Shotgun reconstruction in the hypercube

Mossel and Ross raised the question of when a random coloring of a graph can be reconstructed from local information, namely, the colorings (with multiplicity) of balls of given radius. In this article, we are concerned with random 2-colorings of the vertices of the -dimensional hypercube, or equivalently random Boolean functions. In the worst case, balls of diameter are required to reconstruct. However, the situation for random colorings is dramatically different: we show that almost every 2-coloring can be reconstructed from the multiset of colorings of balls of radius 2. Furthermore, we show that for , almost every -coloring can be reconstructed from the multiset of colorings of 1-balls.     

Electrodynamics of Magnetoelectric Media and Magnetoelectric Fields

The relationship between magnetoelectricity and electromagnetism is a subject of a strong interest and numerous discussions in microwave and optical wave physics and material sciences. The definition of the energy and momentum of the electromagnetic (EM) field in a magnetoelectric (ME) medium is not a trivial problem. The question of whether electromagnetism and magnetoelectricity can coexist without an extension of Maxwell's theory arises when the effects of EM energy propagation are studied and the group velocity of the waves in an ME medium is considered. The energy balance equation reveals unusual topological structure of fields in ME materials. Together with certain constraints on the constitutive parameters of a medium, definite constraints on the local field structure should be imposed. Analyzing the EM phenomena inside an ME material, the question “what kind of the near fields arising from a sample of such a material can be measured?” should be answered. The visualization of the ME states requires an experimental technique that is based on an effective coupling to the violation of spatial as well as temporal inversion symmetry. To observe the ME energy in a subwavelength region, it is necessary to assume the existence of first-principle near fields—the ME fields. These are non-Maxwellian near fields with specific properties of violation of spatial and temporal inversion symmetry. A particular interest to the ME fields arises in studies of metamaterials with “artificial-atoms” ME elements.     

Stochastic sensitivity analysis of coupled acoustic-structural systems under non-stationary random excitations

This paper presents a new sensitivity analysis method for coupled acoustic–structural systems subjected to non-stationary random excitations. The integral of the response power spectrum density (PSD) of the coupled system is taken as the objective function. The thickness of each structural element is used as a design variable. A time-domain algorithm integrating the pseudo excitation method (PEM), direct differentiation method (DDM) and high precision direct (HPD) integration method is proposed for the sensitivity analysis of the objective function with respect to design variables. Firstly, the PEM is adopted to transform the sensitivity analysis under non-stationary random excitations into the sensitivity analysis under pseudo transient excitations. Then, the sensitivity analysis equation of the coupled system under pseudo transient excitations is derived based on the DDM. Moreover, the HPD integration method is used to efficiently solve the sensitivity analysis equation under pseudo transient excitations in a reduced-order modal space. Numerical examples are presented to demonstrate the validity of the proposed method.     

Phase transitions in graphs on orientable surfaces

Let be the orientable surface of genus and denote by the class of all graphs on vertex set with edges embeddable on . We prove that the component structure of a graph chosen uniformly at random from features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd?s‐Rényi random graph chosen uniformly at random from all graphs with vertex set and edges. It takes place at , when the giant component emerges. The second phase transition occurs at , when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from and has only been observed for graphs on surfaces.     

The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

Information percolation and cutoff for the random‐cluster model

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.     

Measurement of Residual Dipolar Couplings of Organic Molecules in Multiple Solvent Systems Using a Liquid-Crystalline-Based Medium

Residual dipolar coupling (RDC), a robust anisotropic NMR parameter for structural elucidation of organic molecules, is only accessible in an anisotropic environment. Herein, we introduce a novel alignment medium based on the molecular self-assembly of oligopeptide amphiphile (OPA). This medium is compatible with different intermediate and polar solvent systems, such as CD₃OD, [D₆]DMSO, and D₂O. The preparation of the OPA-based medium is simple and rapid, while only very weak background signals were observed from OPAs. Furthermore, we show that the purity of OPA has only a minor influence on the quality of the RDC data. These advantages allow RDC measurements of organic molecules with different polarities and solubilities with high efficiency and accuracy.     

Convergence rates of the blocked Gibbs sampler with random scan in the Wasserstein metric

ABSTRACT

This paper establishes explicit estimates of convergence rates for the blocked Gibbs sampler with random scan under the Dobrushin conditions. The estimates of convergence in the Wasserstein metric are obtained by taking purely analytic approaches.     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.