6.3.1 Examplary Constructions of the Universal Graph

At the beginning each example contains rather full calculations and discussions. Subsequently we reduce the amount of comments leaving the most important steps only.

Example 6.3   The construction given in Fig. 6.2 a) has one weakness. If the dimensions of an argument and value coincide then the line linking such points will not cross the dimensionless fibre. What shall we do in such a case? Of course, the construction should be modified. Such geometries are depicted in Fig. 6.3

Figure 6.3: The geometrical configuration of dimensional quantities in the dimensional space.

The necessary modification of the construction in Fig. 6.2 a) is very simple. If the dimensions of an argument and of value coincide, then the value can be obtained from an argument by rescaling with a number $\rho$ ( $\rho$   $\in$ $\bf R_{+}^{}$). In fact, it is the same construction as before but the straight line connecting argument and value has reared up.

To be more concrete let us go back to our rectangular box, discussed in Example 6.1, but now let us look for the total field d of the box as the function of the same quantities as before a, b. Thus, we seek now for the dimensional function d (a, b) where the suitable dimensions are listed below:

p(a) = length²,      p(b) = length,      p(d )= length².

The dimensions of a and d coincide as is depicted in Fig. 6.3 a). The point d (the value) can be obtained from a with the multiplication $\rho$a ( $\rho$ $\in$ $\bf R_{+}^{}$) or from b with the previous method (see Fig. 6.2 a)). As regards the orbit structure we have the following equation:

$$\rho \xi_{b}^{-1}$$ (6.8)

Figure: The universal graph $\cal {G}$ of the dimensional function and the orbit structure for the two cases: a) d (a, b) = 2a + 4ba^{${\frac{1}{2}}$} b) f ($\tilde{a}$, b) = 8$\tilde{a}$ + 4b.

Effectivelly each orbit becomes an hyperbola in the $\rho$,$\xi_{b}^{}$-plane (the space of models X). Once the orbit structure is set up, we may construct the universal graph of the dimensional function d (a, b). Generally, the experimental data are necessary for this step but in our case we know the form of d (a, b):

d (a, b) = 2a + 4ba^{${\textstyle\frac{1}{2}}$},

the universal graph is described by the following equation:

$$\xi_{b}^{}$$$$\rho$$ = ($$\rho$$ - 2)².

This situation is sketched in Fig. 6.4 a).

The Fig. 6.3 b) presents the geometry of our next example. Now all three quantities $\tilde{a}$, b, f have the same dimension length. Supposing that $\tilde{a}$ is the length of the base edge of our rectangular box with square base, b denotes the length of the vertical edges and we look for the total length f of all edges of the box. Of course:

f ($$\tilde{a}$$, b) = 8$$\tilde{a}$$ + 4b.

Since all dimensions coincide we may get the point f by scaling transformation of the point $\tilde{a}$ or of the point b. The suitable scale factors are denoted by $\rho_{a}^{}$ (for the point $\tilde{a}$) and by $\rho_{b}^{}$ (for the point b). The orbit structure is spanned by the equation:

$$\rho_{a}^{}$$$$\tilde{a}$$ = $$\rho_{b}^{}$$b.

Thus orbits are straight lines originated at the point 0, 0 in the space of models $\rho_{a}^{}$,$\rho_{b}^{}$. The universal graph of the dimensional function f (a, b) becomes a hyperbola:

($$\rho_{a}^{}$$ - 8)($$\rho_{b}^{}$$ - 4) = 32.

Let us note that below the asymptotes of the above hyperbola we have no points of the universal graph. The geometry of the orbit structure and the universal graph are depicted in Fig. 6.4 b).

Figure: The three universal graphs $\cal {G}$₁, $\cal {G}$₂ and $\cal {G}$₃ drawn in one, common orbit structure. The proper equations are: $\cal {G}$₁   $\rightarrow$  $\zeta_{1}^{2}$$\zeta_{2}^{}$ = 1, $\cal {G}$₂   $\rightarrow$  16$\zeta_{1}^{}$ = $\zeta_{2}^{}$($\zeta_{1}^{}$ - 2)² and $\cal {G}$₃   $\rightarrow$  ($\zeta_{1}^{}$ - 8)($\zeta_{2}^{}$ - 4) = 32.

The known orbit structure and the universal graphs allow us to compare different dimensional functions. It suffices to perform such transformation of the space of models to obtain the same orbit structure. Of course the real meaning of the orbit parameter x will remain different for different models but the geometry of the orbits will be the same. Since the orbits in Fig. 6.4 b) seem to be the simplest ones, we modify all spaces of models to obtain this structure. We begin with Fig. 6.2 b). The suitable transformation has the form:

$$\xi_{a}^{}$$ $$\longrightarrow$$ $$\zeta_{1}^{}$$,      $$\xi_{b}^{4}$$ $$\longrightarrow$$ $$\zeta_{2}^{}$$.

In a similar manner for Fig. 6.4 a) we will get:

$$\rho$$ $$\longrightarrow$$ $$\zeta_{1}^{}$$,      $$\xi_{b}^{-1}$$ $$\longrightarrow$$ $$\zeta_{2}^{}$$,

and Fig. 6.4 b) will remain unchanged:

$$\rho_{a}^{}$$ $$\longrightarrow$$ $$\zeta_{1}^{}$$,      $$\rho_{b}^{}$$ $$\longrightarrow$$ $$\zeta_{2}^{}$$.

In effect all orbits have the same form - the straight lines originated at the point $\zeta_{1}^{}$ = 0,  $\zeta_{2}^{}$ = 0. Now we may draw all universal graphs at once in one Fig. 6.5 Let us note that although the orbit structures become the same, the universal graphs are different due to the different structure of the dimensional functions. The above considerations suggest a new notion of similarity of dimensional dependencies. We may call the two different dimensional functions the similar ones if their universal graphs depicted in the common orbit geometry are similar in a certain. Fig. 6.5 indicates that the universal graphs $\cal {G}$₁, $\cal {G}$₂ and $\cal {G}$₃ are similar although the functional dependencies are very different. This is new notion of similarity connected rather with algebraic topology or geometry than with usual methods of the identification theory.

Example 6.4   Our method of geometrical constructions is not limited to the one-dimensional space $\cal {W}$ only. It works in any dimension but we have begun with the simplest cases. Now we pass to the two-dimensional example (two-dimensional means here, the two dimensional space $\cal {W}$).

Figure: The configuration of quantities t, s, v₁, v₂,$\xi_{2}^{}$ in the dimensional space.

Supposing that we have the two particles moving along one straight line. The velocities of particle equal v₁, v₂ respectively. The change of the distance s between the first and the second particle as the function of time t is equal:

s = (v₁ + v₂)t.

As before we forget this equation and try to restore it from the experimental data. Thus we have the quantity s with dimension length which is assumed to be the function of three quantities: v₁, v₂ with dimension velocity and t with dimension time. The current situation is depicted in Fig. 6.6 a) and b).

Since the dimension of the space $\cal {W}$ has increased to two the previous construction has to be modified. Instead of lines we have to utilize planes but the whole method is essentially the same. The quantities v₁, v₂ have the same dimension, consequently both belong to the same fiber. We may construct the point s with the two different planes v₁t$\xi_{1}^{}$ and v₂t$\xi_{2}^{}$. As before the points labelled by $\xi_{1}^{}$ and $\xi_{2}^{}$ belong to the dimensionless fiber i.e., both are ordinary real numbers from $\bf R_{+}^{}$. The plane v₁v₂s does not cross the dimensionless fiber and contains the whole fiber p(s). This fact is depicted in Fig. 6.6 b).

The plane v₁, t, s has the equation:

v₁^$\eta$t^$\mu$s^{1 - $\eta$ - $\mu$}

and crosses the fiber $\bf 1$ at the point $\xi_{1}^{}$. To derive $\xi_{1}^{}$ we have to accept a vector basis in the space $\cal {W}$ and write dimensions of all quantities in this chosen basis:

p(v₁) = $$\omega_{1}^{a_{1}}$$$$\omega_{2}^{a_{2}}$$,    p(v₂) = $$\omega_{1}^{a_{1}}$$$$\omega_{2}^{a_{2}}$$,    p(t) = $$\omega_{1}^{c_{1}}$$$$\omega_{2}^{c_{2}}$$,    p(s) = $$\omega_{1}^{d_{1}}$$$$\omega_{2}^{d_{2}}$$,

where $\omega_{1}^{}$,  $\omega_{2}^{}$ is the fixed vector basis of $\cal {W}$. Solving the system of equations for $\eta_{0}^{}$,  $\mu_{0}^{}$:

(a₁ - d₁)$$\eta_{0}^{}$$ + (c₁ - d₁)$$\mu_{0}^{}$$ = - d₁,      (a₂ - d₂)$$\eta_{0}^{}$$ + (c₂ - d₂)$$\mu_{0}^{}$$ = - d₂,

we obtain for $\xi_{1}^{}$:

$$\xi_{1}^{}$$ = v₁^$\eta_{0}$t^$\mu_{0}$s^{1 - $\eta_{0}$ - $\mu_{0}$}.

Inverting the last equation we will get:

$$\xi_{1}^{\frac{1}{1 - \eta_{0} -\mu_{0}}} {\frac{- \eta_{0}}{1 - \eta_{0} - \mu_{0}}} {\frac{- \mu_{0}}{1 - \eta_{0} - \mu_{0}}}$$ (6.9)

Similar calculations for the plane v₂ts yield an analogous equation but with $\xi_{2}^{}$ instead of $\xi_{1}^{}$ and with the velocity v₁ replaced by v₂ (the dimensions of v₁ and v₂ are the same).

Figure: The universal graphs of the functions: a) s = (v₁ + v₂)t ( $\xi_{1}^{}$ + $\xi_{2}^{}$ = 1) and b) s = (v₁ + v₂ + v₃)t ( $\xi_{1}^{}$ + $\xi_{2}^{}$ + $\xi_{3}^{}$ = 1).

In effect the orbit equation has the form

$$\xi_{1}^{}$$v₁^{- $\eta_{0}$} = $$\xi_{2}^{}$$v₂^{- $\eta_{0}$},

and after some calculations we will obtain $\eta_{0}^{}$ = 1,  $\mu_{0}^{}$ = 1 (the values of $\eta_{0}^{}$,  $\mu_{0}^{}$ do not depend on the choice of the basis $\omega_{1}^{}$,$\omega_{2}^{}$ in the space $\cal {W}$).

Of course, generally the universal graph of the function s(v₁, v₂, t) may be restored from measurement results only but in our test example we know the exact form of the function, s(v₁, v₂, t) = (v₁ + v₂)t. The simple calculations give:

$$\xi_{1}^{} \xi_{2}^{}$$ (6.10)

for the universal graph of the function s(v₁, v₂, t).

The resulting orbit structure as well as the universal graph are depicted in Fig. 6.7 a). For simplicity, we have chosen the number x = v₁v₂^-1 as the orbit parameter. The more symmetric choice in v₁, v₂ can be x^$\prime$ = $\sqrt{(v_{1}v_{2})}$/(v₁ + v₂). Here, the orbit O(x) is the set defined below

O(x) = {v₁, v₂, t   :  v₁ = xv₂}.

Thus t is completely free inside one orbit. In the picture the orbits are represented as half-lines originating at point (0, 0) in the $\xi_{1}^{}$,$\xi_{2}^{}$ - plane (the space of models X in the current case).

