Random Point Sets Research Articles

Constrained simulations and excursion sets: understanding the risks and benefits of ‘genetically modified’ haloes

Constrained realisations of Gaussian random fields are used in cosmology to design special initial conditions for numerical simulations. We review this approach and its application to density peaks providing several worked-out examples. We then critically discuss the recent proposal to use constrained realisations to modify the linear density field within and around the Lagrangian patches that form dark-matter haloes. The ambitious concept is to forge `genetically modified' haloes with some desired properties after the non-linear evolution. We demonstrate that the original implementation of this method is not exact but approximate because it tacitly assumes that protohaloes sample a set of random points with a fixed mean overdensity. We show that carrying out a full genetic modification is a formidable and daunting task requiring a mathematical understanding of what determines the biased locations of protohaloes in the linear density field. We discuss approximate solutions based on educated guesses regarding the nature of protohaloes. We illustrate how the excursion-set method can be adapted to predict the non-linear evolution of the modified patches and thus fine tune the constraints that are necessary to obtain preselected halo properties. This technique allows us to explore the freedom around the original algorithm for genetic modification. We find that the quantity which is most sensitive to changes is the halo mass-accretion rate at the mass scale on which the constraints are set. Finally we discuss constraints based on the protohalo angular momenta.

The Material Point Monte Carlo model: A discrete, off-lattice method for microstructural evolution simulations

A new kinetic Monte Carlo model, termed the Material Point Monte Carlo (MPMC) method, is developed to simulate microstructural evolution under a variety of driving forces. The MPMC operates on a spatially random set of discrete material points; at each time step, points are relocated and their material data are updated according to a statistical mechanics scheme. Because material points are randomly distributed, and remain so under affine deformations, the MPMC eliminates artifacts of the simulation grid that occur in lattice-based methods such as the Monte Carlo Potts model. For model systems the MPMC is verified to correctly reproduce grain boundary motion caused by bulk and interfacial driving forces. It is then applied to larger and more complex problems, including grain growth during plastic deformation.

Incidences between points and generalized spheres over finite fields and related problems

Abstract Let 𝔽 q ${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and Q = a 1 ⁢ x 1 c 1 + ⋯ + a d ⁢ x d c d ∈ 𝔽 q ⁢ [ x 1 , … , x d ] , ${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$ where 2 ≤ c i ≤ N ${2\leq c_{i}\leq N}$ , gcd ⁡ ( c i , q ) = 1 ${\gcd(c_{i},q)=1}$ , and a i ∈ 𝔽 q ${a_{i}\in\mathbb{F}_{q}}$ for all 1 ≤ i ≤ d ${1\leq i\leq d}$ . A Q -sphere is a set of the form { 𝒙 ∈ 𝔽 q d ∣ Q ⁢ ( 𝒙 - 𝒃 ) = r } , ${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$ where 𝒃 ∈ 𝔽 q d , r ∈ 𝔽 q ${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$ . We prove bounds on the number of incidences between a point set 𝒫 ${{{\mathcal{P}}}}$ and a Q-sphere set 𝒮 ${{{\mathcal{S}}}}$ , denoted by I ⁢ ( 𝒫 , 𝒮 ) ${I({{\mathcal{P}}},{{\mathcal{S}}})}$ , as the following: | I ⁢ ( 𝒫 , 𝒮 ) - | 𝒫 | ⁢ | 𝒮 | q | ≤ q d / 2 ⁢ | 𝒫 | ⁢ | 𝒮 | . $\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$ We also give a version of this estimate over finite cyclic rings ℤ / q ⁢ ℤ ${\mathbb{Z}/q\mathbb{Z}}$ , where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q-sphere set in 𝔽 q d ${\mathbb{F}_{q}^{d}}$ . We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

Pattern-Search-Based Nonconvex Cooperative Sensing in Multiband Cognitive Radio Systems

In this paper, a pattern search (PS)-based solution is proposed for nonconvex multiband cooperative sensing (NCMCS) problem in cognitive radio systems. This problem consists of maximizing cumulative throughput subject to constraints on cumulative interference, probability of detection, and probability of false alarm. Initially in existing literature, this problem was solved under constraints that make it convex. However, removing the conditions for convexity and solving the NCMCS problem have been shown to improve performance. A two-step PS-based solution is presented: The first step uses uniformly distributed random sets of input points to find a solution. The set of points that gives the maximum throughput is chosen as input to the PS algorithm. Numerical examples show the improvement of the proposed method over existing genetic-algorithm-based solution, as well as PS-algorithm-based solution that uses a single set of random points as inputs. The proposed two-step solution gives higher cumulative throughput and is not sensitive to the choice of input, unlike the PS-based solution using a single set of random points as input.

