Proper Notion Research Articles

Smooth and proper maps with respect to a fibration

Abstract This paper explain how the geometric notions of local contractibility and properness are related to the $\Sigma$ -types and $\Pi$ -types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Majorization requires infinitely many second laws

Majorization is a fundamental model of uncertainty with several applications in areas ranging from thermodynamics to entanglement theory, and constitutes one of the pillars of the resource-theoretic approach to physics. Here, we improve on its relation to measurement apparatuses. In particular, after discussing what the proper notion of second law in this scenario is, we show that, for a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. Moreover, we provide an analogous result for a variation of majorization known as thermo-majorization which, in fact, does not require any constraint on the state space provided the equilibrium distribution is not uniform. Finally, we discuss the applicability of our results to molecular diffusion and catalytic majorization. In this regard, we consider a variation of majorization used in plasma physics as a model of molecular diffusion and show that no finite family of entropy-like functions constituting a second law of molecular diffusion exists. Moreover, we show how our results are useful when dealing with a conjecture regarding catalytic majorization (i.e. trumping). In particular, we show that the sort of characterizations of trumping that have been considered before require an infinite family of real-valued functions.

On cone-based decompositions of proper Pareto-optimality in multi-objective optimization

In recent years, research focus in multi-objective optimization has shifted from approximating the Pareto optimal front in its entirety to identifying solutions that are well-balanced among their objectives. Proper Pareto optimality is an established concept for eliminating Pareto optimal solutions that exhibit unbounded tradeoffs. Imposing a strict tradeoff bound in a classical definition of proper Pareto optimality allows specifying how many units of one objective one is willing to trade in for obtaining one unit of another objective. Recent studies have shown that this notion shows favorable convergence properties. One of the aims of this paper is to translate the proper Pareto optimality notion to an ordering relation, which we denote by M-domination. The mathematical properties of M-domination are thoroughly analyzed in this paper yielding key insights into its applicability as decision making aid and in designing population-based algorithms for solving multi-objective optimization problems. We complement our work by providing four different geometrical descriptions of the M-dominated space given by a union of polyhedral cones. A geometrical description does not only yield a greater understanding of the underlying tradeoff concept, but also allows a quantification of the hypervolume dominated by a particular solution or an entire set of solutions. These descriptions shall enable researchers to formulate hypervolume-based approaches for finding approximations of the Pareto front that emphasize regions that are well-balanced among their tradeoffs in subsequent works.

Assessment Literacy: Secondary School Teachers’ Conception of Formative Assessment

Formative assessment has been proved to be one of the means of improving quantity and quality of students learning. However, it hasn’t been implemented effectively in classrooms may be because of misconceptions. The purpose of this research is to find out how English language teachers conceptualize formative assessment, and if their conceptions vary with level of their qualification. A total of 252 randomly selected secondary school English language teachers with different level of qualification filled in Teachers’ Conception of Formative Assessment (TCFA), questionnaire adapted from Teachers’ Conception of Assessment (TCOA III). They were also interviewed in groups of their qualification. The data from the questionnaire was analyzed running one-sample t-test and The One-way between groups ANOVA. The interview data was analyzed thematically. The result indicated that the participants’ conception is mixed and partially deviates from proper notion of formative assessment; It also revealed that the higher the level of qualification, the better conception of formative assessment. This implied poor conceptualization of the relation of teaching-learning and assessment that can impede its implementation.

On α-type (equivariant) cohomology of Hom-pre-Lie algebras

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context.

On slow minimal reals I

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real η ∈ ∏ i > ω n i ∗ \eta \in \prod _{i>\omega } n^*_i , which is eventually different from any old real in ∏ i > ω n i ∗ \prod _{i>\omega } n^*_i , where the sequence ⟨ n i ∗ ∣ i > ω ⟩ \langle n^*_i \mid i>\omega \rangle grows slowly.

