Coincidence Set Research Articles

Element fracture technique for hypervelocity impact simulation

Hypervelocity impact dynamics is the theoretical support of spacecraft shielding against space debris. The numerical simulation has become an important approach for obtaining the ballistic limits of the spacecraft shields. Currently, the most widely used algorithm for hypervelocity impact is the smoothed particle hydrodynamics (SPH). Although the finite element method (FEM) is widely used in fracture mechanics and low-velocity impacts, the standard FEM can hardly simulate the debris cloud generated by hypervelocity impact. This paper presents a successful application of the node-separation technique for hypervelocity impact debris cloud simulation. The node-separation technique assigns individual/coincident nodes for the adjacent elements, and it applies constraints to the coincident node sets in the modeling step. In the explicit iteration, the cracks are generated by releasing the constrained node sets that meet the fracture criterion. Additionally, the distorted elements are identified from two aspects – self-piercing and phase change – and are deleted so that the constitutive computation can continue. FEM with the node-separation technique is used for thin-wall hypervelocity impact simulations. The internal structures of the debris cloud in the simulation output are compared with that in the test X-ray graphs under different material fracture criteria. It shows that the pressure criterion is more appropriate for hypervelocity impact. The internal structures of the debris cloud are also simulated and compared under different thickness-to-diameter ratios (t/D). The simulation outputs show the same spall pattern with the tests. Finally, the triple-plate impact case is simulated with node-separation FEM.

Simulation of triple coincidences in PET

Although current PET scanners are designed and optimized to detect double coincidence events, there is a significant amount of triple coincidences in any PET acquisition. Triple coincidences may arise from causes such as: inter-detector scatter (IDS), random triple interactions (RT), or the detection of prompt gamma rays in coincidence with annihilation photons when non-pure positron-emitting radionuclides are used (β+γ events). Depending on the data acquisition settings of the PET scanner, these triple events are discarded or processed as a set of double coincidences if the energy of the three detected events is within the scanner’s energy window. This latter option introduces noise in the data, as at most, only one of the possible lines-of-response defined by triple interactions corresponds to the line along which the decay occurred. Several novel works have pointed out the possibility of using triple events to increase the sensitivity of PET scanners or to expand PET imaging capabilities by allowing differentiation between radiotracers labeled with non-pure and pure positron-emitting radionuclides. In this work, we extended the Monte Carlo simulator PeneloPET to assess the proportion of triple coincidences in PET acquisitions and to evaluate their possible applications. We validated the results of the simulator against experimental data acquired with a modified version of a commercial preclinical PET/CT scanner, which was enabled to acquire and process triple-coincidence events. We used as figures of merit the energy spectra for double and triple coincidences and the triples-to-doubles ratio for different energy windows and radionuclides. After validation, the simulator was used to predict the relative quantity of triple-coincidence events in two clinical scanners assuming different acquisition settings. Good agreement between simulations and preclinical experiments was found, with differences below 10% for most of the observables considered. For clinical scanners and pure positron emitters, we found that around 10% of the processed double events come from triple coincidences, increasing this ratio substantially for non-pure emitters (around 25% for 124I and > 50% for 86Y). For radiotracers labeled with 18F we found that the relative quantity of IDS events in standard acquisitions is around 18% for the preclinical scanner and between 14 and 22% for the clinical scanners. For non-pure positron emitters like 124I, we found a β+γ triples-to-doubles ratio of 2.5% in the preclinical scanner and of up to 4% in the clinical scanners.

On the regularity of the free boundary for quasilinear obstacle problems

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the p(x) -Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth p(x)>1 , we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for p(x) -Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of C^1 hypersurfaces: i) by extending directly the finiteness of the (n-1) -dimensional Hausdorff measure of the free boundary to the case of heterogeneous p -Laplacian type operators with constant p, 1 < p <\infty ; ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth p(x)>1 .

