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Economic and financial phenomena are highly complex and‎ ‎non-linear‎. ‎However‎, ‎surprisingly‎, ‎in many cases‎, ‎these phenomena‎ ‎are accurately described by linear models‎ -- ‎or‎, ‎sometimes‎, ‎by‎ ‎piecewise linear ones‎. ‎In this paper‎, ‎we show that fuzzy‎ ‎techniques can explain the unexpected efficiency of linear and‎ ‎piecewise linear models‎: ‎namely‎, ‎we show that a natural‎ ‎fuzzy-based precisiation of imprecise (``fuzzy'') expert knowledge‎ ‎often leads to linear and piecewise linear models‎.‎We show this by applying invariance ideas to analyze which membership functions‎, ‎which fuzzy ``and''-operations (t-norms)‎, ‎and which‎ ‎fuzzy implication operations are most appropriate for applications to economics and finance‎. ‎We also discuss which expert-motivated nonlinear models should be‎ ‎used to get a more accurate description of economic and financial‎ ‎phenomena‎: ‎specifically‎, ‎we show that a natural next step is to add cubic‎ ‎terms to the linear (and piece-wise linear) expressions‎, ‎and‎, ‎in general‎, ‎to consider polynomial (and piece-wise polynomial) dependencies‎.

Experts usually express their degrees of belief in‎ ‎their statements by the words of a natural language (like ``maybe''‎, ‎``perhaps''‎, ‎etc.) If an expert system contains the degrees of‎ ‎beliefs $t(A)$ and $t(B)$ that correspond to the statements $A$‎ ‎and $B$‎, ‎and a user asks this expert system whether ``$A\,\&\,B$'' is‎ ‎true‎, ‎then it‎ ‎is necessary to come up with a reasonable estimate for the‎ ‎degree of belief of $A\,\&\,B$‎. ‎The operation that processes $t(A)$‎ ‎and $t(B)$ into such an estimate $t(A\,\&\,B)$ is called an $\&$-operation‎. ‎Many‎ ‎different $\&$-operations have been proposed‎. ‎Which of them to‎ ‎choose? This can be (in principle) done by interviewing experts and‎ ‎eliciting a $\&$-operation from them‎, ‎but such a process is very‎ ‎time-consuming and therefore‎, ‎not always possible‎. ‎So‎, ‎usually‎, ‎to choose a $\&$-operation‎, ‎we extend the finite‎ ‎set of actually possible degrees of belief to an infinite set‎ ‎(e.g.‎, ‎to an interval [0,1])‎, ‎define an operation there‎, ‎and‎ ‎then restrict this operation to the finite set‎. ‎In this paper‎, ‎we consider only this original finite set‎. ‎We show that a‎ ‎reasonable assumption that an $\&$-operation is continuous (i.e.‎, ‎that gradual change in $t(A)$ and $t(B)$ must lead to a gradual‎ ‎change in $t(A\,\&\,B)$)‎, ‎uniquely determines $\min$ as an‎ ‎$\&$-operation‎. ‎Likewise‎, ‎$\max$ is the only continuous‎ ‎$\vee$-operation‎. ‎These results are in good accordance with the‎ ‎experimental analysis of ``and'' and ``or'' in human beliefs‎.

It is known that, to more adequately describe expert knowledge, it is necessary to go from the traditional (type-1) fuzzy techniques to higher-order ones: type-2, probably type-3 and even higher. Until recently, only type-1 and type-2 fuzzy sets were used in practical applications. However, lately, it turned out that type-3 fuzzy sets are also useful in some applications. Because of this practical importance, it is necessary to design efficient algorithms for data processing under such type-3 (and higher-order) fuzzy uncertainty. In this paper, we show how we can combine known efficient algorithms for processing type-1 and type-2 uncertainty to come up with a new algorithm for the type-3 case.

Among many research areas to which Ron Yager contributed are decision making under uncertainty (in particular, under interval and fuzzy uncertainty) and aggregation—where he proposed, analyzed, and utilized ordered weighted averaging (OWA). The OWA algorithm itself provides only a specific type of data aggregation. However, it turns out that if we allow several OWA stages, one after another, we obtain a scheme with a universal approximation property—moreover, a scheme which is perfectly equivalent to modern ReLU-based deep neural networks. In this sense, Ron Yager can be viewed as a (grand)father of ReLU-based deep learning. We also recall that the existing schemes for decision making under uncertainty are also naturally interpretable in OWA terms.

