A hallmark of chaos is extreme sensitivity both to initial conditions and to model parameters. A...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
A hallmark of chaos is extreme sensitivity both to initial conditions and to model parameters. A very simple hypothetical mechanical model exhibiting such extreme sensitivity involves a very small "perturbation" of an idealized system involving a marble of mass m "elastically" bouncing back-and-forth between two larger spheres. Let the two spheres each be of exactly 1-m diameter with their surfaces centers exactly 1-m apart, and let the marble have mass m= 1.00g be bouncing back & forth at a velocity v (perhaps ve1.0m/s) initially along the direction of the x-axis joining the sphere's centers. See the figure: to Sirius ~8 light years The assumption that the collisions with the spheres are (perfectly) "elastic" means that if m hits a sphere S with a velocity v at an angle 0 to the normal to the surface at the point of collision, then the velocity after collision has exactly the same magnitude v but is directed away from the surface so as to make the same angle 0 on the other "side" of this normal. Thus if no other forces act on an exactly velocity-aligned marble, then it bounces back and forth - forever, if we accept our assumptions, & Newtonian mechanics. But we wish to contemplate what happens with either a small angle error off from the x- axis, or else an incredibly small perturbative force, say due to the gravitational force exerted on the marble m by a mosquito -8 light years away (off to where the star Sirius would be, though we assume it isn't, so that neither Sirius nor anything else exerts a gravitational force on m ). The sensitivity of the system to such an extremely small perturbation (or velocity misorientation of the marble m) is to be gauged by estimating the number N of bounces it should take till m completely misses one of the spheres. Thus let us consider what happens when m leaves from its nth bounce B, going onto its (n+1) th bounce Bi . We denote the offset (from the x-axis) at B, to be a small amount An and have a velocity v, leaving B, at a small angle 9, to the x-axis direction. The situation is indicated in a figure: surface B An surface of 1 Here dotted lines indicate perfectly horizontal lines parallel to the x-axis, a dashed line indicates a normal to the surface of the impacted sphere (where B1 occurs), the solid line indicates the trajectory of the marble m, and o indicates the angle that vp-1 (coincident with an associated surface normal) makes with the x-axis. If the off-sets & angles are small, then the problem may be linearized (whence we might expect matrices and such). We use small-angle formulas so that the off-set at B. is A Er.p, and further also An+1 =4, +2r· 9, which is one of the equations we desire. Still looking at the angles at B,, we see that n+11 941 =9+(@+9,) where o=A /r, so that n+1 r91 = 2(A, +2r9,) +r9, Thus we have a simple matrix recursion for what happens at a single bounce 2A, n+1 If we imagine that the mosquitos influence is felt only during the first transit (between the two spheres) to give a non-zero off set, and thereafter it is just the bounces which produce further off-set, then we simply iterate the bounce-matrix, to give 'n+1 r9 n+1 Evidently the problem is a fair distance along toward solution after a little matrix analysis, at least if we can make an estimate of the initial values for A, & 9. Maybe we should begin with a little eigen-analysis in anticipation that it will be useful. Exercise (a) - Determine the eigenvalues to the matrix appearing above. Evidently the (n+1)th stage in the recursion above is a combination of the two (right) eigenvectors, say vi & v2. And the combinations at one stage differ from those at the previous stage just by factors which are the corresponding eigenvalues, say A1 & A2. That is, if we denote the recursion by Dn+1 = T Dn then Dn+1 = T"D1 = a1A" v1 + a2 A2" V2 Thus for large n the larger of these two eigenvalues (abbreviated just to A) dominates, and D1 →a.2"v n+1 where, we can if we wish identify a= (uD,), with u the biorthonormalized left A- eigenvector corresponding to the right eigenvector v. But anyway once n reaches a value such that 2" A, (or 2" -8,) exceeds a value x1, the offset for the trajectory of the mass m should have reached (to within a bounce or so) a stage where the mass m misses hitting the spheres. Now to deal with the perturbation we need a little physics. The gravitational force due to the distant mass u has a magnitude given (in terms of Newton's universal law of gravitation) as f = Gmµ/ R? \where G = 6.7x10-"Nm?