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Energy invariant for shallow water waves and the Korteweg – de Vries equation.Is energy always an invariant?

# Energy invariant for shallow water waves and the Korteweg – de Vries equation. Is energy always an invariant?

Anna Karczewska Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
Piotr Rozmej Institute of Physics, Faculty of Physics and Astronomy
University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
Eryk Infeld National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa, Poland
August 20, 2019August 20, 2019
August 20, 2019August 20, 2019
###### Abstract

It is well known that the KdV equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum and energy. Here we try to answer the question of how this comes about, and also how these KdV quantities relate to those of the Euler shallow water equations. Here Luke’s Lagrangian is helpful. We also consider higher order extensions of KdV. Though in general not integrable, in some sense they are almost so, these with the accuracy of the expansion.

Soliton, shallow water waves, nonlinear equations, invariants of KdV
###### pacs:
02.30.Jr, 05.45.-a, 47.35.Bb, 47.35.Fg

## I Introduction

There exists a vast number of papers dealing with the shallow water problem. Aspects of the propagation of weakly nonlinear, dispersive waves are still beeing studied. Last year we published two articles KRR ; KRI in which Korteveg–de Vries type equations were derived in weakly nonlinear, dispersive and long wavelength limit. The second order KdV type equation was derived. The second order KdV equation MS ; BS , sometimes called "extended KdV equation", was obtained for the case with a flat bottom. In derivation of the new equation we adapted the method described in BS . In KRI , an analytic solution of this equation in the form of a particular soliton was found, as well.

It is well known, see, e.g. Miura ; MGK ; DrJ ; Newell85 , that for the KdV equation there exists an infinite number of invariants, that is, integrals over space of functions of the wave profile and its derivatives, which are constants in time. Looking for analogous invariants for the second order KdV equation we met with some problems even for the standard KdV equation (which is first order in small parameters). This problem appears when energy conservation is considered.

In this paper we reconsider invariants of the KdV equation and formulas for the total energy in several different approaches and different frames of reference (fixed and moving ones). We find that the invariant  , sometimes called the energy invariant, does not always have that interpretation. We also give a proof that for the second order KdV equation, obtained in KRR ; KRI ; MS ; BS ,   is not an invariant of motion.

There are many papers considering higher-order KdV type equations. Among them we would like to point out works of Byatt-Smith B-S87 , Kichenassamy and Olver SK_PO , Marchant MS ; MS96 ; Mar99 ; Mar02 ; Mar02a , Zou and Su ZouSu , Tzirtzilakis et.al. TMAB and Burde Burde . It was shown that if some coefficients of the second order equation for shallow water problem (1) are diferent or zero then there exists a hierarchy of solition solutions. Kichenassamy and Olver SK_PO even claimed that for second order KdV equation solitary solutions of appropriate form can not exist. This claim was falsified in our paper KRI where the analytic solution of the second order KdV equation (1) was found. Concerning the energy conservation there are indications that collisions of solitons Hirota72 ; Hir which are solutions of higher order equations of KdV type can be inelastic ZouSu ; TMAB .

The paper is organized as follows. In Section II several frequently used forms of KdV equations are recalled with particular attention to transformations between fixed and moving reference frames. In Section III the form of the three lowest invariants of KdV equations is derived for different forms of the equations. In Section IV we show that the energy calculated from the definition has no invariant form. Section V describes the variational approach in a potential formulation which gives a proper KdV equation but fails in obtaining second order KdV equations. In the next section the proper invariants are obtained from Luke’s Lagrangian density. Section VII summarizes conclusions on the energy for KdV equation. In section VIII we apply the same formalism to calculate energy for waves governed by the extended KdV equation (second order). We found that energy is not conserved neither in fixed coordinate system nor in the moving frame.

## Ii The extended KdV equation

The geometry of shallow water waves is presented in Fig. 1.

In KRR ; KRI we derived an equation, second order in small parameters, in the fixed reference system and with scaled nondimensional variables containing terms for bottom fluctuations. They will not be considered here.

For a flat bottom that equation reduces to the second order KdV type equation, identical with (BS, , Eq. (21)) for , that is,

 ηt+ηx+α32ηηx+β16η3x+α2(−38η2ηx) (1) +αβ(2324ηxη2x+512ηη3x)+β219360η5x = 0.

Subscripts denote partial differentiation. Small parameters   are defined by ratios of the wave amplitude  , the average water depth   and mean wavelength

 α=ah,β=(hl)2.

Equation (1) was earlier derived in MS and called "the extended KdV equation".

Limitation to the first order in small parameters yields the KdV equation in a fixed coordinate system

 ηt+ηx+α32ηηx+β16η3x=0. (2)

Transformation to a moving frame  in the form

 ¯x=(x−t),¯t=t,¯η=η, (3)

allows us to remove the term   in the KdV equation in a frame moving withthe velocity of sound

 ¯η¯t+α32¯η¯η¯x+β16¯η3¯x=0. (4)

The explicit form of the scaling leading to equations (1) – (4) is given by (29).

Problems with mass, momentum and energy conservation in the KdV equation were discussed in Kalisch recently. In this paper the authors considered the KdV equations in the original dimensional variables. Then the KdV equatios are

 ηt+cηx+32chηηx+ch26ηxxx=0, (5)

in a fixed frame of reference and

 ηt+32chηηx+ch26ηxxx=0, (6)

in a moving frame. In both, , and (6) is obtained from (5) by setting and dropping the prime sign.

In our present paper we discuss the energy formulas obtained both in fixed and moving frames of reference for KdV (2), (4), (5), (6) . There seem to be some contradictions in the literature because the form of some invariants and the energy formulas differ in different sources because of using different reference frames and/or different scalings. In this paper we address this problems.

The second goal is to present some invariants for a KdV type equation of the second order (1).

## Iii Invariants of KdV type equations

What invariants can be attributed to equations (1) – (2) and (5) – (6) ?

It is well known, see, e.g. (DrJ, , Ch. 5), that an equation of the form

 ∂T∂t+∂X∂x=0, (7)

where neither   (an analog to density) nor   (an analog to flux) contain partial derivatives with respect to  , corresponds to some conservation law. It can be applied, in particular, to KdV equations (where there exist an infinite number of such conservation laws) and to the equations of KdV type like (1). Functions   and   may depend on   but not . If both functions   and   are integrable on   and const (soliton solutions), then integration of equation (7) yields

 ddt(∫∞−∞Tdx)=0or∫∞−∞Tdx=const. , (8)

since

 ∫∞−∞Xxdx=X(∞,t)−X(−∞,t)=0. (9)

The same conclusion applies for periodic solutions (cnoidal waves), when in the integrals (8), (9) limits of integration are replaced by , where is the space period of the cnoidal wave (the wave length).

### iii.1 Invariants of the KdV equation

For the KdV equation (2) the two first invariants can be obtained easily. Writing (2) in the form

 ∂η∂t+∂∂x(η+34αη2+16βηxx)=0. (10)

one immediately obtains the conservation of mass (volume) law

 I(1)=∫∞−∞ηdx=const. (11)

Similarly, multiplication of (2) by    gives

 ∂∂t(12η2)+∂∂x(12η2+12αη3−112βη2x+16βηηxx)=0, (12)

resulting in the invariant of the form

 I(2)=∫∞−∞η2dx=const. (13)

In the literature of the subject, see, e.g. Kalisch ; DrJ ,   is attributed to momentum conservation.

Invariants have the same form for all KdV equations (2), (4), (A), (5), (6).

Denote the left hand side of (2) by   and take

 3η2×KDV(x,t)−23βαηx×∂∂xKDV(x,t). (14)

The result, after simplifications is

 ∂∂t(η3−13βαη2x)+∂∂x(98αη4+12βη2xη2 (15) −βη2xη+η3+118β2αη22x−19β2αηxη3x−13βαη2x) = 0.

Then the next invariant for KdV in the fixed reference frame (2) is

 I(3)fixed frame=∫∞−∞(η3−13βαη2x)dx=const. (16)

The same invariant is obtained for the KdV in the moving frame (4). The same construction like (14) but for equation (4) results in

 ∂∂t(η3−13βαη2x)+∂∂x(98αη4+12βη2xη2 (17) −βη2xη+η3+118β2αη22x−19β2αηxη3x) = 0.

Then the next invariant for KdV equation in moving reference frame (2) is

 I(3)moving frame=∫∞−∞(η3−13βαη2x)dx=const. (18)

The procedure similar to those described above leads to the same invariants for both equations (5) and (6) where KdV equations are written in dimensional variables. To see this, it is enough to take  , where   is the lhs either of (5) or (6). For equation (5) the conservation law is

 ∂∂t(η3−h33η2x)+∂∂x(cη3−9c8hη4−13ch3η2x (19) −ch2ηη2x+12ch2η2ηxx+118ch5η2xx−19ch5ηxηxxx) = 0,

whereas for equation (6) the flux term is different

 ∂∂t(η3−h33η2x)+∂∂x(9c8hη4−ch2ηη2x (20) +12ch2η2ηxx+118ch5η2xx−19ch5ηxηxxx) = 0.

But in both cases the same invariant is obtained as

 I(3)dimensional=∫∞−∞(η3−h33η2x)dx=const. (21)

Conclusion  Invariants have the same form for fixed and moving frames of reference when the transformation from fixed to moving frame scales   and in the same way (e.g. and ). When scaling factors are different, like in (116), then the form of in the moving frame differs from the form in the fixed frame, see Appendix A.

For those solutions of KdV which preserve their shapes during the motion, that is, for cnoidal solutions and single soliton solutions, integrals of any power of   and any power of arbitrary derivative of the solution with respect to   are invariants. That is,

where  , and   is an arbitrary real number. Then an arbitrary linear combination of   is an invariant, as well.

### iii.2 Invariants of the second order equations

Can we construct invariants for KdV type equations of the second order? Let us try to take for equation (1). Then we find that all terms, except  , can be written as , as

 ∫[ηx+α32ηηx+β16η3x+α2(−38η2ηx) +αβ(2324ηxη2x+512ηη3x)+β219360η5x]dx = η+34αη2+16βη2x−18α2η3 +αβ(1348η2x+512ηη2x)+19360β2η4x.

As (III.2) depends on    and space derivatives and also since all those functions vanish when  , the conservation law for mass (volume)

 ∫∞−∞η(x,t)dx=const., (24)

holds for the second order equation.

(Conservation law (24) holds for the equation with an uneven bottom, as well.)

Until now we did not find any other invariants for the secod order equations. Moreover, we can show that the integral   (13)  is no longer an invariant of the second order KdV equation (1).

Upon multiplication of equation (1) by   one obtains

 0 = ∂∂t(12η2)+∂∂x[12η2+12αη3+16β(−12η2x+ηη2x) −332α2η4+19360β2(12η2xx−ηxη3x+ηη4x) +512αβη2η2x]+18αβηηxη2x.