Let us fix the orbit (i.e., x = x₀) and let v₁₀, v₂₀, t₀ belong to the chosen orbit O(x₀). If we know the suitable value s₀ of s then both $\xi_{1}^{}$ and $\xi_{2}^{}$ are fixed. Moreover, any scaling transformation of v₁₀, v₂₀, t₀, s₀ changes neither the orbit O(x₀) nor the values of $\xi_{1}^{}$,$\xi_{2}^{}$. Thus inside one orbit the values of $\xi_{1}^{}$ and $\xi_{2}^{}$ are uniquely determined. Repeating the same construction for all orbits we arrive at the universal graph (D.8) of the functional dependence.

However the above method has one weak point. For any orbit parameter x we have to determine the two numbers $\xi_{1}^{}$,$\xi_{2}^{}$ because the space of models is two dimensional. If we accept the orbit structure as a part of the coordinate system then only one number should suffice. The second family of lines spanning the coordinate system can be easily constructed from the equations of planes (6.9) in the dimensional space. We have

$$\xi_{1}^{} \xi_{2}^{}$$ (6.11)

Adding up these two equations and setting the sum to equal ys we obtain

s($$\xi_{1}^{}$$ + $$\xi_{2}^{}$$) = ys,

which gives the second family of the coordinate lines. Now we may represent any dimensional dependence in the functional form y(x). This step is somewhat analogical to the dimensionless function involved in Theorem $\pi$. In our example the function y(x) becomes simply the constant one

y(x) $$\equiv$$ 1.

The number of velocities need not be restricted to two. If we have the three velocities v₁, v₂, v₃ then the space of models becomes three-dimensional and the equation (D.1) will be replaced by

s = (v₁ + v₂ + v₃)t.

The same method gives us the orbit equations (now we have two equations instead of one)

$$\xi_{1}^{}$$v₂ = $$\xi_{2}^{}$$v₁,      $$\xi_{2}^{}$$v₃ = $$\xi_{3}^{}$$v₂.

As we see the orbits (half-lines originated at the point $\xi_{1}^{}$ = $\xi_{2}^{}$ = $\xi_{3}^{}$ = 0) fill the whole three dimensional space of models. The universal graph, resulting from the equation

$$\xi_{1}^{}$$ + $$\xi_{2}^{}$$ + $$\xi_{3}^{}$$ = 1,

becomes a fragment of the plane. However it does not mean that the dimension (in the geometrical sense) of the universal graph is directly related to the dimension of the space of models.

The appropriate coordinate system in the space of models can be constructed also in our new case when the number of velocities approaches three. As before we accept the half-lines of orbits as one family of coordinate lines. The second coordinate is constructed similarly to preceeding example. We add all plane equations and set the sum to be equal to ys

s($$\xi_{1}^{}$$ + $$\xi_{2}^{}$$ + $$\xi_{3}^{}$$) = ys.

This allows us to represent any dimensional function in the form y(x₁, x₂) (where for simplicity the orbit parameters are chosen as x₁ = v₁v₂^-1 and x₂ = v₂v₃^-1). Our universal graph $\cal {G}$ is then equivalent to the constant function y(x₁, x₂) $\equiv$ 1. The whole geometrical configuration in the space of models is depicted in Fig. 6.7 b). The universal graph $\cal {G}$ becomes a plane (the dashed one in the picture).

Of course this is rather an artificial example, specially prepared to present our ideas. But it presents the main advantages of the universal graph method. We are able to construct the representation of the dimensional function independent of the particular choice of the dimensional basis. It is significant that the approximation of the universal graph requires the same expenses as the method based on the Theorem $\pi$. In both cases we have to identify the numerical function depending on the same number of numerical arguments.

Example 6.5   Let us slightly change the problem discussed in the previous example. The distance covered by the particle moving along the straight line with constant acceleration has to be estimated. Thus the quantities v, t, s will remain the same but a is now the acceleration with dimension $\it length/time^{2}$. The dimensional function a(a, v, t) is now:

$${\textstyle\frac{1}{2}}$$ (6.12)

Here v is the initial velocity, t is the time and s denotes the distance. This is a well known test example from various papers [17]. In similar manner as before we may construct the point s with planes vts, ats. The calculation method is exactly the same.

For the crossing points with the dimensionless fiber we have:

vts :    $$\xi_{v}^{}$$ = vts^-1,          ats :    $$\xi_{a}^{}$$ = a^{${\textstyle\frac{1}{2}}$}ts^{- ${\textstyle\frac{1}{2}}$}.

This gives the first equation for the orbit structure:

$$\xi_{v}^{-1}$$vt = $$\xi_{a}^{-2}$$at².

However we have also another plane which may be used in construction.

The third plane avs is of a different nature. It follows from its equation

v^wa^uS^{1 - w - u}

that the plane avs contains the whole fiber p(s).

Choosing the vectors $\omega_{1}^{}$ = $\it length$ and $\omega_{2}^{}$ = $\it time$ as the vector basis for $\cal {W}$ we have:

length  time^{(- w - 2u)}

for the dimensions of the plane points of the plane avs. It is evident that the fiber $\omega$ = $\bf 1$ is unobtainable but for w = - 2u we obtain the dimension length i.e., the dimension of s. Thus all points (v^-2a^us^{1 + u}) belong to the fibre p(s). In this particular case, for u = - 1 we have the point v²a^-1 on the fiber p(s) which does not depend on s and belongs to the line va defined by the pair of points. This is the new geometrical configuration absent in previous examples. Instead of the plane avs we may construct the point s from the point v²a^-1 of the line av with the equation:

v^ua^{1 - u},         for    u $$\in$$ R.

The point s is then achieved as the rescaled point v²a^-1 i.e.,

$$\rho$$ (6.13)

for some $\rho$ $\in$ $\bf R_{+}^{}$.

Figure: The orbits fill the two-dimensional manifold (surface) in the space of models X with axes $\zeta_{1}^{}$,$\zeta_{2}^{}$,$\zeta_{3}^{}$. The universal graph $\cal {G}$ and orbits (labelled by x) lay in this surface.

Effectively we have for the orbit structure the equations

$$\rho$$v²a^-1 = $$\xi_{v}^{-1}$$vt = $$\xi_{a}^{-2}$$at²,

or equivalently:

$$\xi_{v}^{-1} \xi_{a}^{-2} \rho \xi_{v}^{2} \xi_{a}^{2}$$ (6.14)

Our space of models becomes three-dimensional. However let us notice that the second equation is completely general: it does not depend on any concrete value of the incoming quantities a, v, t. This means that each orbit is completely described by the first equation and also the universal graph depends really on the one orbit parameter x = at/v only. The value of $\rho$ is given by the values of $\xi_{v}^{}$ and $\xi_{a}^{}$.

Effectively the orbits fill the two dimensional manifold (some kind of a surface) embedded in the three dimensional space of models. The universal graph $\cal {G}$ is the line lying in this surface.

For simplicity let us change of variables:

$$\zeta_{1}^{}$$ = $$\xi_{v}^{}$$,    $$\zeta_{2}^{}$$ = $$\xi_{a}^{2}$$,    $$\zeta_{3}^{}$$ = $$\rho^{-1}_{}$$.

After this transformation, the resulting geometry will become quite simple. The whole configuration is depicted in Fig. 6.8 The orbit equations in the variables $\zeta_{1}^{}$,$\zeta_{2}^{}$,$\zeta_{3}^{}$ are:

$$\zeta_{2}^{} \zeta_{3}^{} \zeta_{2}^{} \zeta_{3}^{} \zeta_{1}^{2}$$ (6.15)

The second equation describes the surface of the second order and the first represents the plane. The orbit with the parameter x is just the intersection of the plane and the surface. As it should be any orbit is a line. The universal graph is given by the equation

$$\zeta_{1}^{} {\textstyle\frac{1}{2}} \zeta_{2}^{}$$ (6.16)

Geometrically (6.16) represents the plane and the universal graph is also the intersection of this plane with the surface (6.15).

The space of models geometry of this example is depicted in Fig. 6.8 The projection of this geometrical structure onto the $\zeta_{1}^{}$,$\zeta_{2}^{}$-plane creates a configuration similar to the previous example depicted in Fig. 6.7 This geometry suggests that the current example is the modification of the previous task (Fig. 6.7)

Generally, when orbits do not fill the whole space of models, the given problem is reducible. The dimensional function depends only on some specific combinations of incoming arguments. We may then diminish the number of initial arguments.

The information that the above simplification is possible does not mean that we know how to perform it. In general there are many ways to reduce the number of incoming quantities. Let's look at the universal graph in form (6.16). We see that this equation does not depend on $\zeta_{3}^{}$. On the other hand $\zeta_{3}^{}$ corresponds to the geometrical construction based on points a, v. According to the simple form of (6.16) we want to leave $\zeta_{1}^{}$ and $\zeta_{2}^{}$. Therefore we have to maintain vt (due to $\zeta_{1}^{}$). The only possible replacement is v, t   $\longrightarrow$  v^$\prime$ = vt^-1, t. The orbit structure will then be spanned by the equation

$$\xi_{v^{\prime}}^{-2}$$v^$\prime$t² = $$\xi_{a}^{-2}$$at².

The second pair a, t gives another possibility. We may replace the pair a, t by a^$\prime$ = at, t to kill the axis $\zeta_{3}^{}$. The orbit structure is now given by the equation

$$\xi_{v}^{-1}$$vt = $$\xi_{a^{\prime}}^{-1}$$a^$\prime$t.

The reader will verify that the same simplified geometrical picture may be obtained after each substitution. Therefore an analysis of the universal graph manifold enables us to eliminate the undesired variables of the current task. However in many complicated cases the above method of elimination will entail serious combinatorial problems. There are many possible ways of reduction and our symmetry approach does not favor any.

Example 6.6   The molecular polarizability described by the Debye's equation is our next example. The dipole momentum P of one mole of some substance depends on the absolute temperature according to the equation

P = a + $${\frac{b}{T}}$$,

where a, b are constants and T denotes the temperature (see [2] for more details). As before, we disregard the known dependence and try to restore it from the real data.

Figure: a) The universal graphs in the two dimensional space of models X. $\cal {G}$₁ corresponds to (6.18), $\cal {G}$₂ to (6.19) and $\cal {G}$₃ comes from (6.20). b) The universal graph $\cal {G}$ for the Debye's equation. It is given by (6.22).

If $\omega_{1}^{}$ $\in$ $\cal {W}$ denotes the dimension of T and $\omega_{2}^{}$ $\in$ $\cal {W}$ the dimension of P then the dimensions of a, b are

p(a) = $$\omega_{2}^{}$$,      p(b) = $$\omega_{1}^{}$$$$\omega_{2}^{}$$.

The value P can be constructed: as the rescaled with $\rho$ point a or from the two points b, T. Let $\xi$ mark the intersection of the plane given by P, b, T with the dimensionless fiber (the space $\cal {W}$ is two dimensional). The orbit structure is then spanned by the equation

$$\rho \xi$$ (6.17)

Accordingly we have the one orbit parameter. Orbits also have standard form of half-lines originating at $\rho$ = 0,$\xi$ = 0. The universal graph $\cal {G}$₁ "found from numerical data" has the form

$$\rho \xi \rho \xi$$ (6.18)

i.e., it is the hyperbola with asymptotes $\rho$ = 1,$\xi$ = 1. This configuration is rather uninteresting. However let us change the problem somewhat.

We shift the zero at the temperature scale from 0 to some constant T₀. Moreover for simplicity (to not increase the dimension of the space of models) we include a into P and define

P^$\prime$ = P - a = $${\frac{b}{T^{\prime}-T_{0}}}$$.

