Domain Monotonicity Research Articles

Learning bayesian network parameters from limited data by integrating entropy and monotonicity

High-accuracy parameter learning in Bayesian Networks (BNs) is a key challenge in real-time decision support applications, particularly when the available data are limited. Prior/Expert knowledge was introduced to eliminate the drawbacks of insufficient information; however, this method is subjective. In this study, we explored the use of monotonicity constraints to control the causal relationships between the nodes and their parents-nodes in BNs and proposed a new learning algorithm called global domain monotonicity based on maximum information entropy (GDM-MIE), which was designed for parameter learning in uncertain discrete BNs with nonlinear equality constraints when only finite data are available. In the proposed algorithm, a class of monotonicity is encoded as a constraint on information entropy and the parameter learning problem is transformed into constraints among the node parameters based on a known network. Furthermore, we considered the parameters as uncertain entropy information and discussed the monotonicity among parameters in the global spatial domain, proving the accuracy of the logical relationships of the model, system reliability, and time complexity. Finally, the proposed method was validated using standard BNs, and its performance was analyzed by comparing the proposed method with the existing learning algorithms. The results showed that the proposed method is more accurate and has better Kullback–Leibler divergence. To revalidate the rationality of the proposed method, the Alarm and Asia networks were employed as special cases. The GDM-MIE was found to achieve the intended goal of observing the estimated parameters by closely approximating the original real parameters with a small sample size, indicating that the proposed algorithm can served as an efficient and feasible method for learning Bayesian parameter.

Strict domain monotonicity of the principal eigenvalue and a characterization of lower boundedness for the Friedrichs extension of four-coefficient Sturm–Liouville operators

Using the variational characterization of the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension \( T_F\) of the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In the more general singular context, these four-coefficient differential expressions act according to $$\tau f= \frac{1}{r}(-(f^{[1]})'+sf^{[1]}+qf)\, {\rm with}\, f^{[1]}=p[f'+sf]\, {\rm on}\, (a,b) \subseteq \mathbb{R}, $$ where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p > 0, r > 0 a.e. on (a, b), and \( p^{-1},q,r,s \in L^{1}_{loc}((a,b);dx)\), and \( f\) is supposed to satisfy $$f\in AC_{loc}((a,b)),p[f'+sf] \in AC_{loc}((a,b)). $$ This setup is sufficiently general so that \(\tau \) permits certain distributional potential coefficients q, including potentials in \(H^{-1}_{loc}((a,b)) \).As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result, we characterize all lower bounds of \(T_F \) as those \(\lambda \in \mathbb{R} \) for which the differential equation \(\tau u =\lambda u \) has a strictly positive solution u > 0 on (a, b).

A Variational Characterization of Fluid Sloshing with Surface Tension

We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in an open container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed to the zero surface tension case, where the problem reduces to a partial differential equation for the velocity potential, we obtain a coupled system for the velocity potential and the free surface displacement. We derive a new variational formulation of the coupled problem and establish the existence of solutions using the direct method from the calculus of variations. We prove a domain monotonicity result for the fundamental sloshing eigenvalue. In the limit of zero surface tension, we recover the variational formulation of the mixed Steklov-Neumann eigenvalue problem and give the first-order perturbation formula for a simple eigenvalue.

Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian

Abstract We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.

An eigenvalue optimization problem for the p-Laplacian

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

$$A$$ A -harmonic Picone’s identity with applications

Picone-type identity for a class of nonlinear differential operators which includes the Finsler \(p\)-Laplacian as special case is established and various applications of this formula (such as the nonexistence of positive supersolutions, the uniqueness and simplicity of principal eigenvalues for problems involving \(A\)-harmonic operators, domain monotonicity property, Barta-type inequality, Sturmian comparison principle) are given.

Lower estimates for hyperbolic metrics on subdomains

A result concerning lower estimates for hyperbolic metrics on subdomains is proved. The method of proof involves a standard technique based on the classical Schwarz–Pick lemma that yields the domain monotonicity of the hyperbolic metric. The information we obtained could be used to estimate, from below, the behavior of the metric near the boundary. We will also investigate the situation when equality holds to show the sharpness of our lower bound.