The last term in (III.2) can not be expressed as  . Therefore    is not a conserved quantity.

## Iv Energy

The invariant is usually referred to as the energy invariant. Is this really the case?

### iv.1 Energy in a fixed frame as calculated from the definition

The hydrodynamic equations for an incompressible, inviscid fluid, in irrotational motion and under gravity in a fixed frame of reference, lead to a KdV equation of the form

 ~η~t+~η~x+α32~η~η~x+β16~η3~x=0. (26)

We will find the function

 ~f~x=~η−14α~η2+13β~η~x~x, (27)

obtained as a byproduct in derivation of KdV, useful in what follows. For more details see Appendix B or (EIGR, , Chapter 5). Tildas denote scaled dimensionless quantities.

Now construct the total energy of the fluid from the definition.

The total energy is the sum of potential and kinetic energy. In our two-dimensional system the energy in original (dimensional coordinates) is

 E=T+V = ∫+∞−∞(∫h+η0ρv22dy)dx

Equations (26) and (27) are obtained after scaling BS ; KRR ; KRI . We now have dimesionless variables, according to

 ~ϕ=hla√ghϕ,~x=xl,~η=ηa,~y=yh,~t=tl/√gh, (29)

and

 V=ρgh2l∫+∞−∞∫1+α~η0ρ~yd~yd~x, (30)
 T=12ρgh2l∫+∞−∞∫1+α~η0(α2~ϕ2~x+α2β~ϕ2~y)d~yd~x. (31)

Note, that the factor in front of the integrals has the dimension of energy.

In the following, we omit signs  , having in mind that we are working in dimensionless variables. Integration in (30) with respect to   yields

 V = 12gh2lρ∫∞−∞(α2η2+2αη+1)dx = 12gh2lρ[∫∞−∞(α2η2+2αη)dx+∫∞−∞dx].

After renormalization (substraction of constant term ) one obtains

 V=12gh2lρ∫∞−∞(α2η2+2αη)dx. (33)

In kinetic energy we use the velocity potential expressed in the lowest (first) order

 ϕx=fx−12βy2fxxxandϕy=−βyfxx, (34)

where was defined in (27). Now the bracket in the integral (31) is

 (α2ϕx2+α2βϕy2)=α2(f2x+βy2(−fxfxxx+f2xx)). (35)

Inegration with respect to the vertical corrdinate   gives, up to the same order,

 T = 12ρgh2l∫+∞−∞α2[f2x(1+αη) +β(−fxfxxx+f2xx)13(1+αη)3]dx = 12ρgh2l∫+∞−∞α2[f2x+αf2xη+13β(f2xx−fxfxxx)]dx.

In order to express energy through the elevation funcion   we use (27). We then substitute   in terms of the third order and    in terms of the second order

 T = 12ρgh2l∫+∞−∞α2[(η2−12αη3+23βηηxx) = 12ρgh2lα2[∫+∞−∞(η2+12αη3)dx

The last term vanishes as

 ∫+∞−∞(η2x+ηηxx)dx=∫+∞−∞η2xdx+ηηx|+∞−∞−∫+∞−∞η2xdx=0. (38)

Therefore the total energy in the fixed frame is given by

 Etot = T+V=ρgh2l∫∞−∞(αη+(αη)2+14(αη)3)dx(39) = ρgh2l(αI(1)+α2I(2)+14α2I(3)+112α2β∫∞−∞η2xdx)

The energy (39) in a fixed frame of reference has non invariant form. The last term in (39) results in small deviations from energy conservation only when changes in time in soliton’s reference frame, what occurs only during soliton collision. This deviations are discussed and illustrated in Section VI E.

The result (IV.1) gives the energy in powers of   only. The same structure of powers in   was obtained by the authors of Kalisch , who work in dimensional KdV equations (5) and (6). On page 122 they present a non-dimensional energy density in a frame moving with the velocity . Then, if is set, the energy density in a fixed frame is proportional to as the formula is obtained up to second order in . However, the third order term is , so the formula up to the third order in becomes

 E∼αη+α2η2+14α3η3. (39)

This energy density contains the same terms like (IV.1) and does not contain the term , as well.

Energy calculated from the definition does not contain a proper invariant of motion.

### iv.2 Energy in a moving frame

Now consider the total energy according to (28) calculated in a frame moving with the velocity of sound . Using the same scaling (29) to dimensionless variables we note that in these variables . As pointed by Ali and Kalisch [8,Sect. 3] working in such system one has to replace by the horizontal velocity in a moving frame, that is by . Then repeating the same steps as in the previous subsection yields the energy expressed by invariants

 Etot = ρgh2l∫∞−∞[−12α~η+14(α~η)2+12α3(~η3−13βα~η2~x)]d~x (40) = ρgh2l(−12αI(1)+14α2I(2)+12α3I(3)).

The crucial term   in (40) appears due to integration over vertical variable of the term   supplied by .

## V Variational approach

### v.1 Lagrangian approach, potential formulation

Some attempts at the variational approach to shallow water problems are summarized in G.B. Whitham’s book (Whit, , Sect 16.14).

For KdV as it stands, we can not write a variational principle directly. It is necessary to introduce a velocity potential. The simplest choice is to take  . Then equation (2) in the fixed frame takes the form

 φxt+φxx+32αφxφxx+16βφxxxx=0. (41)

The appropriate Lagrangian density is

 Lfixed frame:=−12φtφx−12φ2x−α4φ3x+β12φ2xx. (42)

Indeed, the Euler–Lagrange equation obtained from Lagrangian (42) is just (41).

For our moving reference frame the substitution   into (4) gives

 φxt+32αφxφxx+16βφxxxx=0. (43)

So, the appropriate Lagrangian density is

 Lmoving frame:=−12φtφx−α4φ3x+β12φ2xx. (44)

Again, the Euler–Lagrange equation obtained from Lagrangian (44) is just (43).

### v.2 Hamiltonians for KdV equations in the potential formulation

The Hamiltonian for the KdV equation in a fixed frame (2) can be obtained in the following way. Defining generalized momentum  , where   is given by (42), one obtains

 H = ∫∞−∞[π˙φ−L]dx=∫∞−∞[12φ2x+α4φ3x−β12φ2xx]dx (45) = ∫∞−∞[12η2+14α(η3−β3αη2x)]dx.

The energy is expressed by invariants   so it is a constant of motion.

The same procedure for KdV in a moving frame (4) gives

 H = ∫∞−∞[π˙φ−L]dx=∫∞−∞[α4φ3x−β12φ2xx]dx (46) = 14α∫∞−∞(η3−β3αη2x)dx.

The Hamiltonian (46) only consists  .

The constant of motion in a moving frame is

 E=14I(3)=const. (47)

The potential formulation of the Lagrangian, described above, is succesful for deriving KdV equations both for fixed and moving reference frames. It fails, however, for the second order KdV equation (1). We proved that there exists a nonlinear expression of , such that the resulting Euler–Lagrange equation differs very little from equation (1). The difference lies only in the value of one of the coefficients in the second order term . Particular values of coefficients in this term also cause the lack of the   invariant for second order KdV equation, (see (III.2)).

## Vi Luke’s Lagrangian and KdV energy

The full set of Euler equations for the shallow water problem, as well as KdV equations (2), (A), and second order KdV equation (1) can be derived from Luke’s Lagrangian Luke , see, e.g. MS . Luke points out, however, that his Lagrangian is not equal to the difference of kinetic and potential energy. Euler–Lagrange equations obtained from do not have the proper form at the boundary. Instead, Luke’s Lagrangian is the sum of kinetic and potential energy suplemented by the term which is necessary in dynamical boundary condition.

### vi.1 Derivation of KdV energy from the original Euler equations according to Eigr

In Chapter 5.2 of the Infeld and Rowlands book the authors present a derivation of the KdV equation from the Euler (hydrodynamic) equations using a single small parameter  . Moreover, they show that the same method allows us to derive the Kadomtsev–Petviashvili (KP) equation KP for water waves IRH ; KP_woda ; L-H&F ; Benj and also nonlinear equations for ion acoustic waves in a plasma KP_plazma ; InRo . The authors first derive equations of motion, then construct a Lagrangian and look for constants of motion. For the purpose of this paper and for comparison to results obtained in the next subsections it is convenient to present their results starting from Luke’s Lagrangian density. That density can be written as (here )

 L=∫1+η0[ϕt+12(ϕ2x+ϕ2z)+z]dz. (48)

In Chapter 5.2.1 of EIGR the authors introduce scaled variables in a movimg frame ( plays a role of small parameter and if  , then KdV equation is obtained). Then (for details, see Appendix B or (EIGR, , Chapter 5.2))

 ϕz = −ε32zfξξ,ϕx=εfξ−ε2z22fξξξ, ϕt = −εfξ+ε2(fτ+z22fξξξ)−ε3z22fξξτ. (49)

Substitution of the above formulas into the expression [ ] under the integral in (48) gives

 [ ] = z−εfξ+ε2(fτ+12f2ξ+z22fξξξ) +ε3z22[−fξξτ+(f2ξξ−fξfξξξ)]+O(ε4).

Remark  The full Lagrangian is obtained by integration of the Lagrangian density (48) with respect to  . Integration limits are   for a soliton solutions, or  , where  –wave length (space period) for cnoidal solutions. Integration by parts and properties of the solutions at the limits, see (9), allow us to use the equivalence .

Therefore

 [ ] = z−εfξ+ε2(fτ+12f2ξ+z22fξξξ) +ε3z22[−fξξτ+2f2ξξ]+O(ε4).

Integration over gives (note that )

 L = (52) +13(1+εη)3[12ε2fξξξ−12ε3fξξτ+ε3f2ξξ].

Write (52) up to third order in

 L=L(0)+εL(1)+ε2L(2)+ε3L(3)+O(ε4).

It is easy to show, that

 L(0) = 12,L(1)=η−fξ, L(2) = fτ+12η2−ηfξ+12f2ξ+16fξξξ, (53) L(3) = ηfτ+12ηf2ξ+12ηfξξξ−16fξξτ+13f2ξξ.

 H = fτ∂L∂fτ+fξξτ∂L∂fξξτ−L = −[12+ε(η−fξ)+ε2(12η2−ηfξ+12f2ξ+16fξξξ) +ε3(12ηf2ξ+12ηfξξξ+13f2ξξ)].

Dropping the constant term one obtains the total energy as

 E = ∫∞−∞[ε(η−fξ)+ε2(12η2−ηfξ+12f2ξ+16fξξξ) (55) +ε3(12ηf2ξ+12ηfξξξ+13f2ξξ)]dξ.

