DETAILED DESCRIPTION OF THE INVENTION

[0054] I. Theory Underlying the Invention

[0055] The invention is based on a new theory of evolution proposed by the inventor, particularly as applied to the evolution of market parameters. The inventor's premise is that the movement of market prices or other market parameters may be described by the laws of physics, and specifically the laws of motion of material objects. The inventor postulates the following:

[0056] The principle of universality:

[0057] The laws governing changes in measured material parameters are universal, recurring laws that are true for all types of matter, material objects and measuring instruments.

[0058] I. 1) Fundamental Laws of Evolution

[0059] From a conceptual point of view, one may consider that an observer receives information on the material world by registering changes in material parameters. The observer generally registers the changes in the material parameters by taking measurements with, for example, instruments. The process of observing material parameter changes is generally objective and is generally carried out by taking measurements. The measurement generally produces a number. The number that reflects a material parameter is generally not exact. The measuring process inevitably entails a measurement error which may be large or small and which depends on the method of measurement and the instrument used. Parameter changes with an amplitude smaller than the measurement error will generally not be registered. When the measurement error appears as a scale unit of the instrument used, the scale unit may be considered to be the increment change that is detected and therefore registered by the instrument used. Thus, any material parameter may be represented as a pair of numbers (e.g., R, r), where R is the value proper and r is a measurement increment. Each time one registers a new parameter value that differs from the preceding one, the registered parameter change is generally equal to the discrete measurement increment such that the value of any material parameter may be represented by an integer number multiplied by a measurement increment r.

[0060] The scale of change determined in this way and calibrated in integer numbers does not generally depend on the instrument and generally complies with the principle of universality. The inventor thus proposes the following:

[0061] First Law of Evolution:

[0062] Registered Change is Always a Measurement Increment

[0063] What this in fact means is that the world that we are cognizing is “discrete”. It is not possible to observe the continuous (non-discrete) changes of material parameters. Thus, the process of change can be described as a sequence of changes of integer numbers in time.

[0064] On the premise that the theory described herein is universal and therefore true for all material objects without exception, a particular case is considered and extended to all others. If one records a change in spatial coordinate of light with an appropriate instrument, the motion is generally composed of identical steps, each equal to a discrete increment of distance. If one redefines “time” as a number of registered changes (hereinafter “Evolution Time”), the clock will always be constructed of the same form of matter as that to which the parameter under examination belongs. On the basis of the principle of universality, the above definition of time may be extended to all forms of matter and material objects as summarized in the following.

[0065] Second Law of Evolution:

[0066] The length of time of change is proportional to the number of successively registered changes.

[0067] In other words, “Evolution Time” stands still when the amplitude of change in the real parameter is less than the specified measurement increment r. One can construct a two dimensional space for which the universality principle holds true, with one coordinate axis representing the parameter value (for example price) as a number of measurement increments r, and the other coordinate axis representing Evolution Time as the number of successively registered changes. A change in parameter is generally registered when the difference between the last registered parameter and the newly measured parameter equals the chosen value of the measurement increment r. Such a two-dimensional space is hereinafter referred to as “Increment-Change Space”.

[0068] It may be noted that the aforesaid Increment-Change Space is dimensionless, since the one axis (e.g., the Y-axis) is a sequence of integers (e.g., representing a number of measurement increments), and the other axis (e.g., the X-axis) is also a sequence of integers (e.g., representing a number of successively registered changes). A parameter in Increment-Change Space is often relative in a double sense: first, the parameter is frequently used as an integer and, second, it is often convenient to set a starting point for the parameter to zero.

[0069] In the present application, notions derived from the quantum theory are generally used to describe the movement of a market parameter in Increment-Change Space. In other words, the movement of a market parameter in Increment-Change Space may be considered analogous to the motion of a wave-particle (electron, photon . . . ) and treated as though subject to the physical laws applying to wave-particles. By analogy, the following terms describing the value of a market parameter over time, after transformation in Increment-Change Space, will be used in this application: 1

[0070] Considering the above, changes of a market parameter (for example the price of a share on the stock market) over real time may be expressed in Increment-Change Space by applying the following system of equations and inequalities:

[0071] Registration of a new parameter value in Increment-Change Space generally takes place when the following condition is met:

| R _real +R _f −R _duka −R _duka(0) |≧r (i)

[0072] where: R _real is the current value of the parameter in real space; R _f is a value which meets the condition |R _f |<r, chosen in such a way as to facilitate splitting into a beam or effecting a “phase shift”; R _duka is the current (latest) registered value of the parameter in Increment-Change Space; the term appearing in the left-hand part of the inequality diminishes abruptly each time a new parameter value is registered; and R _duka(0) is the parameter value by which the parameter scale in Increment-Change Space is shifted in relation to the parameter scale in real space. This makes it possible to combine the starting point of the trajectory with the start (zero point) of the coordinates R _duka =0 in Increment-Change Space. The starting point R _duka =0 is generally used for convenience. At the same time nothing is changed in principle if the starting point is fixed by some other value.