Upper estimates for hyperbolic metrics on subdomains

A result concerning upper estimates for hyperbolic metrics on subdomains is proved. The method of proof involves a standard technique based on the classical Schwarz–Pick lemma that yields the domain monotonicity of the hyperbolic metric. The information we obtain would be used to estimate the hyperbolic length of a closed curve in the subdomain away from the boundary.

EIGENVALUE ESTIMATES ON DOMAIN OF THE POLYDISK

In this paper, we investigate universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem on a bounded domain in an n-dimensional polydisk 𝔻n. Moreover, from the domain monotonicity of the eigenvalue, we can prove that if the first eigenvalue of the Dirichlet Laplacian tends to [Formula: see text] when the domain tends to the polydisk 𝔻n, then all of the eigenvalues tend to [Formula: see text].

Mirror Coupling of Reflecting Brownian Motion and an Application to Chavel's Conjecture

In a series of papers, Burdzy et al. introduced the <em>mirror coupling</em> of reflecting Brownian motions in a smooth bounded domain $D\subset\mathbb{R}^d$, and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplacian on $D$. In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domains $D_1, D_2\subset\mathbb{R}^d$. As applications of the construction, we derive a unifying proof of the two main results concerning the validity of Chavel's conjecture on the domain monotonicity of the Neumann heat kernel, due to I. Chavel ([12]), respectively W. S. Kendall ([16]), and a new proof of Chavel's conjecture for domains satisfying the ball condition, such that the inner domain is star-shaped with respect to the center of the ball.

An eigenvalue problem with mixed boundary conditions and trace theorems

An eigenvalue problem is considered where the eigenvalue appears in the domain and on the boundary. This eigenvalue problem has a spectrum of discrete positive and negative eigenvalues. The smallest positive and the largest negative eigenvalue $\lambda_{\pm 1}$ can be characterized by a variational principle. We are mainly interested in obtaining non trivial upper bounds for $\lambda_{-1}$. We prove some domain monotonicity for certain special shapes using a kind of maximum principle derived by C. Bandle, J.v. Bellow and W. Reichel in [J. Eur. Math. Soc., 10 (2007), 73-104]. We then apply these bounds to the trace inequality.

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ 1 Ω (α) corresponding to Δu = λ (α) u in \(\Omega, \frac{\partial u}{\partial v} = \alpha u \) on ∂Ω, with α a fixed real, and \(\Omega \subset {\mathcal{R}}^n\) a C 0,1 bounded domain. If α > 0 and small, we derive bounds for λ 1 Ω (α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.

Monotonicity results for the principal eigenvalue of the generalized Robin problem

We study domain monotonicity of the principal eigenvalue $\lambda_1^\Omega(\alpha)$ corresponding to $\Delta u=\lambda(\alpha) \, u \text{ in } \Omega, \frac{\partial u}{\partial \nu} =\alpha\, u \text{ on } \partial \Omega$, with $\Omega \subset {\mathcal R}^n$ a $C^{0,1}$ bounded domain, and $\alpha$ a fixed real. We show that contrary to intuition domain monotonicity might hold if one of the two domains is a ball.

Domain Monotonicity and the Performance of Local Solutions Strategies for CDPS-based Distributed Sensor Interpretation and Distributed Diagnosis