/kg? and R (8 light-years) is the distance between m & u. The unit "light year" is defined as the distance light travels in one year, with the speed of light c= 3.0x10®m/s. Thence {1 light year} = (3.0x10* m)(60)(60 in)(24 )(365 )(1.0yr) =9.5×10*m Thus if we take the mass u of our distant mosquito to be 0.02 g, our perturbative gravitational force has magnitude s=(6.7x10" )(1.0x10*kg)(2×10*kg)/(8×9.5x10*m)* =0.2×10 "N -11 Nm2 -50 (where N indicates the mks unit of force, the Newton, equivalent to about 1/5 of a pound). And this incredibly minuscule force should cause the mass m to deviate but ever so very slightly from its unperturbed trajectory (pulling it in the direction of Sirius, since gravitational forces are attractive). Indeed if the mass m starts at the surface of one of the spheres with a 1.00 m/s velocity (exactly) along the x-axis, then by the first bounce it should be pulled off its axis. Elementary physics tells us that m experiences a magnitude of acceleration toward the perturbing mass of a= f/m 0.2x104 m/s? acting while m traverses from one sphere to the next to impact at t= 1.0 s later; so that, m moves a transverse (off-set) distance 0.2x10 5° N 10*kg And m would be given a component of velocity in the direction of the mosquito (a direction normal to our x-axis) of v, = at = 0.2x10 " m/s, and an angle off-set (for the A = } at, = } (1s)' =10 m. velocity at the first bounce) of 9 Ev, /v= 2x 10 4 radians. The perturbing mass 8 light years away continues to exert a force with subsequent bounces, but we might just see what happens neglecting this effect (which in fact only very slightly modifies the answer we are working toward). Exercise (b) - For the perturbed model the incredibly tiny force & consequent off-sets indicated above, roughly estimate the number N of bounces needed before the mass m misses the big spheres altogether. Presumably you have found that this system is super-incredibly sensitive even to utterly ridiculously small perturbations. This includes a similarly ridiculous super- sensitivity to initial-condition off-sets, as that is just what is found when we look at A, & r9 which could be just considered an error in initial conditions (with no perturbing mosquito around, after the first transit). The ridiculous sensitivity signals behavior (for a real system) which is in practice unpredictable, and the scene is set for chaos. You may wonder what happens if the gravitational pull of the mosquito is maintained even past the first transit of the marble from one sphere to the other. Shouldn't this only further shorten the number N of bounces? Let us think about this a bit. The gravitational force from the mosquito continues after the first bounce to pull with essentially exactly the same force, and thence the same acceleration in the transverse direction, The direction of the transverse velocity (v, ) without this continuing force is taken care of in our preceding analysis (leading to the question (b)), using our matrix accounting for what happens at a bounce (when the off-sets are smallI). Thus from the first to the second bounce with the same acceleration, we anticipate an additional displacement of A, beyond that of what we otherwise calculate as A,, and also a similar additional angular off-set of r9, beyond what is in our matrix estimate for r9,. If we let 8, denote our off-set vector after n bounces, our revision now gives 8, = Tổ, +8,, but this additional correction is very nearly exactly the same after each bounce so that 8 = T8, +8, n+1 Now recursion of this gives 8, = Tổ, +8, and so forth, to finally give & 8, = Tổ, +ô, = T(Tổ, +8,)+8, = (T +T+I)8, = (T" + T"+..+T+I)8, But this may also be written as 81 = (T-I) (T"* –I)8, But this may also be written as 8 = (T-I) (T"*! -I)8, n+1 where we have made use of the fact that T & I commute with one another as well as mutually commuting with all their powers and sums (or differences of their powers), so that we can manipulate them just like numbers (including inversing them, if what is being inversed is non-singular). Thence again presuming that only the maximum eigenvalue A survives in the long run, of N bounces when our off-sets come up to someplace around r=1/2 (as expressed in meters), we have 1/2z a (A" -1)/(A-1) = a AN Here a has to do with the overlap of the initial 8, ,while the 1 on the right-hand side should be >1/2 (& <1, again expressed in meters), though given ô, we could compute a (= (A,,) where y, is a suitably normalized maximum-eigenvalue eigenvector). But why don't we just ignore these two factors and write AN -A-1 to estimate the number N of bounces for the marble to be lost. Exercise (c) – Is this estimate much different than what you got in (b). Now though our example seemed rather hypothetical the result here can be viewed to be related to what goes on in a real gas, such as air under ordinary conditions. The constituent gas molecules move (at ordinary temperatures) roughly 102 times faster than m in our example, and they bounce elastically off of one another (at least on the average), with the distance between collisions maybe -104 times less than in our problem. But since the molecules linear dimensions are ~1010 times smaller than r in our example, they have correspondingly higher curvatures (of their "surfaces") for much more severely multiplying collision angle deviations 9 which occur about 106 times more often by each of about 1022 molecules (in a few-liter volume). Though gas molecules are incredibly much less massive than our m &µ, the gravitational forces between nearer neighbor