Nonstandard Extension Research Articles

Multi-model query languages: taming the variety of big data

A critical issue in Big Data management is to address the variety of data–data are produced by disparate sources, presented in various formats, and hence inherently involves multiple data models. Multi-Model DataBases (MMDBs) have emerged as a promising approach for dealing with this task as they are capable of accommodating multi-model data in a single system and querying across them with a unified query language. This article aims to offer a comprehensive survey of a wide range of multi-model query languages of MMDBs. In particular, we first present the SQL-based extensions toward multi-model data, including the standard SQL extensions such as SQL/XML, SQL/JSON, and GQL, and the non-standard SQL extensions such as SQL++ and SPASQL. We then study the manners in which document-based and graph-based query languages can be extended to support multi-model data. We also investigate the query languages that provide native support on multi-model data. Finally, this article provides insights into the open challenges and problems of multi-model query languages.

A new R-matrix module for multi-channel calculations with GECCCOS

A versatile new R-matrix module for multi-channel reaction calculations was introduced into the code GECCCOS (GEneral Coupled-Channel COde System) which has been developed by the nuclear data group at TU-Wien to perform nuclear reaction calculations especially for light nuclear systems. It provides a tool for phenomenological R-matrix analyses of reaction data combined with calculations of a potential-based calculable R-matrix using the Lagrange-mesh technique. In addition it provides a platform for the development of non-standard extensions of R-matrix theory such as Reduced R-matrix analyses and the Hybrid R-matrix. A successful run of the code yields the complete S-matrix (collision matrix) as well as observables for unpolarized beams, angle-differential cross sections, excitation functions and, if existing, angle-integrated cross sections. Recently, extensions to polarization observables for spin-1/2 and spin-1 particles were implemented and tested. For phenomenological R-matrix analyses a separate module assembles calculated and available experimental values, automatically performs transformations with regard to reference frame and matching radii. Furthermore it allows to switch between incident channel and compound nucleus representation and provides the necessary feedback for the chi2 fitting process.

Existence of matching priors on compact spaces

Summary A matching prior at level $1-\alpha$ is a prior such that an associated $1-\alpha$ credible region is also a $1-\alpha$ confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives topological conditions under which matching priors for specific families of credible regions exist. Informally, we prove that, on compact parameter spaces, a matching prior exists if the so-called rejection-probability function is jointly continuous when we adopt the Wasserstein metric on priors. In light of this general result, we observe that typical families of credible regions, such as credible balls, highest-posterior density regions, quantiles, etc., fail to meet this topological condition. We show how to design approximate posterior credible balls and highest-posterior density regions that meet these topological conditions, yielding matching priors. Finally, we evaluate a numerical scheme for computing approximately matching priors based on discretization and iteration. The proof of our main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric that may be of independent interest.

CPP11sort: A parallel quicksort based on C++ threading

SummaryA new efficient implementation of the multithreaded quicksort algorithm called CPP11sort is presented. This implementation is built exclusively upon the threading primitives of the C++ programming language itself. The performance of CPP11sort is evaluated and compared with its mainstream competitors provided by GNU, Intel, and Microsoft. It is shown that out of the considered implementations, CPP11sort mostly yields the shortest sorting times and is the only one that is portable to any conforming C++ implementation without a need of external libraries or nonstandard compiler extensions. The experimental evaluation with various input data distributions resulted in parallel speedup between 16.1 and 44.2 on a 56‐core server and between 6.8 and 14.5 on a 10‐core workstation with enabled hyperthreading.

CosmoReionMC: a package for estimating cosmological and astrophysical parameters using CMB, Lyman-α absorption, and global 21 cm data

ABSTRACT We present a Markov Chain Monte Carlo (MCMC)-based parameter estimation package, CosmoReionMC, to jointly constrain cosmological parameters of the Lambda cold dark matter (ΛCDM) model and the astrophysical parameters related to hydrogen reionization. The package is based on a previously developed physically motivated semi-analytical model for reionization, a similar semi-analytical model for computing the global 21 cm signal during the cosmic dawn and using an appropriately modified version of the publicly available camb for computing the CMB anisotropies. These calculations are then coupled to an MCMC ensemble sampler emcee to compute the posterior distributions of the model parameter. The model has 12 free parameters in total: five cosmological and seven related to the stellar populations. We constrain the parameters by matching the theoretical predictions with CMB data from Planck, observations related to the quasar absorption spectra, and, for the first time, the global 21 cm signal from EDGES. We find that incorporating the quasar spectra data in the analysis tightens the bounds on the electron scattering optical depth τ and consequently the normalization As of the primordial matter power spectrum (or equivalently σ8). Furthermore, when we include the EDGES data in the analysis, we find that an early population of metal-free stars with efficient radio emission is necessary to match the absorption amplitude. The CosmoReionMC package should have interesting future applications, e.g. probing non-standard extensions to the ΛCDM model.

