Misha Koshelev

Fast implementations of fuzzy arithmetic operations using fast Fourier transform (FFT)

Abstract In engineering applications of fuzzy logic, the main goal is not to simulate the way the experts really think, but to come up with a good engineering solution that would (ideally) be better than the experts control. In such applications, it makes perfect sense to restrict ourselves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy numbers, then we can use simple and fast algorithms that are known for operations with simple membership functions. In other applications, especially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership functions that capture every nuance of the experts opinion; these functions are therefore complicated, and fuzzy arithmetic operations with the corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm uses Fast Fourier Transform (FFT) to reduce computation time from O( n 2 ) to O( n log( n )) (where n is the number of points x at which we know the membership functions μ ( x )). To compute FFT even faster, we propose to use special hardware. The results of this paper were announced in the work of Kosheleva et al. [Proc. 1996 IEEE Int. Conf. on Fuzzy Systems, Vol. 3, pp. 1958–1964].

Why monotonicity in interval computations? A remark

Monotonicity of functions has been successfully used in many problems of interval computations. However, in the context of interval computations, monotonicity seems somewhat ad hoc. In this paper, we show that monotonicity can be reformulated in interval terms and is, therefore, a natural condition for interval mathematics.

We must Choose the Simplest Physical Theory: Levin-Li-Vitányi Theorem and its Potential Physical Applications

If several physical theories are consistent with the same experimental data, which theory should we choose? Physicists often choose the simplest theory; this principle (explicitly formulated by Occam) is one of the basic principles of physical reasoning. However, until recently, this principle was mainly a heuristic because it uses the informal notion of simplicity.

Encryption algorithms made natural

Modern cryptographic algorithms such as DES, IDEA are very complex and therefore difficult to learn. Textbooks explain in detail how these algorithms work, but they usually do not explain why these algorithms were designed as they were. In this paper, we explain the why, which hopefully will make cryptographic algorithms easier to learn.

A new method of measuring strong currents by their magnetic field

In many situations, it is difficult to measure strong currents directly, so the currents are measured by their magnetic fields (e.g. by sensors glued onto the cable surface). If we have only one cable, then it is easy to reconstruct the current from the magnetic field. But in many real-life situations, there is a neighboring cable whose current is also unknown. Moreover, the cables are hanging freely, so their distance is not precisely known. We show that in such situations, it is sufficient to have two pairs of sensors that measure magnetic field; the unknown current can be uniquely reconstructed from the corresponding measurement results by applying a simple computer algorithm.

Optimal Enclosure of Quadratic Interval Functions

AbstractIn this paper, we analyze the problem of the optimal (narrowest) approximation (enclosure) of a quadratic interval function

Why it is computationally harder to reconstruct the past than to predict the future

y(x_1 ,...,x_n ) = [y(x_1 ,...,x_n ) \bar y(x_1 ,...x_n )]

Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty

(i.e., an interval function for which both endpoint functions