Now, we need to express and its derivatives by and its derivatives. We use (27) replacing and by , that is,

 fξ=η−14εη2+13εηξξ. (56)

Then the total energy in a moving frame is expressed in terms of the second and the third invariants

 E=−[ε214∫∞−∞η2dx+ε312∫∞−∞(η3−13η2ξ)dx]. (57)

Note that the term    occuring in the third order invariant originates from three terms appearing in  ,    and (see terms    and    in (VI.1)).

### vi.2 Luke’s Lagrangian

The original Lagrangian density in Luke’s paper Luke is

 L=∫h(x)0ρ[ϕt+12(ϕ2x+ϕ2y)+gy]dy. (58)

After scaling as in BS ; KRR ; KRI

 ~ϕ=hla√ghϕ,~x=xl,~η=ηa,~y=yh,~t=tl/√gh, (59)

we obtain

 ϕt=ghα~ϕ~t,ϕ2x=ghα2~ϕ2~x,ϕ2y=ghα2β~ϕ2~y. (60)

The Lagrangian density in scaled variables becomes ()

 L = ρgha∫1+αη0[~ϕ~t+12(~ϕ2~x+α2β~ϕ2~y)]d~y (61) +12ρgh2(1+αη)2.

So, in dimensionless quatities

 Lρgha=∫1+αη0[~ϕ~t+12(α~ϕ2~x+αβ~ϕ2~y)]d~y+12αη2, (62)

where the constant term and the term proportional to   in the expansion of   are omitted. The form (62) is identical with Eq. (2.9) in Marchant & Smyth MS .

The full Lagrangian is obtained by integration over  . In dimensionless variables () it gives

 L=E0∫∞−∞[∫1+αη0[~ϕ~t+12(α~ϕ2~x+αβ~ϕ2~y)]d~y+12αη2]d~x. (63)

The factor in front of the integral,  , has the dimension of energy.

Next, the signs ( ) will be omitted, but we have to remember that we are working in scaled dimensionless variables in a fixed reference frame.

### vi.3 Energy in the fixed reference frame

Express the Lagrangian density by    and  . Now, up to first order in small parameters

 ϕ = f


text:
# Tensor product of modules

by Arian.D
Tags: modules, product, tensor
 Mentor P: 15,921 OK, so then we wish to prove that $$Ker(f_3\circ \psi \circ f_2^{-1})=Im(f_2\circ \varphi \circ f_1^{-1})$$ I'll do one side of the inclusion: Take x in the kernel, then $f_3(\psi(f_2^{-1}(x)))=0$. Since $f_3$ is an isomorphism, its kernel is 0, thus $\psi(f_2^{-1}(x))=0$. Thus $f_2^{-1}(x)\in Ker(\psi)$. By exactness, we have that $Ker(\psi)=Im(\varphi)$, thus there exists a $y\in A$ such that $\varphi(y)=f_2^{-1}(x)$. In other words, $f_2(\varphi(y))=x$. Write $z=f_1(y)$, then $f^{-1}_1(z)=y$. Thus $$f_2(\varphi(f_1^{-1}(z)))=f_2(\varphi(y))=x$$ The left-hand side is certainly in the image, thus $x\in Im(f_2\circ \varphi\circ f_1^{-1})$. Can you do the other inclusion??
P: 101
 Quote by micromass OK, so then we wish to prove that $$Ker(f_3\circ \psi \circ f_2^{-1})=Im(f_2\circ \varphi \circ f_1^{-1})$$ I'll do one side of the inclusion: Take x in the kernel, then $f_3(\psi(f_2^{-1}(x)))=0$. Since $f_3$ is an isomorphism, its kernel is 0, thus $\psi(f_2^{-1}(x))=0$. Thus $f_2^{-1}(x)\in Ker(\psi)$. By exactness, we have that $Ker(\psi)=Im(\varphi)$, thus there exists a $y\in A$ such that $\varphi(y)=f_2^{-1}(x)$. In other words, $f_2(\varphi(y))=x$. Write $z=f_1(y)$, then $f^{-1}_1(z)=y$. Thus $$f_2(\varphi(f_1^{-1}(z)))=f_2(\varphi(y))=x$$ The left-hand side is certainly in the image, thus $x\in Im(f_2\circ \varphi\circ f_1^{-1})$. Can you do the other inclusion??
Sure. It's easy.

Take x in $Im(f_2\circ \varphi \circ f_1^{-1})$. Because it's in image, then there exists y in the domain of $f_2\circ \varphi \circ f_1^{-1}$, i.e. in A', such that $f_2\circ \varphi \circ f_1^{-1}(y)=x$
Since $f_2$ is an isomorphism it's invertible, and we'll have:
$f_2^{-1}(x)=\varphi \circ f_1^{-1}(y)$
Now let's apply $\psi$ to both sides, we'll obtain:
$\psi \circ f_2^{-1}(x) = \psi \circ \varphi \circ f_1^{-1}(y) = 0$
Since $\psi \circ \varphi = o$, because anything that we put into $\varphi$ goes to the kernel of $\psi$ and therefore it becomes zero!
We're done! Since $f_3$ is a homomorphism, it sends 0 to 0, hence:
$f_3 \circ \psi \circ f_2^{-1} (x) = f_3(0) = 0$
and that says $x \in Ker(f_3\circ \psi \circ f_2^{-1})$

It's a very neat theorem, I think it'll make a lot of things easier in future for me, and the good thing is that it works on almost all algebraic structures that I know now, because nowhere in the proof we assumed anything else except the definition of an exact sequence, kernel and image of homomorphisms. All of them stay intact in groups, rings and modules. Thanks for your help micromass.
P: 382
 Quote by Arian.D I'm reading about tensor product of modules, there's a theorem in the book that leaves parts of the proof to the reader. I've attached the file, I didn't put this in HW section because first of all I thought this question was more advanced to be posted in there and also because I want to discuss something else with people on this forum. I'm thinking what happens when a simple tensor becomes zero? I mean suppose that we have $x\otimes y = 0$. Does it mean that x or y must be zero? or it's possible that one of them nonzero but still their tensor product turns out to be zero? If possible, please check my proof in the file as well. I know I'm asking too much, but please help me as quickly as possible because tomorrow I'll have a conference about tensors, the professor is also the head of the math. department of our university and I'm the only under-graduate student in his class. so I'm very determined to have a successful conference tomorrow in the class and I need your help very much guys.
In general, the issue is whether your modules being tensored have torsion or not. If they have torsion elements, then a non-zero product can be zero, and the answer is no otherwise.

 Related Discussions Calculus & Beyond Homework 2 Linear & Abstract Algebra 2 Linear & Abstract Algebra 10 Linear & Abstract Algebra 3 Differential Geometry 5

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TypeScript

Take a function like below… type Combinable = string | number function add(a: Combinable, b: Combinable): Combinable { if (typeof a === 'string' || typeof b === 'string') { return a.toString() + b.toString() } return a + b } If we were to use this function like this: const x = add('black cat', 'white dog') x.split(' ') // TS compiler will complain here TypeScript cannot determine what the return type of add is at this point. ...

## Nullish Coalescing

// say you have some var hangin' around called name let x: string = name ?? '(no name)' ?? is the nullish coalescing operator. It differs from || in that if you had (in the above example) let x: string = name || '(no name)' name equal to an empty string (which is falsey) x would then be assigned the value (no name). The nullish coalescing operator will only use the right-hand assignment for “nullish” values (null or undefined). ...

## Recursive Type Aliases

In TS@4 a type can reference itself, e.g. type JSONValue = | string | number | boolean | null | JSONValue[] | { [k: string]: JSONValue } Previously, this would not be possible without some messy hacks.


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# What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.)

What is the geometric, physical, or other meaning of the third derivative of a function at a point?

If you have interesting things to say about the meaning of the first and second derivatives, please do so.

I’ve found the time, so I deleted one of my original comments in the OP and decided to expand it into a full answer.

One way to geometrically interpret the third derivative is in the notion of the osculating parabola. In much the same way that the first derivative enters into the defining equation for the tangent line (the line that best approximates your curve in the vicinity of a given point), and that the second derivative is involved in the expression for the osculating circle (the circle that best approximates your curve in the vicinity of a given point), the third derivative is required for expressing the osculating parabola, which is the parabola that best approximates… oh, you catch on quick. 😉

More specifically, if you remember the fact that four points uniquely determine a parabola, you can think of the osculating parabola as the limiting case of the parabola through four neighboring points of a given curve when those four points coalesce, or come together. The so-called aberrancy (a translation of the French “déviation“) is the tangent of the angle the axis of the osculating parabola makes with the normal line, and is given by the formula

where $\varrho$ is the radius of curvature and $\phi$ is the tangential angle.

From these considerations, one could derive an expression for the osculating parabola: given a curve represented parametrically as $(f(t)\quad g(t))^T$, the parametric equations for the osculating parabola of the curve at $t=t_0$ are

Here for instance is the cardioid $(2\cos\,t+\cos\,2t\quad 2\sin\,t+\sin\,2t)^T$ and its osculating parabola at $t=2\pi/3$:

and an animation of the various osculating parabolas for the curve $(3\cos\,t-2\cos\,3t\quad 3\sin\,t-2\sin\,3t)^T$:

Further, one could also give a geometric interpretation for the fourth derivative; what one now considers is the osculating conic (the limiting conic through five neighboring points of a curve when those five points coalesce), and one could classify points of a plane curve as elliptic, parabolic or hyperbolic depending on the nature of the osculating conic. In this respect, the discriminant of the osculating conic depends on the first four derivatives.

A lot more information is in these two articles by Steven Schot (who also wrote a nice article on the “jerk”), and the references therein.


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# Super Perfect numbers

A super-perfect number is a number with $\sigma(\sigma (n))=2n$.

How can I prove that every even super perfect number is from the form $n=2^k$ when $2^{k+1}-1$ is prime.

I tried every way please give me some guidance

-
I wanted to ask what is "super perfect numbers"... –  DonAntonio Nov 27 '12 at 4:26

Let $p$ be a prime. We know that $\sigma (p^\alpha)= \dfrac{p^{\alpha+1} -1}{p-1}$.

Now, let $m=p^\alpha r$ be an integer with $p\not| r$.
If $p \not| m$ then $\sigma(p m) = \sigma(p) \sigma(m) =(p+1) \sigma(m) > p \sigma(m)$.
If $p | m$ then we have $\sigma(pm) = \sigma (p^{\alpha +1})\sigma(r)= \dfrac{p^{\alpha +2} -1} {p-1} \sigma(r) >\dfrac{p^{\alpha +2} -p} {p-1} \sigma(r)= p\sigma(p^\alpha)\sigma(r) = p \sigma(m)$.

So, for any product $ab$ we have $\sigma(ab) \ge a \sigma(b)$ and equality holds exactly for $a=1$.

Now, we can conclude: Let $n= 2^ku$ with $u$ odd and $k \ge 1$.

We have $\sigma(n)= (2^{k+1}-1) s$ where $s=\sigma(u) \ge u$.

