Showing papers on "Monotone polygon published in 1977"
TL;DR: This work shows the nonexistence of an algorithm to compute the meet over all paths solution for monotone frameworks, and shows that the maximal fixed point solution exists for every instance of everymonotone framework, and that it can be obtained by Kildall's algorithm.
Abstract: We consider a generalization of Kildall's lattice theoretic approach to data flow analysis, which we call monotone data flow analysis frameworks. Many flow analysis problems which appear in practice meet the monotonicity condition but not Kildall's condition called distributivity. We show that the maximal fixed point solution exists for every instance of every monotone framework, and that it can be obtained by Kildall's algorithm. However, whenever the framework is monotone but not distributive, there are instances in which the desired solution--the "meet over all paths solution" -- differs from the maximal fixed point. Finally, we show the nonexistence of an algorithm to compute the meet over all paths solution for monotone frameworks.
382 citations
TL;DR: It is shown that Ladner's simulation of Turing mac]hines by boolean circuits seems to require an "adequate" set of gates, such as AND and NOT, but the same simulation is possible with monotone circuits using AND and OR gates only.
Abstract: Ladner [3] ]has shown that the circuit value problem (CV) is log space complete for P, adding to the list of such problems found by Cook [iJ and Jones and Laaser [2]. Ladner's simulation of Turing mac]hines by boolean circuits seems to require an \"adequate\" set of gates, such as AND and NOT. We show that the same simulation is possible with monotone circuits using AND and OR gates only. Theorem The monotone circuit value problem (MCV) is log space complete for P. Proof Clearly MCV • P, and the construction below shows that CV -
246 citations
TL;DR: A stochastic matrix is defined to be monotone if its row-vectors are stochastically increasing as discussed by the authors, i.e., the row vectors of the matrix are uniformly increasing.
199 citations
TL;DR: In this article, the set of solutions uE C of the variational inequality is defined as a subset of H x H with the property that whenever [xi, zui] E A for i = 1,2, then (wr wa, x1 XJ > 0.
174 citations
TL;DR: For a set of monotone (and/or convex) data, this article considered the possibility of finding a spline interpolant, of pre-determined smoothness, which is either monotonicity or convex.
Abstract: For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.
112 citations
01 Jan 1977
TL;DR: In this paper, it was shown that if K is a compact convex subset of a linear Hausdorff topological space over the reals and T is a monotone and hemicontinuous mapping of K into E*, then there is a u 0 ∈ K such that (T(u 0), v - u 0) > 0 for all v ∈ k.
Abstract: In this note we have given a direct proof of the result which states that if K is a compact convex subset of a linear Hausdorff topological space E over the reals and T is a monotone and hemicontinuous (nonlinear) mapping of K into E*, then there is a u0 ∈ K such that (T(u0), v - u0) > 0 for all v ∈ K.
90 citations
TL;DR: In this article, a class of monotonely equivalent automorphisms (standard automomorphisms) is studied, which includes all ergodic automorphs with discrete spectrum and most of the well-known examples of automorphism with zero entropy.
Abstract: A class of monotonely equivalent automorphisms (standard automorphisms), which includes all ergodic automorphisms with discrete spectrum and most of the well-known examples of automorphisms with zero entropy, is studied. The basic results are two necessary and sufficient conditions for standardness: the first in terms of periodic approximation and the second in terms of the asymptotic properties of "words" arising from a coding of most trajectories by a finite partition. Also certain monotone invariants are defined and their properties discussed.Bibliography: 39 titles.
71 citations
TL;DR: In this article, sufficient conditions are given for the existence of such a trajectory, couched in terms of a notion of tangency developed by Clarke, which uses techniques of Filippov.
Abstract: : Let a given set be endowed with a preference preordering, and consider the problem of finding a solution to a differential inclusion which remains in the given set and evolves monotonically with respect to the preordering. Sufficient conditions are given for the existence of such a trajectory, couched in terms of a notion of tangency developed by Clarke. No smoothness or convexity is involved in the construction, which uses techniques of Filippov.
55 citations
TL;DR: This paper proved Jackson type estimates for the approximation of monotone non-decreasing functions by splines with equally spaced knots, which is of the same order as the Jackson type estimate for unconstrained approximation by spline with evenly spaced knots.
Abstract: We prove Jackson type estimates for the approximation of monotone nondecreasing functions by monotone nondecreasing splines with equally spaced knots. Our results are of the same order as the Jackson type estimates for unconstrained approximation by splines with equally spaced knots.
46 citations
TL;DR: In this paper, the authors take as a starting point this mapping and obtain results that are applicable to a broad class of problems and apply them to the positive and negative dynamic programming models of Blackwell and Strauch.
Abstract: The structure of many sequential optimization problems over a finite or infinite horizon can be summarized in the mapping that defines the related dynamic programming algorithm. In this paper we take as a starting point this mapping and obtain results that are applicable to a broad class of problems. This approach has also been taken earlier by Denardo under contraction assumptions. The analysis here is carried out without contraction assumptions and thus the results obtained can be applied, for example, to the positive and negative dynamic programming models of Blackwell and Strauch. Most of the existing results for these models are generalized and several new results are obtained relating mostly to the convergence of the dynamic programming algorithm and the existence of optimal stationary policies.
