is most
>> probably not Ross's intention.

> But that is Latex code produced by R, not Rd code produced by Ross.
> The bug is in the Latex production (which I think is done by
> share/perl/R/Rdconv.pm, but I don't know Perl well enough to attempt
> to fix it).

Definitely a problem in Rdconv.

E.g.,

$cat foo.Rd \description{ \eqn{{A}}{B} } hornik at mithrandir:~/tmp$ R-d CMD Rdconv -t latex foo.Rd | grep eqn
\eqn{{A}}{A}{{B}

shows what is going on.

My reading of R-exts would suggest that it is not necessary to escape
braces inside \eqn (and in fact these are not unescaped by Rdconv).

Btw, the conversions of the above example are wrong for at least HTML
and text as well, giving

<i>A</i>{{B}

and

A{{B}

respectively.

-k


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Last Sunday a certain store sold copies of Newspaper A for

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Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. $$\frac{100p}{(125 – p)}$$

B. $$\frac{150p}{(250 – p)}$$

C. $$\frac{300p}{(375 – p)}$$

D. $$\frac{400p}{(500 – p)}$$

E. $$\frac{500p}{(625 – p)}$$

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Re: Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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udaymathapati wrote:
Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?
A. 100p / (125 – p)
B. 150p / (250 – p)
C. 300p / (375 – p)
D. 400p / (500 – p)
E. 500p / (625 – p)

This question can be solved by number plugging: just try some numbers for # of newspaper A sold and the # of newspaper B sold.

Below is algebraic approach:

Let the # of newspaper A sold be $$a$$ and the # of newspaper B sold be $$b$$.

Then:
$$r=\frac{a}{a + 1.25b}*100$$ and $$p=\frac{a}{a+b}*100$$ --> $$b=\frac{a}{p}*100-a=\frac{a(100-p)}{p}$$ --> $$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$ --> reduce by $$a$$ and simplify --> $$r=\frac{100p}{p+125-1.25p}=\frac{100p}{125-0.25p}$$ --> multiply by 4/4 --> $$r=\frac{100p}{125-0.25p}=\frac{400p}{500-p}$$.

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Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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Let's talk strategy here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. As a result, I might write out some steps that I would normally just do in my head on the GMAT, but I want to make sure everyone sees the complete approach. Ready? Here is the full "GMAT Jujitsu" for this question:

The first thing we need to do is setup our equations. The equation for a percent is simple: $$N=\frac{\%}{100}T$$. Alternately, $$100N=\%T$$.
The problem states that “r percent of the store’s revenue from newspaper sales was from Newspaper A.” Revenue is equal to the number of items sold, multiplied by the sales price of each item. So, let’s plug revenue into our percent formula:

$$100(Revenue_A)=\%(Revenue_{total})$$
$$100(1*A)=r(1*A+\frac{5}{4}B)$$

Notice that I translated $$1.25$$ into its fractional form. I call this strategy, “Fractions Are Your Friends” in my classes. Fractions are almost always easier to mathematically manipulate than decimals are.

The problem also states that “p percent of the newspapers that the store sold were copies of Newspaper A.” Thus,

$$100(Copies_A )=\%(Copies_{total})$$
$$100(A)=p(A+B)$$

We now have 2 equations with 4 unknowns, but the problem isn’t asking us to solve for a number. Our target is “r in terms of p”, which means we need to get rid of variables $$A$$ and $$B$$, while simultaneously solving for $$r$$. At this point, we have two options: (1) just doing the dang algebra, or (2) a strategy I call in my classes “Easy Numbers.” Let's take a look at both strategies, starting with “Easy Numbers”:

Tactic 1: Easy Numbers

Since the problem gives us ratios without ever giving us totals, we can actually chose the “total” values in a way that makes the math simple. The reasons why we can pick values for totals (if the totals aren’t given) are two-fold: first, doing this turns abstract formulas into very concrete, easy-to-understand equations. Second, since the total values clearly drop out of the final solutions – after all, $$A$$ and $$B$$ aren’t in the answer choices – assigning them values that will disappear by the end is totally fine.

With percentage questions without totals, a common “Easy Number” we could pick is $$100$$. The equation $$100A = p(A+B)$$ demonstrates this. If we say that $$A+B=100$$, then $$100A = p(100)$$ and $$A=p$$, allowing us to eliminate $$A$$. But we also know that if $$A=p$$ and $$A+B = 100$$, then $$B = 100–p$$. Plugging in these values into the revenue equation above gets us a single equation in terms of $$r$$ and $$p$$:

$$100(p)=r(1p+\frac{5}{4}(100-p)$$

From this point forward, it’s just math, though our goal shouldn’t be just to randomly move things around – instead we use the answer choices to inform what “shape” the GMAT wants the math to be in. Since the question tells us that we are looking for $$r$$ in terms of $$p$$, we isolate $$r$$ in the equation by dividing both sides by the same value:

$$r=\frac{100(p)}{p+\frac{5}{4}(100)-\frac{5}{4}p}=\frac{100(p)}{\frac{5}{4}(100)-\frac{1}{4}p}$$

Multiplying the fraction by $$\frac{4}{4}$$ (a strategy I call in my classes “Multiply by 1”), keeps the fraction equivalent, but simplifies the denominator so it looks like the answer choices:

$$r=\frac{400p}{500-p}$$

We have our answer. It’s “D”.

Tactic 2: Do the Algebra

Since the GMAT is a critical-thinking test that rewards mental flexibility, there is often more than one way to solve a question. (I guess this means that since the GMAT is a Computer-Adaptive Test, there is more than one way to skin a CAT!) Here is what it would look like algebraically…

First, we know we need to eliminate both $$a$$ and $$b$$ out of the equation, leaving only $$r$$ and $$p$$. One quick way to eliminate variables is to substitute. Solving the first equation for $$r$$, we get:

$$100(1*A)=r(A+\frac{5}{4}B)$$

$$r=\frac{100A}{A+\frac{5}{4}B}$$

Solving the second equation for $$b$$, we get:

$$100(A)=p(A+B)$$

$$100A-pA=pB$$

$$B=\frac{100A-pA}{p}=A(\frac{100-p}{p})$$

Now, substituting the second equation into the first allows us to eliminate $$b$$, while simultaneously getting rid of $$a$$, through a tactic I like to call in my classes “Divide and Conquer.” Watch this:

$$r=\frac{100A}{A+\frac{5}{4}B}=\frac{100A}{A+\frac{5}{4}(A(\frac{100-p}{p}))}$$

Now, our job is to turn what we have into one of the answer choices, so use the answer choices as a guide for how you should think about the math. I call this “Stay on Target.” We know we still need to get rid of $$A$$. Because we are dealing with a fraction, let’s see if we can factor out an $$A$$ out of the top and bottom, thereby cancelling it:

$$r=\frac{100A}{A(1+\frac{5}{4}(\frac{100}{p}-1))}=\frac{100}{1+\frac{5}{4}(\frac{100}{p}-1)} =\frac{100}{1+\frac{500}{4p}-\frac{5}{4}}=\frac{100}{\frac{500}{4p}-\frac{1}{4}}$$

We can multiply both the top and bottom of the fraction by $$4p$$, eliminating the messy fractions in the bottom of the denominator. This simplifies it down to:
$$r=\frac{400p}{500-p}$$

Now, let’s look back at this problem from the perspective of strategy. For those of you studying for the GMAT, it is far more useful to identify patterns in questions than to memorize the solutions of individual problems. This problem can teach us a solid pattern seen throughout the GMAT. If you are given percentages or ratios without totals, one way to cut through this abstraction is to plug in “Easy Numbers” for the totals. Otherwise, you could also solve such problems algebraically, but whatever you do, don't think haphazardly. Your job is to turn what you have been given into what you want to have. Use the answer choices as a guide for what you should do. And that is how you think like the GMAT.
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Re: Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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15 Jun 2012, 20:02
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zaarathelab wrote:
Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?
A. 100p / (125 – p)
B. 150p / (250 – p)
C. 300p / (375 – p)
D. 400p / (500 – p)
E. 500p / (625 – p)

What is the simplest way to solve this??

