Consistency Theorem Research Articles

A consistent semantics of self-adjusting computation

AbstractThis paper presents a semantics of self-adjusting computation and proves that the semantics is correct and consistent. The semantics introduces memoizing change propagation, which enhances change propagation with the classic idea of memoization to enable reuse of computations even when memory is mutated via side effects. During evaluation, computation reuse via memoization triggers a change-propagation algorithm that adapts the reused computation to the memory mutations (side effects) that took place since the creation of the computation. Since the semantics includes both memoization and change propagation, it involves both non-determinism (due to memoization) and mutation (due to change propagation). Our consistency theorem states that the non-determinism is not harmful: any two evaluations of the same program starting at the same state yield the same result. Our correctness theorem states that mutation is not harmful: Self-adjusting programs are compatible with purely functional programming. We formalize the semantics and its meta-theory in the LF logical framework and machine check our proofs using Twelf.

Factorization theorem for high-energy scattering near the end point.

A consistent factorization theorem is presented in the framework of effective field theories. Conventional factorization suffers from infrared divergences in the soft and collinear parts. We present a factorization theorem in which the infrared divergences appear only in the parton distribution functions by carefully reorganizing collinear and soft parts. The central idea is extracting the soft contributions from the collinear part to avoid double counting. Combining it with the original soft part, an infrared-finite kernel is obtained. This factorization procedure can be applied to various high-energy scattering processes.

Partial Observability and Its Consistency for Linear PDEs

In this paper, a quantitative measure of partial observability is de_ned for PDEs. The quantity is proved to be consistent in well-posed approximation schemes. A first order approximation of an unobservability index using empirical gramian is introduced. For linear systems with full state observability, the empirical gramian is equivalent to the observability gramian in control theory. The consistency theorem is exemplified using a Burgers' equation.

On Modified Mellin Transform of Generalized Functions

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain and range spaces where the extended modified transform is well defined. Consistency, convolution, analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given.

Global sensitivity measures from given data

Simulation models support managers in the solution of complex problems. International agencies recommend uncertainty and global sensitivity methods as best practice in the audit, validation and application of scientific codes. However, numerical complexity, especially in the presence of a high number of factors, induces analysts to employ less informative but numerically cheaper methods. This work introduces a design for estimating global sensitivity indices from given data (including simulation input–output data), at the minimum computational cost. We address the problem starting with a statistic based on the L1-norm. A formal definition of the estimators is provided and corresponding consistency theorems are proved. The determination of confidence intervals through a bias-reducing bootstrap estimator is investigated. The strategy is applied in the identification of the key drivers of uncertainty for the complex computer code developed at the National Aeronautics and Space Administration (NASA) assessing the risk of lunar space missions. We also introduce a symmetry result that enables the estimation of global sensitivity measures to datasets produced outside a conventional input–output functional framework.

Conserving/dissipative algorithms and designs for a system of N particles: Total energy framework and single-field form

Conserving and dissipative implicit algorithm designs in the single-field form via the total energy framework for N-body systems are summarized in this paper which provides new avenues with improved physical interpretation, and can also explain analogous past efforts. We show various algorithm designs via two distinct approaches based on the algorithmic time level consistency theorem, namely, the discrete total energy conservation method, and the classical and normalized time weighted residual methodologies; and therein draw comparisons among the various time-stepping algorithms and features. A simple numerical simulation is shown to demonstrate the pros/cons for the conserving properties of various algorithms.

Structure of divergences in the Drell-Yan process with small transverse momentum

We consider the structure of divergences in Drell-Yan process with small transverse momentum. The factorization proof is not trivial because various kinds of divergences are intertwined in the collinear and soft parts at high orders. We prescribe a method to disentangle the divergences in the framework of the soft-collinear effective theory. The rapidity divergence is handled by introducing the $\delta$ regulator in the collinear Wilson lines. The collinear part, which consists of the transverse-momentum-dependent parton distribution function (TMDPDF), is free of the rapidity divergence after the soft zero-bin subtraction. There still remains the problem of mixing between the ultraviolet and infrared divergences, which forbids the renormalization group description. We show that the mixing is cancelled by the soft function. This suggests that the collinear and soft parts should be treated as a whole in constructing a consistent factorization theorem. The renormalization group behavior of the combined collinear and soft parts is presented explicitly at one loop. We also show that the integrated PDF can be obtained by integrating the TMDPDF over the transverse momentum.