[0073] The values of the parameter R _duka allowed in Increment-Change Space may be determined in accordance with the following equation:

R _duka =±ir (ii)

[0074] where: i=0, 1, 2, 3 . . . is a series of integers, and r>0 is the increment or the absolute value of the difference between any two adjacent parameter values successively registered in Increment-Change Space.

[0075] The time interval t _duka in Increment-Change Space is also a discrete sequence of values that may be expressed by the equation:

t _duka =τN (iii)

[0076] where: N=0, 1, 2, 3 . . . is the number of registered changes of the parameter in Increment-Change Space during the time interval t _duka , and τ>0 is the constant time interval between any two adjacent parameter values successively registered in Increment-Change Space.

[0077] The transformation described above is generally referred herein as “parameter-normalization” since the changes in the market parameter are registered at each change of the parameter by the increment r. It is however also possible to effect a transformation from real space to Increment-Change Space by considering real time as the parameter and the real parameter as successive increases or decreases in registered changes. Such a transformation is generally referred to herein as “time-normalization” and is generally governed by the system of equations and inequalities set out below.

[0078] The registration of a new parameter value in time-normalized Increment-Change Space may be determined by the following equation:

t _real =t _real(0) +τN+t _f (iv)

[0079] where t _real is the current time value in real space at the moment of registration of the parameter in time-normalized Increment-Change Space (any interruptions in the de facto existence of the parameter in real space, e.g. non-working days, are left out of account if they impede the regular reflection of the real-time data in Increment-Change Space); t _real(0) is the initial moment of time in real space (corresponds to N=0); N=0, 1, 2, 3 . . . is the serial number of the parameter changes registered in time-normalized Increment-Change Space; τ>0 is the time interval between any two adjacent parameter values successively registered in time-normalized Increment-Change Space; and t _f is a value which meets the condition |t _f |<τ, chosen in such a way as to permit splitting into a beam or effecting a “phase shift”.

[0080] Every change in the value of the parameter in time-normalized Increment-Change Space may be determined in accordance with the following formula: 1 $\begin{array}{cc}\Delta R_{\mathrm{duka}}=\frac{r\Delta R_{\mathrm{real}}}{\left|\Delta R_{\mathrm{real}}\right|}& \left(v\right)\end{array}$

[0081] where ΔR _duka is the change in the parameter when a new value is registered in Increment-Change Space relative to the preceding value in time-normalized Increment-Change Space (e.g., if ΔR _real =0, then ΔR _duka equals its preceding value, or else it is determined by some other reasonable method chosen at will); r>0 is the absolute value of the difference between any two adjacent parameter values successively registered in time-normalized Increment-Change Space; and ΔR _real is the parameter change in real space during the time that has elapsed since the preceding registration.

[0082] Finally, the scale of permitted time values in time-normalized Increment-Change Space may be expressed by the following equation:

t _duka =t _duka(0) +τN (vi)

[0083] where: t _duka is the scale of the permitted time values in time-normalized Increment-Change Space, and t _duka(0) is the time value set at N=0, which makes it possible (if desired) to combine the starting point of the trajectory with the zero point of the time count (or any other point fixed as the starting point) in time-normalized Increment-Change Space.

[0084] The pattern of change of any market parameter in Increment-Change Space may be presented, in one example, as a broken (or dashed) line in which the segments have the same angle of inclination with respect to the time axis. A physical analog with which we are familiar is the trajectory of the motion of a light ray along one axis, subject to the condition that “U-turns” are possible only at “specially marked” points on the respective axis, (e.g., points located at identical intervals equal to the value of the increment of measurement). The analogy to light is somewhat idealized but extremely useful for our further investigations. Following the principle of universality, the physical laws of motion of a light ray may be extended to the change of market parameters. Since the physical analog has been determined to exist in a stable manner in the conditions described above only as a wave with a length equal to double the measurement increment r, in Increment-Change Space the motion of any parameter may be interpreted as a wave process with the same wavelength. Such an interpretation generally establishes the basis for applying techniques and methods of wave mechanics when analyzing the process of change of parameters in Increment-Change Space, and generally represents the third law of evolution.