Therefore $\sigma(\sigma(n))= \sigma((2^{k+1}-1)s) \ge \sigma((2^{k+1}-1) s \ge 2^{k+1} u = 2n$.

So, for a super-perfect number, equality has to hold in this chain of inequalities.

The first inequality is true iff $s=1$ and the second inequality is true iff $2^{k+1}-1$ is prime (because the two trivial divisors already give $2^{k+1}$ and the number is clearly bigger than 1, this is where we use $k \ge 1$) and $s=u$.

So, in total $u=1$ and $2^{k+1}-1$ is prime and $n=2^k$.

Just to add a word of motivation:

Perfect numbers have the property $\sigma(n)=2n$, so with $\rho(n)=\sigma(n)-n$, this becomes $\rho(n)=n$.

The concatenation that is expected to stay the same size "on average" would be $\rho(\rho(n))=n$ which is related to the search for amicable numbers.

Now, instead we apply $\sigma$ twice, so we can expect that the number grows too much most of the time. This motivates the search for an inequality chain of the type $\sigma(\sigma(n))\ge 2n$ which then turns out to only work for even $n$.

-
Well, the average value of $\rho(n)$ is $(\pi^2/6-1)n$, so I don't know that I'd say that $\rho(\rho(n))$ is $n$ on average -- I would expect it to be more like 0.3n to 0.5n. –  Charles Nov 28 '12 at 16:28
@Charles: I know that it is not the mean, I'm open to a suggestion of a better formulation. My point is that if you start with the concept of perfect numbers (and abundant numbers, etc), you will expect that concatenation of sigmas to be too large and search for an inequality. This is more linked to "median" than to "mean", but motivations is not about being the exact median, either. –  Phira Nov 28 '12 at 17:54
I think that finding the right formulation of 'staying the same size' is hard and interesting. In my past experience I find $\rho$ an awkward function and much prefer $\sigma_{-1}$ which has quite elegant properties. But the inner $\rho$ is forced on us in this problem, which causes some difficulties since $\rho(n)$ can't be assumed to act like ordinary numbers. –  Charles Nov 28 '12 at 18:24


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# Why is the interpolation between two connections related via a gauge transformation still a connection?

+ 1 like - 0 dislike
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I am studying the theory of anomalies in gauge field.

Let $A$ be a gauge field (or a connection for mathematicians). Let $A_{U}$ be an equivalent gauge related via a local gauge transformation
$$A_{U}=U^{-1}dU+U^{-1}AU$$
Then, there is a interpolating between the two
$$A(s)=sA+(1-s)A_{U}$$
where $s\in[0,1]$.

Why is $A(s)$ a connection? If $A$ is a flat connection, then this interpolation $A(s)$ between two flat connections isn't even flat. What are the geometric and physical meaning of such an interpolation?

Closed by author request


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# Coincidence set closed equivalent to diagonal closed

Prove

For any topological space $Y$ and any continuous maps $f, g : Y → X$, the set $\{y ∈ Y : f(y) = g(y)\}$ is closed in $Y$

is equivalent to

The diagonal $∆ = \{(x, x) : x ∈ X\}$ is a closed subset of $X × X$, in the product topology.

I've proved that the diagonal being closed is equivalent to $X$ being Hausdorff, and also proved that the diagonal closed implies the coincidence set is closed but can't do the reverse implication, any hints? Have tried to show the complement is open, ie there is an open neighbourhood about each point, but can't seem to find a useful way to use the coincidence set being closed. The functions $f, g$ seem to be mapping the 'wrong way' as it were, so not sure how their continuity can be used.

Note that $\Delta$ is the coincidence set of the two projections from $X\times X$ to $X$. The projections are $p_1 : X\times X\to X,\;(x,x')\mapsto x$ and $p_2 : X\times X\to X,\;(x,x')\mapsto x'$.

Indeed, it's the set of points $(x,x')$ s.t. $x=x'$...

Thus the diagonal is closed.

Consider the map $p_1:X\times X\rightarrow X$ given by $(x,y)\mapsto x$ and $p_2:X\times X\rightarrow X$ given by $(x,y)\mapsto y$.

$p_1$ and $p_2$ are continuous maps and $\Delta=\{(x,y)\in X\times X\mid p_1=p_2\}$.

Consider $\pi_1:X_1$x$X_2 \rightarrow X_1$ and $\pi_2:X_1$x$X_2 \rightarrow X_2$ where $X_1 = X = X_2$, $\forall i \in \left\{1, 2\right\}$, $\pi_i$ is the natural projection, i.e. $\pi_i(x_1, x_2) = x_i$.

We know that $\pi_i$ is continue $\forall i \in \left\{1, 2\right\}$, and $\Delta(X)$ is the coincidence set of $\pi_1$ and $\pi_2$.

Now we use the hypothesis and we have the thesis.


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# Cool Quadratic

Algebra Level 3

Find the sum of all values of $$x$$ which satisfy the equation $$\sqrt{x^2+1} + x^2+1=90$$.

###### Source: $$MA\Theta$$ 1991.
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# Solving Initial Value problem ignoring the time-derivative

I am looking at a heat initial value problem \begin{align} \frac{\partial u}{\partial t}-\nabla^2u = f\quad&\text{in}\quad \Omega\times(0,T)\\ u = g \quad&\text{on}\quad \partial\Omega\times(0,T)\\ u = u_0\quad&\text{in}\quad \Omega\times\{0\} \end{align} which I have succesfully solved with the finite element method along with a time-stepping scheme, so that is not my question. My question is however, it seems to work to just solve the boundary value problem \begin{align} -\nabla^2u = f\quad&\text{in}\quad \Omega\\ u = g \quad&\text{on}\quad \partial\Omega\\ \end{align} for various fixed times, e.g. $f(x,y,0),f(x,y,0.1),$ etc. This seems to give a good approximation of the solution to the initial value problem, and I don't understand why. This simply ignores the time-derivative it seems to me (of course, the functions $f$ are the same for both problems so the "information" of the time-derivative is not lost, but it still confuses me). Is this simply the method of lines?

Thanks

• If you remove the time dependence in the boundary conditions, how quickly does $u(x,y,t)$ approach a stead state solution? It could be that this time scale is much shorter than the longer time scale of changes in the boundary conditions. – Brian Borchers Jan 28 '15 at 20:29
• @BrianBorchers I don't quite understand what you mean. It could be that $u(x,y,t)$ does not converge to anything steady as $t\to\infty$ for example for $u(x,y,t) = \sin(2\pi x t)\cos(2\pi y)$. – Eff Jan 28 '15 at 20:33
• @Elf. Yes of course. What I'm asking is if for your particular problem the solution does reach a steady state very rapidly compared to changes in the boundary conditions. e.g. if for any reasonable initial condition, the time dependent equation effectively reachies a steady state in 1.0e-15 seconds, while the boundary conditions are changing slowly on a time scale of 0.1 seconds, then you'd see the behavior you've described. – Brian Borchers Jan 28 '15 at 22:07

If $f$ and $g$ are functions that depend on the spatial variable, but not on time, then $u(x,t)$ converges to a function $\bar u(x)$ that satisfies the Poisson equation $$-\Delta u=f,$$ together with $u|_{\partial\Omega}=g$. It does so regardless of the initial conditions. In other words, the solution of the heat equation is, in a sense, not very interesting if you consider the "long-term" behavior.
• Thank you for the answer and for giving me something new to study (the last paragraph). I have generally (for testing) chosen a solution a priori, e.g. $u(x,y,t) = \text{exp}(-100(x-t)^2-100(y-0.5)^2)$, and then computed $f$ from the known solution. So it will depend on time. – Eff Jan 29 '15 at 5:34


text:
# The Maxwell operator with periodic coefficients in a cylinder

@article{Filonov2018TheMO,
title={The Maxwell operator with periodic coefficients in a cylinder},
author={Nikolai Filonov and Andrei Prokhorov},
journal={St. Petersburg Mathematical Journal},
year={2018}
}
• Published 31 January 2018
• Mathematics
• St. Petersburg Mathematical Journal
In the paper we consider the Maxwell operator in a three-dimensional cylinder with coefficients periodic along the axis of a cylinder. It is proved that for cylinders with circular and rectangular cross-section the spectrum of the Maxwell operator is absolutely continuous.
3 Citations
• N. Filonov
• Computer Science
St. Petersburg Mathematical Journal
• 2022
The Schrödinger operator in the Euclidean space whose potential is periodic in some variables and decays in the remaining variables faster than power is proved.

## References

SHOWING 1-10 OF 10 REFERENCES

We prove that the spectrum of Maxwell operator, (B,E)⟼(−rot(E),a(x)rot(b(x)B)), with non-negative and periodic functions a(x) and b(x), is absolutely continuous.
• Mathematics
• 2005
The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to
We consider the periodic Schrödinger operator on a d-dimensional cylinder with rectangular section. The electric potential may contain a singular component of the form σ(x, y)δΣ(x,y), where Σ is a
An example of the stationary Maxwell system, which has a nontrivial smooth solution with a compact support is given; the coefficients ε and μ belong to Cα for all α < 1. Our example shows that the
• Mathematics
• 2015
In this paper, a “strong” Maxwell operator defined on fields from the Sobolev space W21, and a “weak” Maxwell operator defined on the natural domain are considered. It is shown that in a convex
AbstractThe Maxwell operator in a layer $${\mathbb{R}}^2 \times \left( {0,T} \right)$$ is studied. It is assumed that the electric permittivity μ(x) and the magnetic permeability ε(x)are periodic
• Mathematics
• 2006
We give a simple proof of the absence of the singular continuous component in spectra of self-adjoint operators that are representable as analytic direct integrals. Bibliography: 7 titles.
CONTENTSIntroduction Chapter I. Preparatory results ??1. The cokernel of a Fredholm morphism in spaces of sections ??2. The cokernel of a Fredholm morphism in spaces of sections with bounds Chapter

• Anal. Appl
• 2013

### The selfadjoint Maxwell operator in arbitrary domains

• Algebra i Analiz
• 1989


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# Drawings verses specification

## Drawings verses specification

(OP)
If there is a conflict between the drawing and the specification. Which do you use?

### RE: Drawings verses specification

That shoulkd be addressed by the project specs.  Normally it goes... Contract, Specifications, Drawings... if descending order of priority.

Joe Tank

### RE: Drawings verses specification

"The plans, specifications, shop drawings and other contract documents are intended to completely describe the work. Conflicts among them easily can, and do, arise. Many specification writers attempt to resolve these in advance by declaring an order of precedence among them. Typically, this may read something like this:

"If there is a conflict between contract documents, the document highest in precedence shall control. The precedence shall be: first, permits from other agencies as may be required by law, second , Special Provisions, third, Plans, fourth……………………………………. Sixth, reference specifications."

Recently the trend has been away from such specific requirements for one simple reason. We do not know, before they are revealed, what problems will arise. Declaring a specific order presumes we know how best to solve a problem, before we know what the problem is!