Spectral Experiments+

ABSTRACTWe describe extensive computational experiments on spectral properties of random objects—random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere. We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.

APLICAÇÃO DE UM MODELO LINEAR LOCAL NA DETERMINAÇÃO DE ALTURAS ORTOMÉTRICAS REFERIDAS AO SISTEMA GEODÉSICO BRASILEIRO - DOI 10.5216/bgg.v36i1.40378

O uso de sistemas de posicionamento global por satélites (GNSS) possibilita obtenção rápida de alturas elipsoidais (h) relativamente precisas. Contudo, nas aplicações de engenharia, é necessário determinar as correspondentes alturas ortométricas (H) referidas ao geóide ou ao Sistema Geodésico Brasileiro (SGB), cujos referenciais são o campo gravitacional terrestre e a maré média no marégrafo localizado em Imbituba (SC), respectivamente. A determinação de H em relação à superfície geoidal exige técnicas gravimétricas ou o usode modelos geopotenciais globais ou regionais. Em relação ao SGB aplica-se nivelamento topográfico clássico, partindo-se de marcos da Rede Altimétrica de Alta Precisão (RAAP) até o local de interesse. Em tais métodos, as operações são usualmente laboriosas, de altos custos e despendem muito tempo. Este artigo apresenta a determinação de H referida ao SGB por meio de um modelo linear de primeira ordem calibrado localmente. A metodologia consistiu em utilizar um conjunto de 17 referências de nível (RNs) da RAAP para calibração do modelo e 7 RNs para sua validação. O modelo foi aplicado na avaliação das alturas de dois conjuntos de pontos aleatórios em uma bacia hidrográfica urbana. O modelo pode ser aplicado em regiões onde não há transiçãode zona UTM.Palavras-chave: GPS, Altura Elipsoidal, Bacia Hidrográfica.

A Dimension Spectrum for SLE Boundary Collisions

We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

Large Area Convex Holes in Random Point Sets

Let $K, L$ be convex sets in the plane. For normalization purposes, suppose that the area of $K$ is 1. Suppose that a set $K_n$ of $n$ points is chosen independently and uniformly over $K$, and call a subset of $K$ a hole if it does not contain any point in $K_n$. It is shown that with high probability the largest area of a hole homothetic to $L$ is $(1+o(1)) \log{n}/n$. We also consider the problems of estimating the largest area convex hole and the largest area of a convex polygonal hole; with vertices in $K_n$. For these two problems we give an answer that is asymptotically tight within a factor of 4.

Stochastic collocation-based finite element of structural nonlinear dynamics with application in composite structures

Stochastic analysis of structures having nonlinearity by means of sampling methods leads to expensive cost in term of computational time. In contrast, non-sampling methods based on the spectral representation of uncertainty are very efficient with comparable accurate results. In this pa- per, the application of spectral methods to nonlinear dynamics of structures with random parameters is investigated. The impact of the parameter randomness on structural responses has been consid- ered. To this end, uncertain parameters and the structure responses are represented using the gPC expansions with unknown deterministic coefficients and random orthogonal polynomial basis. The deterministic finite element model of the structure is used as black-box and it is executed on a set of random collocation points. As the sample structure responses are estimated, a nonlinear optimization process is employed to calculate the unknown coefficients. The method has this main advantage that can be used for complicated nonlinear structural dynamic problems for which the deterministic FEM model has been already developed. Furthermore, it is very time efficient in comparison with sampling methods, as MC simulations. The application of the method is applied to the nonlinear transient analysis of composite beam structures including uncertain quadratic random damping. The results show that the proposed method can capture the large range of uncertainty in input parameters as well as in structural dynamic responses while it is too time-efficient.

