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Accurate light-time correction due to a gravitating mass

# Accurate light-time correction due to a gravitating mass

Neil Ashby
Department of Physics
University of Colorado, Boulder, Co. (USA)

Bruno Bertotti
Dipartimento di Fisica Nucleare e Teorica
Università di Pavia (Italy)
###### Abstract

This technical paper of mathematical physics arose as an aftermath of Cassini’s 2002 experiment [6], in which the PPN parameter was measured with an accuracy and found consistent with the prediction of general relativity. The Orbit Determination Program (ODP) of NASA’s Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression (8) for the gravitational delay which differs from the standard formula (2); this difference is of second order in powers of – the gravitational radius of the Sun – but in Cassini’s case it was much larger than the expected order of magnitude , where is the distance of closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat’s action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. This method is closely connected with the much more general approach of [18], which is based on Synge’s world function. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the required quantities, in particular the delay and the deflection, are easily obtained. We also show that in a close superior conjunction, when is much smaller than the distances of A and B from the Sun, of order , say, the second-order correction has an enhanced part of order , which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source when observed from a finite distance from the mass is obtained up to .

## 1 Introduction

In the framework of metric theories of gravity and the PPN formalism, the main violations of general relativity – those linear in the masses – are described by a single dimensionless parameter . The question, at what level and how general relativity is violated, in particular how much differs from unity, Einstein’s value, is still moot. No definite and consistent prediction about it are available, except for the inequality , which must be fulfilled in a scalar-tensor theory, in particular those arising as the low-energy limit of certain string theories. To date, the best measurement of has been obtained with Cassini’s experiment, which has provided the fit (at 1-)

 γ−1=(2.1±2.3)×10−5. (1)

Einstein’s prediction is still acceptable, but more accurate experiment are needed and planned.

While controls also other relativistic effects, in particular those related to gravito-magnetism, it mainly affects electromagnetic propagation. The differential displacement of the stellar images near the Sun historically was the first experimental effect to be investigated and is now of great importance in accurate astrometry. The bending of a light ray also increases the light-time between two points, an important effect usually named after its discoverer I. I. Shapiro [27]. Several experiments to measure this delay have been successfully carried out, using wide-band microwave signals passing near the Sun and transponded back, either passively by planets, or actively, by space probes (see [31], [24]).

Cassini’s 2002 experiment has implemented a third way to measure [4], in which coherent microwave trains sent from the ground station to the spacecraft (at that time about 7 AU far away) were transponded back continuously. The use of high-frequency carriers (in K band, 34 and 32 GHz) and the combination with standard X-band carriers (about 8 GHz) allowed successful elimination of the main hindrance, dispersive effects due to the solar corona traversed by the beam. The tracking was carried out around the 2002 superior conjunction; the minimum value of the impact parameter of the beam was , but in effect only 18 passages have been used, with a minimum impact parameter of . The two-way total amount of phase between the time of emission and the time of arrival has been continuously measured in each passage. In effect, however, NASA’s Deep Space Network provides the phase count in a given integration time . Mathematically, in the limit this would give the received frequency, in which Doppler effects and gravitational frequency shift are mixed up (Sec. 4). Cassini’s observable, therefore, can also be assessed in terms of the predicted change in frequency, as in [4]; but in practice, taking small would introduce unacceptable high-frequency noise. The change in light-time in a given integration time is the correct, theoretically available observable.

In the standard formulation for a superior conjunction, and taking the Sun at rest, the (one-way) light-time from an event A to an event B is:

 tB−tA=rAB+Δt=rAB+(1+γ)mlnrA+rB+rABrA+rB−rAB, (2)

where km is the gravitational radius of the Sun, are, in Euclidian geometry (See Fig. 1 left), the distances of A and B from the Sun and their distance. The velocity of light is unity. , the increase of the light-time over , is the gravitational delay.

In a close superior conjunction A and B are on the opposite sides of the mass and the Euclidian distance of the straight line AB from the mass fulfils , say. In this approximation eq. (2) reduces to

 tB−tA=rAB+Δt=rAB+(1+γ)mln(4rArBb20), (3)

with a logarithmic enhancement over the formal order of magnitude .111As stated in the supplementary material, in eq. (2) of [6] the two terms in the right-hand side should obviously be multiplied by a factor 2. This error, of course, had no consequence on the computer fit. Taking the logarithm equal to 10, this provides an estimate of the timing accuracy in terms of the error in :

 σΔt=1.43σγ×106cm, (4)

corresponding, in Cassini’s case, to 30 cm. (3) embodies also the one-way frequency change induced by gravity between A and B. Their motion makes (and the distances) change with time, so that, for a one-way experiment,

 Δνν=dΔtdt=−2(γ+1)mb0db0dt. (5)

The basic geometric setup is straightforward: a point mass at rest at the origin in an asymptotically flat space generates a line element with rotational symmetry. An invariant Killing time is defined; events on each constant surface are ‘simultaneous’ and the metric components are constant. The proper time of a static observer differs from by the red-shift factor . A null geodesic runs from the event A (with radial coordinate and time ) to the event B (with radial coordinate and time ); it stays on a plane, taken here as the equatorial plane . The (invariant) longitude difference completes the setup. In the PPN formalism and isotropic coordinates the metric reads:

 ds2 = A(r)dt2−B(r)dℓ2= (6) = (1−2mr+2βm2r2+…)dt2−(1+γ2mr+3ϵ2m2r2+…)dℓ2,

where

 dℓ2=dr2+r2(dθ2+sin2θdϕ2)=dr2+r2dΩ2

is the Euclidian line element. The parameters , and are equal to 1 in general relativity; while and are accurately known, currently no information is available about .

In our case the best mathematical tool to deal with electromagnetic propagation is not null geodesics, but the theory of eikonal. It is known (e.g., [19]) that in this problem Fermat’s Principle holds, corresponding to the refractive index

 N(r)=√B(r)A(r); (7)

we develop ab initio the eikonal and solve for it by separation of variables (Sec. 4). The radial part provides Fermat’s action as a radial integral containing and the impact parameter ; when computed at the true value , such action is just the required light-time. The solution can be obtained recursively, using appropriate expansions in powers of : the expansion for begins with , the distance of the straight line AB from the origin. In this way the variational nature of the problem brings about a great conceptual and algebraic simplification. At the linear approximation in one would expect that the light-time contains , the correction in the impact parameter linear in the mass; as one can see from (2), this is not the case. This property is generally true: the correction to the light-time does not contain (Sec. 6).

Cassini’s and many other space experiments have been analyzed using NASA’s Orbit Determination Program (ODP), developed by NASA at Jet Propulsion Laboratory in the 60’s and steadily improved since; a new version called MONTE is under development. The ODP, whose theoretical formulation is due to T. D. Moyer [21], integrates the equations of motion of the relevant bodies and provides their trajectories in the ephemeris time. This task is carried out in a reference system – called BCRS (Barycentric Coordinate Reference System) – in which the centre of gravity of the solar system is at rest and the Sun moves around with a velocity . As discussed in [2], the light-time in this this frame differs from the rest frame of the Sun essentially due to Lorentz time dilatation; being of order , this difference is quite below the sensitivity of Cassini’s experiment. We do not discuss this point any more; is just Killing time.

The ODP uses a fictitious Euclidian space , which corresponds to the isotropic coordinates of (6). This space is just a computational convenience and should not be considered as a physical background in which gravity acts. For example, replacing , the Euclidian distance from the origin, with , where is an arbitrary constant, is fully legitimate in a covariant theory, but it destroys the conformal flatness of space, introduces a gravitational potential and adds a second-order term to the delay . Strictly speaking, the word ‘delay’ is inappropriate: we just have a light-time and there is nothing with respect to which a delay can be reckoned. The object of the measurement is the time change of the delay. The arbitrariness of the radial coordinate also affects gravitational bending: its second-order approximation up to , depends on which radial coordinate is used (see [13], [10] and [25]) [7]).

It should also be noted that the spacetime coordinates of the end events are not directly provided in the experimental setup and depend on the gravitational delay , the very quantity one sets out to measure. The trajectories and are given by the numerical code; the starting time is just a label of the ray, but the arrival time is greater than . The way out is to take for the end point

 rB(rB)=rB(tA+rAB)+ΔtuB(tA+rAB),

where For a typical velocity the correction is of order cm, and the a priori accuracy in is sufficient.

