First-order Properties Research Articles

First-order properties of bounded quantifier depth of very sparse random graphs

We say that a random graph obeys the zero-one -law if every property expressed by a first-order formula with quantifier depth at most holds with probability tending to or . It is known that the random graph obeys the zero-one -law for every and every positive irrational , as well as for all rational which are not of the form (for any positive integer ). It is also known that for all other rational positive , the random graph does not obey the zero-one -law for sufficiently large . In this paper we put and obtain upper and lower bounds for the largest such that the zero-one -law holds.

Envelope theorem in dynamic economic models with recursive utility

This paper investigates the first-order differentiability properties of the value function in dynamic economic models with recursive preferences where the optimal policy may lie at the boundary of the feasible set under several regular assumptions originating from the static optimization theory plus an additional asymptotic condition.

Improved Color Mapping Methods for Multiband Nighttime Image Fusion

Previously, we presented two color mapping methods for the application of daytime colors to fused nighttime (e.g., intensified and longwave infrared or thermal (LWIR)) imagery. These mappings not only impart a natural daylight color appearance to multiband nighttime images but also enhance their contrast and the visibility of otherwise obscured details. As a result, it has been shown that these colorizing methods lead to an increased ease of interpretation, better discrimination and identification of materials, faster reaction times and ultimately improved situational awareness. A crucial step in the proposed coloring process is the choice of a suitable color mapping scheme. When both daytime color images and multiband sensor images of the same scene are available, the color mapping can be derived from matching image samples (i.e., by relating color values to sensor output signal intensities in a sample-based approach). When no exact matching reference images are available, the color transformation can be derived from the first-order statistical properties of the reference image and the multiband sensor image. In the current study, we investigated new color fusion schemes that combine the advantages of both methods (i.e., the efficiency and color constancy of the sample-based method with the ability of the statistical method to use the image of a different but somewhat similar scene as a reference image), using the correspondence between multiband sensor values and daytime colors (sample-based method) in a smooth transformation (statistical method). We designed and evaluated three new fusion schemes that focus on (i) a closer match with the daytime luminances; (ii) an improved saliency of hot targets; and (iii) an improved discriminability of materials. We performed both qualitative and quantitative analyses to assess the weak and strong points of all methods.

Low-scaling first-order properties within second-order Møller-Plesset perturbation theory using Cholesky decomposed density matrices.

An efficient implementation of energy gradients and of hyperfine coupling constants in second-order Møller-Plesset perturbation theory (MP2) is presented based on our fully atomic orbital (AO)-based approach. For the latter, an unrestricted AO-based MP2 formulation is introduced. A reduction in the dependency of the computational efficiency on the size of the basis set is achieved by a Cholesky decomposition and the prefactor is reduced by the resolution-of-the-identity approximation. Significant integral contributions are selected based on distance-including integral estimates (denoted as QQR-screening) and its reliability as a fully controlled screening procedure is demonstrated. The rate-determining steps are shown via model computations to scale cubically in the computation of energy gradients and quadratically in the case of hyperfine coupling constants. Furthermore, a significant speed-up of the computational time with respect to the canonical formulation is demonstrated.

Compact First-Order Probe for Spherical Near-Field Antenna Measurements at Low Frequencies

Guidelines for designing compact and lightweight first-order probes for spherical near-field antenna measurements at frequencies below 1 GHz that exploit first-order properties of electrically small self-resonant radiators combined into superdirective endfire arrays are established theoretically, exemplified numerically, and validated experimentally. A prototype of the probe designed to operate at a central frequency of 435 MHz exhibits the impedance bandwidth of 15 MHz with a directivity of more than 9 dBi and parasitic $|\mu |\ne 1$ spherical modes suppressed to at least −42 dB. The probe height is just 343 mm above a 720-mm circular ground plane and weighs about 5 kg.

Deciding First-Order Properties of Nowhere Dense Graphs

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [2010, 2011], form a large variety of classes of “sparse graphs” including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes (parameterized by the length of the input formula). At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterization of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a “rank-preserving” version of Gaifman’s locality theorem.

Specular motion and 3D shape estimation.

Dynamic visual information facilitates three-dimensional shape recognition. It is still unclear, however, whether the motion information generated by moving specularities across a surface is congruent to that available from optic flow produced by a matte-textured shape. Whereas the latter is directly linked to the first-order properties of the shape and its motion relative to the observer, the specular flow, the image flow generated by a specular object, is less sensitive to the object's motion and is tightly related to second-order properties of the shape. We therefore hypothesize that the perceived bumpiness (a perceptual attribute related to curvature magnitude) is more stable to changes in the type of motion in specular objects compared with their matte-textured counterparts. Results from two two-interval forced-choice experiments in which observers judged the perceived bumpiness of perturbed spherelike objects support this idea and provide an additional layer of evidence for the capacity of the visual system to exploit image information for shape inference.

