Unbounded Case Research Articles

Unbounded behaviorally consistent stopping rules

In this article we study behaviorally consistent stopping rules in an unbounded search from a known distribution with no recall and with positive search cost. We show that if the searcher's preferences are quasi-convex in the probabilities, then behaviorally consistent search strategies in the unbounded case are obtained as limits of the corresponding bounded search strategies and are characterized by reservation levels property. Unlike optimal stopping rules under expected utility theory, however, the reservation levels may not be monotonically increasing in the number of permissible stages of the search process, and, in the unbounded case, may not be unique.

A potential of fuzzy relations with a linear structure: The unbounded case

This paper is a sequel to Yoshida et al. (1993), in which the potential theory for linear fuzzy relations on the positive orthant R+n is considered in the class of fuzzy sets with a compact support under the contractive assumption. In this paper, potential treatment for unbounded fuzzy sets is developed without the assumption of contraction and compactness. The objective of this paper is to give the existence and the characterization of potentials for linear fuzzy relations under some reasonable conditions. Also, introducing a partial order in fuzzy sets, we prove Riesz decomposition theorem in the fuzzy potential theory. The proofs are shown by using only the linear structure and the monotonicity of fuzzy relations. In the one-dimensional case, the potential and its α-cuts are explicitly calculated. Numerical examples are given to comprehend further discussions.

Validation of a Monte Carlo simulation of the plane Couette flow of a rarefied gas

We report and discuss the results of a direct Monte Carlo simulation of the flow of a rarefied gas flowing between two parallel plates when one of them moves in its own plane. The boundary conditions are assumed to be of the “bounceback” type and the molecules to be Maxwell's. Under this condition the moments can be computed exactly, following a method used by Ikenberry and Truesdell in the unbounded case. This allows a comparison of the Monte Carlo methods with the exact solution and an evaluation of its accuracy.

Generalized solution of TE modes in fifth-order nonlinear dielectric media

We present the general solution of Maxwell's equations for TE modes in a dielectric medium characterized by an intensity dependent dielectric constant of the form ε = ε₁ α|<i>E</i>|² + γ|<i>E</i>|⁴. The solution is obtained for a bounded fifth-order nonlinear dielectric medium (i.e., for a film) that has been shown to lead in the limiting forms to the known solutions for the unbounded case (i.e., the medium forming a cladding or a substrate) and the solutions for the Kerr case (both bounded and unbounded cases).

Growth and Indeterminancy in Dynamic Models with Externalities

We study the indeterminacy of equilibria in infinite horizon capital accumulation models with technological externalities. Our investigation encompasses models with bounded and unbounded accumulation paths, and models with one and two sectors of production. Under reasonable assumptions we find that equilibria are locally unique in one-sector economies. In economies with two sectors of production it is instead easy to construct examples where a positive external effect induces a two-dimensional manifold of equilibria converging to the same steady state (in the bounded case) or to the same constant growth rate (in the unbounded case). For the latter we point out that the dynamic behavior of these equilibria is quite complicated and that persistent fluctuations in their growth rates are possible.

Bounding and Dominating Number of Families of Functions on ω

AbstractWe pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to their bounding and dominating numbers, the traditional scheme for gaps. Many open questions remain.Mathematics Subject Classification: 03E35, 04A20.

Lack of generation of strongly continuous semigroups by the damped wave operator on H x H (or: The little engine that couldn't)

We show that (in the unbounded case) the wave operator, or more generally, a wave operator with damping strictly less than the basic operator does not generate a strongly continuous semigroup on the product space H x H, contrary to a common belief.