Since for electromagnetic propagation and in (6) are almost equal, (2) is the correct approximation to the delay to ; one would expect this to be the first term in an expansion in powers of , so that the next term should be

 ≈mmb0=mmR⊙R⊙b0=0.3R⊙b0cm,

quite below Cassini’s sensitivity. The present paper arose because the ODP (eq. (8-54) of [21]), in fact does not use (2), but, in our notation,

 tB−tA=rAB+Δt=rAB+(1+γ)mln(rA+rB+rAB+(1+γ)mrA+rB−rAB+(1+γ)m). (8)

We have not been able to fully reconstruct Moyer’s derivation of this expression. It introduces non linear corrections arising from non linear effects of linear metric terms, but no quadratic metric terms. However, the difference between the two expressions of the delay is much larger than the estimate above; this arises because in Cassini’s case, in (2) the denominator is much smaller than the numerator . Indeed,

 Δt−(Δt)ODP=−2(1+γ)2m2b20rArBrA+rB=−(1+γ)2m2Rb20, (9)

where we have introduced the harmonic mean of the distances

 2R=1rA+1rB=rA+rBrArB. (10)

If, as in Cassini’s experiment, , the correction is about

 1600mmR⊙(R⊙b0)2=500(R⊙b0)2cm.

Even at this correction is somewhat below the sensitivity (4) and it should not have affected the result. However, it cannot be excluded that neglected non linear terms relevant for Cassini’s experiment affect the fit (1). One could say, (8) is mendacious; a full clarification of the problem is needed.

Empirically dropping or keeping ‘small’ terms may lead to inconsistencies and does not work; the rigourous method of asymptotic perturbation theory (see, e. g., [11], [16]) must be used. We briefly sketch it now at a practical level. One begins with a wise choice of a dimensionless ‘smallness’ parameter, and expands every function in the corresponding power series. Our main choice will be , but convenience may suggest using other lengths, like in . An asymptotic series

 G=∑sGs(mb0)s

is a formal object assigned just by the sequence of its coefficients ; arithmetics and calculus follows the obvious rules for sum, multiplication and differentiation. Equality between two asymptotic series just means that the coefficients of the same order are equal. The value of as a function of does not play any role, and even the convergence of the series is irrelevant; what matters is only the truncated value at any order

 G(k)=k∑s=0(mb0)sGs+O(mb0)k+1. (11)

The parameter should not be understood as a fixed number, but as a variable which tends to zero. The symbol means order of infinitesimal; it states how fast the remainder tends to zero as the parameter diminishes. An asymptotic series can be constructed from an ordinary arbitrary function ; but a whole class of functions give rise to the same series; for example, if is the sequence generated by , the same sequence is also generated by

 [1+Pexp(−Qb0/m)]G(m)(Q>0).

In this way any recursive iteration then proceeds automatically and safely, even in the most complex situations.

In our case light-time will be provided as an asymptotic power series

 tB−tA=rAB+m∑s=1Δs(rAb0,rBb0)(mb0)s−1, (12)

with dimensionless coefficients . provides the lowest, standard approximation to (see (2). In principle, asymptotic analysis does not provide a numerical estimate of the remainder in a given situation; this is a physical, not a mathematical question. But when the problem, properly formulated, does not contain small dimensionless quantities other than the smallness parameter itself, one can expect the mathematical operations leading to the result to maintain the order of magnitude and to lead to expansions whose coefficients are numerically of the same order. This is the case of deflection, the angle between the asymptotes of the ray. There is only one length in the problem, the distance of the point of closest approach, or, equivalently, the impact parameter (see Fig. 4); hence in the expansion

 δ=∑sδs(mh)s (13)

the coefficients are dimensionless numbers, solely determined by the PPN parameters and, must be of order unity (see Sec. 9). But in the delay problem the coefficients depend on the geometrical configuration. They are of order unity in the generic (but scarcely interesting) case in which and are of the same order; but in a close superior conjunction – of crucial relevance in experimental gravitation – when , besides , there is another smallness parameter, namely, , and there is no reason to exclude that the increase with beyond the expected order of magnitude unity. This we call enhancement. We already saw in (3) that is enhanced, albeit only logarithmically; the ODP correction (9), formally of second order, is enhanced by . This could place serious limitations on the method and even invalidate the iteration itself. This would occur, for instance, when ; if , this corresponds to AU. The enhancement, which has never been discussed in the literature, has been fully understood and tamed in the present paper (Sec. 8). We have found, indeed, that the second-order terms embodied in the ODP expression (8) which was used in Cassini’s experiment are just the enhanced second-order terms; Cassini’s result (1) is still safe.

The problem can be reduced to one of ordinary optics; due to its variational nature, the eikonal function can be easily solved in an expansion in powers of . The second-order expression of the light-time for a static spacetime has been obtained; extension to third order is also easy. This approach should be compared with the much more general work of [18], who consider Synge’s world function in a generic spacetime for a generic geodesic (not necessarily null) between two events A and B. On the basis of Hamiltonian theory, they develop a method to solve for in a formal power series with respect to the gravitational constant and compute it up to the second order. In the null case the world function vanishes on the solution and becomes the eikonal function. Out method, limited of course to the spherically symmetric case, exploits directly the variational nature of the problem and leads to the second-order expression of the light-time, which agrees with the expression of [18]; extension to third order is also easy.

For a realistic observation of a distant source from a point B at a finite distance , (13) must be generalized to an expansion of the type

 δB=∑sδBs(rBh)(mh)s, (14)

where is the impact parameter. The linear term has been evaluated in [19], \$ 40.3; the quadratic correction will be obtained in Sec.9 .

## 2 Hyperbolic Newtonian dynamics

Newtonian dynamics of a test particle attracted by a point mass , an exactly soluble problem, illustrates these issues. We consider a motion in the equatorial plane , with radial coordinate and azimuthal longitude . The Lagrangian function

 LNew=12[(drdt)2+r2(dϕdt)2]+GMr (15)

keeps the total energy constant; , the ultimate speed of the particle at a large distance, plays a role analogous to the speed of light and will be taken equal to unity. Then

 (drdt)2+r2(dϕdt)2−2mr=1, (16)

where is the gravitational radius. is an ignorable coordinate, so that the angular momentum

 ∂LNew∂(dϕ/dt)=r2dϕdt=h (17)

is constant. Since the velocity at infinity is 1, is also the impact parameter. Eliminating we get:

 rdϕdr=±h√r(r+2m)−h2; (18)

hence

 h=√b(b+2m) (19)

determines , the distance of closest approach where . The sign depends upon whether the ray is ingoing or outgoing. Integrating we get the true anomaly

 f=arccos(b2+2mb−mrr(b+m)). (20)

Alternatively, the motion can be expressed in terms of the semi-major axis and the hyperbolic eccentricity :

 r=a(e2−1)1+ecosf. (21)

The acute angle between the asymptotes is given by

 sinδ=sin(2arccos(−1e))=2mb+m√1−m2(b+m)2 (22)

This angle has a regular expansion in powers of , with no enhancement.

Consider, however, the hyperbola determined by two points and on the opposite sides of the vertex (right side in Fig. 1). As in space navigation – in particular in the ODP – the end points are provided in terms of the initial and final position vectors, or equivalently, in terms of the initial and final distances and and the elongation angle ; the “unperturbed distance of closest approach” may then be calculated from elementary geometry:

 b0=rArBsinΦAB√r2A+r2B−2rArBcosΦAB. (23)

Choosing the angles , and positive, we can express in terms of with the condition

 ΦAB = αA+αB=arccosb0rA+arccosb0rB= (24) = arccos(b(b+2m)rA(b+m)−mb+m)+ + arccos(b(b+2m)rB(b+m)−mb+m).

The symmetric case is sufficient to exhibit the problem. The condition reads:

 bRb+2mb+m−mb+m−b0R=0, (25)

or

 b2+(2m−1)b−m(R+1)=0, (26)

with the solution

 2bb0=b0−mb0+√1+4mb0R+mb0. (27)

Expansion in powers of gives

 bb0=1+(Rb0−1)mb0−((Rb0)2−1)(mb0)2+O(mRb0)3. (28)

The enhancement is clear: when the truncation error at order is , with a coefficient of order unity, as naïvely expected; but when – as in a close superior conjunction – , the error is larger, . Formally this requires introducing another smallness parameter and expanding every coefficient of the primary -expansion in descending powers of . Of course, the condition

 mRb20≪1, (29)

must be fulfilled, lest the whole procedure breaks down. One could say, anchoring the trajectory at far away end points has a lever effect, so that an increase in the mass produces a large increase in closest approach.