Consider using a different approach. The following has been found to work well on many contracts and is suggested.

"If a conflict, error, omission, or lack of detailed description is discovered in the contract documents, the Contractor shall immediately notify the Engineer (Architect) and request clarification. The Engineer (Architect) will resolve the conflict and make any corrections or interpretations necessary to fulfill the intent of the plans and specifications."

good luck

### RE: Drawings verses specification

I agree that somewhere in the contract documents an order or precedence be clearly identified when conflict exists between drawings and specs.  Better way is to define the order of precedence AND have a contact-the-engineer/architect note RWF7437 suggests.  Alternately, one can specify that when conflict is encountered a more restrictive requirement shall govern.

### RE: Drawings verses specification

whyun's advice is good.  The contractor should be required to request clarification.  As to the precedence, I think the drawings should come first as they should always be job specific, with the specifications unfortunately sometimes boilerplate.

When the *&%#@$hits the fan, the attorney's always seem to look to the written word, whether notes on the plan or written specifications. I always try to put a couple notes about conflict & notification on the plans. ### RE: Drawings verses specification Courts will generally accept a reasonable interpretation of the drawings by the party who did not draft the documents. If you have the clause that the engineer interprets, that is probably best. However, generally the contractor would be entitled to a change order if cost or time is affected. I don't like the most restrictive clause. First, it is ocassionly subjective as to which answer is most restrictive, second, it really does not look at what is trying to being accomplished. Generally the specs take precidence. Personally I would like to see the drawings have precidence, because as stated before, they are project specific and get a lot more review than the specs. What would be best would be for the engineer to point out the descrepancy to the engineer, and if the drawing is wrong, then dicuss a change. ### RE: Drawings verses specification .. The question is in practice how to build or construct to the correct solution, and avoid later technical problems and/or conflicts costing time and mony. As suggested by others, one way is to determine the 'rank' of the differentiating underlaying speciifcations. An other is, also as suggested, to specify the problem as 'a minor' technical problem within the contract specification to be solved 'in normal good engineering practice' by the contractor. The best way, in my experience, is to clear any possible misunderstandings immideately, by communicating directly in writing with the counterparts top project responsible person. Not saying that this solution is the best in all cases a formulation something like the one below has worked for me in several cases, if the question cant be solved by a simple mail or telephone conversation: Urgent! Production stopped awaiting your clarification/answer on following: Contract (paper one) stating:xxxx, Drawing (paper two) stating yyyy. Please clarify within zzzz. Consequences of delayed answer: (Production time will be prolonged aaaa, cost will be increased with bbbb / We reserve the right to require compensation for any cost or consequences / etc) This will usually give a prompt response.. ### RE: Drawings verses specification That approach may get a prompt response, but it does not benefit the project in the long run, because threats are always remembered. Just human nature. ### RE: Drawings verses specification when free and open communication is encouraged and facilitated, these issues usually get resolved well, regardless of what language is shown on the plans or specs. that is the goal of partnering. however, partnering is not always a priority and not all contractors, engineers or owners are good communicators. they may also not fully read the plans or specifications. This is where the problems begin. Sometimes the resident engineer is different than the design engineer and in these cases sometimes think they know everything. of course, they did not design it and probably have not read the entire spec. unless it is a really simple project, they can't possibly fully understand the project. ### RE: Drawings verses specification i always tool it as drawing overides spec. As a) drawing is more likely to be revised most recently b) the contractor cannot be expected to look at every drawing and compare with every clause in the spec. Would think it extremely harsh if the contractor is punished in court for following a drawing which if the engineer has carried out his duty properly should be correct. ### RE: Drawings verses specification People on the design side (engineers) tend to assume drawings should take precedence over specifications. When I used to work for a multi-discipline firm, the in-house spec writer was insistent that specs take precedence over drawings. For the time I worked for that firm, I took special care in reviewing the specs to ensure no conflict exists between my drawings and the specs. ### RE: Drawings verses specification Specs always rule from my experience.......... ### RE: Drawings verses specification I worked for uk contractor and drawing was always taken to overrule spec. and was taen that way by our sub contractors and main contractors. Remember the phrase clearly 'drawing overrules spec' being said to me many times. Should be flaged obviously, but would think if you built to drawings and then the designer tried to get you on some obscure spec clause. Think they'd be struggling to prove you liable. Particarly as they'd be admitting their error. Look at it this way, if you were designing something and made it structurally (or whatever your field is) sound and something in a spec that may change your assumptions or design and the contractor builds to that? You can build off drawings, cant build of specs. ### RE: Drawings verses specification A picture is worth 1000 words. Drawings....ALWAYS. Fine, as in its a fine day Fine, as in pay the traffic ticket. You shouldn't have to rely on context when building something. The language of engineering and design is drawings. The language of spec writters and lawyers is (@*%&(*$^#!^\$&^

http://virtualpipeline.spaces.msn.com

"What gets us into trouble is not what we don't know, its what we know for sure" - Mark Twain

### RE: Drawings verses specification

For us, the design drawings take precedence over standard specifications (and this is spelled out in our specs).  Shop drawings, on the other hand, take precedence over nothing in particular, and if there's any doubt, ask.  (The EOR's approval stamp on the shop drawings is very large because it contains long disclaimers that basically translate into "just because I approved this drawing doesn't mean you can assume it's right".)

Hg

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### RE: Drawings verses specification

Lately, Drawings and Specifications are equal on the list.  A statement will be inserted stating where a conflict between the drawings and specifications occur, whichever is of greater value will take precedent.  A tough statement in a competitive market where one is trying to be the low bidder.

Drawings are job specific and specifications tend to be boilerplate.  On some projects, drawings say a certain material is acceptable as structural fill, but the specs say otherwise.  These specs have the same verbage used on thirty or more other projects reviewed and have not been tailored to the project at hand.  I would vote drawings over specs. if I had control over the priority list.

Clarification by the contractor prior to bidding is ideal; however, sometimes questions do not arise until the day before or the day of bidding.  This may not allow for clarification prior to bid submission. Therefore, it is important to emphasize unsusual conditions in the documents and be consistent between plans and specifications.  A costly dispute because of inconsistency in documents will cause a great project to become a nightmare.

### RE: Drawings verses specification

A big part of the problem is that specifications today tend to say how to accomplish something rather than what they are really intended to do, which is stating what an acceptable result should be.

http://virtualpipeline.spaces.msn.com

"What gets us into trouble is not what we don't know, its what we know for sure" - Mark Twain

### RE: Drawings verses specification

The last 4 large contracts I have been involved with has specifications over drawings and both over BOQ description.  Interestingly one party wanted to follow the BOQ rather than the spec and drawings.

### RE: Drawings verses specification

Biginch
I'm sure there are some strong opinions both ways on this but I think there is a place for method specs.  Owners have realized over the years that certain methods result in acceptable quality products and others don't.  They have been burned in the past by contractors saying - "we can do it this easier way and still get good quality".  The end result doesn't always meet expectations.  Then there is the resulting fight at the end of the project about whether it is good enough to be considered acceptable, whether it should be reconstructed, whether the contractor should be short payed for poor quality.  Or perhaps the inspector thinks it is good enough and later on it isn't.

I would rather see the specs (or drawings) list the "unacceptable methods".  Example - many specs disallow "jetting" to densify pipe zone backfill.  While in some parts of the country, it is routinely used and successful.

### RE: Drawings verses specification

I would think those things would be included on something called "assembly instructions", not specs.

Yes that's exactly what I am complaining about, when I have my design engineer hat on.  When I have a construction contractor hat on, I'm always happy to see your kind of specs, because if I do it that way and it doesn't work, I'm off the hook completely and you're in for extras.

http://virtualpipeline.spaces.msn.com

"What gets us into trouble is not what we don't know, its what we know for sure" - Mark Twain

### RE: Drawings verses specification

well, that would be true if "my kind of specs" had no requirements for the quality of the end product.  First, these  specs are generally standards in the industry which we are required to use.  Second, none that I am aware of are strictly method based only.  They all have requirements for the end product as well.  If I could trust a contractor to know how to do everything, I would simply specify the acceptance criteria and leave it at that.  However, I have spent enough time out in the field to see that the low bidder on the job doesn't always hire the sharpest pencil in the box to manage the project.  It starts with the estimator and goes all the way down to the foreman and lead man on the job.  They all contribute to the construction project and frankly, often need to be told not only what you (the engineer) want, but how you want it done.

### RE: Drawings verses specification

I am neither an expert in construction contracts nor case law involving same (and have not as far as I know even been involved in such disputes), however, I believe it probably behooves all to do what they can to minimize conflicts/errors and/or at the very least VERY clearly spell out what is to be done when they occur.  There apparently even exists a “rule of contra proferentem”, that is explained e.g. in the USA decision at http://www.ll.georgetown.edu/federal/judicial/fed/opinions/03opinions/03-5055.html (and the also mentioned in many other places including the reference at http://caselaw.lp.findlaw.com/cgi-bin/getcase.pl?court=Fed&navby=case&;no=035055 ) as follows,

“When a dispute arises as to the interpretation of a contract and the contractor's interpretation of the contract is reasonable, we apply the rule of contra proferentem, which requires that ambiguous or unclear terms that are subject to more than one reasonable interpretation be construed against the party who drafted the document. United States v. Turner Constr. Co., 819 F.2d 283, 286 (Fed. Cir. 1987). However, the court will consider whether the ambiguity or lack of clarity was sufficiently apparent that there arose an obligation on the contractor to inquire as to that provision before entering into the contract.”

But of course there is still at least the “consider”ation of “whether the ambiguity or lack of clarity was sufficiently apparent that there arose an obligation on the contractor to inquire as to that provision before entering into the contract” (and there was of course also a minority dissenting opinion).  I have no idea what would constitute “sufficiently apparent” per this rule, but this phrase at least on the surface sounds quite subjective and may put multiple parties at some risk if things were to get ugly enough to go to court.  However, I suspect not allowing a contractor (or his vendors, subs, etc.) much time in the process to detect and have re solved all conflicts or ambiguities (as I think already mentioned on this thread), and maybe particularly if finding/analyzing what is right or wrong might involve quite complex or time-consuming engineering calculations or testing/determinations etc., I think it is quite possible this might not weigh very heavily in favor of the drafter!

Unfortunately compressed time schedules particularly along with a low bid or price stipulations or mentalities may arguably make it quite difficult to obtain the most consistent and best quality product, whether the product is engineering contracts/services, construction, or building materials for same etc.!

### RE: Drawings verses specification

The EJCDC*, document 00700 - General Conditions of the Construction Contract, does not establish a hierarchy of contract documents. Paragraph 3.01 reads: "The Contract Documents are complementary; what is required gby one is as binding as if required by all."  Paragraph 3.02 futher states: "It is the intent of the contract documents to describe a functionally complete Project...to be constructed in accordance with the Contract Documents...."