pairs (at ordinary pressures) turns out to be more nearly comparable to our minuscule force f, and of course the gas molecules being so much lighter respond much more over-whelmingly to the (electromagnetically based) intermolecular forces. All in all the molecular detail of gas behavior ends up being incredibly much more sensitive to non-ideality (i.e., to mismatches in directions or to perturbations from "spurious" weak forces) than is our simple model problem. That is, the motion of the gas molecules ends up being rather super fantastically incredibly sensitive to minuscule perturbations or errors, and when viewed at this molecular scale the system is chaotic. Amusingly. the word "gas" derives from the Greek word "xáog" for "chaos" (as also does "chaos" with the alternative Greek “ xaoo"). Of course, there is also the problem that at the nanoscale, the appropriate mechanics is "quantum" rather than "classical" – but let's not worry about that. A book of some interest giving a variety of viewpoints (by a variety of authors) is "Chaos / XAOC" ed. D. K. Campbell (Am. Inst. Phys, NY, 1991). As to gases, their nano-chaos somehow or other seems to show regularity on a macro-scale. That is, much of the "static" (i.e., equilibrium) macro-behavior is governed by just a few global variables (such as pressure, volume, and temperature). Indeed even the macro-dynamics of flowing gases (or fluids in general) may be well characterized (seemingly deterministically) in terms of just a few (additional) macro- variables, with position & time dependent pressures & temperatures, in addition to a few extra simple parameters like viscosity & specific heats). These systematics which arise out of the underlying micro-chaos is the subject matter of "statistical mechanics", which is where much of the ideas of stochastic models were first quantitatively applied in the natural sciences. In the case of eco-environmental systems, though there might be chaos at the organismal scale, one can wonder whether overall much simpler systematics might emerge, say for much larger populations. That is indeed what most of our considered models seek – but so far without the fantastic success of the kinetic theory of gases. Perhaps the major part of the eco-environmental difficulty is that the environment in which the eco-environmental populations are viewed to propagate so often changes – and along with it the survivability & fertility parameters. Presumably a way to take this into account is to include more environment within the model, but it is often unclear how much of the overall environment one needs so to take into account. A hallmark of chaos is extreme sensitivity both to initial conditions and to model parameters. A very simple hypothetical mechanical model exhibiting such extreme sensitivity involves a very small "perturbation" of an idealized system involving a marble of mass m "elastically" bouncing back-and-forth between two larger spheres. Let the two spheres each be of exactly 1-m diameter with their surfaces centers exactly 1-m apart, and let the marble have mass m= 1.00g be bouncing back & forth at a velocity v (perhaps ve1.0m/s) initially along the direction of the x-axis joining the sphere's centers. See the figure: to Sirius ~8 light years The assumption that the collisions with the spheres are (perfectly) "elastic" means that if m hits a sphere S with a velocity v at an angle 0 to the normal to the surface at the point of collision, then the velocity after collision has exactly the same magnitude v but is directed away from the surface so as to make the same angle 0 on the other "side" of this normal. Thus if no other forces act on an exactly velocity-aligned marble, then it bounces back and forth - forever, if we accept our assumptions, & Newtonian mechanics. But we wish to contemplate what happens with either a small angle error off from the x- axis, or else an incredibly small perturbative force, say due to the gravitational force exerted on the marble m by a mosquito -8 light years away (off to where the star Sirius would be, though we assume it isn't, so that neither Sirius nor anything else exerts a gravitational force on m ). The sensitivity of the system to such an extremely small perturbation (or velocity misorientation of the marble m) is to be gauged by estimating the number N of bounces it should take till m completely misses one of the spheres. Thus let us consider what happens when m leaves from its nth bounce B, going onto its (n+1) th bounce Bi . We denote the offset (from the x-axis) at B, to be a small amount An and have a velocity v, leaving B, at a small angle 9, to the x-axis direction. The situation is indicated in a figure: surface B An surface of 1 Here dotted lines indicate perfectly horizontal lines parallel to the x-axis, a dashed line indicates a normal to the surface of the impacted sphere (where B1 occurs), the solid line indicates the trajectory of the marble m, and o indicates the angle that vp-1 (coincident with an associated surface normal) makes with the x-axis. If the off-sets & angles are small, then the problem