The quantity (29) gives, in order of magnitude, the ratio between the deflection and the angle which separates the central mass and a distant star, as seen from a distance . Hence the limiting constraint above implies that the geometry of astronomical deflection is the same as in the classical case (see Fig. 4): sources in the sky near the Sun are displaced outward by an amount inversely proportional to the angular distance. The transition through the milestone marks the passage to the gravitational lensing regime, in which the image can appear on both sides.

In Sec. 8 the light-time enhancement is dealt with in the general case and it is shown that the dimensionless coefficients in (12) are

The metric of a spherical body at rest has the general form

 ds2=A(r)dt2−B(r)dr2−C(r)r2dΩ2, (30)

where are of power series the form:

 A(r)=∑sAs(mr)s. (31)

It is asymptotically flat, so that . The radial coordinate is otherwise arbitrary; this is the gauge freedom at our disposal. For consistency, however, any change must become an identity at infinity and have a similar expansion:

 g(r)=r+g1m+g2m2r+…; (32)

the coefficients are not gauge invariant. Two gauges are common. In the isotropic form – the canonical choice in space physics – , so that

 ds2=A(r)dt2−B(r)(dr2+r2dΩ2)=A(r)dt2−B(r)dℓ2; (33)

the space part of the metric is conformally flat. We define

 N(r)=√B(r)A(r)=∑sNs(mr)s=1+N1mr+N2(mr)2+O(mr)3. (34)

In the PPN scheme (e. g., [31])

 N1=γ+1,N2=6−4β+3ϵ+4γ−2γ24. (35)

In ‘Schwarzschild’ gauge and

 ds2=¯A(¯r)dt2−¯B(¯r)d¯r2−¯r2dΩ2;

the area of a sphere of radius is just the Euclidian expression , which defines in an invariant way. In the original Schwarzschild solution . To get the isotropic form one requires

 g2(r)=¯B(g(r))(dgdr)2r2; (36)

to first order

 ¯r=r+γm+…. (37)

In the present paper a third radial coordinate

 ρ=rN(r)=r√B(r)A(r)=r+mN1+m2N2r+… (38)

plays an important role. It is a monotonic function of and ensures . In the linear approximation it was introduced by Moyer in [21] (eq. (8-23)), and boils down to just adding to a constant term, equal to 2.95 km for the Sun.

## 4 Geometrical optics

It is convenient to reduce the problem to geometrical optics using the eikonal function . In a generic spacetime fulfils the eikonal equation

 gμν∂μS∂νS=0; (39)

its characteristics are the null rays (see, e. g., [1]). is the phase of the electromagnetic wave. Let be the trajectories of the end points, given as functions of their proper times ; let

 vμA=drμdsA,vμB=drμdsB

be the corresponding four-velocities. Clocks associated with them measure the proper frequencies

 ωA=−vμA∂μS=dSdsA,ωB=−vμB∂μS=dSdsB. (40)

In the simple case in which the end points are far away from the source, where the metric corrections can be neglected, the contribution to the frequency difference corresponds to the ordinary Doppler effect, and can be evaluated with a slow motion expansion; the change in between A and B is determined by the accumulated gravitational effect along the ray and mainly come from the region near the mass.

 gμν∂μS∂νS=0=N2(r)(∂tS)2−∇S⋅∇S, (41)

where is the Euclidian gradient operator. We are really interested only in the spherically symmetric case, but the reasoning of this Section holds also for an arbitrary .

is the phase; propagation occurs keeping it constant. Separating space and time variables with

 S=St(t)+¯¯¯¯¯Sr(r),

leads to the class of solutions

 S=ω0(¯¯¯¯¯S(r)−t), (42)

where is the spatial part of the phase and is a constant frequency. has the dimension of time and satisfies

 ∇¯¯¯¯¯S⋅∇¯¯¯¯¯S=N2(r). (43)

If a clock is at rest relative to the mass, , and the measured proper frequency includes the appropriate gravitational shift away from the asymptotic value . This is enough to reduce the problem to geometrical optics (see, e. g., [8], Ch. III). A ray , as function of the Euclidian arc length , is orthogonal to the eikonal surfaces and fulfils

 ddℓ(N(r)drdℓ)=∇N(r). (44)

The index of refraction is the rate of increase of the spatial phase along the ray:

 d¯¯¯¯¯Sdℓ=N(r).

Consider now Fermat’s action functional

where the trajectory, any path joining the end points, is expressed in terms of a generic parameter :

 r(λA)=rA,r(λB)=rB. (46)

Since the action is, in fact, independent of the choice of , no generality is lost if , the Euclidean line element. The Euler-Lagrange equation for the action (45) reduces to (44). The actual elapsed time

is just the value of computed at a local minimum – the actual ray (Fermat’s Principle). One should keep in mind the distinction between the action functional, with its argument in square brackets, and the action computed at the extremum, an ordinary function of the end points denoted with . In , but not in , it is allowed to replace the generic independent variable with a more convenient one related to the solution, like . For simplicity, the different functions denoted by the symbol are distinguished by their arguments; below, the quantity will be introduced to denote the action corresponding to a ray anchored at and , but with arbitrary (or ).

## 5 The solution

The eikonal function provides a deep simplification in the evaluation of the light-time. Having already separated out the time, the three-dimensional eikonal equation (43) in spherical symmetry and in the equatorial plane can be solved by separating out the longitude : setting

 ¯¯¯¯¯S(r,ϕ)=¯¯¯¯¯Sr(r)+¯¯¯¯¯Sϕ(ϕ).

It satisfies 222For a function of a single variable a prime indicates the derivative.

 r2(¯¯¯¯¯S′r)2+(¯¯¯¯¯S′ϕ)2=r2N2(r),

so that is a constant. Setting , the eikonal equation reduces to

 (¯¯¯¯¯S′r)2=1r2(r2N2(r)−h2),

with the primitive

 ¯¯¯¯¯S(r)=±∫rdrr√r2N2(r)−h2.

The and the signs correspond, respectively, to an outgoing and an incoming photon. The radial coordinate of closest approach , where , is the solution of

 bN(b)=h; (48)

since , is a real function. In Sec. 9 it will be shown that , just like in the Newtonian case, is the impact parameter (Fig. 4). The total phase is, therefore,

 S=ω0(hϕ±∫rdrr√r2N2(r)−h2−t). (49)

A wavefront propagates keeping constant, so that the time along the ray is

 t=±∫rdrr√r2N2(r)−h2+hϕ. (50)

In the usual case (see Fig. 1), in which the angle is obtuse, the ray has two branches, both taken with the positive sign: an incoming one from to and an outgoing one from to . In the acute case is never reached and we have just an outgoing ray from to . In both cases, in going from A to B the longitude increases by The quantity

 S(h)=∫rBbdrr√r2N2(r)−h2±∫rAbdrr√r2N2(r)−h2+hΦAB (51)

gives the phase change, hence the light-time, between the end points, but the quantity is still arbitrary. The upper (lower) sign corresponds to the case in which the angle is obtuse (acute); in the latter case the two integrals combine in a single one from to , and disappears as a lower limit. (51) is what Fermat’s action functional becomes when its variability is restricted to and the longitude constraint is not imposed; it shall be called reduced action. At the true value it satisfies

 S′(htrue)=0, (52)

keeping the end points fixed.

The present work aims at providing the theoretical foundation for the time delay in all configurations; the sign freedom allows dealing with both cases at the same time, but applications will be mainly given for a conjunction, with the . The origin of longitudes is arbitrary. This general approach is relevant, for ex ample, for a spacecraft on an almost parabolic orbit, as in the Solar Probe concept; with a perihelion as low as , it can have a strong enhancement of the light-time even in the acute configuration.

In the derivative there are no contributions from the lower limits; then (52) provides as an implicit function of the total total elongation :

 ΦAB+∫rBbdrr−h√(rN(r))2−h2±∫rAbdrr−h√(rN(r))2−h2=0. (53)

Hence (51) reads333In a slightly inconsistent notation, we often use to denote both an independent and variable quantity, and the fixed value determined by the elongation. The context should be sufficient to clear the ambiguity.

 S(h)=∫rBbdrN(r)rN(r)√(rN(r))2−h2±∫rAbdrN(r)rN(r)√(rN(r))2−h2. (54)

i

Both integrals are convergent (and in the acute case the singularity at is not even reached). (51) suggests the introduction of the function

 G(r,h)=∫rbdrr√(rN(r))2−h2, (55)

in terms of which

 S(h)=G(rB,h)±G(rA,h)+hΦAB. (56)

(53) reads444The suffix indicates partial derivative.