46 citations
TL;DR: In this paper, the existence of monotone trajectories for a class of discrete and continuous systems sufficiently general to include problems of some interest in economic and biological theory was proved.
Abstract: We prove existence of « monotone trajectories » for a class of discrete and continuous systems sufficiently general to include problems of some interest in economic and biological theory. We prove existence of critical points which are Pareto minima. We study stability properties of Pareto minima.
TL;DR: General inequalities and theorems for the empirical Bayes multiple decision problem with particular applications to a classification problem and a linear-loss monotone multiple decision problems are presented.
Abstract: In the empirical Bayes approach to multiple decision problems, we obtain theorems and lemmas which can be used to obtain asymptotic optimality and rate results in any multiple decision empirical Bayes problem. Applications of these results to a classification problem, a monotone multiple decision, and a selection problem are given. In addition, a special lemma unique to the monotone multiple decision problem gives improved (exact) rate results in that case.
TL;DR: In this paper, a procedure for the construction of a monotone estimator that dominates a given estimator for a class of discrete distributions with monotonous likelihood ratio is given.
Abstract: Summary
A procedure is given for the construction of a monotone estimator that dominates a given estimator for a class of discrete distributions with monotone likelihood ratio. This procedure is applied to some empirical Bayes estimators. Monte Carlo results are given that demonstrate the usefulness of monotonizing.
TL;DR: It is shown that the model theory for monotone quantifiers behaves very much like classical model theory.
Abstract: A generalization of the existential and universal quantifier, the monotone quantifiers, are studied. It is shown that the model theory for monotone quantifiers behaves very much like classical model theory. Completeness theorems, definability theorems and preservation theorems are given. Ultraproducts, reduced products and Back and Forth arguments are studied.
TL;DR: In this paper, a monotone iterative technique for the computation of a solution of a Riccati-type equation relevant to the theory of differential games is presented, and it is shown that the Kleinman algorithm converges under extremely general conditions.
Abstract: We present a monotone iterative technique for the computation of a solution of a Riccati-type equation relevant to the theory of differential games. For this purpose, we show that the Kleinman algorithm for Riccati equation computations converges under extremely general conditions.
01 Jan 1977
TL;DR: In this article, a maximal monotone operator from a real Banach space X to its dual X* may have several extensions into a maximal Monotone Operator from X** to X* by Zorn's lemma.
Abstract: An example is given which shows that a maximal monotone operator from a Banach space X to its dual X* may have several extensions into a maximal monotone operator from X** to X*. Introduction. Let X be a real Banach space with dual X* and let T: X 2x be a maximal monotone operator. Identifying as usual X to a subspace of X**, we look at T as a monotone operator from X** to 2X*; by Zorn's lemma, this operator can be extended into a maximal monotone operator from X** to 2x*. We are interested here in the question whether this extension is unique. There are a number of cases where it is so. Denote by T: X** -* 2x the (monotone) operator whose graph is the closure of the graph of T with respect to the weakest topology on X** x X* which is stronger than u(X**,X*) x u(X*,X**) and such that (x**,x*) * is upper semicontinuous. Since any maximal monotone extension of T to X ** contains T, we see that if T is maximal monotone, then T has a unique maximal monotone extension to X**. An operator T such that T is maximal monotone is called of dense type (a terminology slightly different from that of [2]). This kind of condition arises in the study of monotone operators in nonreflexive Banach spaces (cf. [2], [1], [7]). It is known, for instance, that the subdifferential of a convex function or the monotone operator associated with a saddle function are of dense type (cf. [6], [2], [5], [4]). On the other hand, there are maximal monotone operators which are not of dense type but which have a unique maximal monotone extension to the bidual (cf. the example in [3]: the uniqueness assertion is contained in Proposition I of [3] and the fact that the operator considered there is not of dense type follows easily from relation (1) of [3]). It is our purpose in this note to construct a maximal monotone operator which admits several (actually an infinity) maximal monotone extensions to the bidual. Our construction is based on a refinement of the method used in [3]. Received by the editors February 5, 1976. AMS (MOS) subject classifications (1970). Primary 47H05; Secondary 46B10, 35J60. ? American Mathematical Society 1977
TL;DR: In this paper, exact formulae for the Hausdorff dimensions of the level sets and graph, and of the image of a fixed time set, for a Gaussian process with stationary increments and monotone incremental variance were obtained.
Abstract: The purpose of this paper is twofold. First we obtain exact formulae for the Hausdorff dimensions of the level sets and graph, and of the image of a fixed time set, for a Gaussian process with stationary increments and monotone incremental variance. Inequalities for these dimensions have been obtained by Kahane in the case where the process is the sum of a trigonometric series with random coefficients. Secondly we obtain some precise results for the brownian motion process. We consider when an image set has positive Lebesgue measure and when the zero set has positive capacity with respect to a given kernel. We show that certain conditions, which Kahane has shown to be sufficient to ensure these properties, are also necessary.