may be linearized (whence we might expect matrices and such). We use small-angle formulas so that the off-set at B. is A Er.p, and further also An+1 =4, +2r· 9, which is one of the equations we desire. Still looking at the angles at B,, we see that n+11 941 =9+(@+9,) where o=A /r, so that n+1 r91 = 2(A, +2r9,) +r9, Thus we have a simple matrix recursion for what happens at a single bounce 2A, n+1 If we imagine that the mosquitos influence is felt only during the first transit (between the two spheres) to give a non-zero off set, and thereafter it is just the bounces which produce further off-set, then we simply iterate the bounce-matrix, to give 'n+1 r9 n+1 Evidently the problem is a fair distance along toward solution after a little matrix analysis, at least if we can make an estimate of the initial values for A, & 9. Maybe we should begin with a little eigen-analysis in anticipation that it will be useful. Exercise (a) - Determine the eigenvalues to the matrix appearing above. Evidently the (n+1)th stage in the recursion above is a combination of the two (right) eigenvectors, say vi & v2. And the combinations at one stage differ from those at the previous stage just by factors which are the corresponding eigenvalues, say A1 & A2. That is, if we denote the recursion by Dn+1 = T Dn then Dn+1 = T"D1 = a1A" v1 + a2 A2" V2 Thus for large n the larger of these two eigenvalues (abbreviated just to A) dominates, and D1 →a.2"v n+1 where, we can if we wish identify a= (uD,), with u the biorthonormalized left A- eigenvector corresponding to the right eigenvector v. But anyway once n reaches a value such that 2" A, (or 2" -8,) exceeds a value x1, the offset for the trajectory of the mass m should have reached (to within a bounce or so) a stage where the mass m misses hitting the spheres. Now to deal with the perturbation we need a little physics. The gravitational force due to the distant mass u has a magnitude given (in terms of Newton's universal law of gravitation) as f = Gmµ/ R? \where G = 6.7x10-"Nm?/kg? and R (8 light-years) is the distance between m & u. The unit "light year" is defined as the distance light travels in one year, with the speed of light c= 3.0x10®m/s. Thence {1 light year} = (3.0x10* m)(60)(60 in)(24 )(365 )(1.0yr) =9.5×10*m Thus if we take the mass u of our distant mosquito to be 0.02 g, our perturbative gravitational force has magnitude s=(6.7x10" )(1.0x10*kg)(2×10*kg)/(8×9.5x10*m)* =0.2×10 "N -11 Nm2 -50 (where N indicates the mks unit of force, the Newton, equivalent to about 1/5 of a pound). And this incredibly minuscule force should cause the mass m to deviate but ever so very slightly from its unperturbed trajectory (pulling it in the direction of Sirius, since gravitational forces are attractive). Indeed if the mass m starts at the surface of one of the spheres with a 1.00 m/s velocity (exactly) along the x-axis, then by the first bounce it should be pulled off its axis. Elementary physics tells us that m experiences a magnitude of acceleration toward the perturbing mass of a= f/m 0.2x104 m/s? acting while m traverses from one sphere to the next to impact at t= 1.0 s later; so that, m moves a transverse (off-set) distance 0.2x10 5° N 10*kg And m would be given a component of velocity in the direction of the mosquito (a direction normal to our x-axis) of v, = at = 0.2x10 " m/s, and an angle off-set (for the A = } at, = } (1s)' =10 m. velocity at the first bounce) of 9 Ev, /v= 2x 10 4 radians. The perturbing mass 8 light years away continues to exert a force with subsequent bounces, but we might just see what happens neglecting this effect (which in fact only very slightly modifies the answer we are working toward). Exercise (b) - For the perturbed model the incredibly tiny force & consequent off-sets indicated above, roughly estimate the number N of bounces needed before the mass m misses the big spheres altogether. Presumably you have found that this system is super-incredibly sensitive even to utterly ridiculously small perturbations. This includes a similarly ridiculous super- sensitivity to initial-condition off-sets, as that is just what is found when we look at A, & r9 which could be just considered an error in initial conditions (with no perturbing mosquito around, after the first transit). The ridiculous sensitivity signals behavior (for a real system) which is in practice unpredictable, and the scene is set for chaos. You may wonder what happens if the gravitational pull of the mosquito is maintained even past the first transit of the marble from one sphere to the other. Shouldn't this only further shorten the number N of bounces? Let us think about this a bit. The gravitational force from the mosquito continues after the first bounce to pull with essentially exactly the same force, and thence the same acceleration in the transverse direction, The direction of the transverse velocity (v, ) without this continuing force is taken care of in our preceding analysis (leading to the question (b)), using our matrix accounting for what happens at a bounce (when the off-sets are smallI). Thus from the first to the second bounce with the same acceleration, we anticipate an additional displacement of A, beyond that of what we otherwise calculate as A,, and also a similar additional angular off-set of r9, beyond what is in our matrix estimate for r9,. If we let 8, denote our off-set vector after n bounces, our revision now gives 8, = Tổ, +8,, but this additional correction is very nearly exactly the same after each bounce so that 8 = T8, +8, n+1 Now recursion of this gives 8, = Tổ, +8, and so forth, to finally give & 8, = Tổ, +ô, = T(Tổ, +8,)+8, = (T +T+I)8, = (T" + T"+..