We present improvements to the function representation and generation method used in the Monte Carlo analysis of incomplete ordinary differential equations. Our method widens the scope of the technique to cover cases ...
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We present improvements to the function representation and generation method used in the Monte Carlo analysis of incomplete ordinary differential equations. Our method widens the scope of the technique to cover cases in which no envelopes have been specified for the function under consideration, thereby extending the applicability of the Monte Carlo approach to the full repertoire of models developed for qualitative reasoning algorithms, and paving the ground for the integrated operation of these two highly complementary techniques. Our new representation does not entail unjustified implicit assumptions about the shape of the generated functions, and provides better coverage of the space of models defined by the input specifications. Our simulator (MOCASSIM) also has the capability of imposing additional restrictions (e.g., convexity) on function shapes, which is particularly useful when the Monte Carlo technique is applied for solving system dynamics problems. (c) 2005 Elsevier Ltd. All rights reserved.
We revisit a problem solved in 1963 by Zaanen & Luxemburg in this MONTHLY: What is the largest possible length of the graph of a monotonic function on an interval? Is there such a function that attains this length...
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We revisit a problem solved in 1963 by Zaanen & Luxemburg in this MONTHLY: What is the largest possible length of the graph of a monotonic function on an interval? Is there such a function that attains this length? This is an interesting and intriguing problem with a somewhat surprising answer that should be of interest to a broad spectrum of mathematicians, starting with upper-level undergraduates. Our proof is very elementary, as opposed to the proof of Zaanen & Luxemburg. We are also able to give a pleasing geometric interpretation of our proof that is not possible with the proof of Zaanen & Luxemburg.
In this paper we establish some generalizations of a weighted trapezoidal inequality for monotonic functions and give several applications for the r-moments, the expectation of a continuous random variable and the Bet...
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In this paper we establish some generalizations of a weighted trapezoidal inequality for monotonic functions and give several applications for the r-moments, the expectation of a continuous random variable and the Beta and Gamma functions.
Least-squares monotone regression (also known as least-squares isotone regression) is being used increasingly. Recently, an application arose in which the fitted values are restricted to be integers. We prove that a v...
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Least-squares monotone regression (also known as least-squares isotone regression) is being used increasingly. Recently, an application arose in which the fitted values are restricted to be integers. We prove that a very simple procedure yields the optimum monotonic fit subject to this restriction, or to the more general restriction that the fitted values lie in some specified closed set. This procedure is to perform unrestricted monotone regression, and then to round off each resulting value to the nearest element of the closed set. When there are two nearest elements in the closed set, either one may be chosen, subject to the preservation of monotonicity and to one minor complication. [ABSTRACT FROM AUTHOR]
In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy's type involving monotonic functions on a time scale T in different spaces Lp(T) and Lq(T) when 0<p<...
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In this paper, we prove some new characterizations of weighted functions for dynamic inequalities of Hardy's type involving monotonic functions on a time scale T in different spaces Lp(T) and Lq(T) when 0
Let be a compact Hausdorff space equipped with a closed partial ordering. Let be a linear ordering that either does not have a maximal element or does not have a minimal element. We further assume that has the Tietze ...
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Let be a compact Hausdorff space equipped with a closed partial ordering. Let be a linear ordering that either does not have a maximal element or does not have a minimal element. We further assume that has the Tietze extension property for order preserving continuous functions from to . Denote by the lattice of order preserving continuous functions from to . We generalize a theorem of Kaplanski [K], and show that as a lattice alone, characterizes as an ordered space.
We consider the approximation of a given completely monotonic function $F(t)$ by an exponential sum $Y(t) = a_1 \exp ( - \lambda t) + \cdots + a_n \exp ( - \lambda _n t)$ using the $L_2 $-norm associated with a suit...
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We consider the approximation of a given completely monotonic function $F(t)$ by an exponential sum $Y(t) = a_1 \exp ( - \lambda t) + \cdots + a_n \exp ( - \lambda _n t)$ using the $L_2 $-norm associated with a suitable measure $d\mu (t)$. A best approximation exists and is characterized by a generalized version of the Aigrain–Williams equations. Examples are given to show that a given F may have several best approximations. We apply Rice’s dual algorithm for nonlinear $L_2 $-approximation to this problem and show that it reduces the norm of the error at each step. The algorithm calls for the repeated solution of an associated nonlinear Hermite interpolation problem which involves simple rational functions when $d\mu (t) = dt,0 < t < \infty $. The rate of convergence can be accelerated from first to second order by an inexpensive extrapolation. We discuss the numerical implementation of the algorithm and present numerical results for $F(t) = {1} /{(1 + t)},d\mu (t) = dt,0 < t < \infty $.
The various differential approximation schemes for producing an exponential sum approximation to a given function F are placed within a common mathematical framework, and localization theorems are established in the i...
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The various differential approximation schemes for producing an exponential sum approximation to a given function F are placed within a common mathematical framework, and localization theorems are established in the important case where F is completely monotonic. The replacement of the least squares minimization by a Galerkin orthogonalization leads to a promising new variation of differential approximation which is analyzed and then illustrated by several numerical examples.
We examine the continuity of maps between linearly ordered spaces and apply the results to utility functions representing preference relations. In particular, we show that the continuous function constructed by Debreu...
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