[0085] Third Law of Evolution:

[0086] The process of change may be described as a material wave-particle motion in which the wavelength is equal to double the measurement increment r and the rest mass is equal to zero.

[0087] Thus the motion of a market parameter in Increment-Change Space is physically similar to the motion of light, but it is not light.

[0088] It is important to understand that the properties of a trajectory describing changes of a market parameter in Increment-Change Space are generally related to the value of the measurement increment r, which can take any value in the range from zero to infinity. In other words, waves describing the process of change of parameters in Increment-Change Space (hereinafter “parameter change wave”, or simply “wave”) theoretically have an unlimited spectrum of wavelengths. For example, for any given wavelength a shorter wavelength may be found in which the representation of the process of change is generally more precise and detailed. Thus, the length of a wave is generally not an absolute characteristic—the length is always relative, as is the pattern of the process of change at that wavelength. The essential point here, however, is that development of the process of change at any possible wavelength in the infinitely wide range must be governed by universal laws and must be independent. This independence means that development processes at different wavelengths do not influence each other. Nevertheless the pattern of changes at shorter wavelengths generally supplements and determines the corresponding pattern of long waves.

[0089] I. 2) Application of Physical Laws

[0090] Considering the above, in the following section the laws of Physics are generally applied by analogy to the process of change of a market parameter plotted in Increment-Change Space.

[0091] The wavelength λ and frequency ν of a parameter change wave may be expressed as follows:

λ=2 r (1) 2 $\begin{array}{cc}v=\frac{1}{2\tau }& \left(2\right)\end{array}$

[0092] where τ=r/c and c is the maximum possible velocity of change in Increment-Change Space. τ thus represents the time in Increment-Change Space that it takes to register each change of the parameter by the increment value r.

[0093] In the following development, we shall apply, by analogy, the laws of Quantum Mechanics Theory, which describe the behavior of wave-particles, to the process of change of market parameters in Increment-Change Space.

[0094] The momentum P of a parameter change wave at a selected wavelength equals 3 $\begin{array}{cc}P=\frac{h}{\lambda }=\frac{h}{2r}& \left(3\right)\end{array}$

[0095] where h is an analog of the Planck constant. Considering further that

P=MV (4)

[0096] where M and V are the mass and the velocity of the parameter change particle, respectively, the law of conservation of momentum may be expressed as follows: 4 $\begin{array}{cc}P=\frac{h}{2r}=\mathrm{MV}=\mathrm{const}.& \left(5\right)\end{array}$

[0097] Applying Einstein's law, the following is true for the effective mass of the parameter change particle: 5 $\begin{array}{cc}M=\frac{E}{c^2}& \left(6\right)\end{array}$

[0098] where E represents the energy of the parameter change particle. Moreover, according to Planck, energy can take only the quantum values

E=nhv (7)

[0099] where n=1, 2, 3 . . . , and where the parameter change wave frequency is connected with the wave length of the parameter change wave by the known ratio

λν= c (8)

[0100] Taking expressions (5), (6) and (7) into consideration we arrive at 6 $\begin{array}{cc}\frac{h}{2r}=\frac{{\mathrm{nhvV}}_n}{c^2}=\mathrm{const}& \left(9\right)\end{array}$

[0101] where V _n is the velocity of the parameter change particle corresponding to the quantum number n.

[0102] The rule of the quantization of the velocity of the parameter change particle follows there from and may be expressed by the following equation: 7 $\begin{array}{cc}{V}_n=\frac{c}{n},\mathrm{where}n=1,2,3\dots & \left(10\right)\end{array}$

[0103] We should therefore meet the effect of quantization of the velocity of the parameter change particle, and therefore of the trajectory describing the evolution of a market parameter in Increment-Change Space. This is verified further on when concrete examples are discussed.

[0104] One of the consequences of accepting the quantum hypothesis is the applicability of the Heisenberg uncertainty principle:

ΔRΔP≈h (11)

[0105] where ΔR stands for the uncertainty of the coordinate of the parameter change particle and therefore of the trajectory describing the motion of the particle (localization of the parameter) and ΔP stands for the uncertainty of the parameter change particle momentum.