Let the total copies of newspaper(A+B) sold be 100
so the number of copies of A sold is p
number of copies of B sold is 100-p
thus revenue from A = p*1$= p$
revenue from B = (100-p)5/4; because 1.25 = 5/4
percent of revenue from A = r = p/p+[(100-p)5/4)]= 400p / (500 – p)
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Re: Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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26 Sep 2010, 12:14
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Wow this is a hard question no doubt.
How do you solve this one quickly? I tried pluging numbers instead on r and p but the result was really not comfortable no matter what numbers I used.
Also solving it with pure algebra is far from being simple or done in under 2-3 minutes.

What's the source?
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26 Sep 2010, 15:41
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Note that total revenue will be k*(p + (100-p)*1.25) where k is a constant depending on actual number of papers sold

The contribution of type A is kp

So r=100 * kp/k(p + 125 -1.25p)
= 100p/(125-.25p)
= 400p/(500-p)
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22 May 2012, 10:53
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The algebra way is not time taking even..if we proceed as below:
(News A) A= $1 (News B) B =$1.25 or $5/4 Total newspaper sold= x No. of A Newspaper sold = p/100 *x is r% of total revnue Total revenue: p/100*x*$1 + (100-p)/100*x*$5/4 Equation: px/100=r/100(px/100+(500/4-5p/4)x/100) px/100=r/100(4px+500x-5px/400) removing common terms as 100 and x out and keeping only r on RHS p=r(500-p)/400 or r=400p/(500-p)..Answer..D Manager Joined: 05 Nov 2012 Posts: 63 Concentration: International Business, Operations Schools: Foster '15 (S) GPA: 3.65 Re: Last Sunday a certain store sold copies of Newspaper A for [#permalink] Show Tags Updated on: 16 May 2013, 11:21 81 1 59 udaymathapati wrote: Last Sunday a certain store sold copies of Newspaper A for$1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? A. 100p / (125 – p) B. 150p / (250 – p) C. 300p / (375 – p) D. 400p / (500 – p) E. 500p / (625 – p) This problem can be easily solved by picking numbers. The explanation given in the OG can be very laborious. Lets say the number of newspaper A sold = 20, so revenue from A = 20 and the number of newspaper sold from B = 80, so revenue from B = 100. Now total revenue =120 out of which 20 came from A. So r (A) = 20/120 = 1/6 = 16.7% and p (A) = 20/100 *100 = 20 A) 100*20/(125-20) -> Incorrect B) 150*20/(250-20) -> Incorrect C) 300*20/(375-20) -> Incorrect D) 400*20/(500-20) = 8/48 = 1/6*100 = 16.7% - > Correct E) 500*20/(625-20) -> Incorrect So Ans D Originally posted by pikachu on 02 Mar 2013, 13:09. Last edited by pikachu on 16 May 2013, 11:21, edited 1 time in total. Manager Joined: 04 Dec 2011 Posts: 55 Schools: Smith '16 (I) Re: Last Sunday a certain store sold copies of Newspaper A for [#permalink] Show Tags 03 May 2013, 10:49 6 pikachu wrote: udaymathapati wrote: Last Sunday a certain store sold copies of Newspaper A for$1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? A. 100p / (125 – p) B. 150p / (250 – p) C. 300p / (375 – p) D. 400p / (500 – p) E. 500p / (625 – p) This problem can be easily solved by picking numbers. The explanation given in the OG can be very laborious. Lets say the number of newspaper A sold = 20, so revenue from A = 20 and the number of newspaper sold from B = 80, so revenue from B = 100. Now total revenue =120 out of which 20 came from A. So r = 20/120 = 1/6 = 16.7% and p = 20 A) 100*20/(125-20) -> Incorrect B) 150*20/(250-20) -> Incorrect C) 300*20/(375-20) -> Incorrect D) 400*20/(500-20) = 8/48 = 1/6*100 = 16.7% - > Correct E) 500*20/(625-20) -> Incorrect So Ans D I did tried picking smart nos...mmm..ok may be not smart as yours but basically here is my pick p=5 (5 papers of A sold) so revenue from A = 5 20 papers of B sold so 20*1.25 so revenue from paper B = 25 Total revenue R = 25+5 = 30 no of A paper sold = P = 5 so revenue = 5/30 or around 16.6% percent-------------->>> till this part I got it right now try plugin the answer choice D $$\frac{400*5}{500-5}$$ = $$\frac{2000}{495}$$ is not equal 16.6%.. what's wrong here? Manager Joined: 05 Nov 2012 Posts: 63 Concentration: International Business, Operations Schools: Foster '15 (S) GPA: 3.65 Re: Last Sunday a certain store sold copies of Newspaper A for [#permalink] Show Tags 16 May 2013, 11:23 3 nikhil007 wrote: pikachu wrote: udaymathapati wrote: Last Sunday a certain store sold copies of Newspaper A for$1.00 each and copies of Newspaper B for $1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p? A. 100p / (125 – p) B. 150p / (250 – p) C. 300p / (375 – p) D. 400p / (500 – p) E. 500p / (625 – p) This problem can be easily solved by picking numbers. The explanation given in the OG can be very laborious. Lets say the number of newspaper A sold = 20, so revenue from A = 20 and the number of newspaper sold from B = 80, so revenue from B = 100. Now total revenue =120 out of which 20 came from A. So r = 20/120 = 1/6 = 16.7% and p = 20 A) 100*20/(125-20) -> Incorrect B) 150*20/(250-20) -> Incorrect C) 300*20/(375-20) -> Incorrect D) 400*20/(500-20) = 8/48 = 1/6*100 = 16.7% - > Correct E) 500*20/(625-20) -> Incorrect So Ans D I did tried picking smart nos...mmm..ok may be not smart as yours but basically here is my pick p=5 (5 papers of A sold) so revenue from A = 5 20 papers of B sold so 20*1.25 so revenue from paper B = 25 Total revenue R = 25+5 = 30 no of A paper sold = P = 5 so revenue = 5/30 or around 16.6% percent-------------->>> till this part I got it right now try plugin the answer choice D $$\frac{400*5}{500-5}$$ = $$\frac{2000}{495}$$ is not equal 16.6%.. what's wrong here? nikhil, the error you are making is in terms of p, since p is the % of A newspapers sold P = 5/30*100 not 5 as you are using. hope that helps Senior Manager Joined: 10 Mar 2013 Posts: 458 Location: Germany Concentration: Finance, Entrepreneurship Schools: WHU MBA"20 (A$)
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Re: Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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21 Jul 2013, 09:12
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............A..... B
Price:.... 1.... 1,25
Amount:. P.... 100-P
------------------------------
Revenue: P+1,25*(100-p) ---> P + 125 - 1,25P= 125 - 0,25P

---> Revenue of A / Total Revenue: P / 125 - 0,25P = P/ ((500-p)/4)) = 4P/500-P
---> r/100 = 4P/500-P --> r = 400P / 500-P ...................Correct Answer is (D)

Hope that helps
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05 Nov 2013, 03:55
Quote:
$$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$ --> reduce by $$a$$ and simplify --> $$r=\frac{100p}{p+125-1.25p}=\frac{100p}{125-0.25p}$$

Hi Bunel, I don't get the reduction. How do you get rid of the $$a+$$ in the denominator?

I only get this:
$$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$ --> reduces to --> $$r=\frac{100 p}{a+1,25*(100-p)}$$
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05 Nov 2013, 06:53
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Marcoson wrote:
Quote:
$$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$ --> reduce by $$a$$ and simplify --> $$r=\frac{100p}{p+125-1.25p}=\frac{100p}{125-0.25p}$$

Hi Bunel, I don't get the reduction. How do you get rid of the $$a+$$ in the denominator?

I only get this:
$$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$ --> reduces to --> $$r=\frac{100 p}{a+1,25*(100-p)}$$

$$r=\frac{a}{a + 1.25*\frac{a(100-p)}{p}}*100$$;

Factor out a from the denominator: $$r=\frac{a}{a(1 + 1.25*\frac{(100-p)}{p})}*100$$.