+T+I)8, But this may also be written as 81 = (T-I) (T"* –I)8, But this may also be written as 8 = (T-I) (T"*! -I)8, n+1 where we have made use of the fact that T & I commute with one another as well as mutually commuting with all their powers and sums (or differences of their powers), so that we can manipulate them just like numbers (including inversing them, if what is being inversed is non-singular). Thence again presuming that only the maximum eigenvalue A survives in the long run, of N bounces when our off-sets come up to someplace around r=1/2 (as expressed in meters), we have 1/2z a (A" -1)/(A-1) = a AN Here a has to do with the overlap of the initial 8, ,while the 1 on the right-hand side should be >1/2 (& <1, again expressed in meters), though given ô, we could compute a (= (A,,) where y, is a suitably normalized maximum-eigenvalue eigenvector). But why don't we just ignore these two factors and write AN -A-1 to estimate the number N of bounces for the marble to be lost. Exercise (c) – Is this estimate much different than what you got in (b). Now though our example seemed rather hypothetical the result here can be viewed to be related to what goes on in a real gas, such as air under ordinary conditions. The constituent gas molecules move (at ordinary temperatures) roughly 102 times faster than m in our example, and they bounce elastically off of one another (at least on the average), with the distance between collisions maybe -104 times less than in our problem. But since the molecules linear dimensions are ~1010 times smaller than r in our example, they have correspondingly higher curvatures (of their "surfaces") for much more severely multiplying collision angle deviations 9 which occur about 106 times more often by each of about 1022 molecules (in a few-liter volume). Though gas molecules are incredibly much less massive than our m &µ, the gravitational forces between nearer neighbor pairs (at ordinary pressures) turns out to be more nearly comparable to our minuscule force f, and of course the gas molecules being so much lighter respond much more over-whelmingly to the (electromagnetically based) intermolecular forces. All in all the molecular detail of gas behavior ends up being incredibly much more sensitive to non-ideality (i.e., to mismatches in directions or to perturbations from "spurious" weak forces) than is our simple model problem. That is, the motion of the gas molecules ends up being rather super fantastically incredibly sensitive to minuscule perturbations or errors, and when viewed at this molecular scale the system is chaotic. Amusingly. the word "gas" derives from the Greek word "xáog" for "chaos" (as also does "chaos" with the alternative Greek “ xaoo"). Of course, there is also the problem that at the nanoscale, the appropriate mechanics is "quantum" rather than "classical" – but let's not worry about that. A book of some interest giving a variety of viewpoints (by a variety of authors) is "Chaos / XAOC" ed. D. K. Campbell (Am. Inst. Phys, NY, 1991). As to gases, their nano-chaos somehow or other seems to show regularity on a macro-scale. That is, much of the "static" (i.e., equilibrium) macro-behavior is governed by just a few global variables (such as pressure, volume, and temperature). Indeed even the macro-dynamics of flowing gases (or fluids in general) may be well characterized (seemingly deterministically) in terms of just a few (additional) macro- variables, with position & time dependent pressures & temperatures, in addition to a few extra simple parameters like viscosity & specific heats). These systematics which arise out of the underlying micro-chaos is the subject matter of "statistical mechanics", which is where much of the ideas of stochastic models were first quantitatively applied in the natural sciences. In the case of eco-environmental systems, though there might be chaos at the organismal scale, one can wonder whether overall much simpler systematics might emerge, say for much larger populations. That is indeed what most of our considered models seek – but so far without the fantastic success of the kinetic theory of gases. Perhaps the major part of the eco-environmental difficulty is that the environment in which the eco-environmental populations are viewed to propagate so often changes – and along with it the survivability & fertility parameters. Presumably a way to take this into account is to include more environment within the model, but it is often unclear how much of the overall environment one needs so to take into account.
