The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spaces W1,p, p > n - 1. It is proven that such weakly monotone functions are continuous except in a singular set of p-capacity...
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The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spaces W1,p, p > n - 1. It is proven that such weakly monotone functions are continuous except in a singular set of p-capacity zero that is empty in the case p = n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.
Pseudo-Boolean monotone functions are unimodal functions which are trivial to optimize for some hillclimbers, but are challenging for a surprising number of evolutionary algorithms. A general trend is that evolutionar...
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Pseudo-Boolean monotone functions are unimodal functions which are trivial to optimize for some hillclimbers, but are challenging for a surprising number of evolutionary algorithms. A general trend is that evolutionary algorithms are efficient if parameters like the mutation rate are set conservatively, but may need exponential time otherwise. In particular, it was known that the (1 + 1)-EA and the (1 + lambda)-EA can optimize every monotone function in pseudolinear time if the mutation rate is c/n for some c < 1, but that they need exponential time for some monotone functions for c > 2.2. The second part of the statement was also known for the (mu + 1)-EA. In this paper we show that the first statement does notapply to the (mu + 1)-EA. More precisely, we prove that for every constant c > 0 there is a constant mu(0) is an element of N such that the (mu + 1)-EA with mutation rate c/n and population size mu(0) <= mu <= n needs superpolynomial time to optimize some monotone functions. Thus, increasing the population size by just a constant has devastating effects on the performance. This is in stark contrast to many other benchmark functions on which increasing the population size either increases the performance significantly, or affects performance only mildly. The reason why larger populations are harmful lies in the fact that larger populations may temporarily decrease the selective pressure on parts of the population. This allows unfavorable mutations to accumulate in single individuals and their descendants. If the population moves sufficiently fast through the search space, then such unfavorable descendants can become ancestors of future generations, and the bad mutations are preserved. Remarkably, this effect only occurs if the population renews itself sufficiently fast, which can only happen far away from the optimum. This is counter-intuitive since usually optimization becomes harder as we approach the optimum. Previous work missed the effect because it focus
Over the last decades, Hardytype inequalities have been heavily investigated (see, e.g., [6, 7]) especially on cones of monotone functions. In this context, we refer to [1–5, 8, 10–15], which stimulated our interest...
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Over the last decades, Hardytype inequalities have been heavily investigated (see, e.g., [6, 7]) especially on cones of monotone functions. In this context, we refer to [1–5, 8, 10–15], which stimulated our interest in the issues discussed below.
A measure of departures of monotonicity of a given function, the L(r)-DIP, 1 less than or equal to r less than or equal to infinity, is introduced. Our analysis is performed to cover two different situations: When the...
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A measure of departures of monotonicity of a given function, the L(r)-DIP, 1 less than or equal to r less than or equal to infinity, is introduced. Our analysis is performed to cover two different situations: When the function is known our interest is related to its behavior in a stochastic model. However, in most cases, the knowledge of the function is obtained through a preliminary estimation of the function. In both situations the aim focuses in the obtainment of strong consistency results.
In this paper, we give some optimal upper bounds for the Sugeno's integral of monotone functions. More precisely, we show that: If g: [0, infinity) -> [0, infinity) is a continuous and strictly monotone functio...
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In this paper, we give some optimal upper bounds for the Sugeno's integral of monotone functions. More precisely, we show that: If g: [0, infinity) -> [0, infinity) is a continuous and strictly monotone function, then the fuzzy integral value p = f(0)(a)gd mu, with respect to the Lebesgue measure mu, verifies the following sharp inequalities: (a) g (a - p) >= p for the increasing case, and (b) g(p) >= p for the decreasing case. Moreover, we show that under adequate conditions, these optimal inequalities provides a powerful tool for solving fuzzy integrals. Also, some examples and application. are presented. (c) 2006 Elsevier Inc. All rights reserved.
We prove a fundamental theorem concerning the existence of coupled maximal and minimal solutions of the dynamical systems involving the difference of two monotone functions. Besides the relevance of such problems in m...
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We prove a fundamental theorem concerning the existence of coupled maximal and minimal solutions of the dynamical systems involving the difference of two monotone functions. Besides the relevance of such problems in mathematical biology, this treatment has several implications to the theory of monotone iterative techniques, as has been pointed in the several remarks made in the paper. (C) 2002 Elsevier Science Inc. All rights reserved.
We investigate whether circuit lower bounds for monotone circuits can be used to derandomize randomized monotone circuits. We show that, in fact, any derandomization of randomized monotone computations would derandomi...
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We investigate whether circuit lower bounds for monotone circuits can be used to derandomize randomized monotone circuits. We show that, in fact, any derandomization of randomized monotone computations would derandomize all randomized computations, whether monotone or not. We prove similar results in the settings of pseudorandom generators and average-case hard functions - that a pseudorandom generator secure against monotone circuits is also secure with somewhat weaker parameters against general circuits, and that an average-case hard function for monotone circuits is also hard with somewhat weaker parameters for general circuits. (C) 2012 Elsevier B.V. All rights reserved.
In this work, a general purpose fuzzy controller which allows the fuzzy set ZE to be used anywhere in decision tables is proposed to handle the class of monotone functions. For guaranteed convergence and accuracy, the...
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In this work, a general purpose fuzzy controller which allows the fuzzy set ZE to be used anywhere in decision tables is proposed to handle the class of monotone functions. For guaranteed convergence and accuracy, the rules on the selection of fuzzy sets based on the defuzzification method mean-of-inversion (MOI) are given are proved. Such a guideline can relieve the users' burden on testing accuracy after design. In addition, these imposed restrictions on the selection of fuzzy sets are not unusual in the design of fuzzy controllers. Thus, the optimization on convergence speed is possible for the proposed fuzzy controller in various applications. (C) 1997 Elsevier Science B.V.
We consider Hardy-type operators on the cones of monotone functions with general positive sigma-finite Borel measure. Some two-sided Hardy-type inequalities are proved for the parameter -infinity < p < infinity....
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We consider Hardy-type operators on the cones of monotone functions with general positive sigma-finite Borel measure. Some two-sided Hardy-type inequalities are proved for the parameter -infinity < p < infinity. It is pointed out that such equivalences, in particular, imply a new characterization of the discrete Hardy inequality for the (most difficult) case 0 < q < p <= 1.
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