Evaluating the Volume of Porous Medium Investigated During Slug Tests

AbstractThis paper presents a methodology for quantifying the volume of porous medium investigated during a slug test in an unbounded porous medium, in the presence of a linear constant‐head or no‐flow boundary, and in the presence of a radial no‐flow boundary. For the unbounded case, type curves are generated for different values of the wellbore storage coefficient, which relate the distance travelled by a given pressure perturbation (1, 5, and 10% of the initial drawdown in the well), to dimensionless time. This distance is found to increase linearly on a log‐log plot until it reaches a maximum which is a function of the wellbore storage coefficient. The appropriate choice of dimensionless groups allows the different curves for each level of perturbation to be collapsed into one curve. Functional relationships offer an alternative to using the type curves graphically. For bounded systems, type curves relate the distance to the boundary to the time of 1 and 5% deviation from the unbounded response. Although these curves cannot be collapsed, the presented range of wellbore storage coefficients covers most practical situations. Developed relationships allow the estimation of the maximum distance travelled by the pressure perturbations in the unbounded case, and the maximum distance at which a linear constant‐head or no‐flow boundary, or a radial no‐flow boundary, still produces a given deviation in the pressure response measured at the well. An application shows that substantial error may result if the distance to a boundary is evaluated while neglecting storage in the well. Finally, application of the methodology developed for a linear no‐flow boundary to a real data set yields a realistic distance to the boundary and a far better match between simulated and measured data than if an unbounded system is considered. The type curves and relationships presented here should be applicable to slug test design and analysis.

ON THE EQUIVALENCE OF TWO-WAY PUSHDOWN AUTOMATA AND COUNTER MACHINES OVER BOUNDED LANGUAGES

It is known that two-way pushdown automata are more powerful than two-way counter machines. The result is also true for the case when the pushdown store and counter are reversal-bounded. In contrast, we show that two-way reversal-bounded push-down automata over bounded languages (i.e., subsets of [Formula: see text] for some distinct symbols a1,…, ak) are equivalent to two-way reversal-bounded counter machines. We also show that, unlike the unbounded input case, two-way reversal-bounded pushdown automata over bounded languages have decidable emptiness, equivalence and containment problems.

Uniform approximation theorems for real-valued continuous functions

For a topological space X, F( X) denotes the algebra of real-valued functions over X and C( X) the subalgebra of all functions in F( X) which are continuous. In this paper we characterize the uniformly dense linear subspaces of C( X) by means of the so-called “Lebesgue chain condition”. This condition is a generalization to the unbounded case of the S-separation by Blasco and Moltó for the bounded case. Through the Lebesgue chain condition we also characterized the linear subspaces of F( X) whose uniform closure is closed under composition with uniformly continuous functions.

An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain

For boundary value problems posed on unbounded domains it is often appropriate to impose a boundary condition at "infinity". For certain classes of boundary value problem obvious numerical difficulties can be avoided by truncating the unbounded domain and solving a sequence of finite domain problems instead. We introduce a novel technique which is straightforward to implement and which exploits information contained in this sequence in order to extrapolate to the unbounded case. The technique introduces a new and interesting application of a variety of convergence acceleration algorithms.

New form for the optimal rendezvous equations near a Keplerian orbit

are removable singularities, they may lead to computational instabilities in the solution. Especially bothersome is the fact that computational problems can occur where the true anomaly is near zero. These problems are avoided in the work of solution that does not involve 7(0), but their work is confined to elliptical orbits. The purpose of this Note is to modify the original form by replacing 1(0) by a related integral 7(0), thereby removing all singularities and computational instabilities. The resulting solution, in terms of /(0), is identical for hyperbolic, parabolic, or noncircular elliptic orbits, but the particular case determines the nature of the closed-form evaluation of 7(0). Application of this work to actual problems usually involves the solution of a two-point boundary-value problem and is not considered here. Although we emphasize the case of bounded thrust, the unbounded thrust case can also be investigated through the use of the simpler equations for unpowered flight, which we also present. If the maximum number of impulses is known for an optimal rendezvous in this case, the two-point boundary-value problem is reduced to a problem of parameter optimization on the velocity increments and their locations. We conjecture that the maximum of impulses for this problem is four. If the number of impulses is restricted to two, the problem is solved by methods similar to those of Weiss et al.11'12 For the case of bounded thrust, a method such as that used for the rendezvous problem near circular orbit16 can be applied with starting iteratives obtained from the related unbounded thrust case.

The integer linear complementarity problem

In this paper we consider the case of the linear complementarity problem where all or some of the variables are required to take integer values. We discuss several applications to economic equilibrium problems and polymatrix games. When the integer variables are bounded, then the problem can be solved using an equivalent linear integer formulation. For the general problem (unbounded case) the problem can be solved using enumeration of the feasible points of a set of mixed zero-one linear inequalities.

