A Residual-Based Message Passing Algorithm for Constraint Satisfaction Problems

Message passing algorithms, whose iterative nature captures well complicated interactions among interconnected variables in complex systems and extracts information from the fixed point of iterated messages, provide a powerful toolkit in tackling hard computational tasks in optimization, inference, and learning problems. In the context of constraint satisfaction problems (CSPs), when a control parameter (such as constraint density) is tuned, multiple threshold phenomena emerge, signaling fundamental structural transitions in their solution space. Finding solutions around these transition points is exceedingly challenging for algorithm design, where message passing algorithms suffer from a large message fluctuation far from convergence. Here we introduce a residual-based updating step into message passing algorithms, in which messages varying large between consecutive steps are given a high priority in updating process. For the specific example of model RB, a typical prototype of random CSPs with growing domains, we show that our algorithm improves the convergence of message updating and increases the success probability in finding solutions around the satisfiability threshold with a low computational cost. Our approach to message passing algorithms should be of value for exploring their power in developing algorithms to find ground-state solutions and understand the detailed structure of solution space of hard optimization problems.

Threshold Phenomena for Random Cones

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index $$d-k$$ d - k when also k tends to infinity.

The Threshold between the Material and Immaterial Light. Notes on Luminescence and Dark Environments

This article discusses the phenomenon of luminescence in the production and visualization of images from an art-practice standpoint. The theoretical argument is developed through an analysis of artistic work that explores, inserts, expands, articulates, and interrogates the internal contradictions of UV light and the transitivity of light-sensitive materials in installation contexts. This investigation explores complexities in the encounter of antagonistic concepts: the threshold phenomena between materiality and immateriality, visibility and invisibility, light and darkness, disclosure and concealment. It aims to articulate a new perspective on contemporary debates on physiological, psychological, and environmental effects of light and darkness, articulated through aesthetic experience and artistic practice. Methods for engaging in the sensation of light and darkness will be introduced and how it unfolds as experiential qualities within installation projects will be considered.

Sharp critical thresholds in a hyperbolic system with relaxation

<p style='text-indent:20px;'>We propose and study a one-dimensional <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic <b>critical thresholds</b> for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.</p>

Random Intersection Graphs and Missing Data

Random-graphs and statistical inference with missing data are two separate topics that have been widely explored each in its field. In this paper we demonstrate the relationship between these two different topics and take a novel view of the data matrix as a random intersection graph. We use graph properties and theoretical results from random-graph theory, such as connectivity and the emergence of the giant component, to identify two threshold phenomena in statistical inference with missing data: loss of identifiability and slower convergence of algorithms that are pertinent to statistical inference such as expectation-maximization (EM). We provide two examples corresponding to these threshold phenomena and illustrate the theoretical predictions with simulations that are consistent with our reduction.

Threshold phenomena for high-dimensional random polytopes

Let [Formula: see text], [Formula: see text] be independent random points in [Formula: see text], distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension [Formula: see text] tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

An Approach to Analysing Erosive Characteristics of Two-Channel Combustion Chambers

To acquire the erosive characteristics of two-channel combustion chambers, a quasi-one-dimensional internal ballistics model is proposed. Combining the model with two different erosive burning models, the modified L-R expression, and the Mukunda-Paul expression, the flow parameters and the internal ballistics performance of a test solid rocket motor are computed. The results show good agreement with experimental data. According to the results, the more severe the erosion is, the earlier, longer, and gentler the tail-off stage becomes. During the tail-off stage, the dramatic drop of pressure leads to a low normal burning rate and makes it easier for erosion burning to occur. For this reason, notable erosive burning might appear during tail-off if the case-to-throat area ratio is extremely low. The results also show that flows in the inner channel and the outer channel are similar but not identical. This leads to different erosive burning behaviours in the two channels. Also, the two erosive burning rate models involved in this paper are compared. It seems that the M-P expression provides better results than the modified L-R expression does, since it reveals the threshold phenomena. Besides, the M-P expression has great advantages for its universality for most propellants and different SRM geometries.