Smoothed complexity of convex hulls by witnesses and collectors

We present a simple technique for analyzing the size of geometric hypergraphs defined by random point sets. As an application we obtain upper and lower bounds on the smoothed number of faces of the convex hull under Euclidean and Gaussian noise and related results.

F -Discrepancy for Efficient Sampling in Approximate Dynamic Programming.

In this paper, we address the problem of generating efficient state sample points for the solution of continuous-state finite-horizon Markovian decision problems through approximate dynamic programming. It is known that the selection of sampling points at which the value function is observed is a key factor when such function is approximated by a model based on a finite number of evaluations. A standard approach consists in generating these points through a random or deterministic procedure, aiming at a balanced covering of the state space. Yet, this solution may not be efficient if the state trajectories are not uniformly distributed. Here, we propose to exploit F -discrepancy, a quantity that measures how closely a set of random points represents a probability distribution, and introduce an example of an algorithm based on such concept to automatically select point sets that are efficient with respect to the underlying Markovian process. An error analysis of the approximate solution is provided, showing how the proposed algorithm enables convergence under suitable regularity hypotheses. Then, simulation results are provided concerning an inventory forecasting test problem. The tests confirm in general the important role of F -discrepancy, and show how the proposed algorithm is able to yield better results than uniform sampling, using sets even 50 times smaller.

Measurement of Nonuniformity of Fracture in Structural Steels with Heterogeneous Structure

Different methods for measurement of nonuniformity of fracture in structural steels with heterogeneous structure are studied including the partition of space into Voronoy polyhedrons for measuring the differences in random sets of points in a plane, synthesis of a 3D-picture from several 2D ones (stereophotogrammetry) and indirect analysis of 3D-peculiarities from a 2D-shot, i.e., a “plane” image, for reconstructing scenarios of crack propagation, and solution of problems of linear regression with the use of the principle of maximum likelihood.

Scatterhoarding rodents favor higher predation risks for cache sites: The potential for predators to influence the seed dispersal process

Scatterhoarding rodents often place caches in the open where pilferage rates are reduced, suggesting that they tradeoff higher risks of predation for more secure cache sites. We tested this hypothesis in two study systems by measuring predation risks inferred from measures of giving-up densities (GUDs) at known cache sites and other sites for comparison. Rodent GUDs were measured with small trays containing 3 L of fine sand mixed with sunflower seeds. In the first experiment, we relied on a 2-year seed dispersal study in a natural forest to identify caches of eastern gray squirrels (Sciurus carolinensis) and then measured GUDs at: (i) these caches; (ii) comparable points along logs and rocks where rodent activity was assumed highest; and (iii) a set of random points. We found that GUDs and, presumably, predation risks, were higher at both cache and random points than those with cover. At the second site, we measured GUDs of eastern gray squirrels in an open park system and found that GUDs were consistently lowest at the base of the tree compared to more open sites, where previous studies show caching by squirrels to be highest and pilferage rates by naïve competitors to be lowest. These results confirm that predation risks can influence scatterhoarding decisions but that they are also highly context dependent, and that the landscape of fear, now so well documented in the literature, could potentially shape the temporal and spatial patterns of seedling establishment and forest regeneration in systems where scatterhoarding is common.

Comparison of 1989 Loma Prieta earthquake damage to mapped lineaments along the range front of the Santa Cruz Mountains, California

Damage to public infrastructure at or below the ground surface (streets, curbs, and water and gas lines) in southwestern Santa Clara Valley, California, associated with the 1989 Loma Prieta earthquake, is used to support the assertion that the series of photointerpreted lineaments are tectonic in origin and related to long-term reverse faulting along the range front of the Santa Cruz Mountains. We quantitatively analyze whether the photointerpreted lineaments are spatially correlated with earthquake-induced damage by examining whether the damage was located preferentially closer to the mapped lineaments than to a spatially random set of points. The analysis confirms that damage related to the Loma Prieta earthquake is located preferentially close to mapped lineaments. This result supports the assertion that the lineaments have a tectonic origin related to range-front faulting along the Santa Cruz Mountains, and that their presence may be related to primary fault rupture, spatially focused shaking damage, or slip triggered by strong motion induced by nearby faults.