Avalanche breakdown and self-stabilization effects in electrically driven transition of carbon nanotube covered VO2 film

Electrical-driven metal–insulator transition (MIT) is quite vital and widespread for both applications and MIT mechanism of VO2. In this report, we discussed the avalanche breakdown and self-stabilization effects behind the electrically driven phase transition in macro-scale carbon nanotube covered VO2 film to further understand the phase transition behaviors as well as explore promising electrical-driven VO2 devices. It was found that the Joule heat was the main source to trigger the phase transition of VO2 film. However, the time-dependent triggering routes were quite different, since the avalanche behavior was observed under the voltage-driven mode, while the gradual self-stabilization existed in the current-driven mode. The simulation results based on the proposed thermodynamics models were in good agreement with the experimental phenomena, which were basically originated from the intrinsic first-order phase transition properties of VO2 film.

A logic for arguing about probabilities in measure teams

We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy-Weinberg Principle of biology and the Bell's Inequalities of quantum physics as examples.

Confidence intervals and point estimators for a normal mean under purely sequential strategies involving Gini’s mean difference and mean absolute deviation

ABSTRACTWe have developed purely sequential methodologies for problems associated with both fixed-width confidence interval estimation and minimum risk point estimation for the normal mean μ when the variance σ2 is assumed unknown. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini’s mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies have been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data. Overall, we empirically feel confident that our newly developed GMD-based or MAD-based methodologies are more robust for practical purposes when we compare them with the sample variance–based methodologies respectively, especially when up to 20% suspect outliers may be expected.

Group theoretical derivation of the minimal coupling principle.

The group theoretical methods worked out by Bargmann, Mackey and Wigner, which deductively establish the Quantum Theory of a free particle for which Galileian transformations form a symmetry group, are extended to the case of an interacting particle. In doing so, the obstacles caused by loss of symmetry are overcome. In this approach, specific forms of the wave equation of an interacting particle, including the equation derived from the minimal coupling principle, are implied by particular first-order invariance properties that characterize the interaction with respect to specific subgroups of Galileian transformations; moreover, the possibility of yet unknown forms of the wave equation is left open.

On asymptotic validity of naive inference with an approximate likelihood

Many statistical models have likelihoods which are intractable: it is impossible or too expensive to compute the likelihood exactly. In such settings, a common approach is to replace the likelihood with an approximation, and proceed with inference as if the approximate likelihood were the true likelihood. In this paper, we describe conditions which guarantee that this naive inference with an approximate likelihood has the same first-order asymptotic properties as that with the true likelihood. We investigate the implications of these results using a Laplace approximation to the likelihood in a simple two-level latent variable model and using reduced-dependence approximations to the likelihood in an Ising model.

Equational formulas and pattern operations in initial order-sorted algebras

Abstract A pattern t , i.e., a term possibly with variables, denotes the set (language) 〚 t 〛 of all its ground instances . In an untyped setting, symbolic operations on finite sets of patterns can represent Boolean operations on languages. But for the more expressive patterns needed in declarative languages supporting rich type disciplines such as subtype polymorphism, untyped pattern operations and algorithms break down. We show how they can be properly defined by means of a signature transformation Σ ↦ Σ # that enriches the types of Σ . We also show that this transformation allows a systematic reduction of the first-order logic properties of an initial order-sorted algebra supporting subtype-polymorphic functions to equivalent properties of an initial many-sorted (i.e., simply typed) algebra. This yields a new, simple proof of the known decidability of the first-order theory of an initial order-sorted algebra.

Design and actualisation of the improved Jakes fading channel

Based on the original Jakes model, the authors have proposed a new design for an inexpensive fading channel simulator that can be easily actualised with fewer low-frequency oscillators than that used in existing structures. The major modifications from the original Jakes model include the generation of path gains under an improved distribution, and presetting the arrival angles and initial phases to be zero. These modifications not only reduce the hardware cost of the channel simulator, but also provide better statistic properties than the model proposed by Zheng and Xiao. Specifically, the improved Jakes fading channel model presented in this study needs only eight low-frequency oscillators. Simulation results have demonstrated that the proposed design exhibits improved first-order and second-order statistical properties.