Concerning conflict between documents, Paragraph 3.03 describes a process for doing so, placing the burden on the Contractor for reporting any discrepencies found to the Engineer, and the discrepency is worked out according to the needs of the project.

I have always conducted my practice that NEITHER the drawings or the specifications govern. If a discrepency is discovered, we work it out according to the needs of the project. The only thing the EJCDC general conditions states concerning hierachy of documents is that the Contract Documents have precedence over any referenced standard.

Best Regards,
NDG

*Engineers Joint Contract Documents Committee

### RE: Drawings verses specification

So if thee is a dicrepency in the contract documents, how are the contractors claims for additional time and or cost handled?

### RE: Drawings verses specification

In Public Works, if party "A" develops the Contract documents and there is a conflict, the interpretation belongs to party "B" of the contract.  This does not mean party "A" cannot designate an alternative interpretation,  but party "B" will be entitled to additional compensation if it can be shown Extra Work is required above and beyond the interpretation originally assumed by party "B" in the bid.

### RE: Drawings verses specification

Lcruiser,

That is an incorrect interpretation. In Public Works Contracts, the Contract language is very specific. In fact, the Contract language is onerous and the Contractor does not have much in his favor. The Contractor basically signs away his rights when he bids the public project.

See the Contractor's representations in Article 8 that the Contractor signs to induce the Owner to enter into the Contract:

"I. Contractor has given Engineer written notice of all conflicts, errors, ambiguities, or discrepancies that Contractor has discovered in the Contract Documents, and the written resolution thereof by Engineer is acceptable to Contractor."

3.03 Reporting and Resolving Discrepancies
A. Reporting Discrepancies
1. Contractor's Review of Contract Documents Before Starting Work:
Before undertaking each part of the Work, Contractor shall carefully study and compare the Contract Documents and check and verify pertinent figures therein and all applicable field measurements. Contractor shall promptly report in writing to Engineer any conflict, error, ambiguity, or discrepancy which Contractor may discover and shall obtain a written interpretation or clarification from Engineer before proceeding with any Work affected thereby.
2. Contractor's Review of Contract Documents During Performance of Work:
If, during the performance of the Work, Contractor discovers any conflict, error, ambiguity, or discrepancy within the Contract Documents or between the Contract Documents and any provision of any Law or Regulation applicable to the performance of the Work or of any standard, specification, manual or code, or of any instruction of any Supplier, Contractor shall promptly report it to Engineer in writing. Contractor shall not proceed with the Work affected thereby (except in an emergency as required by Paragraph 6.16.A) until an amendment or supplement to the Contract Documents has been issued by one of the methods indicated in Paragraph 3.04.
3. Contractor shall not be liable to Owner or Engineer for failure to report any conflict, error, ambiguity, or
discrepancy in the Contract Documents unless Contractor knew or reasonably should have known thereof.
B. Resolving Discrepancies
1. Except as may be otherwise specifically stated in the Contract Documents, the provisions of the Contract Documents shall take precedence in resolving any conflict, error, ambiguity, or discrepancy between the provisions of the Contract Documents and:
a. the provisions of any standard, specification, manual, code, or instruction (whether or not specifically incorporated by reference in the Contract Documents); or
b. the provisions of any Laws or Regulations applicable to the performance of the Work (unless such an interpretation of the provisions of the Contract Documents would result in violation of such Law or Regulation).

http://dnrc.mt.gov/cardd/ResDevBureau/wasact/docs/Funding_Agency_Standard_General_Conditions.pdf

### RE: Drawings verses specification

bimr -

Exactly my point.  See your 3.:

"Contractor shall not be liable to Owner or Engineer for failure to report any conflict, error, ambiguity, or discrepancy in the Contract Documents ..."

Like I said, the Owner can make the Contractor do it their way (within the scope of the Contract) but they don't have to do it for free.

Anything can be done.  All it takes is money.

### RE: Drawings verses specification

One of the problems is design consultants write a specification when all that is needed is a data sheet referencing the performance requirements, applicable standards and any special requirements.

Consultants get paid by the hour for bums on seats. Mosty of their specifications are full of motherhood statements with no validity at law as they are so vague e.g. "shall be in accordance with good engineering practice...' What a crock but it appears all the time.

A specification written based on a particular suppliers product catalogue demonstrates to me the author knows less than the supplier about the product. Yet it is seen commonly in requests for quotation.

That said in my experience the order of precedence is normally specification, data sheets then drawings. If cluases in respect of discrepancies are too one sided the court will generally ignore them. Drawings cannot in themselves be relied upon as they do not necessarily reference standards , or more importantly the options in a particular standard. Unless drawings go to detail design then the overall requirements will not be defined thereon.

Design and construct contracts require different documentation than say a construct only , a fabricate or an EPCM type of contract.

This is why the project engineers and project managers get paid the big bucks. They create a minefield that they in turn have to sought out. Nearest thing to a perpetual motion machine.

### RE: Drawings verses specification

LCruiser (Civil/Environme)

You left off the important part:
"unless Contractor knew or reasonably should have known thereof."

The interpretation is that:
Party "A" develops the Contract documents and if there is a conflict, the interpretation is done by party "A" not party "B" (the Contractor).

### RE: Drawings verses specification

True, if the Contractor knew.  As far as "reasonably should have known", how could the Contractor "reasonably should have known" in his brief time period of developing the bid when the Owner's Engineer didn't know in the months or years he was developing the project?

Like I said, the interpretation can be (and in some cases necessarily must be) that of the party writing the Contract, and usually there are provisions to enforce that - but party "B" - the Contractor in this case (unless it's a design build) is entitled to additional monies.

### RE: Drawings verses specification

I don't think that we are that far apart. But, public works contracts are not friendly to Contractor changes because of the onerous Contract conditions such as the Contractor's Representations. As you state, the onus is on the Contractor to prove his points.

Article 8. CONTRACTOR' S REPRESENTATIONS

8.01    In order to induce OWNER to enter into this Agreement, CONTRACTOR makes the following representations:

A.    CONTRACTOR has familiarized itself with the nature and extent of the Contract Documents, Work, site, locality, and all local conditions and Laws and Regulations that in any manner may affect cost, progress, performance or furnishing of the Work.

B.    CONTRACTOR has studied carefully all reports of explorations and tests of subsurface conditions and drawings of physical conditions which are identified in the Supplementary Conditions as provided in paragraph 4.02 of the General Conditions, and accepts the determination set forth in paragraph SC-4.02 of the Supplementary Conditions of the extent of the technical data contained in such reports and drawings upon which CONTRACTOR is entitled to rely.

C.    CONTRACTOR has obtained and carefully studied (or assumes responsibility for obtaining and carefully studying) all such examinations, investigations, explorations, tests, reports and studies (in addition to or to supplement those referred to in paragraph 8.01 B. above) which pertain to the subsurface or physical conditions at or contiguous to the site or otherwise may affect the cost, progress, performance or furnishing of the Work as CONTRACTOR considers necessary for the performance or furnishing of the Work at the Contract Price, within the Contract Time, and in accordance with the other terms and conditions of the Contract Documents, including specifically the provisions of paragraph 4.02 of the General Conditions; and no additional examinations, investigations, explorations, tests, reports, studies, or similar information or data are or will be required by CONTRACTOR For such purposes.

D.    CONTRACTOR has reviewed and checked all information and data shown or indicated on the Contract Documents with respect to existing Underground Facilities at or contiguous to the site and assumes responsibility for the accurate location of said Underground Facilities. No additional examinations, investigations, explorations, tests, reports, studies or similar information or data in respect of said Underground Facilities are or will be required by CONTRACTOR in order to perform and furnish the Work at the Contract Price, within the Contract Time, and in accordance with the other terms and conditions of the Contract Documents, including specifically the provisions of paragraph 4.04 of the General Conditions.

E.    CONTRACTOR has correlated the results of all such observations, examinations, investigations, explorations, tests, reports, and studies with the terms and conditions of the Contract Documents.

F.    CONTRACTOR has given ENGINEER written notice of all conflicts, errors or discrepancies that he has discovered in the Contract Documents and the written resolution thereof by ENGINEER is acceptable to CONTRACTOR.

### RE: Drawings verses specification

In the way of doing it that I described above, if there is a discrepency, you work it out according to the needs of the project. If the Contract Time and Contract Price are impacted, either Owner or Contractor can request a contract Modification (i.e. change order).

Let's take a hypothetical. The Drawings state that a certain embankment should be compacted to 98 percent of Modified Proctor. The Specifications say compaction of embankments should be to 95 percent Standard Proctor. Which is correct? Only the design engineer knows for sure. If you say "Specs govern over drawings", and it really should be to the higher compaction level, then Contractor probably has a claim for extra costs, because he can legitimately claim he put his bid together believing the specs governed--because the Contract Documents said the specs governed. If you say "Drawings govern over Specs", but really the job only needs the lesser compaction, try getting a price reduction from Contractor!

So in this case, you need to work it out. If the Contract Documents are complementary...etc., as I quoted above, and the discrepency is not noted until compaction has begun, either Contractor or Owner might have a claim for a change in Price, but in reality it will probably not make a big different--unless you are talking about tens of thousands of yards of embankment.

bimr: Please say what document you are quoting from. Is this a paragraph in the Instructions To Bidders? If so, what base document did it come from? AIA? EJCDC? Something other, based on what your firm has been using for years?

### RE: Drawings verses specification

There is a common law argument of contra preferentum (not sure of the Latin exactitude so apologies to any learned scholars??), whereby the most onerous conditions are held against the party that created the documents containing the conflict. Wins hands down in most contractual conflicts. SO if there is no precedence of documents and there is the compaction difference mentioned in the above the contractor gets to make a legitimate claim for time and money. If the project engineer has been smart there are schedules for additional work that have been determined during the bid process.

That is why lawyers developed the concept of precedence of documents.

The contractor still has to demonstrate the value of any claim.

There has been many a strategy when bidding a job where a smart contractor has noticed discrepancies to play the ambush and claim game. It generally only works once and you soon lose clients as your reputation spreads.

Governments are generally susceptible to such strategies as they accept tenders from low bidders and their project engineers are generally woeful when preparing documents. They write too much in the documents, using numerous authors and are bound to make mistakes. Ever noticed a government project that did not have a budget blow out?

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# The average of a, b, and c is 8, and the average of a and b is 10. If

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The average of a, b, and c is 8, and the average of a and b is 10. If [#permalink]

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27 Dec 2017, 23:25
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The average of a, b, and c is 8, and the average of a and b is 10. If the average of a and c is 6, what is the value of a – c ?

A. 8
B. 7
C. 6
D. 5
E. 4
[Reveal] Spoiler: OA

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Re: The average of a, b, and c is 8, and the average of a and b is 10. If [#permalink]

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28 Dec 2017, 03:13
Bunuel wrote:
The average of a, b, and c is 8, and the average of a and b is 10. If the average of a and c is 6, what is the value of a – c ?