Now the orbit structure is given by

P^$\prime$ = $$\xi$$bT^{$\prime$ - 1} = $$\xi_{0}^{}$$bT₀^-1.

Once more we obtain the standard orbit structure but the universal graph becomes less trivial. A small amount of algebra yields $\cal {G}$₂ with the equation:

$$\xi \xi_{0}^{}$$ (6.19)

The resulting geometrical picture is even simpler, the last equation represents the half-line depicted in Fig. 6.9 a). Note however that this line intersects only certain orbits. From the point of view of orbits the universal graph becomes singular since it crosses the orbit $\xi$ = $\xi_{0}^{}$ at infinity.

The last two universal graphs $\cal {G}$₁,$\cal {G}$₂ are presented in Fig. 6.9 a). Changing the sign T₀ we arrive at the equation

P^$\prime$ = P - a = $${\frac{b}{T^{\prime}+T_{0}}}$$

with the same orbit structure but the universal graph $\cal {G}$₃ now has the form

$$\xi \xi_{0}^{}$$ (6.20)

The line coming from the last equation is also depicted in Fig. 6.9 a). Please note that it is similar to the initial graph from the equation (6.18). The similarity here means that we may deform one graph on another with suitably regular transformation of the space of models. The regular transformations are these mappings which do not change the orbit structure. In practice it means shifts along the orbit half-lines. The universal graphs $\cal {G}$₁ and $\cal {G}$₃ from Fig. 6.9 a) are similar but $\cal {G}$₂ is completely different. There is no regular transformation shifting $\cal {G}$₂ onto $\cal {G}$₁ or $\cal {G}$₃.

The transformation replacing $\xi$ + $\xi_{0}^{}$ = 1 with $\xi$$\rho$ = $\xi$ + $\rho$ is f (x) = ${\frac{1}{x}}$. Note however that $\rho$ and $\xi_{0}^{}$ are connected with very different constants. The similarity of the universal graphs is related to the functional dependence only and has nothing in common with physics of a given problem.

Finally let us take into account the constant a and examine the function

P = a + $${\frac{b}{T-T_{0}}}$$.

The underlying space of models becomes three dimensional with the orbit structure given by

$$\rho \xi \xi_{0}^{}$$ (6.21)

Now the universal graph $\cal {G}$ itself is the surface

$$\rho \xi \xi_{0}^{} \xi \xi_{0}^{}$$ (6.22)

depicted in Fig. 6.9 b). Clearly, the manifold presented in Fig. 6.9 a) does not resemble any of the universal graphs from Fig. 6.9 a).

Example 6.7   Many dependencies important in practice are taken directly from an experiment. Sometimes the theoretical background is lacking but the conformance with real, measurement data seems to be the main criterion. The Casson equation describing blood flow may be an example. Denoting the components of the stress tensor by $\tau_{ij}^{}$ and by $\dot{\gamma}_{ij}^{}$ the velocity of the shift deformation we have

Figure: The universal graph $\cal {G}$ for the Casson equation.

$$\sqrt{\tau_{ij}} \sqrt{\tau_{y}} \sqrt{\left(-\dot{\gamma}_{ij}\right)}$$ (6.23)

where $\sqrt{\tau_{y}}$ and s are certain characteristics of blood. The dimensions of the variables $\tau_{ij}^{}$, $\tau_{y}^{}$ is (force/length²) and $\dot{\gamma}_{ij}^{}$ has dimension (time)^-1. We refer the reader to chapter 2 of the book [19], where this example is taken from. Of course the characteristic constant s (connected with viscosity of a liquid) has the dimension (mass/length).

From our point of view, considering relevant geometrical methods, this problem should not be complicated. The effective space of pure dimension $\cal {W}$ can be two - dimensional only. After some calculations we get for the orbit structure

$$\sqrt{\tau_{ij}} \rho \sqrt{\tau_{y}} \xi^{-1}_{} \sqrt{\left(-\dot{\gamma}_{ij}\right)}$$ (6.24)

where $\xi$ lies at the dimensionless fiber and $\rho$ $\in$ $\bf R_{+}^{}$ is the suitable scaling factor from outside of the dimensional space. In effect orbits are the hyperbolas and the universal graph $\cal {G}$ has the simple form

$${\frac{1}{\rho}} \xi$$ (6.25)

The current configuration in the space of models is depicted in Fig. 6.10 Note that we have the two main characteristics of the universal graph $\cal {G}$. The first is the intersection with the axis $\rho$: the point $\rho$ = 1,$\xi$ = 0 in the picture. The second characteristic becomes the asymptote $\xi$ $\equiv$ 1 also sketched in the figure.

However the current usage of the Casson equation is completely different. Assuming the functional dependence in the form (6.25) we seek the constants $\sqrt{\tau_{y}}$ and s to fit the experimental data for $\sqrt{\tau_{ij}}$ and $\sqrt{\left(-\dot{\gamma}_{ij}\right)}$. Let us accept the number

x = $${\frac{s\sqrt{\left(-\dot{\gamma}_{ij} \right)}}{\sqrt{\tau_{y}}}}$$,

as the orbit parameter. The value x $\rightarrow$ 0 corresponds to orbits tending to the axes $\rho$ = 0 and $\xi$ = 0. At the same time in the limit x $\rightarrow$ 0 the function (6.23) reduces to

$$\sqrt{\tau_{ij}}$$ = $$\sqrt{\tau_{y}}$$

and the whole dependence is given by the constant $\sqrt{\tau_{y}}$. We may equally say that this constant fixed the characteristic point $\rho$ = 1,$\xi$ = 0 of the universal graph. The opposite limit x $\rightarrow$ $\infty$ corresponds to the orbits escaping to infinity. For such orbits the universal graph does not differ from its asymptote and the function (6.23) reduces to

$$\sqrt{\tau_{ij}}$$ = s$$\sqrt{\left(-\dot{\gamma}_{ij}\right)}$$.

Therefore the second characteristic constant s designates the asymptote $\xi$ $\equiv$ 1 of $\cal {G}$. Note that although the orbits depend only on the ratio of characteristic constants, the function (6.23) varies separately with each constant.

Consequently we may formulate the popular usage of the Casson equation in the language of the universal graph. The assumed functional form (6.23) is equivalent to the assumed form of the universal graph. We choose the characteristic constants in a way to fit as good as possible the characteristics of $\cal {G}$ with experimental data.

The reader may get confused here. The initial function (6.23) contents the two free parameters which may be chosen in a way to fit well the experimental data. However there is nothing free or unfixed in the universal graph equation (6.25). So it seems that we have transposed the theory with free parameters into the theory without any free parameter. It is an illusion only. To see what has really happened let's look at Fig. 6.10 a) (i.e., at the detail depicting the behavior in the vicinity of the characteristic point $\rho$ = 1). From the real experimental data we may reconstruct the values of the universal graph at the isolated orbits only. Moreover we have always some noise disturbing the exact values. The characteristic point $\rho$ is achieved with the help of some extrapolation procedure. However the line $\cal {G}$ is fixed and we choose the values of free parameters to fit this given line. The only difference is that our free parameters are now included in the definition of the coordinates along axes, but the final form of graph is given. In the traditional approach, the function (6.23) is depicted in axes $\sqrt{\tau_{ij}}$, $\sqrt{-\dot{\gamma_{ij}}}$ and the free parameters are included in the graph. In some sense our approach is opposite to the classical one. We fit the axes to the given graph whereas the classical method fits the graph to the given axes. We induce readers to reexamine the universal graphs from the Example 6.6 in this context.

Example 6.8   Strenuous work has been done to analyze the effect of the base choice on the approximation procedures for the Bernoulli equation

$$\rho {\textstyle\frac{1}{2}} \rho \rho {\textstyle\frac{1}{2}} \rho$$ (6.26)

The results are collected in the paper [13]. For comparison let us examine the same problem but with the help of our geometrical approach.

Figure: The configuration of dimensions of variables from the Bernoulli equation in space $\cal {W}$. For simplicity we have labelled the suitable dimensions of quantities with the same symbols as the quantities themselves.

Endeavours were made to restore the functional dependence of the pressure p₂ at the height h₂ where the flow velocity of the liquid is v₂. The pressure p₁ corresponds to the point at height h₁ and the flow velocity v₁. The density of liquid was assumed to be $\rho$ and the acceleration of gravity was constant and equalled g. In effect p₂ is supposed to be the dimensional function of seven arguments v₁², v₂², h₁, h₂, p₁, $\rho$ and g. The dimensions of arguments and the value are obvious and their configuration in the space of pure dimensions $\cal {W}$ is depicted in Fig. 6.11

We immediately notice that p₂ lies in the plane given by $\rho$ and v₁² (or v₂²). Therefore the value p₂ can be constructed from the two pairs of points $\rho$, v₁² and $\rho$, v₂². Another possibility is a construction determined by triples h₁,$\rho$, g and h₂,$\rho$, g. Of course we may also obtain p₂ as rescaled p₁. The suitable equations for planes are

$$\xi^{1-t-u}_{}$$$$\rho^{t}_{}$$(v²)^u,

and for triples we have

$$\theta^{1-t-u-w}_{}$$$$\rho^{t}_{}$$h^ug^w.

To be concise we have omitted the indices 1, 2 in all variables. Solving simple linear equations we will get the following system of equations for the orbit structure

$$\eta \xi_{1}^{-1} \rho \xi_{2}^{-1} \rho \theta_{1}^{-2} \rho \theta_{2}^{-2} \rho$$ (6.27)

The resulting space of models becomes five - dimensional and it is completely filled with orbits. We conclude that the problem cannot be simplified.

The universal orbit parameters, good in any dimensional base are for example

x₁ = $${\frac{\rho v_{1}^{2}}{p_{1}}}$$,    x₂ = $${\frac{\rho v_{2}^{2}}{\rho v_{1}^{2}}}$$ = $${\frac{v_{2}^{2}}{v_{1}^{2}}}$$,    x₃ = $${\frac{h_{1}\rho g}{\rho v_{2}^{2}}}$$ = $${\frac{h_{1}g}{v_{2}^{2}}}$$,    x₄ = $${\frac{h_{2}\rho g}{h_{1}\rho g}}$$ = $${\frac{h_{2}}{h_{1}}}$$.

However the reader may easily involve orbit parameters more symmetric in incoming arguments. In the above variables $\eta$, $\xi_{1}^{}$,$\xi_{2}^{}$, $\theta_{1}^{}$,$\theta_{2}^{}$ the universal graph $\cal {G}$ has the form

+ $$\left(\vphantom{\theta_{2}^{2}-\theta_{1}^{2}}\right.$$$$\theta_{2}^{2}$$ - $$\theta_{1}^{2}$$ $$\left.\vphantom{\theta_{2}^{2}-\theta_{1}^{2}}\right)$$ + $${\textstyle\frac{1}{2}}$$($$\xi_{2}^{}$$ - $$\xi_{1}^{}$$) = $${\frac{1}{\eta}}$$.

In an obvious way after a change of variables

$$\zeta_{0}^{}$$ = $${\frac{1}{\eta}}$$,    $$\zeta_{1}^{}$$ = $$\theta_{1}^{2}$$,    $$\zeta_{2}^{}$$ = $$\theta_{2}^{2}$$

we will get the linear equation for $\cal {G}$. The functional dependencies in various dimensional bases correspond to the intersection of the universal graph manifold with the orbits. The geometrical picture is quite close to Fig. 6.7 b) but, of course, the space is five - dimensional. Generally we have found the only one obstacle in this problem, the large dimension of the space of models.

