There are at least two kinds of similarity. Relational similarity is correspondence between relations, in contrast with attributional similarity, which is correspondence between attributes. When two words have a high degree of attributional similarity, we call them synonyms. When two pairs of words have a high degree of relational similarity, we say that their relations are analogous. For example, the word pair mason:stone is analogous to the pair carpenter:wood. This paper introduces Latent Relational Analysis (LRA), a method for measuring relational similarity. LRA has potential applications in many areas, including information extraction, word sense disambiguation, and information retrieval. Recently the Vector Space Model (VSM) of information retrieval has been adapted to measuring relational similarity, achieving a score of 47 on a collection of 374 college-level multiple-choice word analogy questions. In the VSM approach, the relation between a pair of words is characterized by a vector of frequencies of predefined patterns in a large corpus. LRA extends the VSM approach in three ways: (1) the patterns are derived automatically from the corpus, (2) the Singular Value Decomposition (SVD) is used to smooth the frequency data, and (3) automatically generated synonyms are used to explore variations of the word pairs. LRA achieves 56 on the 374 analogy questions, statistically equivalent to the average human score of 57 . On the related problem of classifying semantic relations, LRA achieves similar gains over the VSM.
For example, there is an analogy between the solar system and Rutherford's model of the atom @cite_54 . The solar system is the source domain and Rutherford's model of the atom is the target domain. The basic objects in the source model are the planets and the sun. The basic objects in the target model are the electrons and the nucleus. The planets and the sun have various attributes, such as mass(sun) and mass(planet), and various relations, such as revolve(planet, sun) and attracts(sun, planet). Likewise, the nucleus and the electrons have attributes, such as charge(electron) and charge(nucleus), and relations, such as revolve(electron, nucleus) and attracts(nucleus, electron). SME maps revolve(planet, sun) to revolve(electron, nucleus) and attracts(sun, planet) to attracts(nucleus, electron).
@cite_54
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We have discovered 16 Type Ia supernovae (SNe Ia) with the Hubble Space Telescope (HST) and have used them to provide the first conclusive evidence for cosmic deceleration that preceded the current epoch of cosmic acceleration. These objects, discovered during the course of the GOODS ACS Treasury program, include 6 of the 7 highest redshift SNe Ia known, all at z > 1.25, and populate the Hubble diagram in unexplored territory. The luminosity distances to these objects and to 170 previously reported SNe Ia have been determined using empirical relations between light-curve shape and luminosity. A purely kinematic interpretation of the SN Ia sample provides evidence at the greater than 99 confidence level for a transition from deceleration to acceleration or, similarly, strong evidence for a cosmic jerk. Using a simple model of the expansion history, the transition between the two epochs is constrained to be at z = 0.46 ± 0.13. The data are consistent with the cosmic concordance model of ΩM ≈ 0.3, ΩΛ ≈ 0.7 (χ = 1.06) and are inconsistent with a simple model of evolution or dust as an alternative to dark energy. For a flat universe with a cosmological constant, we measure ΩM = 0.29 ± (equivalently, ΩΛ = 0.71). When combined with external flat-universe constraints, including the cosmic microwave background and large-scale structure, we find w = -1.02 ± (and w < -0.76 at the 95 confidence level) for an assumed static equation of state of dark energy, P = wρc2. Joint constraints on both the recent equation of state of dark energy, w0, and its time evolution, dw dz, are a factor of 8 more precise than the first estimates and twice as precise as those without the SNe Ia discovered with HST. Our constraints are consistent with the static nature of and value of w expected for a cosmological constant (i.e., w0 = -1.0, dw dz = 0) and are inconsistent with very rapid evolution of dark energy. We address consequences of evolving dark energy for the fate of the universe. A methodology is described to systematically derive coarse-grained (CG) force fields for molecular liquids from the underlying atomistic-scale forces. The coarse graining of an interparticle force field is accomplished by the application of a force-matching method to the trajectories and forces obtained from the atomistic trajectory and force data for the CG sites of the targeted system. The CG sites can be associated with the centers of mass of atomic groups because of the simplicity in the evaluation of forces acting on these sites from the atomistic data. The resulting system is called a multiscale coarse-grained (MS-CG) representation. The MS-CG method for liquids is applied here to water and methanol. For both liquids one-site and two-site CG representations without an explicit treatment of the long-ranged electrostatics have been derived. In addition, for water a two-site model having the explicit long-ranged electrostatics has been developed. To improve the thermodynamic properties (e.g., pressure and density) for the MS-CG models, the constraint for the instantaneous virial was included into the force-match procedure. The performance of the resulting models was evaluated against the underlying atomistic simulations and experiment. In contrast with existing approaches for coarse graining of liquid systems, the MS-CG approach is general, relies only on the interatomic interactions in the reference atomistic system. We describe a complete set of algorithms for ab initio molecular simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of molecular and materials properties from quantum-mechanical first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute “ab initio molecular simulations” (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on density-functional theory (DFT) in the local and semilocal (generalized gradient) approximations, but an extension to hybrid functionals, Hartree–Fock theory, and MP2 GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio molecular dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs. We review some recently published methods to represent atomic neighbourhood environments, and analyse their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms, and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighbourhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbours increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether new approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps these difficulties by directly defining the similarity between any two neighbourhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal. We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+?j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I?I?Rd,f?TdI?I?Rd,f?Td, p,q?Rnp,q?Rn, t?T1t?T1. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in , and and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0<e«10<e«1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O(1)O(1). This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of and . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I?I?RdI?I?Rd. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from and . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31]—notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in and This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc. We give a detailed description of the geometry of single droplet patterns in a non- local isoperimetric problem. In particular we focus on the sharp interface limit of the Ohta- Kawasaki free energy for diblock copolymers, regarded as a paradigm for those energies modeling physical systems characterized by a competition between short and a long-range interactions. Exploiting fine properties of the regularity theory for minimal surfaces, we extend previous partial results in different directions and give robust tools for the geometric analysis of more complex patterns. where u is the order parameter of a two-phases system confined in ⊂ R n , and and m are two nonnegative numerical parameters. The two terms in the energy mimic attractive short-range and repulsive long-range energies between the phases. More precisely the first term is local, it favors minimal interface area and drives the system toward a partition into few pure phases while the second term, involving a Coulomb-like kernel G, is non-local and favor a fine mixing of the phases. A detailed description of the energy is given in § 1. The competition between these two terms is expected to induce the formation of highly regular mesoscopic patterns, whose geometry strongly depends on the choice of the parameters and m (e.g. spherical spots, cylinders, gyroids and lamellae). 0.1. The Ohta-Kawasaki functional for diblock copolymers. The model we consider arises as a simplification of a Ginzburg-Landau functional proposed by Ohta and Kawasaki in their pioneering paper (41) as a possible description of a diblock copolymers' (DBC) system. Even though it is questionable whether such an energy actually describes DBCs (see Choksi and Ren (15), Muratov (39) and Niethammer and Oshita (40)), nevertheless it is a first, and mathematically non-trivial, attempt to capture some of the main features of these systems. For such a reason it deserved over the last twenty years great attention from both the mathematical and the physical We introduce a neural network architecture that models the physical reasoning process and that can be used to extract simple physical concepts from experimental data without being provided with additional prior knowledge. We apply the neural network to a variety of simple physical examples in classical and quantum mechanics, like damped pendulums, two-particle collisions, and qubits. The network finds the physically relevant parameters, exploits conservation laws to make predictions, and can be used to gain conceptual insights. For example, given a time series of the positions of the Sun and Mars as observed from Earth, the network discovers the heliocentric model of the solar system - that is, it encodes the data into the angles of the two planets as seen from the Sun. Our work provides a first step towards answering the question whether the traditional ways by which physicists model nature naturally arise from the experimental data without any mathematical and physical pre-knowledge, or if there are alternative elegant formalisms, which may solve some of the fundamental conceptual problems in modern physics, such as the measurement problem in quantum mechanics. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. We consider the scattering of an incident electromagnetic radiation on a two-dimensional (2D) electron layer with an imbedded met allic contact. We show that the incident wave excites in the system the 2D plasmon polaritons running along the 2D layer and localized near it, and electromagnetic waves reflected back from the system. The ratio of the energy transformed to the 2D plasmon polaritons and reflected back from the layer depends on the frequency and the value of a retardation parameter, which characterizes the importance of retardation effects. When the retardation parameter is large, the energy of the incident radiation is mainly reflected from the 2D electron system and the excitation of the 2D plasmon polaritons is less effective. The results obtained are discussed in connection with recent experiments on the microwave response of the 2D electron systems. Many AI researchers and cognitive scientists have argued that analogy is the core of cognition. The most influential work on computational modeling of analogy-making is Structure Mapping Theory (SMT) and its implementation in the Structure Mapping Engine (SME). A limitation of SME is the requirement for complex hand-coded representations. We introduce the Latent Relation Mapping Engine (LRME), which combines ideas from SME and Latent Relational Analysis (LRA) in order to remove the requirement for hand-coded representations. LRME builds analogical mappings between lists of words, using a large corpus of raw text to automatically discover the semantic relations among the words. We evaluate LRME on a set of twenty analogical mapping problems, ten based on scientific analogies and ten based on common metaphors. LRME achieves human-level performance on the twenty problems. We compare LRME with a variety of alternative approaches and find that they are not able to reach the same level of performance. A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price’s law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.’s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price’s law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou’s C0-formulation, the conjecture is proven to be