A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

In this paper we consider the fully nonlinear parabolic free boundary problem $$ \left\{\begin{array}{ll} F(D^2u) -\partial_t u=1 & \text{a.e. in}Q_1 \cap \Omega\\ |D^2 u| + |\partial_t u| \leq K & \text{a.e. in}Q_1\setminus\Omega, \end{array} \right. $$ where $K>0$ is a positive constant, and $\Omega$ is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $W_x^{2,n} \cap W_t^{1,n} $ solutions are locally $C_x^{1,1}\cap C_t^{0,1} $ inside $Q_1$. A key starting point for this result is a new BMO-type estimate which extends to the parabolic setting the main result in \cite{CH}. Once optimal regularity for $u$ is obtained, we also show regularity for the free boundary $\partial\Omega\cap Q_1$ under the extra condition that $\Omega \supset \{u \neq 0 \}$, and a uniform thickness assumption on the coincidence set $\{u = 0 \}$,

A general class of free boundary problems for fully nonlinear parabolic equations

In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t} u=1 &{} \text {a.e. in }Q_1 \cap \Omega |D^2 u| + |\partial _{t} u| \le K &{} \text {a.e. in }Q_1{\setminus }\Omega , \end{array} \right. \end{aligned}$$ where $$K>0$$ is a positive constant, and $$\Omega $$ is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $$W_x^{2,n} \cap W_t^{1,n} $$ solutions are locally $$C_x^{1,1}\cap C_t^{0,1} $$ inside $$Q_1$$ . A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result in Caffarelli and Huang (Duke Math J 118(1):1–17, 2003). Once optimal regularity for $$u$$ is obtained, we also show regularity for the free boundary $$\partial \Omega \cap Q_1$$ under the extra condition that $$\Omega \supset \{ u \ne 0 \}$$ , and a uniform thickness assumption on the coincidence set $$\{ u = 0 \}$$ .

Riemann solutions without an intermediate constant state for a system of two conservation laws

We present a class of systems consisting of two conservation laws in one spatial dimension that share an intriguing property: they admit structurally stable Riemann solutions without the standard constant state. This striking phenomenon emerges in sharp contrast to what is known for strictly hyperbolic systems of conservation laws, in which the existence of constant states is necessary for the structural stability of Riemann solutions. We prove that, together, coincidence of characteristic speeds and a certain amount of genuine nonlinearity are sufficient to trigger the aforementioned phenomenon. The proof revolves about the presence of a singular point in the coincidence set that organizes the construction of our Riemann solutions.

Starshapedness in the obstacle problem

We establish the starshapedness (with respect to the origin) of coincidence set in the obstacle problem for second order elliptic equations.

Covering maps of spaces of compact subsets

Mappings generated by covering maps of metric spaces are considered in the spaces of compact subsets of metric spaces. Sufficient conditions for the covering property of maps of spaces of compact sets and sufficient conditions for the existence of coincidence sets of maps of metric spaces are obtained.

Cascade search for preimages and coincidences: Global and local versions

In previous papers of the author, the cascade search principle was proposed, which makes it possible to construct a set-valued self-map of a metric spaceX from a set-valued functional or a collection of set-valued maps of X so that the new map generates a multicascade, i.e., a set-valued discrete dynamical system whose limit set coincides with the zero set of the given functional, with the coincidence set of the given collection, or with the common preimage of a closed subspace under the maps from this collection. Stability issues of cascade search were studied. This paper is devoted to a generalization and local modifications of the cascade search principle and their applications to problems concerning local search and approximation of common preimages of subspaces and coincidence sets for finite collections of set-valued maps of metric spaces.

Technical Note: Calibration and validation of geophysical observation models

Abstract. We present a method to calibrate and validate observational models that interrelate remotely sensed energy fluxes to geophysical variables of land and water surfaces. Coincident sets of remote sensing observation of visible and microwave radiations and geophysical data are assembled and subdivided into calibration (Cal) and validation (Val) data sets. Each Cal/Val pair is used to derive the coefficients (from the Cal set) and the accuracy (from the Val set) of the observation model. Combining the results from all Cal/Val pairs provides probability distributions of the model coefficients and model errors. The method is generic and demonstrated using comprehensive matchup sets from two very different disciplines: soil moisture and water quality. The results demonstrate that the method provides robust model coefficients and quantitative measure of the model uncertainty. This approach can be adopted for the calibration/validation of satellite products of land and water surfaces, and the resulting uncertainty can be used as input to data assimilation schemes.