Some remarks on Arslan’s 2011 paper

It is shown that the main theorem of Arslan’s paper (Theorem 2, 2011), as stated, is incorrect. Under additional conditions, we present a short proof of the corrected version of the theorem. We also give a proof of a theorem of Rao and Shanbhag (1991) [2], employed by Arslan, without the use of the Kolmogorov Consistency Theorem.

On strong Hellinger consistency of posterior distributions

We establish a sufficient condition ensuring strong Hellinger consistency of posterior distributions. We also prove a strong Hellinger consistency theorem for the pseudoposterior distributions based on the likelihood ratio with power 0<α<1, which are introduced by Walker and Hjort [2001 ‘On Bayesian Consistency’, J. R. Statist. Soc., B 63, 811–821]. Our result is an extension of their theorem for α=½.

A consistency theorem for regular conditional distributions

Let be a measurable space, a sub--field and a random probability measure on , . In various frameworks, one looks for a probability P on such that is a regular conditional distribution for P given for all n. Conditions for such a P to exist are given. The conditions are quite simple when is a compact Hausdorff space equipped with the Borel or the Baire -field (as well as under similar assumptions). Applications to Gibbs measures and Bayesian statistics are given as well.

The Consistency of Probabilistic Regresses: Some Implications for Epistemological Infinitism

This note employs the recently established consistency theorem for infinite regresses of probabilistic justification (Herzberg in Stud Log 94(3):331–345, 2010) to address some of the better-known objections to epistemological infinitism. In addition, another proof for that consistency theorem is given; the new derivation no longer employs nonstandard analysis, but utilises the Daniell–Kolmogorov theorem.

Extensivity of Energy and Electronic and Vibrational Structure Methods for Crystals

A pedagogical proof is presented for the extensivity of energies of metallic and nonmetallic crystals that proceeds by elucidating the asymptotic distance dependence of the effective chemical interactions: kinetic, Coulomb, exchange, and correlation. On this basis, a guideline for the size-consistent design of electronic and vibrational methods is proposed. This guideline underscores the significance of the distinct use of the intermediate and standard normalization of wave functions for extensive and intensive quantities, includes the extensive and intensive diagram theorems as the unambiguous criteria for determining size consistency of a method for extensive and intensive quantities, and introduces the extensive-intensive consistency theorem, which stipulates the precise balance between the determinant spaces reached by extensive and intensive operators. Electronic and vibrational methods for crystals are reviewed that are inspired by these formal analyses or developed in accordance with the guideline.

The Bernstein-Von-Mises theorem under misspecification

We prove that the posterior distribution of a parameter in misspecified LAN parametric models can be approximated by a random normal distribution. We derive from this that Bayesian credible sets are not valid confidence sets if the model is misspecified. We obtain the result under conditions that are comparable to those in the well-specified situation: uniform testability against fixed alternatives and sufficient prior mass in neighbourhoods of the point of convergence. The rate of convergence is considered in detail, with special attention for the existence and construction of suitable test sequences. We also give a lemma to exclude testable model subsets which implies a misspecified version of Schwartz’ consistency theorem, establishing weak convergence of the posterior to a measure degenerate at the point at minimal Kullback-Leibler divergence with respect to the true distribution.

Reconstructing Bohr’s Reply to EPR in Algebraic Quantum Theory

Halvorson and Clifton have given a mathematical reconstruction of Bohr's reply to Einstein, Podolsky and Rosen (EPR), and argued that this reply is dictated by the two requirements of classicality and objectivity for the description of experimental data, by proving consistency between their objectivity requirement and a contextualized version of the EPR reality criterion which had been introduced by Howard in his earlier analysis of Bohr's reply. In the present paper, we generalize the above consistency theorem, with a rather elementary proof, to a general formulation of EPR states applicable to both non-relativistic quantum mechanics and algebraic quantum field theory; and we clarify the elements of reality in EPR states in terms of Bohr's requirements of classicality and objectivity, in a general formulation of algebraic quantum theory.