Empty non-convex and convex four-gons in random point sets

LetSbe a set ofnpoints distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points ofSin its interior. We show that the expected number of empty non-convex four-gons with vertices fromSis 12n2logn+o(n2logn) and the expected number of empty convex four-gons with vertices fromSis Θ(n2).

A Contrario 2D Point Alignment Detection.

In spite of many interesting attempts, the problem of automatically finding alignments in a 2D set of points seems to be still open. The difficulty of the problem is illustrated here by very simple examples. We then propose an elaborate solution. We show that a correct alignment detection depends on not less than four interlaced criteria, namely the amount of masking in texture, the relative bilateral local density of the alignment, its internal regularity, and finally a redundancy reduction step. Extending tools of the a contrario detection theory, we show that all of these detection criteria can be naturally embedded in a single probabilistic a contrario model with a single user parameter, the number of false alarms. Our contribution to the a contrario theory is the use of sophisticated conditional events on random point sets, for which expectation we nevertheless find easy bounds. By these bounds the mathematical consistency of our detection model receives a simple proof. Our final algorithm also includes a new formulation of the exclusion principle in Gestalt theory to avoid redundant detections. Aiming at reproducibility, a source code and an online demo open to any data point set are provided. The method is carefully compared to three state-of-the-art algorithms and an application to real data is discussed. Limitations of the final method are also illustrated and explained.

Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

We analyze the stability and accuracy of discrete least squares on multivariate polynomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low-discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrepancy point sets, is stable and accurate under the condition that the number of evaluations is proportional to the square of the dimension of the polynomial space, up to logarithmic factors. This result is analogous to those obtained in Cohen et al. (2013), Migliorati et al. (2014), Migliorati (2013) and Chkifa et al. (in press) for discrete least squares with random point sets, however it holds with certainty instead of just with high probability. The result is further generalized to arbitrary polynomial spaces associated with downward closed multi-index sets, but with a more demanding (and probably nonoptimal) proportionality between the number of evaluation points and the dimension of the polynomial space.

Stochastic Dynamic Analysis of Structures with Spatially Uncertain Material Parameters

This paper investigates the uncertainty quantification in structural dynamic problems with spatially random variation in material and damping parameters. Uncertain and locally varying material parameters are represented as stochastic field by means of the Karhunen–Loève (KL) expansion. The stiffness and damping properties of the structure are considered uncertain. Stochastic finite element of structural modal analysis is performed in which modal responses are represented using the generalized polynomial chaos (gPC) expansion. Knowing the KL expansions of the random parameters, the nonintrusive technique is employed on a set of random collocation points where the structure deterministic finite element model is executed to estimate the unknown coefficients of the polynomial chaos expansions. A numerical case study is presented for a cantilever beam with random Young's modulus involving spatial variation. The proportional damping constants are estimated from the experimental modal analysis. The expected value, standard deviation, and probability distribution of the random eigenfrequencies and the damping ratios are evaluated. The results show high accuracy compared to the Monte-Carlo (MC) simulations with 3000 realizations. It is also demonstrated that the eigenfrequencies and the damping ratios are equally affected from material uncertainties.

The number of empty four-gons in random point sets

Let S be a set of n points distributed uniformly and independently in the unit square. Then the expected number of empty four-gons with vertices from S is Θ(n2log⁡n). A four-gon is empty if it contains no points of S in its interior.

Ab-initio reconstruction of complex Euclidean networks in two dimensions.

Reconstruction of complex structures is an inverse problem arising in virtually all areas of science and technology, from protein structure determination to bulk heterostructure solar cells and the structure of nanoparticles. We cast this problem as a complex network problem where the edges in a network have weights equal to the Euclidean distance between their endpoints. We present a method for reconstruction of the locations of the nodes of the network given only the edge weights of the Euclidean network. The theoretical foundations of the method are based on rigidity theory, which enables derivation of a polynomial bound on its efficiency. An efficient implementation of the method is discussed and timing results indicate that the run time of the algorithm is polynomial in the number of nodes in the network. We have reconstructed Euclidean networks of about 1000 nodes in approximately 24 h on a desktop computer using this implementation. We also reconstruct Euclidean networks corresponding to polymer chains in two dimensions and planar graphene nanoparticles. We have also modified our base algorithm so that it can successfully solve random point sets when the input data are less precise.