The reader is probably surprised why we have underlined a few times the difference between the numbers from the dimensionless fiber and the elements from the set $\bf R_{+}^{}$. Yet it is important. The numbers from $\bf R_{+}^{}$ are treated as elements from the group $\bf R_{+}^{}$ acting along any fiber. Thus, they become the geometrical transformations and have no representation as such. In contrast, the elements at the dimensionless fiber are the ordinary points from W. Therefore writing equations such as (6.8), (6.14), (6.17), (6.21), (6.24) or (6.27) we link the elements from two quite distinct structures. One should be very careful here.

To make this point clear we present other, more advanced examples in which the case of "isolated subspaces" appears first time. This specific configuration closes our list of typical geometrical constructions required for the universal graph method.

Example 6.9   Let us slightly modify the problem discussed in Example 6.8.

Figure: a) The configuration of the variables in the dimensional space W. b) The same as a) but an auxiliary point a in the plane generated by fibers of pressure and length is added. For simplicity we have labelled fibers with the same letters as variables.

Supposing that we have a tank filled with a liquid and the density of the liquid is constant. Moreover the external pressure is missing, i.e., it vanishes. If the pressure p₁ corresponds to the point at depth d₁ and if at the depth d₂ pressure equals p₂ then

$${\frac{p_{1}}{p_{2}}} {\frac{d_{1}}{d_{2}}}$$ (6.28)

This is an usual formula for the hydrostatic pressure in a liquid.

Now having forgotten (6.28) we want to restore it from experimental data. Accordingly, we investigate the form of the dimensional function p₂(p₁, d₁, d₂). The configuration of all quantities d₁, d₂, p₁, p₂ has been depicted in Fig. 6.11 a).

For comparisson we begin with the method relevant to Theorem $\pi$. Assuming that we have chosen d₁, p₁ as the dimensional basis. Then we have to construct the plane $\cal {P}$₀ passing through three points d₁, p₁ and 1 at the dimensionless fiber. It includes all points of the form

$$\cal {P}$$₀ :      d₁^tp₁^u1^{1 - t - u} = d₁^tp₁^u,          t, u $$\in$$ R.

The first parallel plane $\cal {P}$_d passing through d₂ becomes the collection of points

$$\cal {P}$$_d :      $$\lambda_{d}^{}$$d₁^tp₁^u,          t, u $$\in$$ R.

The value of $\lambda_{d}^{}$ comes from the condition

d₂ = $$\lambda_{d}^{}$$d₁^tp₁^u.

Clearly t = 1, u = 0 and $\lambda_{d}^{}$ = d₂/d₁. In a similar way we may construct the plane $\cal {P}$_p which passes through point p₂. The crossing points with the dimensionless fiber are $\lambda_{d}^{}$ for $\cal {P}$_d and $\lambda_{p}^{}$ for $\cal {P}$_p, where

$$\lambda_{d}^{}$$ = $${\frac{d_{2}}{d_{1}}}$$,      $$\lambda_{p}^{}$$ = $${\frac{p_{2}}{p_{1}}}$$.

Accordingly we have transferred the dimensional dependence onto the dimensionless fiber. For the fixed dimensional basis, the function p₂(p₁, d₁, d₂) is spanned by the numerical dependence $\lambda_{p}^{}$($\lambda_{d}^{}$). In our case we easily find

$$\lambda_{p}^{}$$($$\lambda_{d}^{}$$) $$\equiv$$ $$\lambda_{d}^{}$$.

The second dimensional basis p₁, d₂ allows a similar treatment.

Figure: a) The orbit parameterisation with help of x₁, x₂ shown in the space of models X. b) The more complicated choice of an auxiliary point b outside of the plane generated by the fibers of pressure and length. The position of the plane bdp is fixed by the point $\xi$ at the dimensionless fiber.

Our method of geometrical constructions will now be applied to the current configuration from Fig. 6.12 a). Of course the point p₂ can be achieved from p₁ with help of scaling by the factor $\rho$. But what to do next? Any attempt to get a point at the fiber of pressure from the points d₁, d₂ produces no effect. On the other hand it is clear that p₂ depends on d₁ and d₂. Such specific, undesired configuration has occured because p₁, p₂ and d₁, d₂ occupy the dimensionaly independent fibers.

In general, such a case, when a group of arguments cannot affect the value because of dimensional independence, will be called the case of isolated subspaces. In the geometrical sense the two planes (depicted in Fig. 6.12 a)) containing the dimensionless fiber and fibers of pressure and length respectively, are separated in a sense. We cannot shift any construction from one plane into the second because of dimensional independence. Of course the isolated subspace is just the right plane generated by the dimensionless fiber and the fiber of length.

In Fig. 6.12 b) we have depicted the possible solution of the isolated subspaces case. Assuming we have chosen an auxiliary point outside the marked fibres in the plane generated by the fibres of pressure and length. In fact, in geometry there are many different constructions involving some additional, auxiliary points. However the final result must be independent of them.

The line given by the pair of points a, d₁ crosses the fiber of pressure at the point p. Supposing that the dimension of p is $\omega_{p}^{}$ $\in$ $\cal {W}$ and similarly the dimension of d₁, d₂ equals $\omega_{d}^{}$ $\in$ $\cal {W}$. Since the point a belongs to the plane generated by both fibers, its dimension should be

p(a) = $$\omega_{p}^{\alpha}$$$$\omega_{d}^{1-\alpha}$$

for some $\alpha$ $\in$ R. The line passing through a and d₁

a^td₁^{1 - t},        t $$\in$$ R,

approaches the fiber of pressure for t₀ satisfying

$$\omega_{p}^{\alpha t_{0}}$$$$\omega_{d}^{(1-\alpha)t_{0}}$$$$\omega_{d}^{1-t_{0}}$$ = $$\omega_{p}^{}$$.

Hence t₀ = 1/$\alpha$ and

p = a^t₀d₁^{1 - t₀}.

The value point p₂ is then obtained as rescaling point p. Of course, the similar construction also applies to d₂ but the auxiliary point a should be kept the same all the time. We have already met similar configuration in the Example 6.5 (cf. (6.13)). Denoting the suitable scaling factors by $\rho_{1}^{}$ for d₁ and $\rho_{2}^{}$ for d₂ we will get

$$\rho \rho_{1}^{} \rho_{2}^{}$$ (6.29)

We recall that p₂ comes from p₁ with scaling by $\rho$ and t₀ depends exclusively on dimensions.

Our space of models X becomes three dimensional with the coordinates $\rho$,$\rho_{1}^{}$,$\rho_{2}^{}$. The orbits fill the whole X and they are labelled by two parameters. We have chosen

x₁ = $${\frac{p_{1}}{a^{t_{0}}d_{1}^{1-t_{0}}}}$$,        x₂ = $${\frac{a^{t_{0}}d_{1}^{1-t_{0}}}{a^{t_{0}}d_{2}^{1-t_{0}}}}$$ = $$\left(\vphantom{\frac{d_{1}}{d_{2}}}\right.$$$${\frac{d_{1}}{d_{2}}}$$ $$\left.\vphantom{\frac{d_{1}}{d_{2}}}\right)^{1-t_{0}}_{}$$

for simplicity.

According to (6.28) and (6.29) the universal graph $\cal {G}$ has the form

$$\rho^{1-t_{0}}_{} {\frac{\rho_{1}}{\rho_{2}}}$$ (6.30)

On the other hand the solution of this problem cannot depend on the choice of the auxiliary point a. Therefore the universal graph $\cal {G}$ should be constant along orbits with fixed x₁. In general $\cal {G}$ comes from the equation of the form

F($$\rho$$,$$\rho_{1}^{}$$,$$\rho_{2}^{}$$) = F₀ = const.

Equivalently we may write (cf. (6.28))

F$$\left(\vphantom{\frac{p_{2}}{p_{1}},p_{2}a^{-t_{0}}d_{1}^{t_{0}-1}, p_{2}a^{-t_{0}}d_{2}^{t_{0}-1}}\right.$$$${\frac{p_{2}}{p_{1}}}$$, p₂a^-t₀d₁^{t₀ - 1}, p₂a^-t₀d₂^{t₀ - 1}$$\left.\vphantom{\frac{p_{2}}{p_{1}},p_{2}a^{-t_{0}}d_{1}^{t_{0}-1}, p_{2}a^{-t_{0}}d_{2}^{t_{0}-1}}\right)$$ = F₀.

The independence of the value of a means that

$$\rho \rho_{1}^{} \rho_{2}^{} \rho {\frac{\rho_{1}}{\rho_{2}}}$$ (6.31)

consistently with (6.30). Note however that t₀ has appeared in (6.30) but this dependence comes from the dimensions solely. The power t₀ can be removed from (6.30) in one way only. One has to redefine $\rho_{1}^{}$ and $\rho_{2}^{}$

$$\rho_{1}^{}$$ = ($$\rho_{1}^{\prime}$$)^{1 - t₀},          $$\rho_{2}^{}$$ = ($$\rho_{2}^{\prime}$$)^{1 - t₀}.

Inserting this into (6.29) we will get

p₂ = $$\rho$$p₁ = a^t₀$$\left(\vphantom{\rho_{1}^{\prime}d_{1}}\right.$$$$\rho_{1}^{\prime}$$d₁$$\left.\vphantom{\rho_{1}^{\prime}d_{1}}\right)^{1-t_{0}}_{}$$ = a^t₀$$\left(\vphantom{\rho_{2}^{\prime}d_{2}}\right.$$$$\rho_{2}^{\prime}$$d₂$$\left.\vphantom{\rho_{2}^{\prime}d_{2}}\right)^{1-t_{0}}_{}$$.

It means that at first we scale the suitable point d_i and only then form the line passing through a, d_i. Moreover in new variables $\rho_{1}^{\prime }$, $\rho_{2}^{\prime }$ the condition (6.31) has the similar form

$$\rho \rho_{1}^{\prime} \rho_{2}^{\prime} \rho {\frac{\rho_{1}^{\prime}}{\rho_{2}^{\prime}}}$$ (6.32)

Figure: a) The universal graph $\cal {G}$ results from the equation $\rho$ = $\rho_{1}^{\prime }$/$\rho_{2}^{\prime }$. $\cal {G}$ does not depend on orbit parameter x₁ from Fig. 6.13 a). b) The new orbit structure and the related universal graph $\cal {G}$.

Finally we have arrived at few corollaries related to the case of isolated subspaces. Choosing an auxiliary point we have to
- perform scaling at fibers of arguments, not at value fiber,
- impose some conditions to reduce the ambiguity of the universal graph form.

In our case the orbit parametrisation with the help of x₁, x₂ is depicted in Fig. 6.13 a) (cf. Fig. 6.7 b)). Independence of the value of a demands $\cal {G}$ to be constant along each line x₁. The universal graph $\cal {G}$ is shown in Fig. 6.14 a). Being independent of x₁, $\cal {G}$ becomes woven with half-lines x₂ =const.

On the other hand the fixed value of x₂ provides the plane in the space of models X. We may as well interpret this plane as an orbit for the assumed x₂ since x₁ corresponds to the auxiliary point a. This is an important feature in the case of isolated subspaces.

In all previous examples (from 6.3 to (6.8) orbits were represented as curves filled with points. Here we have the first case in which orbits become depicted by more complicated geometrical objects. Each plane from Fig. 6.14 b) represents a single orbit labelled by x₂. However now, such planes do not consist of single points but they become filled with half-lines $\rho$ =const. The universal graph $\cal {G}$ for the new representation of orbits has been drawn in Fig. 6.14 b).