 Gh(rB,h)±Gh(rA,h)+ΦAB=0. (57)

While in (51) is an independent parameter, in (54) it is fixed by (53).

This expression for can also be derived directly from Fermat’s Principle, thus providing its significance. Fermat’s action (47), expressed as a function of , has the Lagrange functional

 LF[ϕ(r)]=N(r)√1+r2(dϕ/dr)2, (58)

with the (positive) constant of the motion

 ∂LF∂(dϕ/dr)=±r2N(r)√1+r2(dϕ/dr)2dϕdr=h. (59)

The upper (lower) holds for the outgoing (incoming) branch. Integrating

 rdϕdr=±h√r2N2(r)−h2=±h√ρ2−h2, (60)

(53) is recovered. Comparison with the Newtonian case (18) shows that the latter corresponds to the exact index of refraction

 NNew(r)=√1+2mr, (61)

corresponding, as expected, to , and , etc.

## 6 A variational argument

At this point one could proceed as follows: using power series, solve (53) for in terms of , a known quantity. The value of , inserted into (54), provides the required light-time. The stationary character of the action (52), however, brings about a deep and important simplification. This is already tacitly applied in the usual derivation of the gravitational delay (2). To first order, the integral of in (33) reads

the second integral can be carried out along the straight path from A to B, leading to the characteristic logarithmic term. In principle, however, the first integral should take into account the (first order) deflection; we should understand as the Euclidian length of the bent arc between A and B. But the length of the straight segment AB is a minimum in the set of all curves joining A and B, so that vanishes to .555A didactical remark is in order here. This minimum property, crucial to the argument, is often omitted in the usual derivation. See, e. g., [19] p. 1107, [9] p. 125; in equation (17.59) of [5], p. 581 the minimum is not mentioned and a factor 4 is missing in the argument of the logarithm. Ray bending is irrelevant here.

In order to exploit the variational nature of the problem it is convenient to apply power expansions before imposing the extremum condition (52). We just need the value of the reduced action (51) at the value which fulfils (52), namely, Setting and expanding, the solution to second order is obtained iteratively:

 S′0(h0) = 0, (62) h1S′′0(h0) + S′1(h0)=0, (63) h2S′′0(h0) + h212S′′′0(h0)+h1S′′1(h0)+S′2(h0)=0. (64)

In the expression

 S(h) = S0(h0)+m(h1S′0(h0)+S1(h0))+ (65) + m2(h2S′0(h0)+h212S′′0(h0)+h1S′1(h0)+S2(h0)),

the effect of the extremum property is clear: since , the first order term does not contain , and the second order term does not contain ; in general, the term in of order does not depend on . This important result is reflected in the general approach of [18]. Referring to the equation numbering of that paper, their world function fulfills the Hamilton-Jacobi equation (30). In the null case , (30) becomes the eikonal equation. Their Theorem 2 proves that the -order can be expressed in terms of integrals along the lowest order Minkowskian path. In our case this variational Lemma clarifies the matter and produces considerable simplifications. Using (63), the light-time to second order reads:

 S(h)=S0(h0)+mS1(h0)+m2(h212S′′0(h0)+h1S′1(h0)+S2(h0)), (66)

where is given by (63). The delay coefficients (12) read

 Δ1=S1(h0),Δ2=h212S′′0(h0)+h1S′1(h0)+S2(h0). (67)

The second-order correction in the impact parameter , given by (64), is needed only at third and higher orders. For the record, note the third-order contribution to the light-time:

 Δ3=h1h2S′′0+h316S′′′0+h2S′1+h212S′′1+h1S′2+S3, (68)

where, for simplicity, the arguments have been understood, and is provided by (64).

## 7 Power series

We now proceed to apply this simple and general Lemma to the light-time. To lowest order, in (55) we use and , so that

 S0(h0) = √r2B−h20±√r2A−h20+ (69) − h0(arccosh0rB±arccosh0rA−ΦAB).

The condition determines with the trigonometric relation (see Fig. 1)

 ΦAB=arccosh0rB±arccosh0rA, (70)

or

 h0=rArBrABsin


text:
# Finding smallest canonical cover from given set of functional dependencies

I did not find any book discussing this but I came across the problem. The given set of FD was:

$F=\{B\rightarrow A,D\rightarrow AC,B\rightarrow C,AB\rightarrow D,BC\rightarrow D\}$

$F_{min}=\{B\rightarrow D,D\rightarrow A,D\rightarrow C\}$

$B\rightarrow D$ wasnt present directly in the given FD set. So I realised I must be finding attribute closures first:

1. $A^+=A$

2. $B^+=BACD$

3. $C^+=C$

4. $D^+=ACD$

The non trivial ones are:

$B^+=ACD$

$D^+=AC$

Its easy from these that the smallest FD set would be $F_{min}=\{B\rightarrow D,D\rightarrow A,D\rightarrow C\}$. However the final non trivial attribute closure set was small and had easy pattern to deduce the smallest canonical cover. I was just guessing what if final non trivial attribute closure was more involved and did not contained any easy pattern, how would I have deduced the smallest canonical cover. Can anyone see any simple set of concrete steps for this?

format: plain


text:
## Stream: maths

### Topic: char_zero and char_p

#### Yury G. Kudryashov (Oct 18 2020 at 18:09):

What do you think about the following refactor?

• make p in char_p an out_param;
• make it work for an add_monoid instead of a semiring;
• use char_p α 0 instead of char_zero?

I'm not going to do it any time soon, but if you like it, then I'll add an issue.

#### Gabriel Ebner (Oct 19 2020 at 07:47):

I was very surprised when I first saw that they're different definitions.

#### Johan Commelin (Oct 19 2020 at 07:47):

char_zero is older

#### Johan Commelin (Oct 19 2020 at 07:48):

I'm still scared of out_params... but if it works, that would be cool!

Last updated: May 10 2021 at 08:14 UTC


text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 The lacunary strong Zweier convergent sequence spaces Commun. Korean Math. Soc. 2010 Vol. 25, No. 1, 51-57 https://doi.org/10.4134/CKMS.2010.25.1.51Printed March 1, 2010 Mehmet Seng\"on\"ul Nev\c{s}ehir University Abstract : In this paper we introduce and study the lacunary strong Zweier sequence spaces $N_{\theta}^0[\mathcal{Z}],~~N_{\theta}[\mathcal{Z}]$ consisting of all sequences $x=(x_k)$ such that $(Zx)$ in the space $N_{\theta}$ and $N_{\theta}^0$ respectively, which is normed. Also, prove that $N_{\theta}^0[\mathcal{Z}],~~N_{\theta}[\mathcal{Z}]$ are linearly isomorphic to the space $N_{\theta}^0$ and $N_{\theta}$, respectively. And we study some connections between lacunary strong Zweier sequence and lacunary statistical Zweier convergence sequence. Keywords : lacunary sequence, Zweier space, statisticial convergence, Banach space, isomorphism MSC numbers : Primary 40C05, 40J05, 46A45 Downloads: Full-text PDF

 Copyright © Korean Mathematical Society. The Korea Science Technology Center (Rm. 411), 22, Teheran-ro 7-gil, Gangnam-gu, Seoul 06130, Korea Tel: 82-2-565-0361  | Fax: 82-2-565-0364  | E-mail: paper@kms.or.kr   | Powered by INFOrang Co., Ltd


text:
# Two metal plates form a parallel plate capacitor. The distance between the plates is given as ‘d’. A metal plate of thickness (d/2), and two dielectric slabs of thickness (d/4) is introduced between the plates as shown in the figure. If the metal plate is removed find the work done in slowly removing it. (The plates of capacitor is connected to a battery having potential difference ‘v’)

$\large\frac{4A \in_0 v^2k_1^2k_2^2}{d(k_1+k_2)(k_1+k_2+2k_1k_2)}$


text:
# Browse Dept. of Biochemistry by Title

• (2013-02-03)
Nature has given nucleic acids irreplaceable roles for all living creatures. It plays important roles in genetic information storage and transfer, as well as regulating various aspects of biology. The discovery of in vitro ...

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• (1985)
The aerobic electron transport chain of Escherichia coli contains a terminal oxidase, cytochrome d, which is part of a two subunit protein complex that is composed of two additional cytochromes, b(,595) and b(,558). The ...