TL;DR: In this article, it was shown that the monotone multivariate failure rates of Brindley and Thompson have no natural analog involving the multivariate fault rate function of Basu for continuous distributions.
Abstract: It is shown that the monotone multivariate failure rates of Brindley and Thompson have no natural analog involving the multivariate failure rate function of Basu for absolutely continuous distributions. Quantities related to the multivariate failure rate function are used to define monotone failure rates. It is shown that these are equivalent to the monotone failure rates of Brindley and Thompson. Based on these quantities, the loss of memory property of Marshall and Olkin is characterized.
TL;DR: In this article, an existence theorem for the optimal control of variational inequalities governed by a pseudomonotone operator is proved for the case where the cost is assumed to be quadratic.
Abstract: We prove an existence theorem for the optimal control of variational inequalities governed by a pseudomonotone operator: the cost is assumed to be quadratic. Then, we give an extension of the theorem to more general costs (assuming the operator to be monotone); we also give a result on a perturbation problem.
TL;DR: In this article, the convergence conditions for the monotone and autonomous iterative algorithm model are studied and two convergence conditions are presented, and it is shown that they are not comparable and that they contain the known convergence conditions.
Abstract: This paper is devoted to the study of convergence conditions for the monotone and autonomous iterative algorithm model. Two convergence conditions are presented, and it is shown that (i) they are not comparable and (ii) they contain the known convergence conditions for the model. The presentation of the results is facilitated by the introduction of the concept of extended characteristic set of an iterative procedure.
01 Jan 1977
TL;DR: In this article, Kenderov et al. showed that a monotone operator T: E -E* is locally bounded at x E E E if there is a neighborhood U of x such that T(U) = U{Ty: y E U} is a bounded subset of E*. This does not demnand that x E D(T) = { y E E: Ty # 0}.
Abstract: If a Banach space E has an equivalent norm such that weak* sequential convergence and norm convergence coincide on the dual unit sphere, then every monotone operator on E is single-valued and norm-norm continuous on a dense Gs subset of E. In particular, this holds for reflexive spaces. Let E be a real Banach space with dual E*. A multivalued mapping T: E -E* is called a monotone operator if 0> whenever x* E Tx and y* E Ty. It is called maximal monotone if, in addition, its graph, {(x, x*): x E E, x* E Tx}, is not properly contained in the graph of any monotone operator on E. We say that a monotone operator T: E -E* is locally bounded at x E E if there is a neighborhood U of x such that T(U) = U{Ty: y E U} is a bounded subset of E*. This does not demnand that x E D(T) = { y E E: Ty # 0}. We say that T is continuous at a point x E D(T) if, whenever n x, x* E Tx,E and x* E Tx, we have x,* x* || 0. If T is continuous at x, then it is necessarily single-valued at x, that is, Tx has exactly one element. We will assume from now on that T is a maximal monotone operator on E, with D(T) = E. This latter hypothesis, while not strictly necessary, simplifies both the statements and the proofs of our results. All the proofs can actually be extended to the case where D(T) # E, provided int conv D(T) # 0. The reason for this stems from the first part of the following result of Rockafellar [9], which will be of further use to us. PROPOSITION A. Let T: E -> E* be a maximal monotone operator with int convD(T) # 0. Then intD(T) is convex, clD(T) = cl intD(T), T is locally bounded at each point of int D(T) and T is not locally bounded at any point of the boundary of D(T). The motivation for studying monotone operators comes from the study of Received by the editors May 11, 1976. AMS (MOS) subject classifications (1970). Primary 47H05. I This work was done under the supervision of Professor R. R. Phelps while the author was a Ph.D. student at the University of Washington. It was supported by a Commonwealth Scientific and Industrial Research Organization Postgraduate Studentship. 2 The author notes that the same result has recently been announced by P. Kenderov and R. Robert, in C. R. Acad. Sci. Paris 282 (1976), No. 16. ? Aiiiericiani Mathemaltical Society 1977
TL;DR: In this paper, a constructive procedure for the generation of the solutions of the Galerkin approximations was developed explicitly and in detail in connection with the problem of constructivity.
TL;DR: In this article, the authors considered the problem of solving semilinear elliptic boundary value problems in unbounded domains, and showed that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies that there exists a weak solution u, and that v ≦ u ≦ w ≦ v.
Abstract: We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.
TL;DR: In this paper, proof is given of Lyusternik-Schnirelmann type theorems for convex functions in a Hilbert space, and a proof is also given for the case of convex function in a Euclidean space.
Abstract: : Proof is given of Lyusternik-Schnirelmann type theorems for convex functions in a Hilbert space
01 Jan 1977
TL;DR: In this paper, it is shown that any Pareto optimal program which is prefered or indifferent to the initial program by every agent can be reached by an abstract monotone planning procedure.
Abstract: It is shown that any Pareto Optimal program which is prefered or indifferent to the initial program by every agent can be “reached by an abstract monotone planning procedure”.