We compare the current configuration with the Example 6.4. According to (6.10), (6.11) the universal graph has the form F($\xi_{1}^{}$,$\xi_{2}^{}$) = 1. The single elements of an orbit were given by $\xi_{1}^{}$ =const, $\xi_{2}^{}$ =const, yielding a point in the space of models X. In the current example the form of $\cal {G}$ is given by (6.32). The single element of an orbit corresponds to $\rho$ =const and $\rho_{1}^{\prime }$/$\rho_{2}^{\prime }$ =const. This yields a half-line in the space of models X. Therefore the pictures Fig. 6.14 b) and Fig. 6.4 b) are closely related. The only difference consists in the idea of a "point" in the space of models X.

Finally we arrive at a very important conclusion concerning the case of isolated subspaces. Applying the constructions based on auxiliary points, we replace the points of the space of models X with some standard linear manifolds (half-lines in the current example). When the structure of X elements is properly chosen then the dependence on the auxiliary points disappears as in Fig. 6.14 b).

The reader is probably surprised why even such a trivial problem as here discussed entails so detailed geometrical considerations. The reason is rather simple. According to (6.28) the value p₂ cannot be obtained merely from d₁ and d₂. Only the suitable ratios are comparable. From the physical point of view the case of isolated subspaces may appear when an important characteristic is missing. It may be even a physical constant. In the previous Example 6.8 there were no such problems by dint of the term $\rho$g (density times acceleration of gravity). In (H.5) the orbit parameter x₄ has appeared in natural way. The auxiliary point substitutes this missing variable. Therefore quite often it becomes more natural to add some specific, related to the given problem, physical constant to the list of arguments instead of a free auxiliary point.

An example of such a situation has been presented in Fig. 6.13 b). However then, the added constant b does not necessarily take place in the plane generated by the fibers of p and d. Treating the point b from Fig. 6.13 b) as an auxiliary one we should modify our construction. The plane including the points b, d, p,$\xi$ then replaces line ad₁ from Fig. 6.12 b). Keeping b constant we fix the position of the plane with the help of $\xi$ (at the dimensionless fiber) and impose the consistency condition at the value fiber. Such a method was described in the Example 6.4. Note however that we always demand the final universal graph $\cal {G}$ to be independent of the chosen auxiliary point.

So far, in all examples including the current one we have kept the same strategy and constructed the value from arguments. The method based on an auxiliary point allows another approach. We may equivalently insert the value into the isolated subspace with the help of an auxiliary point. The effective algorithm derives just from this idea.

Figure: The scaling along the fiber of d with help of $\xi_{1}^{\prime }$ $\in$ $\bf R_{+}^{}$ and the point $\xi_{1}^{}$ fixing the position of the line b₀d₁ at the dimensionless fiber. The auxiliary point b₀ inserted into the isolated subspace occupies the central fiber.

The dimension of the auxiliary point b from Fig. 6.13 b) is

p(b) = $$\omega_{p}^{\alpha}$$$$\omega_{d}^{\beta}$$.

The line p₂b with the equation

p₂^{1 - t}b^t,        t $$\in$$ R

crosses the isolated subspace at the point b₀ for t coming from the equation

(1 - t) + $$\alpha$$t = 0.

Elementary calculations give for the crossing point b₀:

b₀ = p₂^{${\frac{\alpha}{\alpha - 1}}$}b^{${\frac{1}{1-\alpha}}$},

with the dimension p(b₀)

p(b₀) = $$\omega_{d}^{\frac{\beta}{1-\alpha}}$$.

Next we treat this new point b₀ as the value of a new dimensional function with arguments d₁, d₂. We assume b to lay outside of the fiber d. Otherwise we will go back to the configuration depicted in Fig. 6.12 b).

For b₀ and d₁, d₂ occupying different fibers we may apply the methods from previous examples. Constructing the lines b₀d₁ and b₀d₂ as in the Example 6.1 we arrive at the equations:

$${\frac{\alpha}{\alpha - 1}} {\frac{1}{1-\alpha}} \xi_{1}^{\frac{\alpha+\beta-1}{\alpha-1}} {\frac{\beta}{\alpha-1}} \xi_{2}^{\frac{\alpha+\beta-1}{\alpha-1}} {\frac{\beta}{\alpha-1}}$$ (6.33)

or equivalently

p₂ = $$\rho$$p₁ = $$\xi_{1}^{\frac{\alpha+\beta-1}{\alpha}}$$b^{${\frac{1}{\alpha}}$}d₁^{- ${\frac{\beta}{\alpha}}$} = $$\xi_{2}^{\frac{\alpha+\beta-1}{\alpha}}$$b^{${\frac{1}{\alpha}}$}d₂^{- ${\frac{\beta}{\alpha}}$},

where $\rho$ is the same as in (6.29). We require the universal graph $\cal {G}$ to be independent of the choice of b. Therefore $\cal {G}$ is given by the equation of the form

F($$\rho$$,$$\xi_{1}^{}$$,$$\xi_{2}^{}$$) = F$$\left(\vphantom{\rho,\frac{\xi_{1}}{\xi_{2}}}\right.$$$$\rho$$,$${\frac{\xi_{1}}{\xi_{2}}}$$ $$\left.\vphantom{\rho,\frac{\xi_{1}}{\xi_{2}}}\right)$$ = const.

Note however that the ratio

$${\frac{\xi_{1}}{\xi_{2}}}$$ = $$\left(\vphantom{\frac{d_{1}}{d_{2}}}\right.$$$${\frac{d_{1}}{d_{2}}}$$ $$\left.\vphantom{\frac{d_{1}}{d_{2}}}\right)^{\frac{\beta} {\alpha+\beta -1}}_{}$$

depends on b through the power exponents $\alpha$ and $\beta$. To remove this dependence we redefine $\xi_{1}^{}$ and $\xi_{2}^{}$

$$\xi_{1}^{} \xi_{1}^{\prime} {\frac{\beta}{\alpha+\beta-1}} \xi_{2}^{} \xi_{2}^{\prime} {\frac{\beta}{\alpha+\beta-1}}$$ (6.34)

One more we insert these redefined variables in the orbit equation (6.33):

b₀ = $$\left(\vphantom{\xi_{1}^{\prime}d_{1}^{-1}}\right.$$$$\xi_{1}^{\prime}$$d₁^-1$$\left.\vphantom{\xi_{1}^{\prime}d_{1}^{-1}}\right)^{\frac{\beta}{\alpha-1}}_{}$$ = $$\left(\vphantom{\xi_{2}^{\prime}d_{2}^{-1}}\right.$$$$\xi_{2}^{\prime}$$d₂^-1$$\left.\vphantom{\xi_{2}^{\prime}d_{2}^{-1}}\right)^{\frac{\beta}{\alpha-1}}_{}$$.

As before the variable change (6.34) reduces to the scaling at the fiber of arguments. The resulting universal graph $\cal {G}$ does not differ from Fig. 6.14 b). The effective geometrical construction corresponding to the variable $\xi_{1}^{\prime }$ in the isolated subspace is depicted in Fig. 6.15 For comparison we have shown also the line related to $\xi_{1}^{}$. The variable $\xi_{1}^{}$ is represented as a point at the dimensionless fiber whereas $\xi_{1}^{\prime }$ denotes the element from the group $\bf R_{+}^{}$ acting along fibers.

Example 6.10   The variables inside the isolated subspace need not occupy one single fiber. Let us slightly modify the problem from the previous Example 6.9. The quantities p₁, p₂ and d₁ remain unchanged but suppose that the depth d₂ is measured with the help of a sonic finder. Denoting the (constant) sound velocity by v we have

d₂ = vt,

Figure 6.16: The isolated subspaces. a) corresponds to the Example 6.9. In the present example the isolated subspace b) becomes two dimensional.

where 2t is the time gone from the emission to the recording of an acoustic impulse. Now we accept p₁, d, v, t as our new arguments of the dimensional function p₂(p₁, d, v, t) (for sake of simplicity we have omitted the index 1 in d₁). As is explained in Fig. 6.16 the isolated subspace has dimension two now.

Figure: a) The scaling along fibers in the isolated subspace. The position of the plane is fixed with help of the two numbers $\eta_{d}^{-1}$ and $\eta_{t}^{-1}$ form the group $\bf R_{+}^{}$. b) This manifold is the single "point" in the space of models X. This curved surface replaces the lines $\rho$ =const from Fig. 6.14 b).

To be concise we involve the notation for dimensions in $\cal {W}$

length = $$\omega_{L}^{}$$,      time = $$\omega_{T}^{}$$,      pressure = $$\omega_{p}^{}$$.

Then the dimensions of variables are

d $$\rightarrow$$ $$\omega_{L}^{}$$,      t $$\rightarrow$$ $$\omega_{T}^{}$$,      v $$\rightarrow$$ $$\omega_{L}^{}$$$$\omega_{T}^{-1}$$,      p $$\rightarrow$$ $$\omega_{p}^{}$$.

Clearly we cannot repeat the construction from Fig. 6.12 b) but the more general choice of an auxiliary point b as in Fig. 6.13 b) is still applicable.

To compare the universal graph method and the classical approach we begin with the direct application of Theorem $\pi$. There are three dimensional bases (p₁, d, v; p₁, d, t; p₁, v, t) among arguments p₁, d, v, t. In each one the resulting functional form is

$${\frac{p_{2}}{p_{1}}}$$ = f$$\left(\vphantom{\frac{vt}{d}}\right.$$$${\frac{vt}{d}}$$ $$\left.\vphantom{\frac{vt}{d}}\right)$$,

where f denotes some numerical function f : $\bf R_{+}^{}$ $\rightarrow$ $\bf R_{+}^{}$.

The auxiliary point b should allow to achieve the points at the pressure fiber with help of b and two points among the three d, v, t. Consistent with linear algebra point b always exists in isolated subspaces. For the purpose of definiteness we may accept $\rho$ (the density of liquid) as b, but the concrete choice of b is not necessary in general considerations. It suffices to assume the dimension of b in the form

b $$\rightarrow$$ $$\omega_{p}^{\mu}$$$$\omega_{L}^{\lambda}$$$$\omega_{T}^{\theta}$$.

Of course R $\ni$ $\mu$ $\not=$ 0. Next we insert the value p₂ into the isolated subspace with the help of the auxiliary point b. Similarly as in the Example 6.9 the line p₂b pierces the isolated subspace at point b₀,

b₀ = b^{${\frac{1}{1-\mu}}$}p₂^{${\frac{\mu}{\mu-1}}$},

with dimension equalling

p(b₀) = $$\omega_{L}^{\frac{\lambda}{1-\mu}}$$$$\omega_{T}^{\frac{\theta}{1-\mu}}$$.

Now we apply our previous technique to achieve the value b₀ with the help of the arguments d, t, v. There are three independent planes and we fix their position by points $\xi_{1}^{}$,$\xi_{2}^{}$,$\xi_{3}^{}$ at the dimensionless fiber as listed below:

b₀dt   $$\rightarrow$$  $$\xi_{1}^{}$$,        b₀dv   $$\rightarrow$$  $$\xi_{2}^{}$$,        b₀vt   $$\rightarrow$$  $$\xi_{3}^{}$$.