[0106] Let us consider an experiment designed to determine ΔP. Given that in practice we can measure only the trajectory velocity, let us concentrate on the determination of V and ΔV. It is understood that ΔP is functionally related to V and ΔV. As a consequence of P=MV, ΔV may be expressed as follows:

Δ P ={square root}{square root over ( M ² ΔV ² ΔM ² V ² )} (12)

[0107] The mass M of the parameter change particle is generally expressed through V as a consequence of equation (5) 8 $\begin{array}{cc}M=\frac{h}{2\mathrm{rV}}& \left(13\right)\end{array}$

[0108] From which it follows that: 9 $\begin{array}{cc}\Delta M=\left|\frac{M}{V}\right|\Delta V=\frac{h\Delta V}{2{\mathrm{rV}}^2}.& \left(14\right)\end{array}$

[0109] Let us insert equations (13) and (14) in equation (12) 10 $\begin{array}{cc}\Delta P=\frac{h\Delta V}{\mathrm{rV}\sqrt{2}}& \left(15\right)\end{array}$

[0110] Applying the law of quantization of velocities, we can write: 11 $\begin{array}{cc}\Delta V=\left|\frac{V}{n}\right|\Delta n=\frac{c\Delta n}{n^2}& \left(16\right)\end{array}$

[0111] By combining expressions (16), (15) and (10) we derive the following equation: 12 $\begin{array}{cc}\Delta P=\frac{h\Delta n}{\mathrm{rn}\sqrt{2}}& \left(17\right)\end{array}$

[0112] Inserting the result in equation (11), we obtain the uncertainty relation for the parameter change particle in the following form: 13 $\begin{array}{cc}\Delta R\approx \frac{\sqrt{2}\mathrm{rn}}{\Delta n}& \left(18\right)\end{array}$

[0113] It remains to determine Δn. As we know that n is a discretely changing quantum number, it is determined in advance that Δn will generally be close to unity. We cannot, however, state with absolute certainty that Δn=1. Accordingly, on the understanding that Δn is a number of the order of unity, the numerical coefficient q={square root}{square root over (2)}/Δ n may be introduced. With this coefficient, the uncertainty relation may be stated in a more convenient form: 14 $\begin{array}{cc}\Delta R\approx \mathrm{qrn}=\frac{q\lambda n}{2}& \left(19\right)\end{array}$

[0114] This formulation also automatically eliminates the question of the coefficient which, generally speaking, may be put in front of h in expression (11). By tacit assumption we took it to be equal to unity. Even if it is not equal to unity, however, the coefficient q successfully “absorbs” this awkwardness and seems to dispose of it completely. Furthermore, by using the parameter q we avoid yet another awkwardness. The formula for the momentum localization (expression 12) is generally not exclusive. For example, the formula may either be written in linear form ΔP=MΔV+ΔMV or expressed in other ways. But the difference between these approaches entails the emergence of a numerical factor of the order of 1. Clearly this factor may also be absorbed by q.

[0115] The precise definition of q in each particular case is one of the major practical problems of the theory of evolution. Later we shall explore this question in more detail, but for the time being we shall use the value q={square root}{square root over (2)}. It should also be noted that here and further on n must be taken to mean, not a discrete series of integer values, but a continuously changing average value. In fact, by assuming a non-zero Δn, we are obliged to acknowledge the existence of the scatter of n, (e.g., a certain quantum number distribution). Even though n is an integer value distribution, the mean value of n is generally changing continuously.

[0116] The expression (19) thus establishes a direct connection between the wavelength at which the trajectory is observed, the quantum number of the trajectory and the vertical distance ΔR between the borders of the band within which the trajectory moves. Since we are conducting the trajectory analysis in Increment-Change Space we must pay attention to the error in the determination of ΔR. The measurement unit here is r, (e.g., half the length of the parameter change wave). Given that the length of a section of a parameter change trajectory between two points is determined as ΔR=R ₁ −R ₂ , the error of the result is generally related to the errors of the measurements of the coordinates δR ₁ =δR ₂ =r in the following way:

δΔ R={square root}{square root over ((δ R ₁ ) ² +(δ R ₂ ) ² )}=r{square root}{square root over ( 2)} (20)

[0117] This enables us to estimate the relative error of the measurements: 15 $\begin{array}{cc}\frac{\delta \Delta R}{\Delta R}\approx \frac{1}{n}& \left(21\right)\end{array}$

[0118] Hence it may be concluded that in the range of low n, where the error is of the order of 100%, it is unrealistic to expect quantitative correspondence from the measurements. Conversely, we may expect the analysis of concrete examples to yield sound, stable results in the high n range, where the error diminishes as 1/n, as will be shown with the verification of the various results of this theory in the examples section below.

[0119] The uncertainty relation has an important property which may make it easier to conduct an experimental verification. The geometrical representation of the parameter localization error ΔR is generally represented by the distance between the high and low peak values of the parameter change trajectory measured as the distance in the direction of the parameter axis (e.g., Y-axis), between upper (resistance) and lower (support) lines (e.g., lines R 10 a , R 10 b , S 10 A, S 10 B) traced through extreme points as illustrated in FIG. 10 . When n>>1 and the measurement error of ΔR is small, the following is true:

ΔR _n,y ≈ΔR _n′,y′ ≈constant (22)

[0120] Where ΔR _n,r and ΔR _n′,r′ are the magnitudes of the parameter localization for different values r and r′ of the measurement increment respectively. In other words, the value of ΔR is substantially independent of the choice of the measurement increment r for a large number of measured changes.

[0121] Compliance with this requirement is more easily verified by the rule of transformation of the quantum number n of the trajectory points when passing from one Increment-Change Space in which the value of the measurement increment is r, to another in which the value of the measurement increment is r′ different from r, where

rn≈r′n′r≈inv≈constant (23)

[0122] This rule may be confirmed by experimental verification as shown below.

[0123] In concluding this section it is useful to add the following. A correlation interconnecting the magnitudes of the measurement increment r and the quantum number n on the scale of the real space parameter/time chart may be derived from expression (23) by means of a simple transformation using the substitution n=(t/τ)/(R/r) and n′=(t/τ′)/(R/r′). Here τ and τ′ stand for the average intervals in real time taken up by single changes of the parameter r or r′ as the case may be. After the substitution, t and R, which do not depend on the choice of the measurement increment because they are coordinates in real space, may be cancelled out and the following invariant relationship obtained:

r ² /τ≈(r′) ² /τ′≈inv≈const.

[0124] II. Experimental Verification of the Theory

[0125] II. 1) Quantum Effect

[0126] The properties of a parameter change trajectory may now be defined and described. Consider a chart as shown in FIG. 1 representing the changes of a market parameter in real time. In this particular example, the market parameter is the quoted exchange rate of the Euro to the US dollar (i.e. the ratio of Euros per USD) from Apr. 20 ^th 1998 to Jan. 28th 2000, on a daily bar basis, the data being provided by Aspen Research Group.

[0127] FIG. 2 shows the section P 1 a -P 1 b (from Oct. 8, 1998 to Jan. 28, 2000) of the chart of FIG. 1 in Increment-Change Space. The real parameter information has been transformed by applying the expressions (i) to (iii) for parameter normalization. In so doing, the absolute real parameter scale along the vertical axis in FIG. 1 has been transformed into the relative parameter scale (expressed as a number of measurement increments r) along the vertical axis in FIG. 2 . Moreover, the real time (in days) along the horizontal axis in FIG. 1 has been transformed into Evolution Time along the horizontal axis in FIG. 2 (as described in the above section “First Law of Evolution”) and expressed as a number of registered changes N as defined in expression (iii). In this example of transformation into Increment-Change Space, r was given the value 0.005. According to expression (iii), the number of registered changes N is expressed as a number of τ units. It is to be noted that the values along the vertical axis representing the real space market parameter (e.g. price, exchange rate, etc . . . ) shown in FIG. 1 generally correspond essentially to the values along the vertical axis representing the relative parameter in Increment-Change Space, as shown in FIG. 2 , except that the latter is expressed in a number of measurement steps (e.g., r units) and the origin is set at zero. The horizontal axes of the charts of FIGS. 1 and 2 however do not correspond.

[0128] Looking at the pattern of motion as depicted by the trajectory T 2 in FIG. 2 , the existence of a quantum effect is not generally apparent due to the fact that, for the visual observation of quantum properties, simple graphical plotting of the trajectory of one parameter change particle is generally not sufficient. It may be shown that the rule of quantization of the velocity (Equation 10) of the parameter change trajectory (V _n =c/n, where the quantum number n=1, 2, 3 . . . ) is met by examining the changes of a set of parameters (hereinafter called a “beam”) in Increment-Change Space. In other words, the manifestation of the quantization or quantum effect has a statistical character. Independently of discussing exclusively the statistics of a coherent beam, such quantization must be typical even for the statistical drift in space of one parameter change particle if the drift is taking place at a stable average velocity.

[0129] FIG. 3 shows, at an enlarged scale, the section as delineated by the dashed box F. 3 in FIG. 2 . As can be seen in FIG. 3 , the average velocity V _n of the trajectory between points P 3 a and P 3 b may be defined by the slope of the line L 3 , which is equal to R _n /t _n . The maximum velocity c is equal to r/τ. The wavelength λ may be determined at will by choosing the value of r. If τ is given the same magnitude as r, such that c=1, then the quantum value n of the line L 3 is equal to 1/V _n according to expression (10). In other words: n=t _n /R _n . In this particular example, t _n =16τ and R _n =8 r, such that n=2 and V _n =0.5.

[0130] FIG. 4 shows the image obtained as a result of the superposition of 5 different trajectories in Increment-Change Space. The image may include trajectories which appear to be unrelated with one another: the trajectory T 4 a represents, using open circles, the ratio EUR/USD, with r=0.005, i.e. the first 100 points of FIG. 2 starting from initial point P 1 a the date of which is Aug. 10, 1998; the trajectory T 4 b represents, using stars, the Dow Jones Industrial Average or DJIA index, with r=50, from October till November 1999; the trajectory T 4 c represents, using open triangles, the DJIA index, with r=300, from the moment of DJIA birth until 1998; the trajectory T 4 d represents, using asterisks, the GBP/USD ratio, with r=0.0009, from May 21 till May 27, 2001; the trajectory T 4 e represents, using open crosses, the USD/CHF ratio, with r=0.0025, from May 17 until May 31, 2001. The starting points of each of the trajectories T 4 a -T 4 e are aligned (for example, set to zero) in Increment-Change Space, where the relative parameter (expressed as the number of measurement increments r) along the vertical axis is equal to one-half of the wavelength, as expressed by equation (1), and the evolution time measurement unit is the value τ (redefined as τ=r/c as expressed in connection with equation 2 above). A slope (e.g., velocity), which may be set at will, was chosen as “c” for all the graphs thus aligned. Some of the trajectories pointed downwards in real space; accordingly, their direction was reversed before alignment.

[0131] FIG. 5 shows a beam-average curve B 5 determined by averaging, at each point along the Evolution Time axis, the value of the Relative Parameters (i.e. the average of the vertical coordinates) of the five trajectories T 4 a to T 4 e shown in FIG. 4 . The term “average” as used herein is the value defined substantially or approximately by calculating an arithmetical mean. For example, the average may be a weighted mean (where the weighting coefficients may be user-defined). However, any other averaged value may be applied that does not distort the idea of the method. In general, the beam-average may be obtained for any type of trajectory in Increment-Change Space by calculating the average of parameter for each Evolution Time. The indication of velocity quantization in accordance with expression (10) may be observed in the beam-average curve B 5 of FIG. 5 . For example, the beam-average curve B 5 seems to have sections P 5 a -P 5 b , P 5 c -P 5 d , P 5 e -P 5 f and P 5 g -P 5 h that follow to quantum lines n=1, 2, 3, and 4, respectively.

[0132] The foregoing indicates the existence of the effect of quantization of velocities in a randomly composed (i.e. incoherent) beam of trajectories. Although the trajectories may be completely unrelated and refer to different market parameters and historical periods, by operating in dimensionless Increment-Change Space, we have been able to combine in one beam what seemed to be incompatible. For example, the beam B 5 includes the DJIA index trajectory which has been in existence almost a century, the EUR/USD currency correlation over a period of about a year and a half, and a brief spurt, lasting only a few days, of the British pound in relation to the dollar. Particular attention is drawn to this factor to emphasize the importance and universality of the results obtained.

[0133] Since recognition of the quantum effect is a cornerstone of the theory developed herein, let us cite here the results of another experiment. Let us verify the existence of the quantum effect in a coherent beam of trajectories, as opposed to the quantum effect in a randomly composed beam of trajectories (e.g., the trajectories T 4 a to T 4 e as shown in FIG. 4 ). The coherent beam may be obtained by splitting one initial trajectory into a beam of two or more coherent trajectories. According to the properties of Increment-Change Space, all points on the real parameter curve that differ from the last registered change by a value less than the measurement increment r will have the same formal coordinate in Increment-Change Space. One curve in real space may be transformed into two or more trajectories in Increment-Change Space, (e.g., to create a coherent beam of trajectories each with the same wavelength λ as expressed in equation (1) and coinciding points of emission). To create the two or more trajectories, it is sufficient merely to “shift” the measurement starting point on the parameter axis of the real curve by a value less than the measurement increment r.

[0134] An example of the aforementioned beam transformation will now be described with reference to FIGS. 6A and 6B . A coherent beam B 6 B in FIG. 6B consisting of two trajectories T 6 Ba and T 6 Bb may be obtained by transforming one real curve C 6 A as shown in FIG. 6 A, where r was given the value 0.01 and the first trajectory T 6 Ba is phase shifted by r/2 with respect to the second trajectory T 6 Bb. For example, the round dots 1-16 in FIG. 6A are used to plot the first trajectory T 6 Ba in FIG. 6 B and the triangular points 1′-12′ in FIG. 6A are used to plot the second trajectory T 6 Bb in FIG. 6B .

[0135] FIG. 7A shows the beam transformation with a phase shift of r/2, as explained above, of the section P 1 a -P 1 b of the real curve shown in FIG. 1 . Quantum lines n=1 to n=11 have been superimposed on the two trajectories T 7 Aa and T 7 Ab. It is interesting to observe that many of the market rebounds occur when the trajectories touch or are very close to the quantum lines (e.g., at points P 7 Aa, P 7 Ab and P 7 Ac), which would tend to confirm the existence of a quantum effect.

[0136] FIG. 7B shows a beam-average curve B 7 B which is the “center of mass” (or the average relative parameter) of the two trajectories T 7 Aa and T 7 Ab of FIG. 7A . The beam-average curve may be obtained for any number of trajectories. The meaning of “beam-average curve”, “center of mass of trajectories” and “trajectory of center of mass” are absolutely equivalent. As was mentioned above, the quantum effect has a statistical character. From the averaging of two trajectories, it may be seen that the quantum directions roughly followed by the beam-average curve B 7 B are the quantum lines n=5 and n=8.

[0137] FIG. 7C shows a beam-average curve B 7 C for 200 trajectories obtained by the phase shift r/200 for the same section P 1 a -P 1 b of the real curve shown in FIG. 1 . Thus, FIG. 7C differs from FIG. 7B only by the number of trajectories used and the manner of presentation. FIG. 7B is drawn mainly to explain the principle of construction of a beam-average curve while FIG. 7C is a real example of a graph that may be used in market analysis. The curve of FIG. 7C may be obtained by using software written in accordance with the teachings contained herein. The quantum effect may be seen much more clearly in FIG. 7C than in FIG. 5 . The difference may be explained by two factors: a greater number of trajectories used (e.g., two hundred trajectories used in FIG. 7C in comparison to only five used in FIG. 5 ), and the independent character of the trends (see FIG. 4 ) used for averaging in FIG. 5 . Due to the much smoother character of the fluctuations, the curve B 7 C subsequently follows the quantum line n=1 in the section P 7 C 1 -P 7 C 2 , line n=3 in the section P 7 C 3 -P 7 C 4 , line n=6 in the section P 7 C 5 -P 7 C 6 , line n=5 in the section P 7 C 7 -P 7 C 8 , briefly follows line n=7 in the section P 7 C 9 -P 7 C 10 and, finally, the line n=8 in the section P 7 C 11 -P 7 C 12 .

[0138] The Increment-Change Space transformations discussed earlier were generally based on a fixed measurement increment r of the market parameter (e.g. price, exchange rate, etc) axis (or parameter-normalization). However, one may also affect a transformation in time-normalized Increment-Change Space as set forth in expressions (iv)-(vi) that may be described as follows: if the market parameter (e.g. stock market closing price) is measured at equal time intervals τ (for example one day), then irrespective of real rise or fall of the market parameter, the corresponding rise or fall in Increment-Change Space is set at a constant value r. In other words, only the direction of change of the market parameter is reflected. If the change is a rise, the fixed value r is added to the preceding Y coordinate; if it is a fall, the fixed value r is deducted. Of course such a transformation will considerably distort the price axis, but what matters is that motion in such space must obey the same universal laws.

[0139] In the same way, as shown in F