Reduce it: $$r=\frac{1}{1 + 1.25*\frac{(100-p)}{p}}*100$$.

Hope it's clear.
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04 Mar 2014, 12:52
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$$\frac{r}{100} = \frac{a}{(a+1.25b)}$$
$$\frac{p}{100} = \frac{a}{(a+b)}$$
So our problem is how to go about solving with a+b and a+1.25b in the denominator. An easy way out is take the reciprocal.

So,
$$\frac{100}{r} = \frac{(a+1.25b)}{a}$$
and
$$\frac{100}{p} = \frac{(a+b)}{a}$$
So,
$$\frac{100}{r} = 1+ \frac{1.25b}{a}$$ ........(1)
and
$$\frac{100}{p} = 1+ \frac{b}{a}$$
or $$\frac{100}{p} -1 = \frac{b}{a}$$...........(2)
So we have isolated b/a to a corner. Let's substitute for b/a in (1) so that we can have an equation only in r & p which we could solve for r
$$\frac{100}{r} = 1+ \frac{5}{4} * (\frac{100}{p} -1)$$
$$\frac{100}{r} = 1+ \frac{5}{4} * (\frac{100-p}{p})$$
$$\frac{100}{r} = 1+ (\frac{500-5p}{4p})$$
$$\frac{100}{r} = (\frac{500-p}{4p})$$
$$\frac{1}{r} = (\frac{500-p}{400p})$$
Now take reciprocal again to get r:
$$\frac{r}{1} = (\frac{400p}{(500-p)})$$
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09 Apr 2014, 11:18
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Here is simplest & quickest way to reach answer:

Price of A= $1. Price of B=$1.25 i.e. $5/4 We want revenue(r) in terms of percent of A(p) To calculate revenue, Assume, p= 20 So, Revenue = 20(1)+80(5/4)=$ 120

Now, r= $20/$120= 1/6

Now, put p=20 in each option and try to see if you can get 100/6 anywhere.
Just by looking at options, I see only Option(D) can serve my purpose.

400p / (500 – p) = 100 X (4X20)/(480) = 100/6.
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09 May 2014, 06:06
Bunuel is there a reason why you chose to isolate B and not A? I tried doing it by isolating A but cant solve it.
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09 May 2014, 07:05
gciftci wrote:
Bunuel is there a reason why you chose to isolate B and not A? I tried doing it by isolating A but cant solve it.

In $$r=\frac{a}{a + 1.25b}*100$$ we have b only in one place while a there is represented twice. So, it's better to substitute b there. Though you should get the same answer no matter whether you substitute a or b.
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Re: Last Sunday a certain store sold copies of Newspaper A for  [#permalink]

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25 Jun 2014, 20:00
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udaymathapati wrote:
Last Sunday a certain store sold copies of Newspaper A for $1.00 each and copies of Newspaper B for$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125 – p)
B. 150p/(250 – p)
C. 300p/(375 – p)
D. 400p/(500 – p)
E. 500p/(625 – p)

When you see percent, it is good idea to pick 100.

Let's say store sold 100 newspapers and p is 60% --> 60 (also 60%) A and 40 (also 40%) B.
Then, Rev(A) = 60 * $1 =$60, Rev(B) = 40 * $1.25 =$50. Total Rev --> $60 +$50 = $110. Given that r percent of stores revenue is from newspaper A --> r =$60 / $110 (D) --> (400 * 60%) / (500 - 60) => 240 / 440, which is same as 60 / 110. _________________ If you learn to do things that you need to do - then someday - you can do things that you want to do. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 10020 Location: Pune, India Re: Last Sunday a certain store sold copies of Newspaper A for [#permalink] Show Tags 25 Jun 2014, 23:46 8 2 udaymathapati wrote: Last Sunday a certain store sold copies of Newspaper A for$1.00 each and copies of Newspaper B for \$1.25 each, and the store sold no other newspapers that day. If r percent of the store’s revenues from newspaper sales was from Newspaper A and if p percent of the newspapers that the store sold were copies of newspaper A, which of the following expresses r in terms of p?

A. 100p/(125 – p)
B. 150p/(250 – p)
C. 300p/(375 – p)
D. 400p/(500 – p)
E. 500p/(625 – p)

Yes, you can solve this question by assuming a value for p. Note that the options are such that they will involve heavy calculations for most values of p. Easiest should be putting p = 100. Now you might think that two types of newspapers are sold so p = 100 will not be accurate but it is possible that p is approximately equal to 100. Say the store sold 1 million newspapers such that only 1 newspaper was of type B while all others were of type A. In that case, p would be approximately equal to 100%. Of course if almost all newspapers sold were of type A, all the revenue would also come from type A newspapers.
So we are looking for the option which gives 100 when you put p = 100.

A. 100p/(125 – p)
If you put p = 100, you will get 100*100/25 (much more than 100)

B. 150p/(250 – p)
If you put p = 100, you will get 150*100/150 = 100

C. 300p/(375 – p)
If you put p = 100, you will get (300/275)*100 (more than 100)

D. 400p/(500 – p)
If you put p = 100, you will get 400*100/400 = 100

E. 500p/(625 – p)
If you put p = 100, you get (500/525)*100 (less than 100)

So answer should be one of (B) and (D). Put p = 50. If 50% newspapers were A and 50% were B, say 100 type A papers were sold and 100 type B such that fraction of revenue from type A papers = (100/225)* 100 = 400/9

B. 150p/(250 – p)
Put p = 50, we get 150*50/200. There is no 9 in the denominator here so answer must be (D). Just to verify, we can calculate for (D) as well.

D. 400p/(500 – p)
Put p = 50, we get 400*50/450 = 400/9

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26 Jun 2014, 01:41
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 2009-01-06, 21:33 #1 Dougal     Jan 2009 Ireland 2×3×31 Posts MM127 Since there seems to be a large chance that this number is prime compared to other numbers,has anyone tried trial factoring it to a high bit depth?im new enough to this,so is there any way that you can stop in the middle of factoring and pick up where you started later on?
2009-01-06, 22:34   #2
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"Luigi"
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Quote:
 Originally Posted by Dougal Since there seems to be a large chance that this number is prime compared to other numbers,has anyone tried trial factoring it to a high bit depth?im new enough to this,so is there any way that you can stop in the middle of factoring and pick up where you started later on?

Check Will Edgington page: http://www.garlic.com/~wedgingt/MMPstats.txt
MathWorld page http://mathworld.wolfram.com/DoubleMersenneNumber.html

and if you still want to check them, you have the choice of Ernst Mayer's mfactor (if there is a version that can handle large exponents) ready to restart from any k in factors like 2kp+1, and my Factor5 multi-thread aware.

Luigi

 2009-01-06, 23:41 #3 ewmayer ∂2ω=0     Sep 2002 República de California 24·733 Posts FYI, I have tested MM127 up to 175 bits (k = 2^47 = 140737488355328, a few bits higher than the limits listed for Tony Forbes search on the MMPstats.txt page) with no factors found. Once I finish work on my current updates to Mlucas I plan to revisit the double-Mersenne factoring code, perhaps release a multithreaded binary (the current version allows the factoring to be split into up to 16 independent passes, but that requires 16 executable images running separately) and see if we can't organize a search at least up to 180-odd bits. And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime. But it sure would be nice to settle the question once and for all, just to shut up the cranks. [Who would then surely proceed to some other equally-unfounded speculation - but one would hope that at least the cranks would lose enough "credibility" thereby that they would have trouble wasting other folks' times in their future crankish endeavors.]
2009-01-06, 23:50   #4
rogue

"Mark"
Apr 2003
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660310 Posts

Quote:
 Originally Posted by ewmayer FYI, I have tested MM127 up to 175 bits (k = 2^47 = 140737488355328, a few bits higher than the limits listed for Tony Forbes search on the MMPstats.txt page) with no factors found. Once I finish work on my current updates to Mlucas I plan to revisit the double-Mersenne factoring code, perhaps release a multithreaded binary (the current version allows the factoring to be split into up to 16 independent passes, but that requires 16 executable images running separately) and see if we can't organize a search at least up to 180-odd bits. And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime. But it sure would be nice to settle the question once and for all, just to shut up the cranks. [Who would then surely proceed to some other equally-unfounded speculation - but one would hope that at least the cranks would lose enough "credibility" thereby that they would have trouble wasting other folks' times in their future crankish endeavors.]
Geoff Reynold's and I wrote new code for double-Mersenne factoring. I've tried to contact Tony to see if he was interested as our code can be run on many platforms, but Tony never responded to my inquiries. If you are interested I can send the source to you.