A. 8
B. 7
C. 6
D. 5
E. 4

We'll start by writing by writing all our information as equations and then simplify them.
This is a Precise approach.

(a+b+c)/3 = 8 --> a+b+c=24
(a+b)/2 = 10 --> a+b=20
(a+c)/2 = 6 --> a+c=12

We'll subtract the first two equations to get c = 4
and substitute c=4 into our third equation to get a = 8
then a - c = 4

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The average of a, b, and c is 8, and the average of a and b is 10. If [#permalink]

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28 Dec 2017, 04:36
Bunuel wrote:
The average of a, b, and c is 8, and the average of a and b is 10. If the average of a and c is 6, what is the value of a – c ?

A. 8
B. 7
C. 6
D. 5
E. 4

Substituting twice.

$$\frac{a + b + c}{3} = 8$$
$$a + b + c = 24$$

$$\frac{a + b}{2} = 10$$
$$a + b = 20$$

$$20 + c = 24$$
$$c = 4$$

$$\frac{a + c}{2} = 6$$
$$a + c = 12$$
$$c = 4$$
$$a = 8$$

$$(a - c) = (8 - 4) = 4$$

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Re: The average of a, b, and c is 8, and the average of a and b is 10. If [#permalink]

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28 Dec 2017, 04:54
a+b+c/3 =8 ==> a+b+c= 24
add a+b/2 and a+c/2 => 2a+b+c= 32=> a+24=32=>a= 8
8+c/2=6=>c=4
a-c=4

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Re: The average of a, b, and c is 8, and the average of a and b is 10. If   [#permalink] 28 Dec 2017, 04:54
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# Model for regression with multiple binomial predictors?

#### Junes

##### New Member
Hi everyone,

I was wondering what model to pick when all the predictors are binomial distributions.

I have a website that's being visited through several (exclusive) channels. I know how many people enter a page from a certain channel. Then, they can go on to purchase and payment. However, not everyone makes it out to the end; most (95%) exit. I know how many people make a purchase in total but noy by channel, and I want to know the contribution of each channel.

The numbers of purchases are too low to use linear regression, so I want to use some count model. The only thing I can find is negative binomial regression, eg package pscl in R. However, then I'm modelling the exits instead of the visits and that seems to be a rather round-about way. This is how I model the exits right now:

Code:
df$abandon <- df$buy_pag_visits- df$buyers model.nb <- glm.nb(abandon ~ channel1 + channel2 + channel3 , link=sqrt) #sqrt seems to work better than log When I search for "binomial regression" I get stuff on logistic regression, where the outcome is binary. That's not what I want. Is there such a thing as "binomial regression" instead of "negative binomial regression"? It seems like such a logical thing but the only thing I see is the negative variant. By the way, I tried approximating by linear regression. It works okayish, but it has obvious problems with the many zeroes. And some of the coefficients become negative, which is obviously impossible. Last edited: #### Junes ##### New Member Another follow-up question: I have reason to believe that the p of one of the binomial distributions has undergone a sudden shift as a result of my actions. How would I model this? I'm a bit lost here. #### hlsmith ##### Omega Contributor A bunch of binary variables predicting purchase yes/no would be logistic regression. You are saying count, but each encounter can only have a single outcome, so you have a series of Bernoulli trials (if independent), then logistic regression. If the trials are not independent, some of the encounters are for the same people at different times, can you track this, and if so, you possibly have multilevel logistic regression, with purchase status clustered in person. If you can convince me this is count data, I can tell you about those options. Though, given your description, I don't think this is the case. P.S., Can each purchase only have one channel, if so, you may not need to use any sophisticated methods. #### Jake ##### Cookie Scientist Do you really mean that the predictors are "binomial", i.e., they follow a binomial distribution and thus are counts from 1 to N? Or do you instead mean that they are "binary" or "categorical" or something else? If they are truly "binomial" then it's really not clear that anything special needs to be done. If they are categorical then they just need to be appropriately dummy- or contrast-coded. It works okayish, but it has obvious problems with the many zeroes. And some of the coefficients become negative, which is obviously impossible. Obvious problems? Why? What problems? And why do you think the coefficients can't be negative just because the predictors are binomial (or binary or categorical or whatever they are)? #### Junes ##### New Member Thanks for the answers, guys. A bunch of binary variables predicting purchase yes/no would be logistic regression. Yes, but I don't have 0/1 as my outcome. This is a snippet of my data: Code: date byr ch1 ch2 ch3 ch4 ch5 ch6 ch7 ch8 ch9 10-12-16 0 48 13 3 0 7 17 0 2 0 10-13-16 0 50 16 3 0 6 13 0 0 3 10-14-16 0 70 29 2 0 9 23 0 0 1 10-15-16 1 82 44 2 0 6 24 0 2 1 10-16-16 11 136 78 5 0 14 32 0 0 3 10-17-16 4 125 49 1 0 9 24 0 0 3 10-18-16 4 108 52 2 0 9 20 0 0 1 10-19-16 2 124 63 2 0 7 21 0 1 4 10-20-16 5 120 55 6 0 13 26 0 0 4 10-21-16 14 137 66 11 0 3 28 0 1 3 byr = my dependent variable (no of buyers). Furthermore, I have 57% zeroes in my dependent variable. It is very right-skewed. I want to predict buyers by modelling the channels as B(n, p), where the p for each channel is constant over days and to be determined, and n is the number in the table. You are saying count, but each encounter can only have a single outcome, so you have a series of Bernoulli trials (if independent), then logistic regression. If the trials are not independent, some of the encounters are for the same people at different times, can you track this, and if so, you possibly have multilevel logistic regression, with purchase status clustered in person. They are independent, thankfully. Well probably not 100%, but close enough. P.S., Can each purchase only have one channel, if so, you may not need to use any sophisticated methods. Yes, pretty much. There is some cross-talk, not much. Jake said: Do you really mean that the predictors are "binomial", i.e., they follow a binomial distribution and thus are counts from 1 to N? Or do you instead mean that they are "binary" or "categorical" or something else? Yes, I think I should model them as binomial variables. I'm assuming a fixed probability of each user converting for each channel. I have n users on a given day. That sounds to me like B(n, p), but I'm not 100% sure. Obvious problems? Why? What problems? And why do you think the coefficients can't be negative just because the predictors are binomial (or binary or categorical or whatever they are)? Oh, sorry, I was talking about linear regression there. I did a linear regression and it did reasonably well, but it had trouble with the edge cases. If coefficients are negative that means conceptually that a channel leads to a loss of sales, which is not possible. So, a linear regression can only be a rough approximation, I think. Last edited: #### hlsmith ##### Omega Contributor Perhaps you also need to present a case example for an observation - to help us understand the data generating function. Is Y a count variable because you have meta-data for the day. So it is actually single purchased y/n, but what you have is all of them for the day? Can you link actual channels to the outcome or do you have counts for outcome and channels? I still need a little more info, but zero inflated Poisson regression may be an option. #### Jake ##### Cookie Scientist Basically I still don't see what the problem is with a model like Code: glm.nb(byr ~ channel1 + channel2 + channel3) Junes said: Is there such a thing as "binomial regression" instead of "negative binomial regression"? Yes. Binomial regression is regression where the outcome is binomial. So two examples of binomial regression are logistic regression and probit regression. Which are just binomial regression with logit or probit link functions, respectively. Note that virtually all non-Normal variants of regression refer to the distribution of the outcome, not the predictor. Because regression already doesn't make distributional assumptions about the predictors, we don't generally need exotic forms of regression for handling predictors with particular types of distributions. This is partly why I'm a bit skeptical that you really need what you think you need. Now because your outcome is a count with low values, it makes sense that negative binomial regression would be appropriate. But of course this has nothing to do with the predictors. The question is whether, in addition to the negative binomial assumption for the outcome, you also need to do something special because of the distributions of your predictors. Junes said: If coefficients are negative that means conceptually that a channel leads to a loss of sales, which is not possible. Why not? It seems easy to imagine that there is some unmeasured confounding factor that, when active on a particular day, leads respondents to be more likely to visit the site through a particular channel, but that also makes respondents less likely to ultimately buy something. I don't see why a channel having a negative coefficient is a logical impossibility. Also, note that even if the simple/total effect of a channel really is impossible to be negative (although I'm not really convinced), that doesn't preclude the multiple regression coefficient from being negative, since that estimates the effect while holding constant the other predictors. Junes said: I want to predict buyers by modelling the channels as B(n, p), where the p for each channel is constant over days and to be determined, and n is the number in the table. This doesn't really make sense to me. You want to predict buyers by modeling something other than buyers? Suppose you did apply some binomial model to your predictors. How then are you wanting to use those binomial models to predict buyers? Do you want to then use those estimated values of $$n$$ and/or $$p$$ as predictors in the negative binomial regression of buyers? Junes said: Yes, I think I should model them as binomial variables. I'm assuming a fixed probability of each user converting for each channel. I have n users on a given day. That sounds to me like B(n, p), but I'm not 100% sure. What does "converting" mean here? #### Junes ##### New Member Thanks again for the replies. This is helping me clarifying my thinking so much. Jake said: What does "converting" mean here? Sorry, marketing speak. "To convert" is for a user to perform an action of value to the website owner. Hence, you have conversions, conversion rate, etc. Note that virtually all non-Normal variants of regression refer to the distribution of the outcome, not the predictor. Because regression already doesn't make distributional assumptions about the predictors, we don't generally need exotic forms of regression for handling predictors with particular types of distributions. This is partly why I'm a bit skeptical that you really need what you think you need. Now because your outcome is a count with low values, it makes sense that negative binomial regression would be appropriate. But of course this has nothing to do with the predictors. The question is whether, in addition to the negative binomial assumption for the outcome, you also need to do something special because of the distributions of your predictors. Ah, I see. This makes sense. But I'm still a bit puzzled why I need to be modelling the thing I'm not interested in (leaving the channel). Isn't there a model for the thing I'm interested in (conversions)? I realize I can model one and calculate the other (as I'm doing now), but it seems a bit round-about? Why not? It seems easy to imagine that there is some unmeasured confounding factor that, when active on a particular day, leads respondents to be more likely to visit the site through a particular channel, but that also makes respondents less likely to ultimately buy something. I don't see why a channel having a negative coefficient is a logical impossibility. Also, note that even if the simple/total effect of a channel really is impossible to be negative (although I'm not really convinced), that doesn't preclude the multiple regression coefficient from being negative, since that estimates the effect while holding constant the other predictors. You're absolutely right. I was thinking about it causally, it didn't make sense to me that the presence of a user on a channel would make a purchase less likely. But yes, say a channel is used a lot by spammer and that slows down the site, it would make overall sales go down. But still, in my case, I find it hard to see people present on a channel leading to a net negative sale. I'm trying hard to think of realistic confounding factors that would do that. Maybe something like the day of week. The purchases are tickets for a museum, so I can imagine day of week being a confounder factor for channel use (e.g., more through ads on certain days) and ticket sales. Last edited: #### Junes ##### New Member Perhaps you also need to present a case example for an observation - to help us understand the data generating function. Thanks hlsmith. So, the data are about website channels. There is one for organic search (people enter through Google), one for direct use (people enter website directly), several for different advertising campaigns. People are assigned one channel (I'm not 100% sure I didn't make a data error somewhere but the overlap between the channels should be minimal). Also, I captured most of the traffic with these channels. There's probably some minor one I missed, but not much. Now what happens is that a user criss-crosses on the website, lands on the ticket page (my data) - and then I lose them. They go on to another website to make the sale and the museum owner didn't set the tracking properly. Or, they leave the website altogether. I simply don't know. People have to make an account to buy tickets, which is something they are loath to do in this day and age. Only 4.1% of the people landing on the ticket page make a transaction (though that number may be artificially lowered by fraudulent spam, see below). As you say, I only have the aggregated result per day - both in total number of transactions and total number of tickets sold (each transaction concerns an unknown number of tickets sold). Can you link actual channels to the outcome or do you have counts for outcome and channels? The latter. I still need a little more info, but zero inflated Poisson regression may be an option. Thanks, going to check that out. I will also post a bit later after this what I got now using the negative binomial. By the way, this project has changed from attribution to possible fraud detection. I have serious suspicions one of the channels is not "real" traffic. Possibly due to a part of the traffic being bots or from people just clicking without being interested in going to the museum. This is called "click fraud": advertising agencies charge you for traffic to your website that's actually worthless. There are even so-called "click farms" in places like Bangladesh where people do nothing but click websites all day for shady companies. I have some other, quite serious evidence from Google Analytics, but that's pretty restricted to some minor aspect of this fraud - it would also help if I can show that the coefficient for the advertising channel is very low. #### Junes ##### New Member Jake said: Basically I still don't see what the problem is with a model like Code: glm.nb(byr ~ channel1 + channel2 + channel3) Ah wait, I missed this. I can do this? I was under the impression that with negative binomial I need to model the people not making a purchase? So that the negative binomial is the distribution of non-purchases to happen before a purchase? So you are saying I could also model the distribution of purchases before a non-purchase? Even though a purchase is very rare (4%)? Last edited: #### Junes ##### New Member So, I've uploaded the data for anyone who wants to take a look. Code: date (European notation) tickets = tickets sold buyers = number of transactions exits = number of people leaving the ticket page and also the website ticket_page = number of people entering the ticket page organic = organic search (Google) adwords_name = AdWords channel, searched by name museum adwords_other = AdWords channel, searched by something other than name referral = traffic from another website direct = direct traffic (e.g., typed in website url) adv_banner = suspect campaign, via banner adv_page = suspect campaign, via information page adv_search = suspect campaign, via search results My R code: Code: df <- read.csv("attribution.txt") pairs(df[,2:ncol(df)]) The distribution of the suspect campaign channels is not strange. However, you need a lot more visitors to get the same effect. That tells me some of the traffic may be polluted with bots. Note that the most important and expensive one, adv_banner is only active for a few weeks per year, hence the many zeroes. Code: model <- glm.nb(buyers ~ direct + organic + adwords_name + adwords_other + referral + adv_banner + adv_search + adv_page, data=df, link=sqrt) summary(model) plot(df$buyers,predict(model,type="response"))
I'm using the square-root link function because it gives a much better fit. I understand that the curve is less steeper (which fits the data), but I'm not sure if I know what it does conceptually.