After some standard calculations we obtain

$$\xi_{1}^{\frac{1-\mu-\lambda-\theta}{1-\mu}} {\frac{\lambda}{1-\mu}} {\frac{\theta}{1-\mu}} \xi_{2}^{\frac{1-\mu-\lambda}{1-\mu}} {\frac{\lambda+\theta}{1-\mu}} {\frac{\theta}{1-\mu}} \xi_{3}^{\frac{1-\mu-2\lambda-\theta}{1-\mu}} {\frac{\lambda}{1-\mu}} {\frac{\lambda+\theta}{1-\mu}}$$ (6.35)

Finally, the resulting orbit equation has the form

$$\rho \xi_{1}^{\frac{\mu+\lambda+\theta-1}{\mu}} {\frac{1}{\mu}} {\frac{\lambda}{\mu}} {\frac{\theta}{\mu}} \xi_{2}^{\frac{\mu+\lambda-1}{\mu}} {\frac{1}{\mu}} {\frac{\lambda+\theta}{\mu}} {\frac{\theta}{\mu}} \xi_{3}^{\frac{\mu+2\lambda+\theta-1}{\mu}} {\frac{1}{\mu}} {\frac{\lambda}{\mu}} {\frac{\lambda+\theta}{\mu}}$$ (6.36)

As we see the ratios $\xi_{1}^{}$/$\xi_{2}^{}$ and $\xi_{2}^{}$/$\xi_{3}^{}$ depend on the auxiliary point through the power exponents.

Now let us apply the modified method presented in Fig. 6.17 a) with scaling along fibers of arguments. The suitable numbers from $\bf R_{+}^{}$ are $\eta_{d}^{-1}$,$\eta_{t}^{-1}$,$\eta_{v}^{-1}$. Assuming the consistency condition at point b₀ we get (cf. (6.36))

$$\eta_{d}^{-\frac{\lambda}{1-\mu}} \eta_{t}^{-\frac{\theta}{1-\mu}} {\frac{\lambda}{1-\mu}} {\frac{\theta}{1-\mu}} \eta_{d}^{-\frac{\lambda+\theta}{1-\mu}} \eta_{v}^{\frac{\theta}{1-\mu}} {\frac{\lambda+\theta}{1-\mu}} {\frac{\theta}{1-\mu}} \eta_{v}^{-\frac{\lambda}{1-\mu}} \eta_{t}^{-\frac{\lambda+\theta}{1-\mu}} {\frac{\lambda}{1-\mu}} {\frac{\lambda+\theta}{1-\mu}}$$ (6.37)

This gives the following orbit equation

$$\eta_{v}^{}$$$$\eta_{t}^{}$$d = $$\eta_{d}^{}$$vt,

which does not depend on the auxiliary point b. Clearly the numbers $\eta_{d}^{}$,$\eta_{t}^{}$,$\eta_{v}^{}$ come from arguments solely and do not include any information about value b₀. In contrast the points $\xi_{1}^{}$,$\xi_{2}^{}$,$\xi_{3}^{}$ at the dimensionless fiber are found for the assumed value b₀. This constitutes the difference between (6.35) and (6.37).

We want the universal graph $\cal {G}$ to be independent of b. Therefore $\cal {G}$ is given by the equation of the form

F($$\rho$$,$${\frac{\eta_{t}\eta_{v}}{\eta_{d}}}$$) = const.

In our case we obtain

$$\rho$$$$\eta_{d}^{}$$ = $$\eta_{t}^{}$$$$\eta_{v}^{}$$.

The space of models X is four dimensional with coordinates $\rho$,$\eta_{d}^{}$,$\eta_{t}^{}$,$\eta_{v}^{}$. However the "single point" in X corresponds to $\rho$ =const and $\eta_{t}^{}$$\eta_{v}^{}$/$\eta_{d}^{}$ =const. This gives the curved surface in X. For $\rho$ =const (the crosssection) it is depicted in Fig. 6.17 b). Each orbit consist of such points for different values of $\rho$.

Comparing with Fig. 6.14 b) we see that the more complicated structure of an isolated subspace entails the more complex manifold of a single orbit. The half-lines from Fig. 6.14 b) have been replaced by the surfaces depicted in Fig. 6.17 b). The universal graph $\cal {G}$ crosses each orbit at such a surface. However the dimension of the space of models exceeds our possibilities to produce pictures.

We encourage the reader to reexamine this example by replacing the movement with constant velocity (d₂ = vt) by (6.12) (i.e., d₂ = vt + ${\frac{1}{2}}$at²). The resulting orbit structure and the universal graph becomes quite interesting then.

Example 6.11  

Our aim is to demonstrate a method enabling us to obtain the mathematical model of the object shown in Fig. 6.18, on the basis of experimental data (the additional assumption is: we can measure and vary each input and output during the experimental research). The object is formed from the tube with a liquid flowing inside it; the physical properties of the liquid are $\rho$-density, $\rho_{el}^{}$-density of the electrical charge [3]. There is also the gravitational field (it is determined by $\vec{g}\,$) and the electric field that comes from the point charge q (the intensity of this field is determined by $\phi$). The current example demonstrates the universal graph method for the complex dimensional function.

Figure 6.18: The object for identification.

The distance between the point charge and the tube is denoted by d. The space outside the tube is filled with dielectric of dielectric constant $\epsilon$. During the experiment, we measure the pressure p₁, p₂ and the square of the velocity of the flowing liquid $\vec{v}_{1}^{2}$, $\vec{v}_{2}^{2}$ at two different points at height h₁ and h₂ as you see in the figure. Let us assume that the pressure p₂ is a value of the unknown dimensional function $\Phi$. Because the p₂ depends on the p₁, h₁, h₂, $\vec{v}_{1}^{2}$, $\vec{v}_{2}^{2}$, $\rho$, $\rho_{el}^{}$, $\phi_{1}^{}$, $\phi_{2}^{}$ we regard them as variables of the $\Phi$. In fact we know the formula defining the function $\Phi$:

$$\left(\vphantom{h_{1}-h_{2}}\right. \left.\vphantom{h_{1}-h_{2}}\right) \rho \left(\vphantom{\vec{v}_{1}^{2}-\vec{v}_{2}^{2} }\right. \vec{v}_{1}^{2} \vec{v}_{2}^{2} \left.\vphantom{\vec{v}_{1}^{2}-\vec{v}_{2}^{2} }\right) \rho \left(\vphantom{\displaystyle{q \over \epsilon\sqrt{d^{2}+r_{1}^{2}}} - \displaystyle{q \over \epsilon\sqrt{d^{2}+r_{2}^{2}}}}\right. {q \over \epsilon\sqrt{d^{2}+r_{1}^{2}}} {q \over \epsilon\sqrt{d^{2}+r_{2}^{2}}} \left.\vphantom{\displaystyle{q \over \epsilon\sqrt{d^{2}+r_{1}^{2}}} - \displaystyle{q \over \epsilon\sqrt{d^{2}+r_{2}^{2}}}}\right) \rho_{el}^{}$$ (6.38)

Notice that $\phi_{1}^{}$ and $\phi_{2}^{}$ become different when you vary the height h_i, r_i (i = 1, 2) or the $\epsilon$ or the distance d or the quantity of the electrical charge q.

In other words, we want to write the equation of the function $\Phi$, the variables of which are: u_{r + 1}, u_{r + 2}, ..., u_n, $\nu_{1}^{(1)}$, $\nu_{2}^{(1)}$, ..., $\nu_{m_{1}}^{(1)}$, $\nu_{1}^{(2)}$, $\nu_{2}^{(2)}$, ..., $\nu_{m_{2}}^{(2)}$, ..., $\nu_{m_{r}}^{(r)}$ and the value of $\Phi$ is the output y.

The experimental data are written in matrix U and Y:

$$\robtex { U = \left[ \begin{array}{cccccccc} \nu_{l(1)}^{(1)} & ... ...begin{array}{c} y_{(1)} \\ y_{(2)} \\ \ddots \\ y_{(t)} \end{array}\right].$$

To determine the form of the function $\Phi$, the mathematical model of each of r + 1 cases must be known. The main object is determined by the function $\Phi^{(M)}_{}$:

$$\Phi^{(M)}_{}$$ (6.39)

Accordingly, we assume that the rest of cases are characterized by functions $\Phi_{i}^{}$:

$$\Phi_{1}^{} \nu_{1}^{(1)} \nu_{2}^{(1)} \nu_{m1}^{(1)}$$ u₁ =

$$\Phi_{2}^{} \nu_{1}^{(2)} \nu_{2}^{(2)} \nu_{m2}^{(2)}$$ u₂ = (6.40)

$$\vdots \vdots$$

$$\Phi_{r}^{} \nu_{1}^{(r)} \nu_{2}^{(r)} \nu_{mr}^{(r)}$$ u_r =

Inserting (6.40) into (6.39), we obtain:

$$\Phi \left(\vphantom{ \Phi_{1}(\nu_{1}^{(1)},\nu_{2}^{(1)},\ldots,\nu_... ...u_{1}^{(r)},\nu_{2}^{(r)},\ldots,\nu_{mr}^{(r)}), u_{r+1},\ldots, u_{n}}\right. \Phi_{1}^{} \nu_{1}^{(1)} \nu_{2}^{(1)} \nu_{m1}^{(1)} \Phi_{r}^{} \nu_{1}^{(r)} \nu_{2}^{(r)} \nu_{mr}^{(r)} \left.\vphantom{ \Phi_{1}(\nu_{1}^{(1)},\nu_{2}^{(1)},\ldots,\nu_... ...u_{1}^{(r)},\nu_{2}^{(r)},\ldots,\nu_{mr}^{(r)}), u_{r+1},\ldots, u_{n}}\right)$$ (6.41)

The function $\Phi$ is called the complex dimensional function, so our aim is to find the form of the complex function, when matrices U and Y are known. To effect this we must make an approximation of the function, where variables and values are dimensional magnitudes. This problem might be solved with the help of the dimensional analysis, but the main theorem of this theory - the Theorem $\pi$ - necessitates the differentiation of the, so called, dimensional base in that set of variables. One can prove that the result of the approximation of function $\Phi$ depends on the choice of the base. The universal graph method, presented in this book, does not require choosing the base, so it does not mark out any variables in contrast to Buckingham's Theorem. The essence of the universal graph method (UGM) is a geometrical representation of the dimensional function. This function is equivalent to some manifold in specially constructed geometry. Let us now consider the function $\Phi$:

y = $$\Phi$$$$\left(\vphantom{ \Phi_{1}(\nu_{1}^{(1)},\nu_{2}^{(1)},\ldots,\nu_... ...1}^{(r)},\nu_{2}^{(r)},\ldots,\nu_{m_{r}}^{(r)}), u_{r+1},\ldots, u_{n}}\right.$$$$\Phi_{1}^{}$$($$\nu_{1}^{(1)}$$,$$\nu_{2}^{(1)}$$,...,$$\nu_{m_{1}}^{(1)}$$),...,$$\Phi_{r}^{}$$($$\nu_{1}^{(r)}$$,$$\nu_{2}^{(r)}$$,...,$$\nu_{m_{r}}^{(r)}$$), u_{r + 1},..., u_n$$\left.\vphantom{ \Phi_{1}(\nu_{1}^{(1)},\nu_{2}^{(1)},\ldots,\nu_... ...1}^{(r)},\nu_{2}^{(r)},\ldots,\nu_{m_{r}}^{(r)}), u_{r+1},\ldots, u_{n}}\right)$$.