2009-01-07, 00:21   #5
CRGreathouse

Aug 2006

3·1,993 Posts

Quote:
 Originally Posted by ewmayer And the fact that MM127 has no known factors only makes it roughly twice as likely to be prime as a randomly chosen Mersenne number in that size range, which - if we extrapolate the known plausible heuristics regarding the statistical distribution of M-prime exponents - still makes it exceedingly unlikely to be prime.
I'm familiar with the standard heuristic about the probability of 2^p-1 being prime, given that p is prime. But how do you scale that probability given that there are no primes under k that divide it?

2009-01-07, 00:34   #6
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Quote:
 Originally Posted by CRGreathouse I'm familiar with the standard heuristic about the probability of 2^p-1 being prime, given that p is prime. But how do you scale that probability given that there are no primes under k that divide it?
http://web.archive.org/web/200711121...e.org/math.htm
Quote:
 A simple approach is to repeatedly apply the observation that the chance of finding a factor between 2X and 2X+1 is about 1/x. For example, you are testing 210000139-1 for which trial factoring has proved there are no factors less than 264. The chance that it is prime is the chance of no 65-bit factor * chance of no 66 bit factor * ... * chance of no 5000070 bit factor. That is: Code:  64 65 5000069 -- * -- * ... * ------- 65 66 5000070 This simplifies to 64 / 5000070 or 1 in 78126.
Of course, it goes on to say that it's really uncertain and there's some slightly more accurate estimates, (but I don't know if those other ones can be adjusted for varying factor depths as easily) but there's an approximation for you.

 2009-01-07, 07:00 #7 CRGreathouse     Aug 2006 10111010110112 Posts Great, I combined that source with Chris Caldwell's simplified Wagstaff proof to come up with a reasonable heuristic. There is a funny bit about constants (wanting to be e^gamma at one end at 2 at the other), but I'll probably just interpolate smoothly. This gives the odds that MM127 is prime as 1.8 x 10^-36, given that M127 is prime and it has no factors up to 175 bits.
 2009-01-08, 18:02 #8 Dougal     Jan 2009 Ireland BA16 Posts MM127 luigi, i have tried your factoring programs before and they are very good,but there seems to be a problem with the cygwin files so it wont open,could you help me with this problem? with prime95 are you supposed to create your own work to do file?because my download doesnt have one,it also doesnt have a help file. MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it? what other trial factoring programs are there that can be stopprd and restarted where it left off? i dont know a lot about this subject but im looking to learn,does anyone have links to web sites with dumbed down explanations of,lets say, the lucas lehmer test? thanks for all your help. Don
2009-01-08, 18:31   #9
ewmayer
2ω=0

Sep 2002
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Quote:
 Originally Posted by rogue Geoff Reynold's and I wrote new code for double-Mersenne factoring. I've tried to contact Tony to see if he was interested as our code can be run on many platforms, but Tony never responded to my inquiries. If you are interested I can send the source to you.
Thanks, but I have my own code for this. ;)

Be happy to do some comparative timings, though, to see where each of us in the speed ballpark.

2009-01-08, 19:35   #10
10metreh

Nov 2008

44228 Posts

Quote:
 Originally Posted by Dougal MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it?
Then there's MMM127 etc. Just like for Fermat's Last Theorem, proving one case is nothing.

2009-01-08, 20:26   #11
bsquared

"Ben"
Feb 2007

3,617 Posts

Quote:
 Originally Posted by Dougal MM127 is MMMMM2,isnt it?i know essentialy this doesnt make it more likely to be prime but it would be important if it was proved that it doesnt have a factor,wouldnt it?
Proving that it doesn't have a factor by trial division up to some bound is essentially meaningless, as CR Greathouse showed (you could raise the trial factoring bit level several levels and it would still have only an infinitesimal chance of being prime). The hope is to prove that it *does* have a factor. Then the MM127 zealots can go pick a different number to rally around.

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# Sturm’s theorem

This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.

###### Definition 1.

Let $P(x)$ be a real polynomial in $x$, and define the Sturm sequence of polynomials $\big{(}P_{0}(x),P_{1}(x),\ldots\big{)}$ by

 $\displaystyle P_{0}(x)$ $\displaystyle=$ $\displaystyle P(x)$ $\displaystyle P_{1}(x)$ $\displaystyle=$ $\displaystyle P^{\prime}(x)$ $\displaystyle P_{n}(x)$ $\displaystyle=$ $\displaystyle-\mathrm{rem}(P_{n-2},P_{n-1}),n\geq 2$

Here $\mathrm{rem}(P_{n-2},P_{n-1})$ denotes the remainder of the polynomial $P_{n-2}$ upon division by the polynomial $P_{n-1}$. The sequence terminates once one of the $P_{i}$ is zero.

###### Definition 2.

For any number $t$, let $var_{P}(t)$ denote the number of sign changes in the sequence $P_{0}(t),P_{1}(t),\ldots$.

###### Theorem 1.

For real numbers $a$ and $b$ that are both not roots of $P(x)$,

 $\#\{\textrm{distinct real roots of }P\textrm{ in }(a,b)\}=var_{P}(a)-var_{P}(b)$

In particular, we can count the total number of distinct real roots by looking at the limits as $a\rightarrow-\infty$ and $b\rightarrow+\infty$. The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials.

Note that deg $P_{n}<$ deg $P_{n-1}$, and so the longest possible Sturm sequence has deg $P+1$ terms.

Also, note that this sequence is very closely related to the sequence of remainders generated by the Euclidean Algorithm; in fact, the term $P_{i}$ is the exact same except with a sign changed when $i\equiv 2$ or $3\pmod{4}$. Thus, the Half-GCD Algorithm may be used to compute this sequence. Be aware that some computer algebra systems may normalize remainders from the Euclidean Algorithm which messes up the sign.

For a proof, see Wolpert, N., “ http://web.archive.org/web/20050412175929/http://www.mpi-sb.mpg.de/ nicola/Vorlesung/sturm.psProof of Sturm’s Theorem”

Title Sturm’s theorem SturmsTheorem 2013-03-22 14:28:49 2013-03-22 14:28:49 rspuzio (6075) rspuzio (6075) 14 rspuzio (6075) Theorem msc 11A05 msc 26A06 EuclidsAlgorithm Sturm Sequence


text:
# differential-geometry [closed]

For PDE (to compute surface of constant total curvature, Monge form, Monge-Ampere type): $$(rt-s^2)= -(1+p^2+q^2)^2,$$ where to find solution or how to program on Mathematica?