Summary:

Code:
Call:
link = sqrt, init.theta = 0.8022468254)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-2.0358  -1.0030  -0.7609   0.1350   3.0869

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)    0.425533   0.065654   6.481 9.09e-11 ***
direct        -0.005586   0.008421  -0.663   0.5072
organic        0.030517   0.005451   5.599 2.16e-08 ***
referral       0.023910   0.011101   2.154   0.0313 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Negative Binomial(0.8022) family taken to be 1)

Null deviance: 535.10  on 334  degrees of freedom
Residual deviance: 323.05  on 326  degrees of freedom
AIC: 1065.4

Number of Fisher Scoring iterations: 1

Theta:  0.802
Std. Err.:  0.144

2 x log-likelihood:  -1045.427

The fit is not great, but it works.

So the adv coefficients are really low. For the banner especially, the standard error is also relatively low. The upper limit of its confidence interval is 0.0188 (95%) or 0.0225 (99%). That seems low compared to referral, which is probably the best direct comparison. It seems suspiscious, taken together with some other information that's not in this data.

Is it a problem that my predictors are somewhat correlated (mean 0.33, range 0 to 0.80)?

I also realized I have the number of people exiting the ticket page, instead of just entering the page. When people exit, they either exit altogether or they go to the page that takes care the transaction. The number of exits correlates somewhat better with the number of transactions. But I'm not sure how to integrate it in the model.

#### hlsmith

##### Omega Contributor
So your data is still unclear to me. If the people could only take one channel, well you would have dummy codes for the independent variables. Say channel was age group, you can't be 0-10 years old and also 11-20 years old.

If you only have a list of counts, and it is not linked to the dependent variable, how is this relationship connected. So in post #5 first data row of your data frame, there were 0 purchases and 48 people entering site through channel 1. Now for row #4, there was one purchase and 82 people entering from channel #1, 46 from ch #2,...,1 from ch#9. How do you know where the purchase came from given your data. If I know 5% cars are red, 80 blue, and 15% white and there are 10 car accidents, I am working with 2 unlinked pieces of meta-data, I could create a probably model given car accidents are random and not with other cars and figure out the probability of a blue car crashing, but that is based on prevalence and doesn't tell me about say how all blue cars are driven by inexperienced drivers and actually 90% of car accidents include a blue cars. Do you see my repeated inquiry yet. I don't get if how you have two counts you can answer your question unless you know subject specific level data, ob1 used ch#1 and didn't buy anything; ob2 used ch#1 and bought something,...,obs 999 used ch#9 and didn' buy anything. And given that structure channels should be dummy variables.

#### Junes

##### New Member
So your data is still unclear to me. If the people could only take one channel, well you would have dummy codes for the independent variables. Say channel was age group, you can't be 0-10 years old and also 11-20 years old.
Yes, but here it's a count.

If you only have a list of counts, and it is not linked to the dependent variable, how is this relationship connected. So in post #5 first data row of your data frame, there were 0 purchases and 48 people entering site through channel 1. Now for row #4, there was one purchase and 82 people entering from channel #1, 46 from ch #2,...,1 from ch#9. How do you know where the purchase came from given your data.
Well, I don't know yet. That's what I want to find out by modelling.

I don't get if how you have two counts you can answer your question unless you know subject specific level data, ob1 used ch#1 and didn't buy anything; ob2 used ch#1 and bought something,...,obs 999 used ch#9 and didn' buy anything. And given that structure channels should be dummy variables.
Well, for an extreme example:

Code:
byr ch1   ch2
2    999   2
1    0      1
3    20     0
Given these data, where would you say byr comes from? Now imagine the same idea but where the difference is less clear, but where you have more observations.

#### Jake

Ah wait, I missed this. I can do this?
Yeah, I don't see why not.

I was under the impression that with negative binomial I need to model the people not making a purchase? So that the negative binomial is the distribution of non-purchases to happen before a purchase?
No... "negative binomial regression" just means that we assume the outcome follows a negative binomial distribution. For practical purposes in a regression context, you can just think of this as a Poisson distribution except with a flexible dispersion parameter (rather than a fixed mean/variance relationship) so that it can better handle underdispersion or overdispersion relative to Poisson regression. The "negative" does not mean that you have to model non-events, or anything like that.

So you are saying I could also model the distribution of purchases before a non-purchase? Even though a purchase is very rare (4%)?
Yep. In fact, the fact that you have a very low rate of occurrences in your outcome is the main reason why I think it's worth it to use a count model (like NB or Poisson regression) at all. Classical/Normal regression usually works pretty much fine when the occurrence rate is high enough, but not when the occurrence rate is low, because you run into things like negative fitted values (which really are impossible) and because homogeneity of error variance is almost certainly violated in such cases.

#### GretaGarbo

##### Human
It seems like these data are data that sums of events for each day. But that it is not known from which channel each buyer came from. So it would be like multi-level data, with only information of of level 2 and no information about the individuals. Then there is a risk of simpsons paradox or the ecological fallacy (like in the classic study by Robinson with data from 1930:ies about illiteracy and "race". The data were summed by municipality or something, and gave much higher correlation coefficients than what would have been achieved by individual data.)

I guess that these data are a good start, but maybe it is possible to get individual data later.

#### hlsmith

##### Omega Contributor
Greta, this is what I am trying to get at (ecological fallacy). There are 5 blue cars, 10 red cars, and 18 whites cars and a police traffic stop, 5 drivers were arrested for intoxication. You don't know which drivers were arrested. If it was random, then you can construct probabilities. But the fact that this whole question is being asked provides doubt in regards to it merrily being random. There are no numerators.

I got a bill at the restaurant for $100 and I order two pizzas, cola, bread sticks. How much did each item cost? No idea. Now you do this over a bunch of restaurants, you may get a little hit of a pattern but the costs could be totally different at other restaurants (just like people using Google or direct URL can't assume to be the same each or across days). I think trying to use or justify a model still does not negate the overlying issue. #### Junes ##### New Member Thanks guys. And guess what? That channel that didn't lead to sales? I've looked around Google Analytics all day and I've discovered an extensive bot network with servers all over the country. I just wrote up an 11-page report and sent it to my client. They've probably scammed my client out of thousands of euros worth of advertising, possibly tens of thousands. Wow. And I discovered all this just out of curiosity, thinking how I could tackle this statistical problem. Just saying, statistics brings you places #### Junes ##### New Member Greta, this is what I am trying to get at (ecological fallacy). There are 5 blue cars, 10 red cars, and 18 whites cars and a police traffic stop, 5 drivers were arrested for intoxication. You don't know which drivers were arrested. If it was random, then you can construct probabilities. But the fact that this whole question is being asked provides doubt in regards to it merrily being random. There are no numerators. I got a bill at the restaurant for$100 and I order two pizzas, cola, bread sticks. How much did each item cost? No idea. Now you do this over a bunch of restaurants, you may get a little hit of a pattern but the costs could be totally different at other restaurants (just like people using Google or direct URL can't assume to be the same each or across days). I think trying to use or justify a model still does not negate the overlying issue.
Ah, now I see what you mean. Yeah, I guess that's a fair point. I'm not sure how likely ecological fallacy type events are in this context, but that's certainly a possibility.