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• (1988)
Small RNAs of defined length and sequence can be synthesized enzymatically using T7 RNA polymerase and synthetic DNA templates which contain the T7 promoter. The templates are base paired from $-17$ to +1 and contain a ...

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• (1992)
The presequence of CCP, an intermembrane space heme protein, has been proposed to be composed of an amino-terminal basic domain, a stretch of hydrophobic residues and a basic, carboxy-terminal domain (Kaput et al, 1982). ...

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Plasmodium falciparum is the most lethal of four species of protozoa that cause malaria in humans. Since the blood stage of the parasite life cycle is responsible for the clinical manifestations of the disease, studies of ...

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Investigations were carried out on the cytochrome components of the E. coli aerobic electron transport system. This system involves two parallel electron transport chains: (UNFORMATTED TABLE FOLLOWS)

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Signal recognition particle (SRP), the ribonucleoprotein responsible for the recognition and targeting of nascent secretory proteins to the endoplasmic reticulum, is composed of 7SL RNA, along with six polypeptides. Earlier ...

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The focus of this research is the role of conformational flexibility in catalysis by a TIM-barrel enzyme in pyrimidine biosynthesis, orotidine 5’-monophosphate decarboxylase (OMPDC). OMPDC catalyzes the decarboxylation ...

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EcoRI and RsrI endonucleases present rare opportunities to study the mechanisms by which site-specific DNA-binding proteins identify their substrate sequences among an overwhelming number of other sequences to which they ...

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Since the discovery of nucleic acid enzymes and aptamers that can perform functional roles other than storing genetic information; a new paradigm in nucleic acid chemistry has been opened. In 1990 the first artificially ...

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The initiation of translation of eukaryotic mRNA is believed to proceed by a "scanning mechanism" where the ribosome recognizes a 5$\sp\prime$ terminal cap structure (m$\sp7$GpppN ...) and then scans down the ...

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Several features of the initiation of translation of CcTMV-SRNA were investigated. First, the site of initiation of translation was found to be the first AUG in the 5$\sp\prime$-terminal sequence m$\sp7$GpppGUAUUUGAUGAUGGCA ...

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The $\beta\sb1$ integrins are a family of $\alpha\beta$ heterodimeric cell surface receptors which mediate cell adhesion to molecules of the extracellular matrix (ECM), and also span the membrane and associate with the ...

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text:
# Thermodynamics problems with answers

Recent questions in Thermodynamics
Passafaromx 2022-08-14

### I'm currently going through statistical physics, especially on Fermi energy when I came across a term called "quasi-continuum", what exactly is it?

Damon Campos 2022-08-13

### Is there a liquid which boils at room temperature and normal pressure, and can we use it to produce electricity?

Bierlehre59 2022-08-13

### When we are dealing with a gaseous thermodynamic system, in books it's written that state of the system can be described by only two independent variables from the three (p,V,T) . But it's not written why or how? Why we have to choose only two independent variables, why not more or less? I've gone through some answers but still unable to understand.

vrteclh 2022-08-13

### Is there an equation for convective heat transfer?

polynnxu 2022-08-11

### Why must black hole have entropy?

Leypoldon 2022-08-11

### A rigid container holds 3 of air. The air is stirred so that its pressure changes from 500kPa to 2000kPa, while its temperature is held at ${50}^{\circ }\text{C}$ . The heat transferred is 200kJ. Assume the air to behave like an ideal gas with ${c}_{v}=0.714\phantom{\rule{thinmathspace}{0ex}}\frac{\text{kJ}}{\text{kg}\cdot \text{K}}$ . Find the final temperature, the change in internal energy, and the work done.

motsetjela 2022-08-11

### How can the fugacity of this state equation:$pV/\left(nRT\right)=1+\left[b-a/\left(RT\right)\right]p/\left(RT\right)$be calculated?

pominjaneh6 2022-08-09

### A substance at 200-degree Celsius is given some amount of heat to raise its temperature by one degree Celsius and the same substance when at -200 degrees Celsius is given some amount of heat to increase its temperature to -199 degrees Celsius. Is the amount of heat required for both the processes same?

Lexi Mcneil 2022-07-22

### Scientists say that it is impossible to reach a temperature of zero kelvin, because the atoms will stop moving, and the volume of the substance will become zero, But we have reached the pico kelvin temperature, which is almost zero, and there is no difference between it and zero, so why do they say it is impossible? and The laws of physics still work at picokelvin.

Deromediqm 2022-07-20

### How does the refractive index of air rely on the temperature? Is there a theoretical derivation of it?

Bernard Boyer 2022-07-20


text:
# AcuSurf

Extract external surfaces of a mesh, for imposing boundary conditions.

## Syntax

acuSurf [options]

## Type

AcuSolve Preparatory Program

## Description

AcuSurf is a simple utility program that extracts and organizes all external surfaces of a mesh. The generated surface files may be used in the input file for element boundary conditions, surface output, and so on.

The surfaces are collected based on their parent connectivity file, surface shape, and neighboring angles. The collections are written in files problem.srfindex; where index is an index starting from 1 to the number of surface sets. Note that disconnected surfaces are stored in different files.

The algorithm used by AcuSurf is as follows. It first traverses through all element surfaces and collects only those surfaces with elements on one side. These surfaces are the external or boundary surfaces. All surfaces that have elements on both sides are internal surfaces, and hence are not considered.

The algorithm then traverses through the surfaces given in surface_file_list files. Each boundary surface found in these files is tagged as it is read.

It then traverses through the "partial" surfaces given in partial_surface_file_list files. These files must contain surfaces without the parent element numbers. Similar to the surface_file_list case, the boundary surfaces found in these files are tagged as they are read. In addition, the surfaces including the parent element numbers are written in the surface files file.index where file is the original file name; and index is the index of the parent connectivity file starting from 1 in the order specified in connectivity_file_list.

It then traverses through the surface node sets given in surface_node_file_list files. These files must contain the list of nodes on the boundary. For each file, the nodes of each boundary surface are checked against the node list. If all surface nodes are present, the surface is tagged as it is read and is written into the file file.index_shape, where file is the node set file name, index is the index of the parent connectivity file starting from 1 in the order specified in connectivity_file_list, and shape is either tri3, quad4, or tri6.

Finally, if auto_generate_surfaces flag is set, it randomly selects one surface from the non-tagged boundary surface, which starts a new surface set. It then recursively traverses through the neighbors of the selected surfaces. If a neighbor has the same surface shape, that is, 3-node triangle, 4-node quadrilateral, or 6-node triangle, comes from the same connectivity file, and its outward normal direction has an angle less than or equal to max_angle, the neighbor is added to the surface set. If no other surface qualifies to be added to the surface set, then a new non-tagged surface is chosen and the process is repeated.

Choice of the max_angle can significantly affect the collection process. A max_angle of 0 only admits elements that fall exactly on a plane, whereas a max_angle of 180 gives one surface set for each connected set of surfaces with the same shape and parent connectivity file.

This utility also determines whether or not the mesh is constructed by a two dimensional mesh extracted into a third dimension. If so, it determines the axis of extrusion. Any external surface that is perpendicular to this axis is ignored, unless -no_c2d is set, in which case no external surface is ignored. For convenience, AcuSurf also determines the extruded node pairs, which are output in the file problem.pbc. This file is suitable for the nodes parameter of the PERIODIC_BOUNDARY_CONDITION command.