(BC : $y=0, p=0 ; x=0, q=0 ; x=0, y=0, z= 1).$

Regards, Narasimham GL

-
You may want to ask the question there: mathematica.stackexchange.com ... – tomasz Nov 26 '12 at 22:43
Why down voting? – Sigur Nov 26 '12 at 22:44

## closed as off topic by tomasz, draks ..., robjohn♦Nov 26 '12 at 23:38

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text:
#### Philippe G. LeFloch (Paris)

MAIN THEMES OF THE PROGRAM

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts (Cauchy problem, cosmic censorship, asymptotic behavior). These developments have brought into focus the deep connections between the Einstein equation and other important geometric PDE’s, including the wave map equation, Yang-Mills equation, Yamabe problem, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthrough, concerning the uniqueness and stability of the Kerr black hole model, the formation of trapped surfaces, and the bounded L2 curvature problem. Specifically, the themes of mathematical interest that will be developed in the present Program and are currently most active include:

• The initial value problem for Einstein equation and the causal geometry of spacetimes with low regularity, formation of trapped surfaces
• Techniques of Lorentzian geometry: injectivity radius estimates, geometry of null cones; construction of parametrix
• Geometry of black hole spacetimes: uniqueness theorems, censorship principles
• Coupling of Einstein equation for self-gravitating matter models, weakly regular spacetimes with symmetry

General schedule for the Trimester available here

SCIENTIFIC ACTIVITIES during the Trimester

WORKSHOPS AND CONFERENCES

REGISTER HERE

Sept. 14 to 18, 2015 Summer School – INTRODUCTION TO MATHEMATICAL GENERAL RELATIVITY

List of speakers

Greg Galloway (Miami)

Gerhard Huisken (Tuebingen)

Hans Ringstrom (Stockholm)

Sept. 23 to 25, 2015  Workshop – RECENT ADVANCES IN MATHEMATICAL GENERAL RELATIVITY

List of speakers

Spyros Alexakis (Toronto)

Piotr Chrusciel (Vienna)

Joao Costa (Lisbon)

Semyon Dyatlov (Cambridge, USA)

Stefan Hollands (Cardiff)

Alexandru Ionescu (Princeton)

Lionel Mason (Oxford)

Vincent Moncrief (Yale)

Jean-Philippe Nicolas (Brest)

Harvey Reall (Cambridge, UK)

Hans Ringstrom (Stockholm)

Mu-Tao Wang (New York)

Sept. 28 to Oct. 1, 2015  Workshop – GEOMETRIC ASPECTS OF MATHEMATICAL RELATIVITY (Hold in Montpellier and organized by Marc Herzlich and Erwann Delay)

List of speakers

Piotr Chrusciel (Vienna)

Michael Eichmair (Zürich)

Mu-Tao Wang (New York)

Oct. 26 to 29, 2015  Workshop – DYNAMICS OF SELF-GRAVITATING MATTER

List of speakers

Hakan Andreasson (Gothenburg)

Thierry Barbot (Avignon)

Robert Beig (Vienna)

David Fajman (Vienna)

Marc Mars (Salamanca)

David Maxwell (Fairbanks)

Todd Oliynyk (Monash)

Volker Schlue (Toronto)

Bernd Schmidt (Potsdam)

Jared Speck (Cambridge, USA)

Eric Woolgar (Alberta)

List of speakers

Jean-Pierre Bourguignon (Bures-sur-Yvette)

Pieter Blue (Edinburgh)

Demetrios Christodoulou (Zürich & Athens)

Mihalis Dafermos (Princeton)

Thibault Damour (Bures-sur-Yvette)

(Cape Town)

Richard Hamilton (New York) *

Gustav Holzegel (London)

Jonathan Luk (Cambridge, UK)

(Oxford)

Igor Rodnianski (Princeton)

Richard Schoen (Stanford)

Jacques Smulevici (Orsay)

Jérémie Szeftel (Paris)

Robert Wald (Chicago)

Qian Wang (Oxford)

* (to be confirmed)

Dec. 14 to 16, 2015  International Conference-  RELATIVITY AND GEOMETRY – IN MEMORY OF A. LICHNEROWICZ  (Organized by Giuseppe Dito, Jean-Pierre Francoise, Paul Gauduchon, Richard Kerner, Yvette Kosmann-Schwarzbach et Daniel Sternheimer)

List of speakers

Robert Bryant (Durham)

Pierre Cartier (Gif-Sur-Yvette)

Thibault Damour (Gif-Sur-Yvette)

Nathalie Deruelle (Paris 7)

Simon Donaldson  (Stony Brook & London)

Michel Dubois-Violette  (Paris 11)

Edward Frenkel  (Berkeley)

Simone Gutt  (Bruxelles)

James Isenberg  (Eugene)

Sergiu Klainerman  (Princeton)

Maxim Kontsevich  (Gif-Sur-Yvette)

Alan Weinstein  (Berkeley)

Program coordinated by the Centre Emile Borel at IHP. Financial support provided by the Institut Henri Poincaré and the ANR Project “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15/25-326

11h Emmanuel Hebey (Cergy-Pontoise) Systèmes de Kirchhoff critiques stationnaires sur des variétés compactes

14h  Lydia Bieri (Ann Arbor) Gravitational radiation and two types of memory

Abstract.  We are believed to live on the verge of detection of gravitational waves, which are predicted by General Relativity. In order to understand gravitational radiation, we have to investigate analytic and geometric properties of corresponding solutions to the Einstein equations. Gravitational waves leave a footprint in the spacetime regions they pass, changing the manifold – and therefore displacing test masses – permanently. This is known as the memory effect. It has been believed that for the Einstein equations, being nonlinear, there exists one such effect with a small linear’ and a large nonlinear’ part. In this talk, I present some of my joint work with D. Garfinkle showing that these are in fact two different effects.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Thierry Barbot (Avignon) Surfaces polygonales fuchsiennes et espace de Teichmüller décoré

Abstract. Dans l’article “Fuchsian polyhedra in Lorentzian space-forms, Mathematische Annalen 350, 2, pp. 417-453, 2011″, F. Fillastre a montré que toute métrique euclidienne avec singularités coniques d’angles > 2 pi sur une surface compacte se réalise de manière unique comme une surface de Cauchy polygonale dans un espace-temps globalement hyperbolique localement plat radial (i.e. dont le groupe d’holonomie fixe un point de l’espace de Minkowski). Dans cet exposé, j’évoquerai le travail de L. Brunswic dans son travail de thèse sous ma direction, qui vise à reprouver ce résultat et à l’étendre au cas des surfaces polygonales dans un espace-temps localement plat mais admettant des particules massives. Le but est de montrer qu’il y a encore existence et unicité une fois prescrit la masse des particules massives (le cas régulier montré par Fillastre correspondant au cas où l’angle singulier des particules massives est 2pi). Je montrerai aussi que la situation étudiée par R. Penner dans l’article “The Decorated Teichmϋller Space of Punctured Surfaces, Commun. Math. Phys. 113, 299-339 (1987)” est un cas limite de la situation étudiée par Brunswic, et correspond au cas où les particules sont d’angle conique nul. Je montrerai aussi comment répondre positivement à la question dans le cas où il n’y a qu’une singularité.

15h30 Andrea Seppi (Pavia) Convex surfaces in (2+1)-dimensional Minkowski space

Abstract.  It is known that the hyperbolic plane admits an isometric embedding into Minkowski space; in 1983 Hanu and Nomizu first observed the existence of non-equivalent isometric embeddings, thus showing a relevant difference with the Euclidean case. In this talk, I will introduce some natural properties of a convex surface in Minkowski space, concerning causality and asymptotic behavior. I will then explain some new results (jointly with Francesco Bonsante) on the classification of constant curvature surfaces with bounded principal curvatures and on the solvability of Minkowski problem in (2+1)-dimensional Minkowski space. If time permits, I will give the main ideas of the proof and especially the relation to some type of Monge-Ampere equations.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Shiwu Yang (Cambridge) Decay properties of solutions of Maxwell Klein-Gordon equations

Abstract.  I will present some recent progress on the asymptotic behavior of global solutions to Maxwell-Klein-Gordon equations. I will show that the integrated local energy and the energy flux through the outgoing null hypersurfaces decays polynomially in the retarded time in Minkowski space with data merely bounded in some gauge invariant weighted Sobolev space. This in particular includes the case with large charge. One novelty of this work is that these decay estimates precisely capture the asymptotic properties for the non-linear fields with arbitrarily large data. If in addition that the initial data for the scalar field is sufficiently small, then we show the pointwise decay of the solutions. This result improves the previous result of Lindblad and Sterbenz in which smallness is required for both the scalar field and the Maxwell field.