First-order properties of bounded quantifier depth of very sparse random graphs

Говорят, что случайный граф подчиняется $k$-закону нуля или единицы, если для любого свойства, выражаемого формулой первого порядка с кванторной глубиной не более $k$, вероятность выполнения этого свойства стремится либо к $0$, либо к $1$. Известно, что случайный граф $G(n,n^{-\alpha})$ подчиняется $k$-закону нуля или единицы для любого $k\in\mathbb{N}$ и любого положительного иррационального $\alpha$, а также для любого рационального $\alpha>1$, отличного от $1+1/l$ (для любого натурального числа $l$). Известно также, что для всех остальных рациональных положительных $\alpha$ при достаточно больших $k$ случайный граф не подчиняется $k$-закону. В настоящей работе при $\alpha=1+1/l$ получены нижняя и верхняя оценка на наибольшее $k$, при котором выполнен $k$-закон нуля или единицы. Библиография: 20 наименований.

Zero-one laws for random graphs with vertices in a Boolean cube

We study the limit probabilities of first-order properties for random graphs with vertices in a Boolean cube. We find sufficient conditions for a sequence of random graphs to obey the zero-one law for first-order formulas of bounded quantifier depth. We also find conditions implying a weakened version of the zero-one law.

On the zero–one [formula omitted]-law extensions

The presented paper is devoted to the asymptotical behavior of first-order properties of the Erdős–Rényi random graph. In previous works the zero–one k-law was proved. This law describes asymptotical behavior of first-order properties which are expressed by formulae with a quantifier depth bounded by k. The random graph G(N,N−α) obeys the law if α∈(0,1/(k−2)). In this work we find new values of α, which are close to 1, such that G(N,N−α) obeys the zero–one k-law and, therefore, extend the previous result.

Nonparametric covariate-adjusted regression

We consider nonparametric estimation of a regression curve when the data are observed with multiplicative distortion which depends on an observed confounding variable. We suggest several estimators, ranging from a relatively simple one that relies on restrictive assumptions usually made in the literature, to a sophisticated piecewise approach that involves reconstructing a smooth curve from an estimator of a constant multiple of its absolute value, and which can be applied in much more general scenarios. We show that, although our nonparametric estimators are constructed from predictors of the unobserved undistorted data, they have the same first-order asymptotic properties as the standard estimators that could be computed if the undistorted data were available. We illustrate the good numerical performance of our methods on both simulated and real datasets.

Analytic energy derivatives for the calculation of the first-order molecular properties using the domain-based local pair-natural orbital coupled-cluster theory

The domain-based local pair-natural orbital coupled-cluster (DLPNO-CC) theory has recently emerged as an efficient and powerful quantum-chemical method for the calculation of energies of molecules comprised of several hundred atoms. It has been demonstrated that the DLPNO-CC approach attains the accuracy of a standard canonical coupled-cluster calculation to about 99.9% of the basis set correlation energy while realizing linear scaling of the computational cost with respect to system size. This is achieved by combining (a) localized occupied orbitals, (b) large virtual orbital correlation domains spanned by the projected atomic orbitals (PAOs), and (c) compaction of the virtual space through a truncated pair natural orbital (PNO) basis. In this paper, we report on the implementation of an analytic scheme for the calculation of the first derivatives of the DLPNO-CC energy for basis set independent perturbations within the singles and doubles approximation (DLPNO-CCSD) for closed-shell molecules. Perturbation-independent one-particle density matrices have been implemented in order to account for the response of the CC wave function to the external perturbation. Orbital-relaxation effects due to external perturbation are not taken into account in the current implementation. We investigate in detail the dependence of the computed first-order electrical properties (e.g., dipole moment) on the three major truncation parameters used in a DLPNO-CC calculation, namely, the natural orbital occupation number cutoff used for the construction of the PNOs, the weak electron-pair cutoff, and the domain size cutoff. No additional truncation parameter has been introduced for property calculation. We present benchmark calculations on dipole moments for a set of 10 molecules consisting of 20-40 atoms. We demonstrate that 98%-99% accuracy relative to the canonical CCSD results can be consistently achieved in these calculations. However, this comes with the price of tightening the threshold for the natural orbital occupation number cutoff by an order of magnitude compared to the DLPNO-CCSD energy calculations.

First-order and monadic properties of highly sparse random graphs

A random graph is said to obey the (monadic) zero–one k-law if, for any property expressed by a first-order formula (a second-order monadic formula) with a quantifier depth of at most k, the probability of the graph having this property tends to either zero or one. It is well known that the random graph G(n, n–α) obeys the (monadic) zero–one k-law for any k ∈ ℕ and any rational α > 1 other than 1 + 1/m (for any positive integer m). It is also well known that the random graph does not obey both k-laws for the other rational positive α and sufficiently large k. In this paper, we obtain lower and upper bounds on the largest at which both zero–one k-laws hold for α = 1 + 1/m.