A quantitative assessment of the distribution and extent of urban clusters in China

Urban clusters are the expected products of high levels of industry and urbanization in a country, as well as being the basic units of participation in global competition. With respect to China, urban clusters are regarded as the dominant formation for boosting the Chinese urbanization process. However, to date, there is no coincident, efficient, and credible methodological system and set of techniques to identify Chinese urban clusters. This research investigates the potential of a computerized identification method supported by geographic information techniques to provide a better understanding of the distribution of Chinese urban clusters. The identification method is executed based on a geographic information database, a digital elevation model, and socio-economic data with the aid of ArcInfo Macro Language programming. In the method, preliminary boundaries are identified according to transportation accessibility, and final identifications are achieved from limiting city numbers, population, and GDP in a region with the aid of the rasterized socio-economic dataset. The results show that the method identifies nine Chinese urban clusters, i.e., Pearl River Delta, Lower Yangtze River Valley, Beijing-Tianjin-Hebei Region, Northeast China Plain, Middle Yangtze River Valley, Central China Plains, Western Taiwan Strait, Guanzhong and Chengdu-Chongqing urban clusters. This research represents the first study involving the computerized identification of Chinese urban clusters. Moreover, compared to other related studies, the study’s approach, which combines transportation accessibility and socio-economic characteristics, is shown to be a distinct, effective and reliable way of identifying urban clusters.

A Quadratic $C^0$ Interior Penalty Method for the Displacement Obstacle Problem of Clamped Kirchhoff Plates

We study a quadratic $C^0$ interior penalty method for the displacement obstacle problem of Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. Under the conditions that the obstacles are sufficiently smooth and separated from each other and the boundary displacement, we prove that the magnitudes of the errors in the energy norm and the $L_\infty$ norm are $O(h^{\alpha})$, where $h$ is the mesh size and $\alpha>\frac12$ is determined by the interior angles of the polygonal domain. We also address the approximations of the coincidence set and the free boundary. The performance of the method is illustrated by numerical results.

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: − ε Δ p u = u q − 1 f ( a ( x ) − u ) in Ω with u = 0 on ∂ Ω, where ε is a positive parameter, Δ p u = div ( | ∇ u | p − 2 ∇ u ) , 1 < q ⩽ p < ∞ , f ( s ) ∼ | s | θ − 1 s ( s → 0 ) for some θ > 0 and a ( x ) is a positive smooth function satisfying Δ p a = 0 in Ω with inf Ω | ∇ a | > 0 . It is proved in this paper that if 0 < θ < 1 the coincidence set O ε = { x ∈ Ω : u ε ( x ) = a ( x ) } has a positive measure for small ε and converges to Ω with order O ( ε 1 / p ) as ε → 0 . Moreover, it is also shown that if θ ⩾ 1 , then O ε is empty for any ε > 0 . The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension ( k − 1 ) n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface. As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.

Three-point bending tests-Part I: Mathematical study and asymptotic analysis

The goal of this work is to study the static behaviour of a three-dimensional elastic beam when is subjected to a three-point bending test. In the first part, under suitable compatibility conditions on the applied forces and on the geometry of the beam, we will prove the existence of a unique solution for the associated contact elastic problem; these conditions of compatibility on the data come from the absence of a Dirichlet condition on the beam boundary. In the second part, we will study the asymptotic behaviour of this problem; in particular, we will deduce the one-dimensional models associated with the displacement components, and we will give the existence and uniqueness of solution for them. Moreover, we will give an expression for the normal axial stress in the beam which is related to the modulus of rupture of brittle materials. In the final part of the work, we will deal with the regularity of the solution for the bending problem and we will prove some properties of the coincidence set. Copyright © 2011 John Wiley & Sons, Ltd.

Velocity modeling of a complex deep crustal structure across the Mesoproterozoic south Delhi Fold Belt, NW India, from joint interpretation of coincident seismic wide-angle and near-offset reflection data: An approach using unusual reflections in wide-angle records