15h30 Gustav Holzegel (London) Local and global dynamics in asymptotically anti de Sitter spacetimes

Abstract.  Asymptotically anti de Sitter (aAdS) spacetimes play a prominent role in theoretical physics and mathematics.  Due to the presence of a timelike hypersurface at infinity these spacetimes are not globally hyperbolic, a fact that leads to intricate initial boundary value problems when studying global dynamics of hyperbolic equations on these backgrounds. In this talk, I will present several local and global results for the massive wave equation on aAdS spacetimes (including black hole spacetimes) with emphasis on how different boundary conditions (Dirichlet, Neumann or dissipative) influence the global dynamics. In particular, I will outline a recent proof (obtained in collaboration with J. Luk, J. Smulevici and C. Warnick) of linear stability and decay for gravitational perturbations on anti de Sitter space under dissipative boundary conditions. The proof unravels an interesting trapping phenomenon near the conformal boundary which necessarily leads to a degeneration in the decay estimates. Time permitting some future applications will also be discussed.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h François Fillastre (Cergy-Pontoise) Minkowski problem in Minkowski space

Abstract. T. Barbot, F. Beguin and A. Zeghib solved a smooth Lorentzian version of the Minkowski problem in dimension (2+1). More precisely they proved that if M is a flat 3-dimensional maximal globally hyperbolic spatially compact spacetime, then there exists a unique strictly convex smooth space-like surface in M with a prescribed smooth positive Gauss curvature. We will look at this problem for any dimensions. The existence part is solved in a generalized way (a measure is prescribed rather than a function). Concerning the regularity of the solution, the 2+1 case is specific. The arguments are based on tools from the geometry of convex sets. Joint work with Francesco Bonsante (Pisa).

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–326

14h Bruno Premoselli (Cergy-Pontoise) Robustness of the conformal constraints in a scalar-field setting

Abstract. The constraint equations arise in the initial-value formulation of the Einstein equations. The conformal method allows one to rewrite the constraint equations into a determined system of nonlinear, supercritical, elliptic PDE’s. In this talk, we will investigate some stability properties for this elliptic system. The notion of stability under consideration, defined as the continuous dependence of the set of solutions of the conformal constraint system with respect to its coefficients, is reformulated for the conformal method. The analysis of these stability properties involves blow-up techniques concerning defects of compactness for supercritical nonlinear elliptic equations. This is a joint work with Olivier Druet.

15h30 Christophe Bavard (Bordeaux) Points conjugués des tores lorentziens

Abstract. Les points conjugués jouent un rôle important dans l’étude des variétés riemanniennes et lorentziennes, en particulier pour l’étude du rayon d’injectivité. Dans le cadre riemannien, l’absence de points conjugués impose des contraintes assez fortes sur la topologie de la variété, et parfois même sur sa géométrie. Ainsi, un résultat de Hopf (1948), généralisé par Burago et Ivanov (1994), affirme qu’un tore riemannien sans points conjugués est nécessairement plat. Dans cet exposé, je montrerai l’existence de métriques sans points conjugués dans toute composante connexe de l’espace des métriques lorentziennes sur le tore de dimension 2 ; cela prouve en particulier l’existence de tores lorentziens sans points conjugués et non plats. Il s’agit d’un travail conjoint avec Pierre Mounoud.

#### Workshop from January 20 to 23, 2015

Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts, including the Cauchy problem, cosmic censorship, and asymptotic behavior. These developments have brought into focus the deep connections between the Einstein equation and other important geometric partial differential equations, including the wave map equation, Yang-Mills equation, Yamabe equation, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthroughs concerning the uniqueness and stability of black hole models, the formation of trapped surfaces, and the bounded L2 curvature problem. The themes of mathematical interest that will be particularly developed in the present Program include the formation of trapped surfaces and the nonlinear interaction of gravitational waves. The new results are based on a vast extension of the earlier technique by Christodoulou and Klainerman establishing the nonlinear stability of the Minkowski space. This Program will be an excellent place in order to present the recent breakthrough on the bounded L2 curvature problem for the Einstein equation, which currently provides the lower regularity theory for the initial value problem, as well as the recently developed theory of weakly regular Einstein spacetimes with distributional curvature.

Long-term participants

Michael Anderson (Stony Brook)

Piotr Chrusciel (Vienna)

Mihalis Dafermos (Princeton)

Cécile Huneau (Paris)

Alexandru D. Ionescu (Princeton)

James Isenberg (Eugene)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Paris)

Jared Speck (Cambridge, USA)

Jinhua Wang (Hangzhou)

Mu-Tao Wang (New York)

Qian Wang (Oxford)

Willie Wong (Lausanne)

Speakers during the Workshop

• Tuesday January 20
• Sung-Ji Oh (Berkeley) Linear instability of the Cauchy horizon in subextremal Reissner-Nordström spacetime under scalar perturbations
• Volker Schlue (Toronto) Stationarity of time-periodic vacuum spacetimes
• Alexandru D. Ionescu (Princeton) The Euler–Maxwell system for electrons: global solutions in 2D
• Joachim Krieger (Lausanne) Concentration-compactness for the critical Maxwell-Klein-Gordon equation
• Wednesday January 21
• Xianliang An (Piscataway) Two results on formation of trapped surfaces
• Tahvildar-Zadeh (Piscataway) The Dirac electron and the Kerr-Newman spacetime
• Mihalis Dafermos (Princeton)
• Jim Isenberg (Eugene) Asymptotically hyperbolic shear-free solutions of the Einstein constraint equations
• Thursday January 22
• Cécile Huneau (Paris) Stability in exponential time of Minkowski
• Jacques Smulevici (Orsay) Vector field methods for transport equations with applications to the Vlasov-Poisson system
• Mu-Tao Wang (New York) Quasi-local angular momentum and the limit at infinity
• Spyros Alexakis (Toronto) The Penrose inequality for perturbations of the Schwarzschild exterior
• Friday January 23
• Mihai Tohaneanu (Statesboro) Pointwise decay for the Maxwell system on black holes
• Qian Wang (Oxford)
• Peter Blue (Edinburgh) Revisiting decay of fields outside a Schwarzschild black hole
• Philippe G. LeFloch (Paris) Weak solutions to the Einstein equations in spherical or T2 symmetry

Attendees List

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25–104

14h Qian Wang (Oxford)  A geometric approach to the sharp local well-posedness theory for quasilinear wave equations

Abstract. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces Hs with s>2+(2-\sqrt 3)/2 for general quasilinear wave equation in (1+3) spacetime by him and Rodnianski. Via this approach they obtained the local well-posedness with s>2 for (1+3) vacuum Einstein equations. A proof of the sharp H2+ local well-posedness result for general quasilinear wave equation was provided by Smith and Tataru by constructing a parametrix using wave packets. The difficulty of the problem is that one has to face the major hurdle caused by the Ricci tensor of the metric for the quasilinear wave equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in (1+3) spacetime by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

15h30 Jonathan Luk (Cambridge, UK) Stability of the Kerr Cauchy horizon

Abstract. The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 15-25-321

14h Arick Shao (Imperial College) Unique continuation from infinity for linear waves

Abstract. We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects. This is joint work with Spyros Alexakis and Volker Schlue.

15h30 Claude Warnick (Warwick)  Dynamics in anti-de Sitter spacetimes

Abstract. When solving Einstein’s equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike hypersurface. Questions of local well-posedness and global stability are sensitive to the choices of boundary conditions. I will present recent work exploring the effects of non-trivial boundary data for the asymptotically AdS initial-boundary value problem, including a recent result in collaboration with Holzegel. I will also outline some interesting open problems in the area.

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room 1525-321

14h Jan Sbierski (Cambridge, UK) A Zorn-free proof of the existence of a maximal Cauchy development for the Einstein equations

Abstract. In 1969, Choquet-Bruhat and Geroch showed that there exists a unique maximal Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In particular, their proof ensures the existence of the maximal development without actually constructing it. In this talk, we present a proof of the existence of a maximal Cauchy development that avoids the use of Zorn’s lemma and, moreover, provides an explicit construction of the maximal development.

15h15 Sergiu Klainerman (Princeton)  Remarks on the stability of the Kerr solution in axial symmetry

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-321

Abstract. We will show that some curvature operators of Ricci (or Einstein) type are locally invertible, in some weighted Sobolev spaces on Rn, near the euclidian metric. In the smooth case, we then deduce that the image of some Riemann-Christoffel type operators are smooth submanifolds in the neighborhood of the Euclidian metric.