One of the most effective image processing techniques is the use of convolutional neural networks that use convolutional layers. In each such layer, the value of the layer’s output signal at each point is a combination of the layer’s input signals corresponding to several neighboring points. To improve the accuracy, researchers have developed a version of this technique, in which only data from some of the neighboring points is processed. It turns out that the most efficient case—called dilated convolution—is when we select the neighboring points whose differences in both coordinates are divisible by some constant ℓ. In this paper, we explain this empirical efficiency by proving that for all reasonable optimality criteria, dilated convolution is indeed better than possible alternatives.

As a system becomes more complex, at first, its description and analysis becomes more complicated. However, a further increase in the system’s complexity often makes this analysis simpler. A classical example is Central Limit Theorem: when we have a few independent sources of uncertainty, the resulting uncertainty is very difficult to describe, but as the number of such sources increases, the resulting distribution gets close to an easy-to-analyze normal one—and indeed, normal distributions are ubiquitous. We show that such limit theorems often make analysis of complex systems easier—i.e., lead to blessing of dimensionality phenomenon—for all the aspects of these systems: the corresponding transformation, the system’s uncertainty, and the desired result of the system’s analysis.

Different geophysical data sets such as receiver functions, surface wave dispersion measurements, and first arrival travel times, provide complementary information about the Earth structure. To utilize all this information, it is desirable to perform a joint inversion, i.e., to use all these datasets when determining the Earth structure. In the ideal case, when we know the variance of each measurement, we can use the usual Least Squares approach to solve the joint inversion problem. In practice, we only have an approximate knowledge of these variances. As a result, if a geophysical feature appears in a solution corresponding to these approximate values of variances, there is no guarantee that this feature will still be visible if we use the actual (somewhat different) variances. To make the joint inversion process more robust, it is therefore desirable to repeatedly solve the joint inversion problem with different possible combinations of variances. From the mathematical viewpoint, such solutions form a Pareto front of the corresponding multi-objective optimization problem.

One of the most computationally convenient nonredundant ways to describe the dependence between two variables is by describing the corresponding copula. In many applications, a special class of copulas—known as FGM copulas—turned out to be most successful in describing the dependence between quantities. The main result of this paper is that these copulas are the fastest to compute, and this explains their empirical success. As an auxiliary result, we also show that a similar explanation can be given in terms of fuzzy logic.

We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.

In many application problems, F-transform algorithms are very efficient. In F-transform techniques, we replace the original signal or image with a finite number of weighted averages. The use of a weighted average can be naturally explained, e.g., by the fact that this is what we get anyway when we measure the signal. However, most successful applications of F-transform have an additional not-so-easy-to-explain feature: the fuzzy partition requirement that the sum of all the related weighting functions is a constant. In this paper, we show that this seemingly difficult-to-explain requirement can also be naturally explained in signal-measurement terms: namely, this requirement can be derived from the natural desire to have all the signal values at different moments of time estimated with the same accuracy. This explanation is the main contribution of this paper.

The main ideas of F-transform came from representing expert rules. It would be therefore reasonable to expect that the more accurately the membership functions describe human reasoning, the more successful will be the corresponding F-transform formulas. We know that an adequate description of our reasoning corresponds to complicated membership functions—however, somewhat surprisingly, many successful applications of F-transform use the simplest possible triangular membership functions. There exist some explanations for this phenomenon, which are based on local behavior of the signal. In this paper, we supplement these local explanations by a global one: namely, we prove that triangular membership functions are the only one that provide the exact reconstruction of the appropriate global characteristic of the signal.

Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.