Safety Model Comprehensive Judgement Application for Dike Construction Risk Evaluation System Research

Inclusive of the consideration for Solid-Liquid double phase character of dike construction, fuzzy comprehensive judgement system model was set up on the basis of Analytic Hierarchy Process as well as Fuzzy Consistency Theorem. With normal membership of fuzzy function, dike construction safety comprehensive judgement was realized by the help of weighted theorem that had been fabricated in the foregoing contents. With indirect theorem of model discrimination, generalized multi-dimension weighted fuzzy discriminating model was set up on the basis of fuzzy comprehensive evaluation. Furthermore, dike section system safety model discrimination for representative dike section was realized and sensitivity analysis for model system was carried out.

Thermodynamic limit and size-consistent design

This article aims to dispel confusions about the definition of size consistency as well as some incompatibility that exists between different criteria for judging whether an electronic structure theory is size consistent and thus yields energies and other quantities having correct asymptotic size dependence. It introduces extensive and intensive diagram theorems, which provide unambiguous sufficient conditions for size consistency for extensive and intensive quantities, respectively, stipulated in terms of diagrammatic topology and vertex makeup. The underlying algebraic size-consistency criterion is described, which relies on the polynomial dependence of terms in the formalism on the number of wave vector sampling points in Brillouin-zone integrations. The physical meanings of two types of normalization of excitation amplitudes in electron-correlation theories, namely, the intermediate and standard normalization, are revealed. The amplitudes of the operator that describes an extensive quantity (the extensive operator) are subject to the intermediate normalization, while those of the operator that corresponds to an intensive quantity (the intensive operator) must be normalized. The article also introduces the extensive-intensive consistency theorem which specifies the relationship between the spaces of determinants reached by the extensive and intensive operators in a size-consistent method for intensive quantities. Furthermore, a more fundamental question is addressed as to what makes energies extensive and thus an application of thermodynamics to chemistry valid. It is shown that the energy of an electrically neutral, periodic, non-metallic solid is extensive. On this basis, a strictly size-consistent redefinition of the Hartree--Fock theory is proposed.

Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth

For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised which rests on one of two asymptotic scenarios. For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic. In application, this allows to verify the geodesic hypothesis for leaf growth of Canadian black poplars and to discriminate genetically different trees by observations of leaf shape growth over brief time intervals. With a test based on Procrustes tangent space coordinates, not involving the shape space’s curvature, neither can be achieved.

The [formula omitted]-consistency of Dirichlet mixtures in multivariate Bayesian density estimation

Density estimation, especially multivariate density estimation, is a fundamental problem in nonparametric inference. In the Bayesian approach, Dirichlet mixture priors are often used in practice for such problems. However, the asymptotic properties of such priors have only been studied in the univariate case. We extend the L 1 -consistency of Dirichlet mixutures in the multivariate density estimation setting. We obtain such a result by showing that the Kullback–Leibler property of the prior holds and that the size of the sieve in the parameter space in terms of L 1 -metric entropy is not larger than the order of n . However, it seems that the usual technique of choosing a sieve by controlling prior probabilities is unable to lead to a useful bound on the metric entropy required for the application of a general posterior consistency theorem for the multivariate case. We overcome this difficulty by using a structural property of Dirichlet mixtures. Our results apply to a multivariate normal kernel even when the multivariate normal kernel has a general variance–covariance matrix.

Linguistic truth-valued lattice-valued propositional logic system ℓ P( X) based on linguistic truth-valued lattice implication algebra

In the semantics of natural language, quantification may have received more attention than any other subject, and syllogistic reasoning is one of the main topics in many-valued logic studies on inference. Particularly, lattice-valued logic, a kind of important non-classical logic, can be applied to describe and treat incomparability by the incomparable elements in its truth-valued set. In this paper, we first focus on some properties of linguistic truth-valued lattice implication algebra. Secondly, we introduce some concepts of linguistic truth-valued lattice-valued propositional logic system ℓ P( X), whose truth-valued domain is a linguistic truth-valued lattice implication algebra. Then we investigate the semantic problem of ℓ P( X). Finally, we further probe into the syntax of linguistic truth-valued lattice-valued propositional logic system ℓ P( X), and prove the soundness theorem, deduction theorem and consistency theorem.

Plug-in estimators for higher-order transition densities in autoregression

In this paper we obtain root-n consistency and functional central limit theorems in weighted L1-spaces for plug-in estimators of the two-step transition density in the classical station- ary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estima- tor for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit the fact that the innovations have mean zero. If an efficient estimator for the autoregression parameter is used, then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes.