Porous functions

1 Introduction

There are many fields in science and technology that provide strong motivation for the creation of new tools for mathematical modeling of real materials and structures, which includes, for example, granular materials; colloids, liquid crystals, emulsions, foams; polymers, textiles, rubber, glass; rock layers, sediments, oil, soil; biopolymers and biological materials, etc. Most recently, fractional-order elements and controllers are manufactured using various kinds of layered and fractal-like structures [5], graphenes [4], MWCNT-epoxy nanocomposites [6], and MXenes [11].

All real materials are characterized by various types of inhomogeneity, and very many of them are granular or porous. Granularity or porosity is also typical for many real structures that can be observed in nature or in technology. They clearly need an approach different from the idea of continuum.

In the presented paper some classical notions are re-examined from the viewpoint of the historical development of mathematics, and, as a result, the notions of porous intervals and porous functions are introduced and illustrated. Basic operations with such objects are also presented, and possible extensions for multidimensional cases are outlined.

2 Intervals on the real line

The coordinate system, invented by Descartes, was one of the greatest discoveries in the history of mathematics. It allowed to use numbers (associated with the length of intervals) for describing geometric shapes and objects using algebraic equations; and this combination of geometry and algebra is now known under the name of analytical geometry.

At the very same time, a strong link between the length of the interval and the number denoting this length was created. The most obvious manifestation of this link is the uniform ”real axis”, on which the addition of two numbers is performed as the addition of the lengths of the intervals corresponding to those numbers (Fig. 2.1). Of course, the addition of lengths was known and used many thousands of years before Descartes; but it was intuitive, and did not relate to the tools of algebra.

The next step was realizing that images of numbers can be put on the line in a non-uniform manner. On a line with logarithmic spacing, the addition of the lengths of intervals corresponds to the multiplication of the numbers associated through their logarithms with the lengths of their intervals (Fig. 2.2).

Fig. 2.1

Uniform axis: addition of lengths corresponds to addition of numbers

Fig. 2.2

Logarithmic axis: addition of lengths corresponds to multiplication of numbers

Knowing about the uniformly and logarithmically spaced “real axes”, it is clear that the set of real numbers can be mapped to the set of points of a line in infinitely many ways [2]. This can be used, for example, for simplifications of some particular operations or for other useful purposes.

The invention of the coordinate system by Descartes created the foundation for another great invention in the history of mathematics and physics – the geometrization of time introduced by Isaac Barrow and Isaac Newton [12]. From the plots in Newton’s own works [10] it is obvious that he was using the Descartes coordinates and considering the lengths of intervals as equivalents of quantities expressed by numbers (Fig. 2.3). Moreover, as Newton postulated [9] that “absolute, true and mathematical time of itself, and from its own nature, flows equably without relation to anything external’, it is clear that he was using uniformly spaced axes (similar to shown in Fig. 2.1).

The geometrization of time allowed depicting of processes of change in time (dynamical processes) using geometric curves, and since Newton’s time the values of the quantity that is changing with time are linked to the lengths of the corresponding line segments (Fig. 2.3, Fig. 2.4); in the case of logarithmic or semilogarithmic axes, which appeared much later, the lengths of intervals correspond to the logarithms of the quantity that is changing in time.

Fig. 2.3

Relationship between numbers and lengths in Newton’s works

Fig. 2.4

Geometrization of time

3 Porous intervals

Mainly due to Descartes, Barrow, and Newton the link between numbers and the lengths of segments notably influenced further theoretical developments in many directions. One of them was the formulation of the notion of a number. Since the line is continuous and does not contain holes, significant efforts were devoted to “filling the holes” between the rational numbers, and this finally led to the development of the notion of a real number by Cantor [3]. Naturally, before Cantor’s achievement, scientists used rational and irrational numbers and considered them being real numbers without having a formal clear definition; however, for computations they always used rational numbers, exactly like we do nowadays.

At this point, we have to take into account the difference between the real life and real computations, on one side, and the theoretical notion called real numbers, on the other. This difference is very well known to physicists, engineers, and applied scientists in general. In several of his works Vladimir Arnold, one of the great theoretical and applied mathematicians of modern times, reminds us that while the “real axis” is continuous, the real world is not. Arnold recalled [1] a discussion between two famous scientists: Lev Pontryagin and Yakov Zeldovich. In his textbook for students of physics and engineering, Zeldovich defined the derivative of a function as a ratio of the change of the function to the change of the variable, when the latter is sufficiently small. Pontryagin required that the limit of such ratio must be used, but Zeldovich argued that for really small distances and time intervals the structure of space and time at the quantum level is so different from the mathematical continuum, that the mathematical limits will lose any relation to the results of experiments.

Arnold also provides an example by Mikhail Lidov, who was solving the problem of landing lunar modules on the Moon. According to Lidov (and Arnold), although the proof of the uniqueness theorem for ordinary differential equations is mathematically perfectly correct, the landing based on this theorem is impossible in practice. The reason is that the time of landing would be infinite since the process of landing requires the control loop (or feedback); so the landing in the final stage, when the distance to the ground is sufficiently small, is replaced by a soft collision of the lunar module with the ground.

In fact, the examples provided by Arnold indicate that the differential and integral calculus inherited the same view on the link between the numbers and the lengths of line segments, which assumes that everything real is continuous; this also affected some important aspects of mathematical modeling of non-continuous reality, and many important engineering applications, especially automation and control (recall the relationship between the Laplace transform and Z-transform, and their use in control).

A close look at the process of computations on digital computers provides yet another example of the difference between the theory and reality.

Namely, on the “real axis” the Lebesgue measure of the set of rational numbers in [0, 1] is equal to 0, and the measure of the set of irrational numbers in [0, 1] is equal to 1. In computers (the reality!), however, it is reversed: the measure of the set of rational numbers in [0, 1] is equal to 1, and measure of the set of irrational numbers in [0, 1] is equal to 0, because in a computer every rational number represents the surrounding interval and therefore is, in fact, a kind of a particle, or a “grain”, and the size of such a “grain” is determined by the precision that is allowed by the hardware and the compiler in use.

It should also be noted here that on a real computer we have a finite number of rational numbers in [0, 1], so this interval is split into a finite number of “grains” (subintervals), and every number (rational or irrational) between 0 and 1 that we try to input in a computer falls onto some particular “grain”. Such practical issues led to the development of the so-called interval arithmetics, and all modern compilers for programming languages implement interval arithmetics in a hidden manner, not easily detectable by a scientist who does scientific computations.

However, the grains can be found also in the field that is far away from computer computations. Looking closely at the picture (Fig. 3.1a) showing the idea of the Lebesgue integration [7] and rotating it counterclockwise (Fig. 3.1b), we get the idea that one input grain (in this picture, the intervals around y₁ or around y₂) is mapped to a set of output grains. Keeping this idea, we have vertical porous intervals I₁ and I₂ shown in Fig. 3.1c.

Fig. 3.1

From Lebesgue integration to porous intervals

A porous interval, or p-interval, can be constructed as follows. Let us take the interval [a, b] and divide it into N subintervals; for now, let us consider subintervals of equal lengths. Using some probability distribution μ(x) with support in [a, b], generate N random values belonging to [a, b]. If one or more of these random values fall on some subinterval, then this subinterval is considered filled and is a grain, otherwise, this subinterval is considered empty and is a pore. The length δx of those subintervals can be called the pore size or the grain size. An interval containing only grains (not containing pores) is a granular interval.

A possible notation for denoting a porous interval I_p can be

In Fig. 3.1c the interval [0, 10] is divided into ten subintervals with the grains size equal 1 (this gives us a granular interval), and for two of them (the 3rd and the 6th) the vertical porous intervals [0, 20] with the grain size also equal 1 are shown.

In Fig. 3.2 vertical porous intervals for different sizes of grains/pores are shown. The grains can be also of non-rectangular shape (oval, polygon, etc.).

Fig. 3.2

Examples of porous intervals and shapes of grains/pores

Each porous interval is characterized by its size (size = b – a) and by its body (the set of grains). The addition of two porous intervals can be done in two ways: (a) attaching second interval to the first one (like we do with usual line segments); (b) overlapping the second interval with the first one, so some grains of the second interval will get into the pores in the first interval, and the result will have, in general, fewer pores. In both cases, the addition of two porous intervals is not commutative, except for the body addition when both porous intervals have the same size of grains/pores.

The multiplication of a porous interval by a constant K can also be done in two ways: (a) multiplication of the size of the porous interval by K and keeping the same grain/pore size and the probability distribution law; (b) multiplication of the size of grains/pores by K, which can be called zooming or change of scale. The first method is more natural since the size of grains/pores does not change.

4 Porous functions

Having outlined the notion of a porous interval, we can introduce the notion of porous functions, or p-functions.

Let us take a granular interval, that is an interval [a, b] divided into N subintervals G_i (i = 1, …, N) of the length δx = (b – a)/N. Each i-th subinterval is mapped to a (vertical) porous interval {u(ξ_i): δy: v(ξ_i), ρ(x)} with some grain size δy, where ξ_i ∈ G_i.

A possible notation for a porous function F_p on a granular interval I_p can be:

So, a porous function of one variable is defined by the constrains (the bound a and b of its granular domain), the grain sizesδx and δy (which in general can be different), by its lower shapeu(x) and upper shapev(x), and by the probability distribution μ which is used for computing grans and pores. A porous function has in general a porous body comprising grains and pores.

In Fig. 4.1 is shown an example of a porous function in granular interval [a, b], with a = 1 and b = 3 and with the grain/pore size δx = δy = 0.025, with lower shape u(x) ≡ 0 and upper shape v(x)=x, and with uniform distribution U(u(ξ_i), v(ξ_i)) of grains in each porous interval [u(ξ_i), v(ξ_i)].

Fig. 4.1

Porous function: shape, body, constrains, grains and pores

Looking at the objects of nature and at the composite materials, we immediately see numerous examples of porous functions – or, in other words, porous functions are suitable for modeling the structure of various objects of nature, complex materials and systems, etc.

In Fig. 4.2 one can see photos from the “garden of porous functions” – Jardin des Prés Fichaux in Bourges, France.

Fig. 4.2

In the “garden of porous functions” (Jardin des Prés Fichaux, Bourges, France)

In Fig. 4.3 it is shown that each multi-floor building with apartments or offices is yet another example of a porous function. Its granular domain, say [0, 1], consists of 12 subintervals with δx = 1/12, its lower shape is u(x) = 0, its upper shape is v(x) = 3, so δy = 1/6, with some probability distribution for determining the illuminated windows.

Fig. 4.3

Porous function: evaluation of the porous step function

In fact, Fig. 4.3 is an example of a porous step function. It also provides a simple illustration of the computation of a porous function: for each grain in the granular domain, a set of 18 random numbers is generated (since the upper and the lower shapes are constant, we have the same interval), and the window is illuminated if at least one of those 18 numbers falls in the corresponding interval.

If we repeat the evaluation of a porous function, we will see (due to randomness) a different configuration/arrangement of grains and pores within the body of a porous function. Repeating this process, in the limit we will obtain the body completely filled with grains (Fig. 4.4), which is, in fact, the case of a classical notion of function.

Fig. 4.4

The classical function as the limit of the porous function with u(x) = 0, v(x) = |sin x|, and grains/pore size δx = δy = 0.05. On the right, the number of the iteration is shown.

5 Operations with porous functions

For porous functions we can define two kinds of addition; we will call them shape addition and body addition.

In the case of shape addition of two porous functions, which are defined on the same granular interval, the second addend is put on top of the first addend, as illustrated in Fig. 5.1. This means that for each subinterval of the granular domain the corresponding porous interval of the second addend is put on top of the corresponding porous interval of the first addend. Obviously, the shape addition of porous functions is not commutative (see Fig. 5.1). One can think of shape addition as putting whipped cream on top of a coffee or a cake, making layered composite materials, etc.

Fig. 5.1

Porous functions: shape addition

The second kind of addition is body addition. For two porous functions, which are defined on the same granular interval, the grains of the second addend are put between the grains of the first addend, if this is possible. In the Fig. 5.2, the body addition of two porous functions with the same lower shape u(x) ≡ 0 and the same upper shape v(x) = x², but with different grain/pore sizes is illustrated.

Fig. 5.2

Porous functions: body addition

In other words, the grains of the second addend are put where are the pores of the first addend. It is also obvious that the body addition is not commutative, too. A simple illustration of the body addition is putting some small stones and sand in a jar: if the stones are put in first, the sand can fill the pores between the stones; but if the sand is put in first, then the picture will be different. The body addition can also be observed in rocks, minerals, ores, composite materials, etc.

The multiplication of a porous function by a constant can be defined as the multiplication of its shape functions, while preserving the grains/pore size.

6 Interpretation

Interpretations play an important role for various applications. Researchers normally do not think in terms of formulas (formulas come to play at the stage of implementation), but rather in terms of objects, or images, or processes. This is why the geometric interpretations are of great importance. We have already mentioned the geometric interpretations of addition of numbers and their multiplication; multiplication can also be geometrically imagined as the change of scale. Operations of differential and integral calculus also have traditional geometric interpretations. Namely, in all calculus textbooks, the first derivative is related to the slope of a tangent line to the function plot at a given point, and the one-fold integral of a positive-valued function is interpreted as “the area under the curve”. However, if we follow the historical development and realize that the idea of definite integration was known already to Archimedes, who used it for evaluation of the volumes of spheres and cylinders, and that the idea of differentiation appeared due to Newton and Leibniz many hundreds of years later, we can develop different geometric interpretations.

Recalling once again that numbers were strongly associated with the length geometric intervals or segments of lines, we can think of the function value v(x) as of a vertical segment  v(x) 0, which is shown in Fig. 6.1. Then the integral (historically the first notion of calculus) of the function v(x), expressing that famous “area A under the curve v(x)’, can be written as

Fig. 6.1

Function value as a line segment length

Fig. 6.2

Interpretation of the integral of a porous function: ’’amount of material in the porous function body’’

and its derivative (recall that derivatives historically appeared much later) can be written as

In other words, the derivative of an integral of a positive-valued function v(x) with moving upper bound can be interpreted as a segment of line of the length v(x).

Denoting here for simplicity a porous vertical interval as v(x) 0, we can write the integral of a porous function with the lower shape u(x)≡ 0 and the upper shape v(x) as

and interpret the integral of a porous function as “amount of material in the porous function body”. For example, in Fig. 4.1 it is the total area of grains in the porous function body; in Fig. 4.2 and Fig. 6.1 it is the amount of wood between the upper and the lower shape; in Fig. 4.3 it is the total area of lighted windows.

Then the derivative of the integral of a porous function is nothing else but a vertical porous interval:

Of course, since we consider here porous functions on granular intervals, the derivative should be treated in Zeldovich sense, with the increment of x equal to a sufficiently small size of the grains of the considered granular interval.

7 Towards porous functions of several variables

The notion of a porous interval can be easily extended to the notion of a porous domain in two or more dimensions. In Fig. 7.1a, a view on a forest is shown, and the trees in that forest can be considered as grains in the 2D porous domain, so the forest is an illustration of a porous function defined in a 2D porous domain. Similarly, the set of buildings shown in Fig. 7.1b is yet another example of a 2D porous function, where the function values (porous intervals) can represent the height of the buildings, or energy consumption (e.g., illuminated windows), etc.

Fig. 7.1

Examples of 2D porous domains and porous functions

Conclusions

In this paper, a new type of functions has been presented. We have introduced the notion of porous intervals (and granular intervals), the notion of a porous function defined in a porous or granular interval, basic operations with porous intervals and porous functions, and geometric interpretations of integration and differentiation of porous functions. We have demonstrated how porous functions can be visualized (or, so to say, “plotted”). In addition, we have outlined how the notion of porous intervals and porous functions can be extended to two-dimensional case. The distribution of grains/pores in a porous function body can be stochastic (determined by some probability distribution) or deterministic (determined by some rule or algorithm).

For computations, we developed and use the first version of the Porous Functions Toolbox for MATLAB. By their nature, porous functions are efficiently computed using object-oriented programming and parallel algorithms.

The work in progress includes extensions to two- and three-dimensional cases, and using porous functions in applications for modeling structure of real materials and systems. Also, based on the interpretations of the classical calculus operations provided in Section 6, the extension of integer-order differentiation and integration of porous functions to arbitrary orders (the fractional calculus) will be natural.

It must be noted here that, despite the similarity in the name, the porous intervals and porous functions introduced in this paper are not related to the so-called “porous sets” considered in [8] and other works.