In the following, the full name of each option is followed by its abbreviated name and its type. For a general description of option specifications, see Command Line Options and Configuration Files. See below for more individual option details:

help or h (boolean)
If set, the program prints a usage message and exits. The usage message includes all available options, their current values, and the place where each option is set.
problem or pb (string)
The name of the problem is specified via this option. This name is used to generate input file names and extracted surface file names.
coordinates_file or crd (string)
Nodal coordinates file name. The file must contain four columns: node number and x, y, z coordinates. If coordinates_file is set to _auto, problem.crd and problem.crd.B are assumed.
connectivity_file_list or cnn (string)
Comma-separated list of connectivity files. The shape of the elements is extracted from the number of columns in each file. The elements are assumed to be 4-node tetrahedra if the file has 5 columns; 5-node pyramids if 6 columns; 6-node wedges if 7 columns; 8-node bricks if 9 columns; and 10-node tetrahedra if 11 columns. If connectivity_file_list is set to _auto, all problem*.cnn and problem*.cnn.B files are used.
surface_file_list or srf (string)
Comma-separated list of surface files. The shape of the surfaces is extracted from the number of columns in each file. The surfaces are assumed to be 3-node triangles if the file has 5 columns; 4-node quads if 6 columns; and 6-node triangles if 8 columns. The surfaces in these files are removed from the generated surface files. If surface_file_list is set to _none, no surface file is read. If surface_file_list is set to _auto, then problem*.srf, problem*.srf.B, problem*.ebc, and problem*.ebc.B are used.
partial_surface_file_list or psrf (string)
Comma-separated list of surface files, without the parent element numbers. The shape of the surfaces is extracted from the number of columns in each file. The surfaces are assumed to be 3-node triangles if the file has 4 columns; 4-node quads if 5 columns; and 6-node triangles if 7 columns. The boundary surfaces of each file are written in files file.index, where file is the original file name; and index is the index of the parent connectivity file starting from 1 in the order specified in connectivity_file_list. The generated files contain the parent element numbers, which are needed by all ACUSIM programs. All surfaces in these files are removed from the generated surface files. If partial_surface_file_list is set to _none, no surface file is read.
surface_node_file_list or node (string)
Comma-separated list of surface node files. The boundary surfaces of each connectivity file are searched for the nodes in each of these files. If any boundary surface is found, a surface file is created with the name file.index_shape, where file is the node file name; index is the index of the parent connectivity file starting from 1 in the order specified in connectivity_file_list; and shape is either tri3, quad4, or tri6. All extracted surfaces are removed from the generated surface files. If surface_node_file_list is set to _none, no node file is read. If surface_node_file_list is set to _auto, then problem*.nbc and problem*.nbc.B are used.
auto_generate_surfaces or asrf (boolean)
Output boundary surfaces that are not processed due to parameters surface_file_list, partial_surface_file_list, and surface_node_file_list.
max_angle or angle (real)
This option specifies the maximum angle allowed between two adjacent surfaces in order to be considered as part of the same surface set. max_angle must be between 0 and 180 degrees. If max_angle is set to 0, surfaces that fall on a flat plane, and have the same shape and parent connectivity file are collected together. If max_angle is set to 180, all connected surfaces of the same shape and parent connectivity are collected together.
output_surface_nodes or onod (boolean)
Generate a nodal list for every surface file. The nodes of the surfaces are written in surface_file.nod; where surface_file is the name of the surface file.
check_2d_mesh or c2d (boolean)
Do not generate surfaces for the 2D planes of a 2D mesh.
verbose or v (integer)
Set the verbose level for printing information to the screen. Each higher verbose level prints more information. If verbose is set to 0, or less, only warning and error messages are printed. If verbose is set to 1, basic processing information is printed in addition to warning and error messages. This level is recommended. verbose levels greater than 1 provide information useful only for debugging.

## Examples

Consider the mesh of the channel problem. AcuSurf may be executed as:
acuSurf -pb channel -angle 10
or alternatively the options may be put into the configuration file Acusim.cnf as follows:
problem = channel
max_angle = 10
and AcuSurf invoked as
acuSurf
The above example produces four surface sets, stored in files channel.srf1 through channel.srf4, and four node sets, stored in files channel.srf1.nod through channel.srf4.nod. Whereas,
acuSurf -pb channel -angle 91

produces a single surface file channel.srf1 containing all the surfaces, plus its nodes in the -channel.srf1.nod file.

In another example, assume you have the inflow surface file channel.inflow.srf, then:
acuSurf -pb channel -srf channel.inflow.srf

produces the files channel.srf1 through channel.srf3 (plus their nodal files). These files do not contain any of the surfaces in channel.inflow.srf. The file channel.inflow.srf.nod is also generated.

In another example, assume you have the inflow surfaces without the parent element number in the partial file channel.inflow.psrf, then
acuSurf -pb channel -psrf channel.inflow.psrf

produces the files channel.srf1 through channel.srf3, the file channel.inflow.psrf.1, which is the inflow surface file channel.inflow.psrf including the parent element number for the first connectivity file, plus the nodal files channel.srf1.nod through channel.inflow.psrf.1.nod.

In another example, assume you have a list of inflow nodes in the file channel.inflow.nodes, then
acuSurf -pb channel -node channel.inflow.nodes

produces the files channel.srf through channel.srf3, channel.inflow.nodes.1_quad4 containing the inflow surfaces for the first connectivity file, plus the nodal files.

To prevent the nodal files to be generated, issue the command with -no_onod option, or place
output_surface_nodes= FALSE
in the Acusim.cnf configuration file.


text:
I can't believe we ever managed to make such a hash of cables, but to refresh everyone, back three years ago, I wrote 2018 The State of USB C Charging and then again back in 2017 USB C Cable Recommendations, so time now for another review of USB C and how to get cables and chargers.

Well the biggest and nicest thing is that prices have really come down on lots of these things. But, the big problem is that identical looking cables can be really different, so here is what you need to know because you need to know the power rating and speed rating for cables and you really need to label make to remember what is what:

1. Chipped cables. The first thing is that you want a cable which is actually USB IF blessed. This will have a chip inside of it (and not surprisingly, these are the expensive ones). If you buy a cheap clone, you can actually one that has passed some certification and has a legit control chip.
2. What data capacity. The lowest is 480Mbps (USB 2.0), the USB 3 Gen 1 (5Gbps), USB 3 Gen 2 (10Gbps) and finally Thunderbolt 3 (20Gbps). Right now the sweet spot in pricing is USB 3 Gen 2. A two meter cable costs about $16 for that but$35 for a Thunderbold 3 cable. The most common "no name cable" is USB 2.0 which is mainly used for powering Android phones. Personally, I would just get the $16 and try to get a uniform fleet. 3. What power capacity. Yes, this is a thing, you really want a cable that is marked as able to handle the full 100W specification. There is another problem that not all chargers put out all voltage and amperage combinations. 100W is 20W at 5A as an aside, so you have to read the side of the chargers to get the right one. It is pretty common for cheap cables to only support 60W. Net, net, how can you tell what is what, well with the Mac it is pretty easy, connect the device and then go to About > System Preferences > System Report and go to the section called Power and see what the Power rating is, it should say 87W if you have a 100W Power Delivery (PD) cable and that is the maximum that a Mac can call on. Similarly go to the USB section and connect a device and see how it connects. You need to make sure it is a USB device that supports up to USB Gen 2 and then you can tell if it supports Gen 1 or USB 2. And then label your stuff! So some good recomemdnations for 2 meter cables (I like to get these and they are just out for Thunderbolt are): 1. Plugable Thunderbolt 3, 100W, 2m. This is$35 and if you don't need something light and small, it is fully future proof. Call it the Cadillac of cables.
2. NIMASO 6g USB C 3.1 Gen 2 10Gbps, 100W PD. This is a no name brand and is right now a super cheap \$14.
3. Anker 6 foot USB C to Lightning Powerline II. This is a specialty cable, but it supports the iPhone X and above with higher power charging up to 15W and 480Mbps USB 2. It's half the price of the Apple cable which is a pretty delicate beast and breaks alot.


text:
• This event has passed.

Boram Park (박보람), Odd coloring of sparse graphs

April 25 Monday @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)

We introduce an odd coloring of a graph, which was introduced very recently, motivated by parity type colorings of graphs. A proper vertex coloring of graph $G$ is said to be odd if for each non-isolated vertex $x \in V (G)$ there exists a color $c$ such that $c$ is used an odd number of times in the neighborhood of $x$. The recent work on this topic will be presented, and the work is based on Eun-Kyung Cho, Ilkyoo Choi, and Hyemin Kown.

Details

Date:
April 25 Monday
Time:
4:30 PM - 5:30 PM KST
Event Category:
Event Tags:
, ,

Room B232


text:
+0

weird fraction

0
93
1

What is the numerical value of x = $$\frac{1}{4-\frac{1}{4-\frac{1}{4-\ldots}}}$$?

Jun 22, 2022

#1
+2448
0

We have the equation: $$x = {1 \over { 4 - x}}$$.

Multiplying both sides by $$4 - x$$ gives us: $$-x^2 + 4 x = 1$$

Subtracting 1 from both sides gives us: $$-x^2 + 4x - 1 = 0$$

Using the quadratic formula, we find $$x = \color{brown}\boxed{2 \pm \sqrt 3}$$

Jun 23, 2022

#1
+2448
0

We have the equation: $$x = {1 \over { 4 - x}}$$.

Multiplying both sides by $$4 - x$$ gives us: $$-x^2 + 4 x = 1$$

Subtracting 1 from both sides gives us: $$-x^2 + 4x - 1 = 0$$

Using the quadratic formula, we find $$x = \color{brown}\boxed{2 \pm \sqrt 3}$$

BuilderBoi Jun 23, 2022


text:
Abstract vs. Concrete Syntax

Moderator: Konzepte der Programmiersprachen

topracer
Mausschubser
Beiträge: 53
Registriert: 10. Jan 2014 19:14

Abstract vs. Concrete Syntax

Hello,

I noticed that in the slides the terms "abstract" and "concrete syntax" are used inconsistently... On the definition slide (p. 4) of V02, Abstract syntax seems to refer to the <AE> ::= ... notation:
VL02_Abstract_Concrete_Syntax.png (87.29 KiB) 177 mal betrachtet
However, in V03 e.g. on p. 8 the same notation is named "concrete syntax" and the term "abstract syntax" refers to the Scala syntax:
VL03_Abstract_Concrete_Syntax.png (104.61 KiB) 177 mal betrachtet
So, what term refers to which syntax, actually?

Thank you for your response!

nicolas_m
Erstie
Beiträge: 11
Registriert: 14. Jun 2013 16:13

Re: Abstract vs. Concrete Syntax

I'm confused by this too.

On slide 40 of V04 the

Code: Alles auswählen

<FWAE> ::= <num> ...
syntax is referred to as concrete and the

Code: Alles auswählen

sealed abstract class FLAE ...
is referred to as abstract. (also here FWAE should be FLAE I believe, which I suppose was a typo).

topracer
Mausschubser
Beiträge: 53
Registriert: 10. Jan 2014 19:14

Re: Abstract vs. Concrete Syntax

According to what I found here the non-Scala syntax seems to be the concrete syntax and the Scala representation the abstract one. (That also roughly matches the definition slide in the lecture, except for the example being in the wrong place.)

But I would really appreciate an official statement on this, though.
nicolas_m hat geschrieben:
25. Feb 2019 15:53
also here FWAE should be FLAE I believe, which I suppose was a typo
I think this is the same naming inconvenience which is discussed here. The "W" stands for "with", which is the equivalent to "let" that is used in the PLAI book.

mirko-koehler
Moderator
Beiträge: 66
Registriert: 16. Okt 2017 12:28

Re: Abstract vs. Concrete Syntax

Concrete syntax is how the code looks to the programmer. The abstract syntax are the syntax trees that result from parsing the concrete syntax.

In our interpreters, the abstract syntax are the expression case classes (e.g. App(funName: Symbol, ...) ). The expressions build expression trees that are then used by the interpreters and type checkers. We did not care much about concrete syntax: the programs are written with the abstract syntax.

In Slide V02 p.4, we use a grammar to define how the abstract syntax trees of some language look like (we then introduce case classses to model these trees in Scala). In Slide V03 p.8, we use a grammar to define how the programmer is writing code. A programmer would write

Code: Alles auswählen

f 2

Code: Alles auswählen

App('f, 2)

In essence:
Concrete syntax = what the programmer writes
Abstract syntax = what the interpreter sees (result of parsing concrete syntax)

topracer
Mausschubser
Beiträge: 53
Registriert: 10. Jan 2014 19:14

Re: Abstract vs. Concrete Syntax

Okay, thank you for the good explanation!


text:
# Existence of a BCH-code?

I would like to ask about existence of a $[31,11,12]$ binary BCH-code. How to prove its existence, if it does i have to find the generator polynomial? Can i use some specific bound that could show or deny the existence of this code?

Since the BCH-codes are cyclic codes, we have dimension $11$, so the degree of the generator polynomial must be $20$.

Can anybody help me with this problem, please? Thank you in advance!

• Do you know the definition of a BCH code? Do you know how to determine the conjugates of $\alpha^i$ where $\alpha$ is a primitive element of $GF(32)$ a.k.a. $\mathbb F_{2^5}$? – Dilip Sarwate Nov 12 '14 at 15:05
• Yes, i know how to compute the conjugates. – Lullaby Nov 12 '14 at 18:11
• Good. Now, what is the definition of a binary BCH code? – Dilip Sarwate Nov 12 '14 at 19:49
• Hint: $\alpha^5$ and $\alpha^9$ share the minimal polynomial. – Jyrki Lahtonen Nov 21 '14 at 20:48

A restrictive but mostly adequate definition of a binary BCH code of designed distance $2t+1$ is of a cyclic code of length $n$ whose generator polynomial $g(x)$ has roots $\alpha, \alpha^2, \ldots, \alpha^{2t}$ where $\alpha$ is an $n$-th root of unity in some extension field GF$(2^m)$ (also called $\mathbb F_{2^m}$) of GF$(2)$ or $\mathbb F_2$. Since $g(x) \in \mathbb F_2[x]$, the conjugates of these $2t$ elements also are roots of $g(x)$. If we proceed systematically, beginning by making a list of the conjugates of $\alpha$ (which are $\alpha^2, \alpha^4, \cdots$), we see that some elements that we would take up later have already been accounted for.
The generator polynomial of a binary BCH code of length $31$ with designed distance $11$ has roots $\alpha, \alpha^2, \alpha^3, \ldots, \alpha^{10}$ and so we start listing the conjugates as
$$\underline{\alpha},\quad \underline{\alpha^2},\quad \underline{\alpha^4},\quad \underline{\alpha^8},\quad \alpha^{16}\\ \underline{\alpha^3}, \quad \underline{\alpha^6}, \quad \alpha^{12}, \quad \alpha^{24},\quad\alpha^{17}\\ \underline{\alpha^5}, \quad \underline{\alpha^{10}}, \quad \alpha^{20}, \quad \underline{\alpha^{9}},\quad\alpha^{18}\\ \underline{\alpha^7}, \quad {\alpha^{14}}, \quad \alpha^{28}, \quad {\alpha^{25}},\quad\alpha^{19}$$
where the elements on the $2t$ list are underlined. Note that we would have ended up with the same result if we had been looking for a code with minimum distance $9$; the fact that $\alpha^9$ is a conjugate of $\alpha^5$ (as Jyrki noted) and so the designed distance works out to be $11$ instead of $9$ is a serendipitous bonus that we were not seeking.
We conclude that $\deg(g(x)) = 20$ and so there does exist a $[n,k,d] = [31,11,11]$ BCH code, but not a $[31,11,12]$ code. There is a $[31,10, 12]$ BCH code which is a subcode (consisting of the even-weight codewords) of the $[31,11,11]$ BCH code described above. (It is a BCH code under a more general definition of BCH code, where any $d-1$ successive powers of $\alpha$ as roots of $g(x)$ gives a code of designed distance $d$: just include $\alpha^0 = 1$ as a root of $g(x)$ in the list displayed above).