Quantum state readout, collapses, probes, and signals

Theories involving localized collapse allow the possibility that classical information could be obtained about quantum states by mechanisms other than the standard quantum dynamical maps defined by positive operator valued measurements, but nonetheless without allowing superluminal signaling. We can model this by extending quantum theory to include hypothetical devices that read out information about the local quantum state at a given point, which is defined by considering unitary evolution modified only by the effects of collapses in its past light cone. Like Popescu-Rohrlich boxes, these hypothetical devices would have practical and scientific implications if realizable. These include signaling through opaque media, probing the physics of distant or opaque systems without needing a reflected signal and giving detailed information about collapse dynamics without requiring direct observation of the collapsing system. These potential applications add to the motivation for systematic searches for possible signatures of these nonstandard extensions of quantum theory, and in particular for relevant gravitational effects, such as the validity of semiclassical gravity on small scales.

Testing the binomial fixed effects logit model, with an application to female labour supply

Regression models for proportions are frequently encountered in applied work. The conditional expectation function is bounded between 0 and 1 and therefore must be nonlinear, requiring nonstandard panel data extensions. One possible approach is the binomial panel logit model with fixed effects (Machado in J Econom 119:73–98, 2004). We propose a new and simple implementation of this conditional maximum likelihood estimator for standard software. We investigate the properties of the estimator under misspecification and derive a new test for overdispersion. Estimator and test are applied in a study of contracted working volumes, measured as proportion of full-time work, for women in Switzerland.

More about divisibility in βN

AbstractWe continue the research of an extension of the divisibility relation to the Stone‐Čech compactification . First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in . Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type in . Finally, we consider ultrafilters without divisors in and among them find the greatest class.

CONGRUENCE OF ULTRAFILTERS

AbstractWe continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$ , defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$ -equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$ . Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation.

Nonstandard analysis of uniform spaces

There are nonstandard characterizations of classical topological concepts like total boundedness, completeness, comapactness etc., in metric spaces. In this article we present nonstandard extensions of these characterizations to uniform spaces. We also furnish completion of uniform spaces. We accomplish this by extending the methods in metric spaces by way of passing from sequences to filters and nets.

Hyperfinite logics and non-standard extensions of Boolean algebras

Infinitary propositional logics, i.e., propositional logics with infinite conjunction and disjunction, have some deficiencies, e.g., these logics fail to be compact or complete, in general. Such kind of infinitary propositional logics are introduced, called hyperfinite logics, which are defined in a non-standard framework of non-standard analysis and have hyperfinite conjunctions and disjunctions. They have more nice properties than infinitary logics have, in general. Furthermore, non-standard extensions of Boolean algebras are investigated. These algebras can be regarded as algebraizations of hyperfinite logics, they have several unusual properties. These Boolean algebras are closed under the hyperfinite sums and products, they are representable by hyperfinitely closed Boolean set algebras and they are omega-compact. It is proved that standard Boolean algebras are representable by Boolean set algebras with a hyperfinite unit.

Elements of XPath-like languages in problems of constructing semantic XML models of natural language texts

The subject of the research is the possibility of using XPath-like micro-languages of programming in the generation systems of programs of the PGEN ++ class for the selection and completion of XML-models describing the plan for solving the original problem, according to which the solver program is generated. It is supposed to build such models according to the description of the problem in natural language, thus, we are talking about elements of artificial intelligence technologies. XPath-like language works in the layer of regular-logical expressions (highlighting elements of the primary XML document), performing primary processing of the data obtained in the layer of grammatical parsing of the source text. In addition, XPath-like elements are used to render the final XML models. The standard natural language parsing libraries are used. Non-standard XPath query language extensions are used. For the first time, the idea of expanding the XPath query language to the level of an algorithmic language by introducing the minimum required number of syntactic elements is proposed. It is also proposed to use syntactic elements with an XPath-like structure as both generating and controlling weak constraints of the process of direct inference of final semantic XML models.

On the road to percent accuracy: non-linear reaction of the matter power spectrum to dark energy and modified gravity

ABSTRACT We present a general method to compute the non-linear matter power spectrum for dark energy (DE) and modified gravity scenarios with per cent-level accuracy. By adopting the halo model and non-linear perturbation theory, we predict the reaction of a lambda cold dark matter (ΛCDM) matter power spectrum to the physics of an extended cosmological parameter space. By comparing our predictions to N-body simulations we demonstrate that with no-free parameters we can recover the non-linear matter power spectrum for a wide range of different w0–wa DE models to better than 1 per cent accuracy out to k ≈ 1 $h \,{\rm Mpc}^{-1}$. We obtain a similar performance for both DGP and f(R) gravity, with the non-linear matter power spectrum predicted to better than 3 per cent accuracy over the same range of scales. When including direct measurements of the halo mass function from the simulations, this accuracy improves to 1 per cent. With a single suite of standard ΛCDM N-body simulations, our methodology provides a direct route to constrain a wide range of non-standard extensions to the concordance cosmology in the high signal-to-noise non-linear regime.

Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a standard way. We also find some non-standard extensions of several well-known $\mathcal{B}_n$-representations such as the Burau and LKB representations. Moreover, we prove that any local representation of $\mathcal{B}_n$ (i.e. coming from a braided vector space) can be extended to $\mathcal{NB}_n$, in contrast to the situation with $\mathcal{LB}_n$. We also discuss some directions for future study from categorical and physical perspectives.

Mass-jump and mass-bump boundary conditions for singular self-adjoint extensions of the Schrödinger operator in one dimension

Physical realizations of non-standard singular self-adjoint extensions for one-dimensional Schrödinger operator in terms of the mass-jump are considered. It is shown that corresponding boundary conditions can be realized for the Hamiltonian with the position-dependent effective mass in two qualitatively different profiles of the effective mass inhomogeneity: the mass-jump and the mass-bump. The existence of quantized magnetic flux in a case of the mass-jump is proven by explicit demonstration of the Zeeman-like splitting for states with the opposite projections of angular momentum.

Testing the Binomial Fixed Effects Logit Model; with an Application to Female Labor Supply

Regression models for proportions are frequently encountered in applied work. The conditional expectation function is bounded between 0 and 1 and therefore must be non-linear, requiring nonstandard panel data extensions. One possible approach is the binomial panel logit model with fixed effects (Machado, 2004). We propose a new and simple implementation of the conditional maximum likelihood estimator for standard software. We investigate the properties of the estimator under misspecification and derive a new test for overdispersion. Estimator and test are applied in a study of contracted working volumes, measured as proportion of full-time work, for women in Switzerland.

Magnetizing the Cosmic Web during Reionization

Increasing evidence suggests that cosmological sheets, filaments, and voids may be substantially magnetized today. The origin of magnetic fields in the intergalactic medium (IGM) is, however, currently uncertain. It seems well known that non-standard extensions to the physics of the standard model can provide mechanisms susceptible of magnetizing the universe at large. Perhaps less well known is the fact that standard, classical physics of matter–radiation interactions actually possesses the same potential. We discuss a magnetogenesis mechanism based on the exchange of momentum between hard photons and electrons in an inhomogeneous IGM. Operating in the neighborhood of ionizing sources during the epoch of reionization, this mechanism is capable of generating magnetic seeds of relevant strengths over scales comparable to the distance between ionizing sources. In addition, summing up the contributions of all ionizing sources and taking into account the distribution of gas inhomogeneities, we show that this mechanism leaves the IGM, at the end of reionization, with a level of magnetization that might account, when amplification mechanisms take over, for the magnetic fields strengths in the current cosmic web.

BOUNDED POLYNOMIALS AND HOLOMORPHIC MAPPINGS BETWEEN CONVEX SUBRINGS OF *

AbstractUsing convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

SSpace: A Toolbox for State Space Modeling

SSpace is a MATLAB toolbox for state space modeling. State space modeling is in itself a powerful and flexible framework for dynamic system modeling, and SSpace is conceived in a way that tries to maximize this flexibility. One of the most salient features is that users implement their models by coding a MATLAB function. In this way, users have complete flexibility when specifying the systems, have absolute control on parameterizations, constraints among parameters, etc. Besides, the toolbox allows for some ways to implement either non-standard models or standard models with non-standard extensions, like heteroskedasticity, time-varying parameters, arbitrary nonlinear relations with inputs, transfer functions without the need of using explicitly the state space form, etc. The toolbox may be used on the basis of scratch state space systems, but is supplied with a number of templates for standard widespread models. A full help system and documentation are provided. The way the toolbox is built allows for extensions in many ways. In order to fuel such extensions and discussions an online forum has been launched.

Choice sequence and nonstandard extension of type theory

Partant de son travail Mathematics of Infinity (1989), Martin-Löf a développé l'idée d'un lien conceptuel profond entre les notions de suite de choix et d'objet mathématique nonstandard. Précisément, il a défini une extension nonstandard de la théorie des types en ajoutant une série d'axiomes nonstandard conçue comme une sorte de suite de choix. Enfin, dans une communication de 1999, il a présenté les grandes lignes d'une théorie des types nonstandard plus générale et munie d'un fort contenu computationnel. Le présent travail est une tentative de donner un développement complet d'une théorie de ce genre. Cependant, dans le but de garder un fort contrôle sur la théorie résultante et notablement pour éviter quelques problèmes en rapport avec l'égalité définitionnelle, le champ des axiomes nonstandard est moins général que ceux proposés dans sa communication de 1999. L'étude présente est poussée jusqu'à l' introduction d'une notion de proposition externe qui joue le même rôle que les propriétés externes si utiles dans l'analyse nonstandard usuelle. Du fait que ce texte débute par une introduction à la théorie des types de Martin-Löf, il peut intéresser les mathématiciens non familiers avec ce sujet.