Dimensions of $\nu_{i}^{}$, u_j are given by:

$$\left(\vphantom{x^{1}(y),\ldots, x^{k}(y)}\right. \left.\vphantom{x^{1}(y),\ldots, x^{k}(y)}\right)$$ p(y) =

$$\nu_{1}^{} \left(\vphantom{x^{1}(\nu_{1}),\ldots,x^{k}(\nu_{1})}\right. \nu_{1}^{} \nu_{1}^{} \left.\vphantom{x^{1}(\nu_{1}),\ldots,x^{k}(\nu_{1})}\right)$$ =

$$\vdots \vdots$$

$$\left(\vphantom{x^{1}(u_{m}),\ldots,x^{k}(u_{m})}\right. \left.\vphantom{x^{1}(u_{m}),\ldots,x^{k}(u_{m})}\right)$$ p(u_m) =

For instance, let us consider the velocity v; the dimension of v is: length to the power of 1, time to the power of -1, so p(v) = (1, - 1).

We divide the complex function $\Phi$ into r + 1 simple functions: $\Phi^{(M)}_{}$, $\Phi_{1}^{}$, ..., $\Phi_{r}^{}$. Then we construct one universal graph for each of these functions. The next step is to join r + 1 graphs, so we get the manifold, which represents complex function $\Phi$. At first the simple function $\Phi^{(M)}_{}$ will be examined (see the equation (6.39)).

If there are quantities u_i (for i $\leq$ n), that have the following property:

p(u_i1) = p(u_i2) =...= p(u_is),

we can say that u_i has the multiplication factor s(u_i) equal t. Next, we create the set $\bf A_{1}^{(M)}$, defined by:

$$\bf A_{1}^{(M)}$$ = $$\left\{\vphantom{ u_{1},\ldots,u_{n_{1}}: p(u_{i_{1}})\not=p(u_{j... ...for}\;\;\; i_{1}\not=j_{1}\;\;\;\mbox{\rm and}\;\;\; i_{1},j_{1}\leq n }\right.$$u₁,..., u_n1 : p(u_i1) $$\not=$$p(u_j1)       for      i₁ $$\not=$$j₁       and      i₁, j₁ $$\leq$$ n$$\left.\vphantom{ u_{1},\ldots,u_{n_{1}}: p(u_{i_{1}})\not=p(u_{j_... ...or}\;\;\; i_{1}\not=j_{1}\;\;\;\mbox{\rm and}\;\;\; i_{1},j_{1}\leq n }\right\}$$.

In other words, the set $\bf A_{1}^{(M)}$ consists of quantities of different dimensions. In the first step of the elimination procedure we examine if there are u_i1 ( i₁ = 1, 2,..., n₁) for which:

$$\in \bf R$$ (6.42)

or

$$\underbrace{(0,\ldots,0)}_{k-times}^{}\, \in \bf R \setminus$$ (6.43)

If the (6.42) or (6.43) equation is valid for fixed u_i1, then:

$$\xi_{1_{t}}^{a_{i_{1}}}$$ (6.44)

where a_i1, b_i1 depend on p(u_i1) and p(y), $\xi_{1_{t}}^{}$ $\in$ $\bf R_{+}^{}$. Next, we create the set $\bf B_{1}^{(M)}$:

$$\bf B_{1}^{(M)}$$ = $$\left\{\vphantom{u_{i_{1}}: u_{i_{1}}\;\;\; \mbox{\rm satisfy (\ref{ben10}) or (\ref{ben11})}}\right.$$u_i1 : u_i1       satisfy (6.42) or (6.43) $$\left.\vphantom{u_{i_{1}}: u_{i_{1}}\;\;\; \mbox{\rm satisfy (\ref{ben10}) or (\ref{ben11})}}\right\}$$,

and the set $\bf A_{2}^{(M)}$:

$$\bf A_{2}^{(M)}$$ = $$\left\{\vphantom{ (u_{i_{2}},u_{j_{2}}): p(u_{i_{2}})\not=p(u_{j_... ...;\; i_{2},j_{2}\leq n,\;\;u_{i_{2}},u_{j_{2}}\not\in {\bf B}_{1}^{(M)} }\right.$$(u_i2, u_j2) : p(u_i2) $$\not=$$p(u_j2)    i₂ $$\not=$$j₂    i₂, j₂ $$\leq$$ n,    u_i2, u_j2 $$\not\in$$$$\bf B_{1}^{(M)}$$$$\left.\vphantom{ (u_{i_{2}},u_{j_{2}}): p(u_{i_{2}})\not=p(u_{j_{... ...\; i_{2},j_{2}\leq n,\;\;u_{i_{2}},u_{j_{2}}\not\in {\bf B}_{1}^{(M)} }\right\}$$.

Secondly we examine if there are (u_i2, u_j2) $\in$ $\bf A_{2}^{(M)}$ for which:

$$\in \bf R$$ (6.45)

$$\underbrace{(0,\ldots,0)}_{k-times}^{}\, \in \bf R \setminus$$ (6.46)

If the (6.45) or (6.46) holds for (u_i2, u_j2) $\in$ $\bf A_{2}^{(M)}$ then:

$$\xi_{2_{t}}^{a_{i_{2}}j_{2}}$$ (6.47)

where a_i2j₂, b_i2j₂, c_i2j₂ depend on p(u_i2), p(u_j2) and p(y), $\xi_{2_{t}}^{}$ $\in$ $\bf R_{+}^{}$.

We repeat our reasoning k times, consequently we get the sequence of sets $\bf A_{1}^{(M)}$, $\bf B_{1}^{(M)}$, $\bf A_{2}^{(M)}$, $\bf B_{2}^{(M)}$, ..., $\bf B_{k-1}^{(M)}$, $\bf A_{k}^{(M)}$, $\bf B_{k}^{(M)}$ and the multiplication factors of elements of these sets.

Next, we introduce the space of models S^(M), supplied by the following Cartesian product:

S^(M) = $$\xi_{1_{1}}^{}$$ x $$\xi_{1_{1}}^{}$$ x...$$\xi_{1_{t_{1}}}^{}$$ x $$\xi_{2_{1}}^{}$$ x...$$\xi_{k_{t_{k}}}^{}$$.

For fixed u and y equations (6.44), (6.47), ... define some single-dimensional manifolds in S^(M):

$$\xi_{1}^{a_{1}} \xi_{t_{1}}^{a_{t_{1}}} \xi_{T}^{a_{T_{1}}}$$ (6.48)

where T = $\sum_{i=1}^{k}$t_i. Note that the sequence $\xi_{1_{1}}^{}$, $\xi_{1_{2}}^{}$, ..., $\xi_{1_{t_{1}}}^{}$, $\xi_{2_{1}}^{}$, ..., $\xi_{k_{t_{k}}}^{}$ is represented by a single point in the space of models. On the basis of matrices U and Y, we can calculate coordinates of t points in this space. Each of these points lies on a separate curve (6.48). If there are points that do not satisfy this condition, the function depends on additional arguments we have not taken into consideration. Next we construct the manifold lying "close" to these points. Let us assume that the following equation yields the manifold:

$$\left(\vphantom{\xi_{1_{1}}, \xi_{1_{2}}, \ldots, \xi_{1_{t_{1}}}, \xi_{2_{1}}, \ldots, \xi_{k_{t_{k}}} }\right. \xi_{1_{1}}^{} \xi_{1_{2}}^{} \xi_{1_{t_{1}}}^{} \xi_{2_{1}}^{} \xi_{k_{t_{k}}}^{} \left.\vphantom{\xi_{1_{1}}, \xi_{1_{2}}, \ldots, \xi_{1_{t_{1}}}, \xi_{2_{1}}, \ldots, \xi_{k_{t_{k}}} }\right)$$ (6.49)

The (6.49) is called the universal graph equation. One can prove that determining the dimensional function $\Phi^{(M)}_{}$, it suffices to know the real function f^(M).

The presented universal graph method may be applied to remaining functions: $\Phi_{1}^{}$, $\Phi_{2}^{}$, ..., $\Phi_{r}^{}$, (see (6.41)). From this, we get r spaces of models:

S_i = $$\xi_{1_{1}}^{(i)}$$ x $$\xi_{1_{2}}^{(i)}$$ x...$$\xi_{1_{t_{1_{i}}}}^{(i)}$$ x $$\xi_{2_{1}}^{(i)}$$ x...$$\xi_{k_{t_{k_{i}}}}^{(i)}$$,          i = 1, 2,..., r.

On the basis of information obtained by way of experiment (matrix U and Y) we know the coordinates of t points in each of these spaces, so we can write the universal graph equation for each of the function $\Phi_{i}^{}$ ( i = 1, 2,..., r):

$$\left(\vphantom{\xi_{1_{1}}^{(i)}, \xi_{1_{2}}^{(i)}, \ldots, \xi... ..._{1_{i}}}}^{(i)}, \xi_{2_{1}}^{(i)}, \ldots, \xi_{k_{t_{k_{i}}}}^{(i)} }\right. \xi_{1_{1}}^{(i)} \xi_{1_{2}}^{(i)} \xi_{1_{t_{1_{i}}}}^{(i)} \xi_{2_{1}}^{(i)} \xi_{k_{t_{k_{i}}}}^{(i)} \left.\vphantom{\xi_{1_{1}}^{(i)}, \xi_{1_{2}}^{(i)}, \ldots, \xi... ..._{1_{i}}}}^{(i)}, \xi_{2_{1}}^{(i)}, \ldots, \xi_{k_{t_{k_{i}}}}^{(i)} }\right)$$ (6.50)

The complex dimensional function $\Phi$ is represented by the manifold given by the system of real functions resulting from equations (6.49) and (6.50) in the space of models S, where:

S = s^(M) x S₁ x S₂ x...x S_r (6.51)

The set (6.51) is called the universal graph equation for complex function. The universal graph is the manifold in a specially constructed space, the geometry of which depends only on dimensions of arguments of examined function. This why it is difficult to consider additional intuitions linked with the function.

The object shown in the Fig. 6.18 has been chosen as an example. It follows that

p₂ = $$\Phi$$$$\left(\vphantom{ p_{1},h_{1},h_{2},\vec{v}_{1}^{2},\vec{v}_{2}^{2... ...,\rho_{el},g,\phi_{1}(\epsilon,q,d,r_{1}),\phi_{2}(\epsilon, q,d,r_{r})}\right.$$p₁, h₁, h₂,$$\vec{v}_{1}^{2}$$,$$\vec{v}_{2}^{2}$$,$$\rho$$,$$\rho_{el}^{}$$, g,$$\phi_{1}^{}$$($$\epsilon$$, q, d, r₁),$$\phi_{2}^{}$$($$\epsilon$$, q, d, r_r)$$\left.\vphantom{ p_{1},h_{1},h_{2},\vec{v}_{1}^{2},\vec{v}_{2}^{2... ...,\rho_{el},g,\phi_{1}(\epsilon,q,d,r_{1}),\phi_{2}(\epsilon, q,d,r_{r})}\right)$$.