Stochastic product integral w.r.t. infinite dimensional semimartingale:ii–uniform operator topology case

While stochastic product integration w.r.t finite rank Hilbert Schmidt operator (K2- ) valued semimartingales has received reasonable attention, there is no work, to date, dealing with infinite dimensional semimartingales. We initiate here a theory of stochastic product integration of K2- predictable processes w.r.t the exponential of a K2- Brownian motion. To understand the estimations involved, we start with a finite rank Brownian motion β while keeping the integrand in K2 and then we continue with infinite dimellsioilal Brownian motion. In order to apply our stochastic product integral to solve Doleans–Dade – Protter type linear stochastic equations, we also need to, and therefore, define the Ito integral in the space HS. We finally give the stochastic product integral construction of the solution of a linear stochastic equation. All this is done in the uniform operator topology case. The more difficult unbounded operator case is the subject matter of the final part of this series

On the complexity of finding the chromatic number of a recursive graph II: the unbounded case

A recursive graph is a graph whose edge set and vertex set are both recursive. Although the chromatic number of a recursive group G (denoted χ( G)) cannot be determined recursively, it can be determined if queries to the halting set are allowed. We show that the problem of determining the chromatic number of a recursive graph with a minimum number of queries to the halting set, is closely related to the unbounded search problem. In particular if ƒ is a non-decreasing function such that ∑ i⩾02 −ƒ(i) is effectively computable, then there is an algorithm to determine χ( G) with ƒ(χ(G)) queries to K iff ∑ i⩾02 −ƒ(i)⩽1 (i.e., ƒ satisfies Kraft's inequality). We also investigate recursive chromatic numbers (which require queries to a set much harder than the halting set, namely ∅ M ), the effect of allowing queries to a weaker set, and the effect of being able to ask p queries at a time. Most of our results are also true for highly recursive graphs (graphs where one can determine the neighbors of a given node recursively), though there are some interesting differences when queries to K are allowed for free in the computation of a recursive chromatic number.

Darling-Erd?s theorems for martingales

This paper considers Darlin-Erdős theorems for sums of martingale differences. Our main theorem provides an optimal result for the case of bounded martingale difference sequences. A number of other results are presented, which deal with the unbounded case and which specialize to the case of independent summands. Previous related work on this problem has been based on deep strong approximation theorems. One of the novel features of our approach is that our methods rely on the more easily accessible Skorokhod-type embeddings.

Microscopic model for hysteresis and phase equilibria of fluids confined between parallel plates.

The hysteresis effects of fluids confined in porous media are studied by means of a new microscopic model. We consider a slitlike pore of infinite area and finite height. In contrast to the unbounded parallel-plate case, the new model allows the formation and the disappearance of curved menisci at the open ends of the pore. This effect has a profound influence on the extent of the hysteresis loops. On the other hand, we find that the presence of the menisci does not play a significant role in locating the equilibrium first-order transition in long pores. The present treatment reconciles the traditional picture of capillary condensation with the modern microscopic treatment based on the density-functional formalism.

An unbounded generalization of Hilbert algebras

A generalization of Hilbert algebras to the unbounded case is considered. While such constructions have been considered earlier, the present one accommodates a richer class of algebras of unbounded operators and is more applicable to examples.

The generalized strauss compactness lemma and its applications on nonlinear elliptic problems

To PROVE the existence of solutions of partial differential equations, usually it needs some compactness properties. For the bounded domain case, there is a useful one-Rellich-Kondrakov theorem. However for the unbounded domain case, we know quite a few of such results. Strauss [I I] proved a compactness lemma for the entire space RN. In this article we generalized Strauss’ result to the case of an infinite strip Q = RN x (0, l), N L 2. Several applications on elliptic equations are also given. In this section we introduce some notations, concepts, and equations which will be used later. In Section 2 we prove the compactness lemma. In Section 3 we give some applications on nonlinear elliptic equations, as one of the continuing studies of Lions [8] and Berestycki-Lions [41. Consider (0, 1) as the circle group R/Z, and Q = RN x (0, I) the locally compact abelian group with the Haar measure dxdt where dx and dt are Lebesque measure on R’ and (0, l), respectively. We therefore may consider the convolution of two functions or generalized functions on the group !A. Let f: R + R be a measurable function. The distribution function f,~: Rf R+ off is defined by, for CY > 0, f&) = Ilx E 4lf(.@l > 41

On the asymptotic distribution of eigenvalues of banded matrices

We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.