Abstract. The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-103

###### 14h  Florian Beyer  (Dunedin)  Graceful exit from inflation for minimally coupled Bianchi A scalar field models

Abstract. We consider the dynamics of Bianchi A scalar field models which undergo inflation. The main question is under which conditions does inflation come to an end and is succeeded by a decelerated epoch. This so-called ‘graceful exit’ from inflation is an important ingredient in the standard model of cosmology, but is, at this stage, only understood for restricted classes of solutions. We present new results obtained by a combination of analytical and numerical techniques.

###### 15h30  Cécile Huneau (ENS, Paris)Vacuum constraint equations for asymptotically flat space-times with a translational Killing field

Abstract. In the presence of a space-like translational Killing field, vacuum Einstein equations in 3+1 dimensions reduces to 2+1 Einstein equations with a scalar field. Minkowski space-time is a trivial solution of vacuum Einstein equation with a translational Killing field. A natural question is therefore the nonlinear stability of Minkowski solution in this setting. A first step in addressing this problem is the study of the constraint equations. The main difficulty in that case is due to the delicate inversion of the Laplacian. In particular, we have to work in the non constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.

### of Liquid-Vapor Flows

#### Thursday February 27, 2014 at 1:00pm

____________________________

### Subway station: Jussieu

#### Schedule and abstracts here !

____________________________

INVITED SPEAKERS

#### ____________________________

CONTRIBUTING SPEAKERS

#### ____________________________

Main organizer

Philippe G. LeFloch (Paris)

Co-organizers

Benjamin Boutin (Rennes)

Frédéric Coquel (Palaiseau)

#### ____________________________

PRACTICAL INFORMATIONS

How to come to the Laboratoire Jacques-Louis Lions

Hotels near the University Pierre et Marie Curie

#### ____________________________

EARLIER WORKSHOPS “Micro-Macro Modeling and Simulation of Liquid-Vapor Flows”

Eight Workshop, Berlin, February 2013

Seventh Workshop, Paris, February 2012

Sixth Workshop, Stuttgart, January 2011

Fifth Workshop, Strasbourg, April 2010

Fourth Workshop, Aachen, February 2009

Third Workshop, Strasbourg, January 2008

Second Workshop, Bordeaux, November 2007

Opening Workshop, Kirchzarten, November 2005

## Nonlinear Wave Equations at IHP

### Institut Henri Poincaré, Paris

Schedule available here

Further informations available here

Poster of the conference  here

INVITED SPEAKERS

Stefanos Aretakis (Princeton)

Nicolas Burq (Paris-Sud)

Pieter Blue (Edinburgh)

Mihalis Dafermos (Princeton)

Jean Marc Delort (Paris-Nord)

Gustav Holzegel (London)

Alexandru Ionescu (Princeton)

Joachim Krieger (EPFL)

Jonathan Luk (UPenn)

Franck Merle (Cergy & IHES)

Sung-Jin Oh (Princeton)

Fabrice Planchon (Nice)

Pierre Raphael (Nice)

Igor Rodnianski (MIT)

Chung-Tse Arick Shao (Toronto)

Jacques Smulevici (Paris-Sud)

Jacob Sterbenz (San Diego)

## Mathematical General Relativity

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

#### Lecture room  1525-103

###### 14h  Florian Beyer  (Dunedin)  Asymptotics and conformal structures of solutions to Einstein’s field equations

Abstract. Roger Penrose’s idea that the essential information about the asymptotics of solutions of the Einstein’s field equations is contained in the conformal structure and the associated conformal boundary has led to astonishing successes. In his original work, he provided several examples which made the importance of his idea evident. However, the question whether general solutions of Einstein’s field equations are compatible with this proposal remained unanswered. Motived by this, Helmut Friedrich has initiated a research programme to tackle this problem based on his so-called conformal field equations. In this talk I report on the status of this work and some of Friedrich’s results, but also on joint work with  collaborators at the University of Otago.

###### 15h30  Julien Cortier (IHES, Bures-sur-Yvette)  On the mass of asymptotically hyperbolic manifolds

Abstract. By analogy with the ADM mass of asymptotically Euclidean manifolds, a set of global charges can be defined for asymptotically hyperbolic manifolds. We will review their various definitions and , in particular, focus on the notion of “mass aspect” tensor, which gives rise to the  energy-momentum vector and arises  in the hyperbolic formulation of the positive mass theorem. We will compute these quantities for examples such that the Schwarzschild-anti de Sitter metrics, and we will present a family of counter-examples with “non-positive” mass when completeness is not assumed.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

#### .

###### 11h (Room  15-25- 104)  Sergiu Klainerman (Princeton)   On  the formation of trapped surfaces

Abstract. I will talk about a new result obtained in collaboration with J. Luk and I. Rodnianski in which we relax significantly Christodoulou’s main condition for the formation of trapped surfaces in vacuum.

###### 14h (Room 15-25-326)Chung-Tse Arick Shao (Toronto)   Null cones to infinity, curvature flux, and Bondi mass

Abstract. In general relativity, the Bondi mass in an asymptotically flat spacetime represents, roughly, the mass remaining in the system after some has radiated away. In this talk, we make sense of and control the Bondi mass for a single null cone in an Einstein-vacuum spacetime under minimal assumptions. In terms of regularity, we assume only small weighted curvature flux along the null cone and small data on an initial sphere of the cone. Furthermore, we make no global assumptions on the spacetime, as all our conditions deal only with the single null cone under consideration. This work is joint with S. Alexakis.

###### 15h30  (Room 15-25-326)Gustav Holzegel  (Princeton) Existence of dynamical vacuum black holes

Abstract. This is joint work with Mihalis Dafermos and Igor Rodnianski. We prove the existence of a large class of non-stationary vacuum black holes whose exterior geometry asymptotes in time to a fixed Schwarzschild or Kerr metric. The spacetimes are constructed by solving a backwards scattering problem for the vacuum Einstein equations with characteristic data prescribed on the horizon and at null infinity. The data admits the full functional degrees of freedom to specify data for the Einstein equations. An essential feature of the construction is that the solutions converge to stationarity exponentially fast with their decay rate intimately related to the surface gravity of the horizon and hence to the strength of the celebrated redshift effect which, in our backwards construction, is seen as a blueshift.

## Mathematical General Relativity

### Speaker

###### 11h15 –  José A. Font  (Valencia)  Simulations of neutron star mergers and black hole-torus systems

Abstract. Merging binary neutron stars are among the strongest sources of gravitational waves and have features compatible with the events producing short–hard gamma-ray bursts. Numerical relativity has reached a stage where a complete description of the inspiral, merger and post-merger phases of the late evolution of binary neutron star systems is possible. This talk presents an overview of numerical relativity simulations of binary neutron star mergers and the evolution of the resulting black hole–torus systems. Such numerical work is based upon a basic theoretical framework which comprises the Einstein’s equations for the gravitational field and the hydrodynamics equations for the evolution of the matter fields. The most well-established formulations for both systems of equations are briefly discussed, along with the numerical methods best suited for their numerical solution, specifically high-order finite-differencing for the case of the gravitational field equations and high-resolution shock-capturing schemes for the case of the relativistic Euler equations. A number of recent results are reviewed, namely the outcome of the merger depending on the initial total mass and equation of state of the binary, as well as the post-merger evolution phase once a black hole–torus system is produced. Such system has been shown to be subject to non-axisymmetric instabilities leading to the emission of large amplitude gravitational waves.

## Mathematical General Relativity

### Speakers

###### 10h – Miguel Sánchez Caja (Granada) Recent interrelated progress in Lorentzian, Finslerian and Riemannian geometry

Abstract.    Recently, a correspondence between the conformal structure of a class of Lorentzian manifolds (stationary spacetimes) and the geometry of a class of Finsler manifolds (Randers spaces) has been developed. This correspondence is useful in both directions. On one hand, it allows a sharp description of geometric elements on stationary spacetimes in terms of Finsler geometry. On the other hand, the geometry of spacetimes suggests, both, new geometric elements and new results, for any Finsler manifold, including the Riemannian case. Here, three levels of this correspondence will be explained: (1) Causal structure of spacetimes: properties of Finslerian distances: 0903.3501.  (2) Visibility and gravitational lensing: convexity of Finsler hypersurfaces: 1112.3892, 0911.0360. (3) Causal boundaries: Cauchy, Gromov, and Busemann boundaries in Riemannian and Finslerian settings: arXiv:1011.1154.