Using the involved notation, we can write:

$$\Phi^{(M)}_{} \left(\vphantom{ p_{1},h_{1},h_{2},\vec{v}_{1}^{2},\vec{v}_{2}^{2}, \rho,\rho_{el},g,\phi_{1},\phi_{2}}\right. \vec{v}_{1}^{2} \vec{v}_{2}^{2} \rho \rho_{el}^{} \phi_{1}^{} \phi_{2}^{} \left.\vphantom{ p_{1},h_{1},h_{2},\vec{v}_{1}^{2},\vec{v}_{2}^{2}, \rho,\rho_{el},g,\phi_{1},\phi_{2}}\right)$$ p₂ =

$$\phi_{1}^{} \Phi_{1}^{} \epsilon$$ =

$$\phi_{2}^{} \Phi_{2}^{} \epsilon$$ =

Sets $\bf A^{(M)}_{i}$, $\bf B^{(M)}_{i}$ ( i = 1, 2, 3, 4) for the function $\Phi^{(M)}_{}$ yield the orbit equations:

$$\xi_{1}^{-1} \rho \vec{v}_{1}^{2} \xi_{2}^{-1} \rho \vec{v}_{2}^{2} \xi_{3}^{-1} \rho_{el}^{} \phi_{1}^{} \xi_{4}^{-1} \rho_{el}^{} \phi_{2}^{} \xi_{5}^{-2} \rho \xi_{6}^{-2} \rho \xi_{7}^{}$$ (6.52)

Applying the elimination procedure to functions $\Phi_{1}^{}$ and $\Phi_{2}^{}$ we obtain:

$$\phi_{1}^{}$$ = $${(\xi_{1}^{(1)})^{2} q\over d\epsilon}$$ = $${(\xi_{2}^{(1)})^{2} q\over r_{1}\epsilon}$$,

$$\phi_{2}^{}$$ = $${(\xi_{1}^{(2)})^{2} q\over d\epsilon}$$ = $${(\xi_{2}^{(2)})^{2} q\over r_{2}\epsilon}$$.

Let us assume that the range of each argument ( p₁, $\vec{v}_{1}^{2}$, $\vec{v}_{2}^{2}$, h₁, h₂, $\rho$, $\rho_{el}^{}$, $\epsilon$, g, q, d, r₁, r₂) is fixed. We calculate the pressure p₂ applying the formula (6.38) and assuming the error of this calculus. In this way we simulate the experiment and as a result we get matrix U and Y.

The universal graph is the manifold in the space of models S, where:

S = $$\xi_{1}^{}$$ x $$\xi_{2}^{}$$ x $$\xi_{3}^{}$$ x $$\xi_{4}^{}$$ x $$\xi_{5}^{}$$ x $$\xi_{6}^{}$$ x $$\xi_{7}^{}$$ x $$\xi_{1}^{(1)}$$ x $$\xi_{2}^{(1)}$$ x $$\xi_{1}^{(2)}$$ x $$\xi_{2}^{(2)}$$.

In our case we can write:

$$\left\{\vphantom{ \begin{array}{r c l} f^{(M)}\left(\xi_{1},\xi_{... ... f_{2}\left(\xi_{1}^{(2)},\xi_{2}^{(2)} \right) & = & 1 . \end{array} }\right. \begin{array}{r c l} f^{(M)}\left(\xi_{1},\xi_{2},\xi_{3},\xi_{4}... ...= & 1 \\ f_{2}\left(\xi_{1}^{(2)},\xi_{2}^{(2)} \right) & = & 1 . \end{array}$$ (6.53)

where $\xi_{1}^{}$,$\xi_{2}^{}$,$\xi_{3}^{}$,$\xi_{4}^{}$,$\xi_{5}^{}$,$\xi_{6}^{}$,$\xi_{7}^{}$,$\xi_{1}^{(1)}$,$\xi_{2}^{(1)}$,$\xi_{1}^{(2)}$,$\xi_{2}^{(2)}$ $\in$ $\bf R_{+}^{}$ and we get these numbers from experimental data and orbit equations (6.52). We treat the set ($\xi_{1}^{}$,$\xi_{2}^{}$,$\xi_{3}^{}$,$\xi_{4}^{}$,$\xi_{5}^{}$,$\xi_{6}^{}$,$\xi_{7}^{}$,$\xi_{1}^{(1)}$,$\xi_{2}^{(1)}$,$\xi_{1}^{(2)}$,$\xi_{2}^{(2)}$) as a single point in a space of models, thus we have 650 points and the set of these points is situated close to the universal graph (these points do not lie in this manifold, because we can not measure without error).

Our task is to find the equation of manifold that has the following property: the distance between this manifold and the set of points ($\xi_{1}^{}$,$\xi_{2}^{}$,$\xi_{3}^{}$,$\xi_{4}^{}$,$\xi_{5}^{}$,$\xi_{6}^{}$,$\xi_{7}^{}$,$\xi_{1}^{(1)}$,$\xi_{2}^{(1)}$,$\xi_{1}^{(2)}$,$\xi_{2}^{(2)}$) is "small". Although we do not know the formula yielding the (6.53), one can approach it by any other functions; we choose for instance:

$$\left\{\vphantom{ \begin{array}{r c l} a_{1}\xi_{7}^{-1}+ a_{2}\... ..._{1}(\xi_{1}^{(2)})^{2} + c_{2}(\xi_{2}^{(2)})^{2} & = & 1. \end{array}}\right.$$$$\begin{array}{r c l} a_{1}\xi_{7}^{-1}+ a_{2}\xi_{1}+ a_{3}\xi_{2... ... 1\\ c_{1}(\xi_{1}^{(2)})^{2} + c_{2}(\xi_{2}^{(2)})^{2} & = & 1. \end{array}$$

We can find coefficients a_i, ( i = 1, 2,..., 7) b_j, c_j, (j = 1, 2) applying the smallest square method. Thus, we get:

$$\left\{\vphantom{ \begin{array}{r c l} 0.98\xi_{7}^{-1}+ 0.51\xi... ...\\ -(\xi_{1}^{(2)})^{2} + 6.3(\xi_{2}^{(2)})^{2} & = & 1. \end{array}}\right.$$$$\begin{array}{r c l} 0.98\xi_{7}^{-1}+ 0.51\xi_{1} -0.48\xi_{2} +... ... & = & 1\\ -(\xi_{1}^{(2)})^{2} + 6.3(\xi_{2}^{(2)})^{2} & = & 1. \end{array}$$

Thus

$$\rho \vec{v}_{1}^{2} \rho \vec{v}_{2}^{2} \rho \rho \rho_{el}^{} {q \epsilon \over -0.01 r_{1}+4.6 d} \rho_{el}^{} {q \epsilon \over - r_{2}+6.3 d}$$ (6.54)

Of course, the formula (6.54) differs from the (6.38), but it should be remembered that the form of functions f^(M), f₁ and f₂ was arbitrarily chosen. The approximation error equals Q=25820 [Pa].

The example here described has been intentionally chosen. The dependence of the result of approximation on the choice of the dimensional base for function (6.26) has been carefully tested in [13]. The UGM is not equivalent to the representation of equations; we have got as a result the approximation in all bases (note, that the dimension of the space of models equals eleven whereas we have to examine 79 bases only for function (6.38)).

It seems that the universal graph method is worse numerically conditioned than algorithms based on dimensional analysis. Using the Buckingham theorem, we often do not concern ourselves with the influence of the choice of the dimensional base on the final solution. Let us apply the Theorem $\pi$ to the problem of the approximation of the function $\Phi^{(M)}_{}$:

$$\phi_{1}^{} \phi_{2}^{} \phi_{3}^{} \phi_{4}^{} \phi_{5}^{} \phi_{6}^{} \prod_{i=1}^{4}$$ (6.55)

where $\phi_{j}^{}$ $\in$ $\bf R_{+}^{}$ ( j = 1, 2,..., 6) and b_i fulfil:

det$$\left[\vphantom{ \begin{array}{c} p(b_{1})\\ p(b_{2})\\ p(b_{3})\\ p(b_{4}) \end{array}}\right.$$$$\begin{array}{c} p(b_{1})\\ p(b_{2})\\ p(b_{3})\\ p(b_{4}) \end{array}$$ $$\left.\vphantom{ \begin{array}{c} p(b_{1})\\ p(b_{2})\\ p(b_{3})\\ p(b_{4}) \end{array}}\right]$$ $$\not=$$0.

The base might be chosen in many ways (in this particular case there are 79 different bases). Let us assume that:

g($$\phi_{1}^{}$$,$$\phi_{2}^{}$$,$$\phi_{3}^{}$$,$$\phi_{4}^{}$$,$$\phi_{5}^{}$$,$$\phi_{6}^{}$$) = $$\sum_{j=1}^{6}$$d_j$$\phi_{j}^{}$$.

It is important to know that the value of d_j depends on the base, thus we can classify bases. To choose the best one we should calculate coefficients d_j and the criterion Q for each base using the smallest square method. Of course, for the optimal base the equation (6.55) minimizes the criterion Q. After calculating the formula (6.55) for 79 bases, we get the equation that minimizes the Q:

$$\overline{p}_{2}^{} \left(\vphantom{ \displaystyle{p_{1} \over \vec{v}_{2}^{2}\rho} ... ...}\rho} -1.02\displaystyle{\phi_{2}\rho_{el} \over \vec{v}_{2}^{2}\rho} }\right. {p_{1} \over \vec{v}_{2}^{2}\rho} {h_{2} \over h_{1}} {\vec{v}_{1}^{2} \over \vec{v}_{2}^{2}} {g h_{1} \over \vec{v}_{2}^{2}} {\phi_{1}\rho_{el} \over \vec{v}_{2}^{2}\rho} {\phi_{2}\rho_{el} \over \vec{v}_{2}^{2}\rho} \left.\vphantom{ \displaystyle{p_{1} \over \vec{v}_{2}^{2}\rho} ... ...}\rho} -1.02\displaystyle{\phi_{2}\rho_{el} \over \vec{v}_{2}^{2}\rho} }\right) \vec{v}_{2}^{2} \rho$$ (6.56)

Next, we apply the Theorem $\pi$ to functions $\Phi_{1}^{}$ and $\Phi_{2}^{}$:

$$\phi_{1}^{} {q \over \epsilon(r_{1}+d)} \phi_{2}^{} {q \over \epsilon(r_{2}+d)}$$ (6.57)

Putting (6.57) into (6.56), we get the equation of the complex dimensional function:

$$\overline{p}_{2}^{} \left(\vphantom{ \displaystyle{p_{1} \over \vec{v}_{2}^{2}\rho} ... ...ver \vec{v}_{2}^{2}} +0.47 \displaystyle{g h_{1} \over \vec{v}_{2}^{2}}}\right. {p_{1} \over \vec{v}_{2}^{2}\rho} {h_{2} \over h_{1}} {\vec{v}_{1}^{2} \over \vec{v}_{2}^{2}} {g h_{1} \over \vec{v}_{2}^{2}} \left.\vphantom{ +1.4\displaystyle{q\rho_{el} \over \epsilon(r_{1... ...25\displaystyle{q\rho_{el} \over \epsilon(r_{2}+d)\vec{v}_{2}^{2}\rho} }\right. {q\rho_{el} \over \epsilon(r_{1}+d)\vec{v}_{2}^{2}\rho} {q\rho_{el} \over \epsilon(r_{2}+d)\vec{v}_{2}^{2}\rho} \left.\vphantom{ +1.4\displaystyle{q\rho_{el} \over \epsilon(r_{1... ...25\displaystyle{q\rho_{el} \over \epsilon(r_{2}+d)\vec{v}_{2}^{2}\rho} }\right) \vec{v}_{2}^{2} \rho$$ (6.58)

Nevertheless, to do so, we had to examine 79 bases. The criterion Q equals 433300 [Pa] for (6.58). It is more than the criterion we obtained earlier using the UGM. In fact, the generalization of the UGM for the complex dimensional function here presented is equivalent to the approximation of real functions. We have no intuitions, which help us to choose the class of functions for approximation. The example has been chosen to describe the main object by the polynomial. This should not be expected in more complicated cases.

Now, having completed all necessary tools, we may build an effective algorithm. As we have shown the list of arguments of any dimensional function divides into two groups. The first gives the orbits and space of models but it has no effect on the points of orbits. The second group of arguments, inside the isolated subspace, changes the dimension of the manifold of orbit. Consequently, the space of models becomes filled with some subsets treated as new points of X.