[1] Seismic wide-angle record sections from favorably placed source and limited-aperture recording spreads, particularly across collision boundaries, reveal bright unusual high-velocity contrast reflections of limited lateral coherence. The arrival times of the unusual reflections for recording updip of the reflecting plane align as high apparent velocity phases at wide-angle offsets. Coincident near-offset reflection images from similar structural features provide additional control for modeling the unusual phases on the wide-angle record sections. We propose an approach for joint interpretation of these coincident seismic data sets and modeling the unusual phases on the processed wide-angle records to delineate both the velocity stratification and the geometry of the deep crustal structures consistent with the near-offset reflection images. Processed and migrated near-offset reflection images reveal several steeply dipping isolated reflections on the western margin of the south Delhi Fold Belt (DFB), NW India, interpreted as the south Delhi thrust fault. Coincident limited-aperture, wide-angle records reveal bright unusual reflections, which are also found to be associated with the same thrust fault. Observations of similar dipping reflection patterns in various other regions lend support to consider them as genuine in-plane reflections resulting from a structural fabric imprinted on horizontally oriented geological structures. We use the reflectivity structural model from the near-offset data set that provides necessary constraints for modeling and simulating the travel times and amplitudes of the unusual reflections recognized in the wide-angle records. The unusual reflections are effectively used as part of the wide-angle data set for Gaussian beam synthetic seismogram modeling, thus delineating for the first time a complex 2-D model of the seismic velocity structure of the deep crust and uppermost mantle across the Marwar Basin (MB) and the DFB. The dipping events revealed in the wide-angle data set are successfully modeled and interpreted as reflections from the thrust fault, consistent with the near-offset reflection image. The 2-D velocity model, parameterized as a stack of isolated dipping reflectors embedded in deep crustal layers, is successively refined by including the structural features recognizable both in the near-offset reflection image and in the wide-angle records. The unusual reflections are modeled as a set of discrete reflectors representing local velocity variations imprinted on the background of a deep crustal velocity structure. The lamellar structure revealed by short, isolated dipping and the subhorizontal middle to lower crustal reflections suggests heterogeneities at scale lengths on the order of at least the Fresnel zone diameter as resolved here. The local velocity variations in the deep crust can be considered analogous, on a different scale, to the models in which velocity heterogeneities on scale lengths of the seismic wavelength constitute a medium of random distribution of scatterers. The crustal-scale south Delhi thrust fault on the western margin of the DFB, delineated as the stack of reflectors dipping 25°–30°SE, probably developed during the Mesoproterozoic collisional episode related to the Delhi orogeny, contemporaneous with the formation of Rodinia. It is concluded that complex structural features developed during collisional episodes can survive through geological periods, longer than generally expected, depending on the accretionary characteristics at the time of orogenic evolution. The 2-D velocity model further reveals significant variations of crustal seismic structure, with a relatively sharp and shallow Moho (at 39–40 km) in the MB and a deeper Moho (at 45–47 km) underlying a 12 km thick subhorizontal lamella of high-velocity (6.9–7.5 km/s) layers in the DFB regions.

Parametrized Borsuk–Ulam problem for projective space bundles

Let $\pi: E \to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\pi^{'}: E^{'} \to B$ be vector bundle such that $\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\pi^{'}:E^{'} \to B$. For a fiber preserving $\mathbb{Z}_2$-equivariant map $f:E \to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \{x \in E | f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E | f(x) = f(T(x)) \}$ of a fiber preserving map $f:E \to E^{'}$.

Berezin transform in polynomial bergman spaces

AbstractFix a smooth weight function Q in the plane, subject to a growth condition from below. Let Km,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n − 1 of finite L2‐norm with respect to the measure e−mQ dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure ${\rm d}B_{m,n}^{\langle z_0\rangle}(z)=K_{m,n}(z_0,z_0)^{-1}|K_{m,n}(z,z_0)|^2{\rm e}^{-mQ(z)}\,{\rm d}A(z)$ for the point z0 is a probability measure that defines the (polynomial) Berezin transform for continuous $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}f\in L^\infty(\C)$. We analyze the semiclassical limit of the Berezin measure (and transform) as m → + ∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z0 converges weak‐star to the unit point mass at the point z0 provided that Δ Q(z0) &gt; 0 and that z0 is contained in the interior of a compact set ${\cal S}_\tau$, defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}z_0\in \C\setminus{\cal S}_\tau$, the Berezin measure cannot converge to the point mass at z0. In the model case Q(z) = |z|2, when ${\cal S}_\tau$ is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z0 relative to $\font\open=msbm10 at 10pt\def\C{\hbox{\open C}}\C\setminus{\cal S}_\tau$.Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L2‐estimates for the equation ${\bar \partial} u=f$ when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc.

Stability of the coincidence set of a solution to a parabolic variational inequality with an obstacle

In this paper we propose a new technique for the stability analysis of the coincidence set of a solution to a parabolic variational inequality with an obstacle inside the domain. It is based on the reformulation of the initial inequality in the form of a parabolic initial boundary value problem with an exact penalty operator.

Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations

We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular $L^2$-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.