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You measure … This list of the nearest galaxies to Earth is ordered by increasing distance. You can see how big it appears to be, but since you do not know its distance, you cannot tell how big it really is. They found that the FRB came from a medium-sized galaxy at a cosmological distance. The Hydra Supercluster is about 100 million light years long, and the galaxy clusters within it range from about 105 to 160 million light years away from us. The Sun is in a finger called the Orion Spur. Science, this issue p. 565; see also p. 546. The nearest star is 4.2 light-years away, while the nearest spiral galaxy lies about 2.5 million light-years from Earth. [/caption] The distance to the Andromeda Galaxy is 2.54 million light-years, or 778 kiloparsecs. Specifically, Earth's movement toward the supermassive black hole at the center of the Milky Way Galaxy just got faster by 7 kilometers per second. Just like your fingertip, stars that are closer to us shift positions relative to more-distant stars, which appear fixed. It's a big flat disk of stars measuring up to 120,000 light years across. A light-year is the distance light travels in one year, which is about 5.88 trillion miles or almost 800 times the diameter of the solar system. It takes eight minutes for light to get from the Sun to the Earth, this super long distance. We're embedded deep within our galaxy, the Milky Way. The Hubble Space Telescope has calculated the distance to the most far-out galaxy ever measured — GN-z11 — providing scientists with a look deep into the history of the universe. Astronomers have done their best to replicate what they think it looks like and where we sit within it. The diameter of this galaxy is 220000 light years. Mystery Object Comet Holmes in the Kuiper Belt. while the Andromeda galaxy is 2.537 million light years away. Some are known to repeat, but most are single bursts. Astronomers nail down distance to the earliest galaxy yet The galaxy EGS-zs8-1, the most distant galaxy yet seen, was discovered in images from the Hubble and Spitzer space telescopes. So one thing, this huge distance wouldn't even show up on this picture. Distance Information. Remember that, whatever we use to measure these vast distances, it must be very luminous. The Andromeda Galaxy located at a distance of 2 million light years away is the nearest major galaxy and is of comparable size to our own galaxy. On Earth, the only information we could get from that galaxy is what we could see. Earth is inching closer to a supermassive black hole in the Milky Way galaxy at 7 km per second, a study has found out. Andromeda is the nearest spiral galaxy.LMC is the nearest galaxy. It is a spiral type galaxy. Localizing FRBs will help determine their causes and allow them to be used as cosmological probes. The earth is on the Milky Way Galaxy. At first glance it looks like the Milky Way is at the center of the universe and it committed some galactic social blunder because all of the other galaxies are rushing away from it … So beyond this, individual galaxy redshifts yield distances to accuracies better than 10% and obviously this improves as we go to greater distance. distance scale for the Galaxy would lead to better estimates of the ages of globular clusters, Ho, and the age of the Universe. The Kubasik Galaxy does not exist in real life. The Earth is a tiny planet on a galaxy known as Milky Way galaxy, which is just one of the many billions of galaxies that form the universe. Graphic view of our Milky Way Galaxy. The spiral-shaped Milky Way galaxy is enormous and consists of 100 to 400 billion stars including the Sun as one of the stars. Of these very bright objects, 3C 273 is the brightest in our skies. The Andromeda Galaxy (IPA: / æ n ˈ d r ɒ m ɪ d ə /), also known as Messier 31, M31, or NGC 224 and originally the Andromeda Nebula (see below), is a barred spiral galaxy approximately 2.5 million light-years (770 kiloparsecs) from Earth and the nearest major galaxy to the Milky Way. Some extra info: according to this article, the Sun oscillates through the plane of the galaxy with an amplitude of about 230 light-years, crossing the plane every 33 million years. image from www.daviddarling.info This is one representation of what our galaxy may look like. And just remember that orbit of the Earth, that was at a huge distance. In the solar system "A" is the distance from Earth to the Sun, the astronomical unit, 150,000,000 km or 93,000,000 miles. Many of the distances are only estimates, and some may be incorrect by 50% or more. $\endgroup$ – … Andromeda galaxy distance from Earth is 2.5 million light years. The nearest galaxy to Earth, Andromeda is about 2.5 million light years away. It is 8.178 ± 0.035 kiloparsecs (26,670 ± 110 ly) away from Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest. It is hard to give an exact distance for a supercluster, because the galaxy clusters within it are often millions of light years from each other! You must consider one thing however, there is no way for us to look down upon our own galaxy as we might with the Andromeda galaxy for example. Suppose you see a mystery object like this one, and you want to learn where and what it is. The extremely large redshifts of various quasars suggest that they are moving away from the Earth at tremendous velocities (i.e., approximately 90 percent the speed of light) and thereby constitute some of the most distant objects in the universe. A parsec is a unit of distance equal to 3.26 light-years. Distances are given in light years (ly). ), provided p is small (which it is for all stars), that the distance D to the near star is given by 206265 AU / p, where AU is the astronomical unit mentioned above (i.e., the distance from the earth to the sun, 150 million km or 93 million miles), and p is the parallax angle measured in seconds of arc. Fast radio bursts (FRBs) are brief radio emissions from distant astronomical sources. As Earth orbits the Sun, astronomers invoke this same principle to determine the distance to nearby stars. According to a map issued by Japanese radio astronomy project VERA, earth … For example, if a galaxy has a redshift of 20,000 km/sec and H is set to 70 km/sec/Mpc, then the galaxy's distance = (20,000 km/sec)/(70 km/sec/Mpc) = 20,000/70 × [(km/sec)/(km/sec)] Mpc = 286 megaparsecs. The Magellanic Clouds are irregular dwarf galaxies that are even nearer. Distance Information. The galaxies nearest to the Earth include Andromeda, Segue 1, Sagittarius Dwarf Spheroidal, and Canis Major Dwarf Galaxies. “From this, they derived a refined distance of estimate of 7.6 to 8.2 kpc – which works out to about 24,788 to 26,745 light years.” Which is off by a factor of 10. When we saw the larger image, we would have no way of knowing the galaxy actually was larger: we would assume that it was simply closer to us. The Milky Way Galaxy is organized into spiral arms of giant stars that illuminate interstellar gas and dust. The distance ranking in this list should be seen only as a guess. But even a small fraction of mis-classified stars with very wrong distances could screw up our whole view of the Galaxy. Scientists believe that there are 1 trillion stars in this galaxy. Suppose a certain galaxy is twice as large as an average galaxy. The Galactic Center (or Galactic Centre) is the rotational center of the Milky Way galaxy; it is a supermassive black hole of 4.100 ± 0.034 million solar masses, which powers the compact radio source Sagittarius A*. Although a lot of people list Andromeda as our closest neighbouring Galaxy, they are in fact, incorrect. Our Solar System is located in the middle of this galactic disk. Its distance is 158, 200 light years to the Earth. Otherwise, we won't be able to see it in very distant galaxies, halfway across the universe. If you look at a cluster of galaxies then you can also improve matters by $\sqrt{n}$ by averaging over the individual galaxy peculiar motions. The closest known galaxy to us is the Canis Major Dwarf Galaxy, at 236,000,000,000,000,000 km (25,000 light years) from the Sun. In fact, we need to know half of the angle, not the entire one. (NASA) The mass of this galaxy is equal to 400 to 700 billion suns. Abstract. So, if our distance to the LMC is uncertain by 10%, then every single distance to another galaxy will be uncertain by at least 10%. The Andromeda Galaxy is speeding toward us, but it will take 4 billion years to get here. It can easily be seen, using basic trigonometry (try it! The important thing about clusters is that all the stars in the cluster are at (nearly) the same distance from Earth. The Small and Large Magellanic Clouds are located 165,000 light years and 195,000 light years away respectively. Quasars are capable of emitting hundreds or even thousands of times the entire energy output of our galaxy, making them some of the most luminous and energetic objects in the entire Universe. If you shot a bullet at the Sun from Earth, it would take you that 17 years to actually get to the Sun. 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# Phase-field approximation for a class of cohesive fracture energies with an activation threshold

created by crismale on 21 Feb 2019

[BibTeX]

preprint

Inserted: 21 feb 2019

Year: 2018

ArXiv: 1812.05301 PDF

Abstract:

We study the $\Gamma$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$.

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### Automated Analysis of Cryptographic Assumptions in Generic Group Models

Gilles Barthe, Edvard Fagerholm, Dario Fiore, John Mitchell, Andre Scedrov, and Benedikt Schmidt

##### Abstract

We initiate the study of principled, automated, methods for analyzing hardness assumptions in generic group models, following the approach of symbolic cryptography. We start by defining a broad class of generic and symbolic group models for different settings---symmetric or asymmetric (leveled) k-linear groups---and by proving "computational soundness" theorems for the symbolic models. Based on this result, we formulate a very general master theorem that formally relates the hardness of a (possibly interactive) assumption in these models to solving problems in polynomial algebra. Then, we systematically analyze these problems. We identify different classes of assumptions and obtain decidability and undecidability results. Then, we develop and implement automated procedures for verifying the conditions of master theorems, and thus the validity of hardness assumptions in generic group models. The concrete outcome of this work is an automated tool which takes as input the statement of an assumption, and outputs either a proof of its generic hardness or shows an algebraic attack against the assumption.

##### Metadata
Available format(s)
Category
Foundations
Publication info
Published by the IACR in CRYPTO 2014
Keywords
generic group modelmultilinear mapsgraded encoding schemesautomated cryptographic proofs
Contact author(s)
benedikt schmidt @ imdea org
History
2014-06-15: received
Short URL
https://ia.cr/2014/458
License

CC BY

BibTeX

@misc{cryptoeprint:2014/458,
author = {Gilles Barthe and Edvard Fagerholm and Dario Fiore and John Mitchell and Andre Scedrov and Benedikt Schmidt},
title = {Automated Analysis of Cryptographic Assumptions in Generic Group Models},
howpublished = {Cryptology ePrint Archive, Paper 2014/458},
year = {2014},
note = {\url{https://eprint.iacr.org/2014/458}},
url = {https://eprint.iacr.org/2014/458}
}

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