Expert Answer:
Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
Posted Date:
Students also viewed these accounting questions
-
Two small spheres with mass m = 15.0 g are hung by silk threads of length L = 1.20 m from a common point (Fig. 21.44). When the spheres are given equal quantities of negative charge, so that q1 = q2...
-
A particle of mass m1 experienced a perfectly elastic collision with a stationary particle of mass m2. What fraction of the kinetic energy does the striking particle lose, if? (a) It recoils at right...
-
Two spheres A and B have an equal mass M and are electro statically charged such that the repulsive force acting between them has magnitude F and is directed along line AB. Determine the angle...
-
Attlee Ltd holds 28% of the issued shares of Nehru Ltd. Attlee Ltd acquired these shares on 1 July 2019 and on this date all the identifiable assets and liabilities of Nehru Ltd were recorded at...
-
What is the minimum risk portfolio? Why is this portfolio usually not the portfolio chosen by FIs to optimize the return-risk tradeoff?
-
Gavin Petrosino, a building contractor, builds houses in tracts, often building as many as 20 homes simultaneously. Petrosino has budgeted costs for an expected number of houses in 20X0 as follows:...
-
Based on the following pedigree for a trait determined by a single gene (affected individuals are shown as filled symbols), state whether it would be possible for the trait to be inherited in each of...
-
Air pollution control specialists in southern California monitor the amount of ozone, carbon dioxide, and nitrogen dioxide in the air on an hourly basis. The hourly time series data exhibit...
-
The demand curve for cookies is a rightward curve and the quantity demanded is 100 when the price of cookies is $2.00. What happens to consumer surplus when the price is $3.00? What happens to...
-
Kaia Mechanics has the following accounts: Create a chart of accounts for Kaia Mechanics using the standard numbering system. Each account is separated by a factor of 10. For example, the first asset...
-
A water cooler using R12 refrigerant works between 30C to 9C. Assuming the volumetric and mechanical efficiency of the compressor to be 80 and 90% respectively, and the mechanical efficiency of motor...
-
You are a junior sell-side analyst working for Deutsche Bank and you have been assigned to cover Dettol. On the back of extreme profits in the consumer hand sanitizer market, you believe that Dettol...
-
Two fish travel to a particular location on the reef to forage for food. One fish travels from a further location from the reef, the other fish travels from a location that is nearer to the reef....
-
After its first month of operations, the following amounts were taken from the accounting records of West Coast Dreams Realty Inc. as of June 30, 20Y9. Account Amount Cash $82,000 Common stock...
-
Carter Company's return on common equity is 29%. Its sales are $68,000,000, its debt ratio is 45%, and its total liabilities are $15,000,000. What is the firm's return on total assets? Question 4...
-
The following information is related to ABC Co.: 1. Accounts receivable of 60,000 (gross) are factored with BFO Credit without guarantee at a financing charge of 9%. Cash is received for the...
-
I dont know what am I getting wrong. please excel explanationbecause I did them with BA II plus calc and got them wrong coupletimes 13. Ann is looking for a fully amortizing 30 -year Fixed Rate...
-
Find a least expensive route, in monthly lease charges, between the pairs of computer centers in Exercise 11 using the lease charges given in Figure 2. a) Boston and Los Angeles b) New York and San...
-
Creating New Problems (b) Find a 3 x 3 matrix with a triple eigenvalue and two linearly independent eigenvectors.
-
Show that if y1(t) and y2(t) are solutions of y" + p(t) y' +q(t)y = 0, then so is c1y1(t) + c2y2 (t) for any constants c1 and c2.
-
Geometric Vector Operations For 1. A + C 2. ½ A + B 3. A - 2B and C=1-2
-
KK Pty Ltd is a small manufacturing business. For the year ending 30 June 2019, the company achieved sales of $2 772 000 and a gross profit margin of 30%. Although satisfied with this result,...
-
The following actual balance sheet was prepared for Colombo Clocks Ltd as at 31 March 2020. At 31 March you are also provided with the following information. 1. Sales forecasts available for 2020:...
-
Fab and Fast Ltd buys and sells motor vehicle accessories. The firms estimated sales and expenses for the first 4 months of 2020 are shown below. Actual sales for December 2019 were $900 000 and...
Study smarter with the SolutionInn App