false. This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the PDs is given by an atomic configuration and the output is expressed as 2D histograms. Then, specific distributions such as curves and islands in the PDs identify meaningful shape characteristics of the atomic configuration. Although the method can be applied to a wide variety of disordered systems, it is applied here to silica glass, the Lennard-Jones system, and Cu-Zr met allic glass as standard examples of continuous random network and random packing structures. In silica glass, the method classified the atomic rings as short-range and medium-range orders and unveiled hierarchical ring structures among them. These detailed geometric characterizations clarified a real space origin of the first sharp diffraction peak and also indicated that PDs contain information on elastic response. Even in the Lennard-Jones system and Cu-Zr met allic glass, the hierarchical structures in the atomic configurations were derived in a similar way using PDs, although the glass structures and properties substantially differ from silica glass. These results suggest that the PDs provide a unified method that extracts greater depth of geometric information in amorphous solids than conventional methods. Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point... What happens if one pushes a cup sitting on a table toward the edge of the table? How about pushing a desk against a wall? In this paper, we study the problem of understanding the movements of objects as a result of applying external forces to them. For a given force vector applied to a specific location in an image, our goal is to predict long-term sequential movements caused by that force. Doing so entails reasoning about scene geometry, objects, their attributes, and the physical rules that govern the movements of objects. We design a deep neural network model that learns long-term sequential dependencies of object movements while taking into account the geometry and appearance of the scene by combining Convolutional and Recurrent Neural Networks. Training our model requires a large-scale dataset of object movements caused by external forces. To build a dataset of forces in scenes, we reconstructed all images in SUN RGB-D dataset in a physics simulator to estimate the physical movements of objects caused by external forces applied to them. Our Forces in Scenes (ForScene) dataset contains 65,000 object movements in 3D which represent a variety of external forces applied to different types of objects. Our experimental evaluations show that the challenging task of predicting long-term movements of objects as their reaction to external forces is possible from a single image. The code and dataset are available at: http: allenai.org plato forces. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. We propose an approach to find and describe objects within broad domains. We introduce a new dataset that provides annotation for sharing models of appearance and correlation across categories. We use it to learn part and category detectors. These serve as the visual basis for an integrated model of objects. We describe objects by the spatial arrangement of their attributes and the interactions between them. Using this model, our system can find animals and vehicles that it has not seen and infer attributes, such as function and pose. Our experiments demonstrate that we can more reliably locate and describe both familiar and unfamiliar objects, compared to a baseline that relies purely on basic category detectors. We present algorithms to reconstruct the planar cross-section of a simply connected object from data points measured by rays. The rays are semi-infinite curves representing, for example, the laser beam or the articulated arms of a robot moving around the object. This paper shows that the information provided by the rays is crucial (though generally neglected) when solving 2-dimensional reconstruction problems. The main property of the rays is that they induce a total order on the measured points. This order is shown to be computable in optimal time O(n log n). The algorithm is fully dynamic and allows the insertion or the deletion of a point in O(log n) time. From this order a polygonal approximation of the object can be deduced in a straightforward manner. However, if insufficient data are available or if the points belong to several connected objects, this polygonal approximation may not be a simple polygon or may intersect the rays. This can be checked in O(n log n) time. The order induced by the rays can also be used to find a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, we have shown that 3n-3 probes are necessary and sufficient if the object has n non-colinear edges. Each probe can be determined in O(log n) time yielding an O(n log n)-time 0(n)-space algorithm. When each probe outcome consists of a contact point and a ray measuring that point but not the normal, the same strategy can still be applied. Under a mild condition, 8n-4 probes are sufficient to discover a shape that is almost surely the actual shape of the object. In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere ( S ^ n-1 R ^ n ) with ( n ), and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Cote, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Cote, Commun. Pure Appl. Math. 69(4):609–612, 2016; Cote, Am. J. Math. 137(1):139–207, 2015; , Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Riviere (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). We present a novel computational model to explore the relatedness of objectness and saliency, each of which plays an important role in the study of visual attention. The proposed framework conceptually integrates these two concepts via constructing a graphical model to account for their relationships, and concurrently improves their estimation by iteratively optimizing a novel energy function realizing the model. Specifically, the energy function comprises the objectness, the saliency, and the interaction energy, respectively corresponding to explain their individual regularities and the mutual effects. Minimizing the energy by fixing one or the other would elegantly transform the model into solving the problem of objectness or saliency estimation, while the useful information from the other concept can be utilized through the interaction term. Experimental results on two benchmark datasets demonstrate that the proposed model can simultaneously yield a saliency map of better quality and a more meaningful objectness output for salient object detection.