###### 11h30 Vladimir Matveev (Jena) Geodesic degree of mobility of Lorentzian metrics

Abstract. The degree of mobility of a metric can be defined as the dimension of the space of solutions of a certain linear PDE system of finite type whose coefficients depend on the metric, and, for a given metric, there are standard algorithms to determine it. The standard algorithms strongly depend on the metric and in most cases it is possible to find the maximal and sub-maximal values of the degree of mobility, only. I will show that the degree of mobility of a manifold is closely related to the space of parallel symmetric tensor fields on the cone over the manifold. In the case the metric is Einstein, it is essentially the tractor cone. I will use it to describe all possible values of the degree of mobility (on a simply connected manifold) for Riemannian and Lorentzian metrics. I will also consider the case when the metric is Einstein and, as a by-product, solve the classical Weyl-Petrov-Ehlers conjecture, and also show applications. Most these results are based on joint projects with  A. Fedorova and S. Rosemann.

###### 14h30 Philippe LeFloch (Paris)  Injectivity radius and canonical foliations of Einstein spacetimes

Abstract. I will discuss recent results on the local geometry of spacetimes with low regularity, when no assumption on the derivatives of the curvature tensor is made, obtained in collaboration with Bing-Long. Chen. Specifically, I will establish that, under geometric bounds on the curvature and injectivity radius, only, there exist local foliations by CMC (constant mean curvature) hypersurfaces, as well as CMC–harmonic coordinates. Importantly, these coordinates are defined in geodesic balls whose radii depend on the assumed bounds, only, and the components of the Lorentzian metric have the best possible regularity.

###### 15h00 Ghani Zeghib (Lyon) Actions on the circle and isometry groups of globally hyperbolic Lorentz surfaces (after D. Monclair)

Abstract. Let M be a globally hyperbolic spatially compact spacetime with dimension 1+1. A Cauchy surface in it is diffeomorphic to the circle and, more canonically, its family of lightlike geodesics is diffeomorphic to two copies of the circle and, under mild conditions, M embeds as an open set of the 2-torus.  The isometry group G of M acts naturally on these circles, so that G is a subgroup of Diff(S1). We will establish here that G tends to be included in PSL(2, R), the group of projective transformations of the circle S1, up to a global conjugacy by an element of the circle.

###### 15h30 Eduardo Garcia-Rio (Santiago de Compostela) Quasi-Einstein and Ricci soliton Lorentzian metrics

Abstract. Quasi-Einstein metrics are natural generalizations of Einstein metrics and gradient Ricci solitons.  Moreover, they are closely related to the existence of warped product Einstein metrics. Such metrics are defined by an overdetermined equation involving the Ricci curvature and the Hessian of a potential function. I will present some results on the geometry of Lorentzian quasi-Einstein metrics by focusing primary on those which are locally conformally flat. In this setting the Ricci curvature determines the whole curvature tensor and thus the different possibilities depend on the geometry of the level sets of the potential function: warped product metrics and pp-waves appear in a natural way.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

#### .

###### Jean-Philippe Nicolas (Brest) Perspectives in conformal scattering

Abstract. The origins of conformal scattering are to be found in a paper by Friedlander in 1980 “Radiation fields and hyperbolic scattering theory”, in which he realized that the Lax-Phillips theory was in fact providing an interpretation of scattering theory as the well-posedness of the characteristic Cauchy problem for the conformally rescaled wave-equation on null infinity. He clearly saw that the method provided an interesting geometrical short-cut to define a scattering operator with the advantage that all the analytical structure can be recovered a posteriori. The true power of the conformal approach to scattering theory lies in its complete indifference to time dependence. This talk will review the essential features of Lax-Phillips theory and its intimate link with conformal infinity via the Radon transform and the Whittaker formula, then describe the pinciples of conformal scattering with the crucial importance of the precise resolution of the Goursat problem; we will present some results (actual scattering constructions and studies of the Goursat problem) and explain the necessary steps of the extension of the method to black hole spacetimes, which is currently under development.

###### Jérémie Szeftel (Paris) The bounded L2 curvature conjecture in general relativity

Abstract.  In order to control locally a spacetime which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need an L2 bound on the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

## Mathematical General Relativity

### Université Pierre et Marie Curie, Paris

#### Decay for the Maxwell field outside a Kerr black hole

Abstract.  This talk will repeat some of the material from last year on the same topic (January 12, 2011) and present some new results. The goal of this talk is to prove uniform energy bounds and Morawetz (integrated decay) estimates.  In the exterior of a Kerr black hole, one of the components of the Maxwell system satisfies a wave equation with a complex potential. Trapping and the complex potential interact to provide surprisingly difficult challenges. Pseudodifferential techniques can treat a model problem with both features. However, because of the structure of the original Maxwell system, a new idea suggests classical derivatives alone should be sufficient.

#### Global structure of spherically symmetric spacetimes

Abstract. At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, an example of which is given by Einstein-Maxwell-Klein-Gordon (self-gravitating charged scalar fields). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary. In particular, for any ‘strongly tame’ Einstein-matter model, a ‘first singularity’ must emanate from a spacetime boundary to which the area-radius r extends continuously to zero.


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Browse Questions

# Find the linear functions that maps $[-1,1]$ onto $[0,2]$.

$\begin{array}{1 1} x-1,x+1 \\ x-1,-x-1 \\ x+1,-x+1 \\ x-1,-x+1 \end{array}$

Let $f(x)=ax+b,$ which is linear.
Since $f(x)$ is linear, it should be either increasing or decreasing.
Case (i); Let $f(x)$ be increasing.
$\Rightarrow\:f'(x)\geq 0$
$\Rightarrow\:a\geq 0$
$\Rightarrow\: f(-1)=0\:and\:f(1)=2$ (Since $f$ is onto,increasing in $[-1,1]\rightarrow\:[0,2]$).
$\Rightarrow\:-a+b=0\:\:and\:\:a+b=2$
Solving these two equations we get $a=b=1$
$\Rightarrow f(x)=x+1$
Case (ii): Let $f(x)$ be decreasing .
$\Rightarrow\:f'(x)\leq 0$
$\Rightarrow \:a\leq 0$
$\Rightarrow f(-1)=2\:\:and\:\:f(1)=0$ (Since $f$ is onto, decreasing in $[-1,1]\rightarrow [0,2])$.
$\Rightarrow -a+b=2\:\:and\:\:a+b=0$.
Solving these two equations we get
$a=-1,\:b=1$
$f(x)=-x+1$
The two linear equations are
$x+1\:\:and\:\:-x+1$


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Generating Functionals of Random Packing Point Processes: From Hard-Core to Carrier Sensing

Nguyen Tien Viet, Francois Baccelli

In this paper we study the generating functionals of several random packing processes: the classical Mat\'ern hard-core model; its extensions, the $k$-Mat\'ern models and the $\infty$-Mat\'ern model, which is an example of random sequential packing process. We first give a sufficient condition for the $\infty$-Mat\'ern model to be well-defined (unlike the other two, the latter may not be well-defined on unbounded spaces). Then the generating functional of the resulting point process is given for each of the three models as the solution of a differential equation. Series representations and bounds on the generating functional of the packing models are also derived. Last but not least, we obtain moment measures and Palm distributions of the considered packing models departing from their generating functionals.

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## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

$P = P_0 + \rho~g~h$ $P$ is the pressure $P_0$ is the atmospheric pressure $\rho$ is the density of the liquid $h$ is the depth below the surface For all six points A to F, $P_0$ is the same. Therefore, a point that is located deeper under the surface will have a higher pressure because the value of $h$ is greater. Since points A, B, and C are located at the same depth, the pressure is the same at all three points.