In engineering design problems, we want to make sure that a certain quantity c of the designed system lies within given bounds - or at least that the probability of this quantity to be outside these bounds does not exceed a given threshold. We may have several such requirements - thus the requirement can be formulated as bounds [Fc(x), Fc(x)] on the cumulative distribution function Fc(x) of the quantity c; such bounds are known as a p-box. The value of the desired quantity c depends on the design parameters a and the parameters b characterizing the environment: c = f(a, b). To achieve the design goal, we need to find the design parameters a for which the distribution Fc(x) for c = f(a, b) is within the given bounds for all possible values of the environmental variables b. The problem of computing such a is called backcalculation. For b, we also have ranges with different probabilities - i.e., also a p-box. Thus, we have backcalculation problem for p-boxes. For p-boxes, there exist efficient algorithms for finding a design a that satisfies the given constraints. The next natural question is to find a design that satisfies additional general, the problem of finding such a design is computationally difficult (NP-hard). We show that this problem is NP-hard already in the simplest possible linearized case, when the dependence c = f(a, b) is linear. We also provide an example when an efficient algorithm is possible.En los problemas de diseño de ingeniería, en los que quiere asegurarse de que una determinada cantidad c del sistema se encuentra dentro de los límites dados - o por lo menos que la probabilidad de esa cantidad fuera de estos límites no superen un determinado umbral. Es posible que haya varios requisitos - la exigencia puede formularse como límites [Fc(x), Fc(x)] en la función de distribución acumulada Fc(x) de la cantidad de c, esos límites son conocidos como p-caja. El valor de la cantidad deseada c depende de los parámetros de diseño a y los parámetros b caracterizar el medio ambiente: c = f(a, b). Para lograr el objetivo de diseño, tenemos que encontrar los parámetros de diseño a para que la distribución de Fc(x) para c = f(a, b) esé dentro de los límites dados por todos los valores posibles de las variables ambientales b. El problema de la informática se llama a retrocálculo. Por b, también tienen rangos con diferentes probabilidades, es decir, también una p-box. Por lo tanto, tenemos un problema de retrocálculo para p-cajas. Para p-cajas, existen algoritmos eficientes para encontrar un diseño a que satisface las restricciones dadas. La pregunta lógica es encontrar un diseño que satisfaga restricciones adicionales: el coste, la eficiencia, etc. En este trabajo, demostramos, en general, el problema de encontrar un diseño que es computacionalmente difícil (NPhard). Se demuestra que este problema es NP-hard ya en el caso lineal más simple posible, cuando la dependencia c = f(a, b) es lineal. También ofrecemos un ejemplo, cuando un algoritmo eficiente es posible.

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms—of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

It is well known that the ancient Egyptians represented each fraction as a sum of unit fractions - i.e., fractions with unit numerators; this is how they, e.g., divided loaves of bread. What is not clear is why they used this representation. In this paper, we propose a new explanation: crudely speaking, that the main idea behind the Egyptian fractions provides an optimal way of dividing the loaves. We also analyze the related properties of fractions.

In many practical situations like weather prediction, we are interested in large-scale (averaged) value of the predicted quantities. For example, it is impossible to predict the exact future temperature at different spatial locations, but we can reasonably well predict average temperature over a region. Traditionally, to obtain such large-scale predictions, we first perform a detailed integration of the corresponding differential equation and then average the resulting detailed solution. This procedure is often very time-consuming, since we need to process all the details of the original data. In our previous papers, we have shown that similar quality large-scale prediction results can be obtained if, instead, we apply a much faster procedure—first average the inputs (by applying an appropriate fuzzy transform) and then use these averaged inputs to solve the corresponding (discretization of the) differential equation. In this paper, we provide a general theoretical explanation of why our semiheuristic method works, that is, why fuzzy transforms are efficient in large-scale predictions.

In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have functions A1(x)≥0,...,An≥0, with ∑i=1nAi(x)=1, we can then represent each function f(x) by the coefficients Fi=(∫f(x)⋅Ai(x)dx)/(∫Ai(x)dx). Once we know the coefficients Fi, we can (approximately) reconstruct the original function f(x) as ∑i=1nFi⋅Ai(x). The original motivation for this transformation came from fuzzy modeling, but the transformation itself is a purely mathematical transformation. Thus, the empirical successes of this transformation suggest that this transformation can be also interpreted in more traditional (nonfuzzy) mathematics as well. Such an interpretation is presented in this paper. Specifically, we show that the 2002 probabilistic interpretation of fuzzy modeling by Sánchez et al. can be modified into a natural probabilistic explanation of fuzzy transform formulas.