The growth in computer networks has created the potential to harness a great deal of computing power, but new models of distributed computing are often required. Cooperative distributed problem solving (CDPS) is the subfield of multi-agent systems (MAS) that is concerned with how large-scale problems can be solved using a network of intelligent agents working together. Building CDPS systems for real-world applications is still very difficult, however, in large part because the effects that domain and strategy characteristics have on the performance of CDPS systems are not well understood. This paper reports on the first results from a new simulation-based analysis system that has been created to study the performance of CDPS-based distributed sensor interpretation (DSI) and distributed diagnosis (DD). To demonstrate the kind of results that can be obtained, we have investigated how the monotonicity of a domain affects the performance of a potentially very efficient class of strategies for CDPS-based DSI/DD. Local solutions strategies attempt to limit communications among the agents by focusing on using the agents' local solutions to produce global solutions. While these strategies have been described as being important for effective CDPS-based DSI/DD, they need not perform well if a domain is nonmonotonic. We had previously suggested that the reason they have performed well in several research systems was that many DSI/DD domains are what we termed nearly monotonic. In this paper, we will provide quantitative results that relate the performance of local solutions strategies to the monotonicity of a domain. The experiments confirm that domain monotonicity can be important to consider, but they also show that it is possible for these strategies to be effective even when domains are relatively nonmonotonic. What is required is that the agents receive a significant fraction of the data that is relevant to their subproblems. This has important implications for the design of DSI/DD systems using local solutions strategies. In addition, while the work indicates that many DSI/DD domains are likely to be “nearly monotonic” according to our original definitions, it also shows that these measures are not as predictive of performance as other measures we define. This means that near monotonicity alone does not explain why local solutions strategies have performed well in previous systems. Instead, a likely explanation is that these systems typically involved only a small number of agents.

Uniform estimates for the hyperbolic metric and Euclidean distance to the boundary.

Throughout this article,D is a proper subdomain of the complex plane C possessing at least two finite boundary points, usually termed a hyperbolic domain. Each suchD carries constant negative curvature metrics, and we let λD denote the scale factor or density for the maximal constant curvature −1 metric. We call λD the Poincare hyperbolic metric for D; it can be defined by λD(z) = λB(ζ)/|p′(ζ)| = 2/(1− |ζ|2)|p′(ζ)|, where z = p(ζ) and p : B→ D is any holomorphic covering projection from the unit disk B = {|ζ| < 1} onto D. See [BP; HM; M1; M2] and their references for basic properties of the Poincare metric. An elementary exercise using Schwarz’s lemma shows thatλD satisfies a domain monotonicity property, from which we easily conclude that

Reflecting Brownian Motions and Comparison Theorems for Neumann Heat Kernels

We consider the challenging problem of Chavel′s conjecture on the domain monotonicity of the fundamental solution of the Neumann problem. It says that if D ⊂ D̃ are open convex domains and if p t ( x, y) and p̃ t ( x, y) denote the corresponding parabolic heat kernels for the respective Neumann problems, then: p t(x, y) ≥ p ̃ t(x, y) ∀x, y ∈ D, ∀t > 0. The quantities p t ( x, y) and p̃ t ( x, y) can be interpreted as the transition densities of the reflecting Brownian motions X t and X̃ t in D and D̃, respectively. The conjecture can be restated as a comparison problem for Brownian motions with reflecting boundary conditions. We give a detailed analysis of the small time asymptotics of the Brownian paths reflected on the boundary of a polyhedron and we give a probabilistic proof of Chavel′s conjecture for small time t uniformly for all x and y in D when the closure of D is contained in D̃. An interesting by-product of our proof is that it does not require the large domain to be convex.

On Domain Monotonicity of the Neumann Heat Kernel

Some examples are given of convex domains for which domain monotonicity of the Neumann heat kernel does not hold.

A domain monotonicity theorem for graphs and Hamiltonicity

A necessary condition for the existence of long cycles in a graph involving the Laplacian spectrum of graphs is derived. It is based on two theorems which relate the eigenvalues of a graph and the eigenvalues of its induced subgraphs.

Coupled Brownian motions and partial domain monotonicity for the Neumann heat kernel

Let D and Ω be two convex domains (with smooth boundary) and with corresponding Neumann heat kernels η D ( x, y, t) and η gW ( x, y, t). Chavel has conjectured that if x, y ϵ D ⊂ Ω then the following monotonicity result holds for all t: η D(x, y, t) ⩾ η Ω(x, y, t) . He has proved this in the special case when Ω is a ball centred at either x or y. By exploiting the connection of the Neumann kernel to reflecting Brownian motion it is possible to prove the result when Ω is general and D is a ball centred at either x or y. The proof depends on a careful coupling construction of the reflecting Brownian motions for D and Ω using the same probability space for both processes.

On the domain monotonicity of the first eigenvalue of a boundary value problem

This paper is concerned with the monotonicity of the first eigenvalue λ1 (D) of (1) as a functional of the domainD.