Proper adaptive filtering in four-dimensional Cayley–Dickson algebras

A family of hypercomplex algebras in four dimensions (4D) is proposed to devise adaptive filters. Such a family, called β-quaternions, has multiplication rules for the complex units that depend on a parameter β, and this family contains, as particular cases, both standard Hamilton quaternions and split quaternions. In this framework, two notions of properness for random vectors are introduced and their implications on the statistical processing involved are analyzed. Then, statistical tests to check properness in practice and a method to select the best algebra where the properness conditions could hold are provided. Also, proper adaptive filters are suggested and row and column updating problems are studied. The main advantage of the techniques proposed compared with the standard ones is that a notable reduction in the computational burden is achieved. Finally, simulation examples validate the proper adaptive filters and demonstrate that our scheme performs better than the traditional quaternion domain.

Palatini variation in generalized geometry and string effective actions

We develop the Palatini formalism within the framework of generalized Riemannian geometry of Courant algebroids. In this context, the Palatini variation of a generalized Einstein–Hilbert–Palatini action - formed using a generalized metric, a Courant algebroid connection (in contrary to the ordinary case, not necessarily a torsionless one) and a volume form - leads naturally to a proper notion of a generalized Levi-Civita connection and low-energy effective actions of string theory.

Learning With Proper Partial Labels.

Partial-label learning is a kind of weakly supervised learning with inexact labels, where for each training example, we are given a set of candidate labels instead of only one true label. Recently, various approaches on partial-label learning have been proposed under different generation models of candidate label sets. However, these methods require relatively strong distributional assumptions on the generation models. When the assumptions do not hold, the performance of the methods is not guaranteed theoretically. In this letter, we propose the notion of properness on partial labels. We show that this proper partial-label learning framework requires a weaker distributional assumption and includes many previous partial-label learning settings as special cases. We then derive a unified unbiased estimator of the classification risk. We prove that our estimator is risk consistent, and we also establish an estimation error bound. Finally, we validate the effectiveness of our algorithm through experiments.

On the wave equation on moving domains: regularity, energy balance and application to dynamic debonding

We revisit some issues about existence and regularity for the wave equation in noncylindrical domains. Using a method of diffeomorphisms we show, through increasing regularity assumptions, how the existence of weak solutions, their improved regularity and an energy balance can be derived. As an application, we give a rigorous definition of dynamic energy release rate density for some problems of debonding, and we formulate a proper notion of solutions for such problems. We compare the consistence of such a formulation with that of previous ones, given in the literature for particular cases.

Further Addenda and Corrigenda to Annotations of Boswell’s Hypochondriack

Further Addenda and Corrigenda to Annotations of Boswell’s <i>Hypochondriack</i>

Consistent Approximation of Interpolating Splines in Image Metamorphosis

This paper investigates a variational model for splines in the image metamorphosis model for the smooth interpolation of key frames in the space of images. The Riemannian manifold of images based on the metamorphosis model defines shortest geodesic paths interpolating two images as minimizers of the path energy which measures the viscous dissipation caused by the motion field and dissipation caused by the material derivative of the image intensity along motion paths. In this paper, we aim at smooth interpolation of multiple key frame images picking up the general observation of cubic splines in Euclidean space which minimize the squared acceleration along the interpolation path. To this end, we propose the spline functional which combines quadratic functionals of the Eulerian motion acceleration and of the second material derivative of the image intensity as the proper notion of image intensity acceleration. We propose a variational time discretization of this model and study the convergence to a suitably relaxed time continuous model via varGamma -convergence methodology. As a byproduct, this also allows to establish the existence of metamorphosis splines for given key frame images as minimizers of the time continuous spline functional. The time discretization is complemented by effective spatial discretization based on finite differences and a stable B-spline interpolation of deformed quantities. A variety of numerical examples demonstrates the robustness and versatility of the proposed method in applications. For the minimization of the fully discrete energy functional, a variant of the iPALM algorithm is used.

Two-dimensional signature of images and texture classification

We introduce a proper notion of two-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a two-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here, we implement a simple procedure for texture classification. In this context, we show that a low-dimensional set of features based on signatures produces an excellent accuracy.

Determinacy and regularity properties for idealized forcings

AbstractWe show under that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under if we additionally assume that the set of Borel codes for I‐positive sets is . If we do not assume , the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch [2], we show under without using that every set of reals is I‐regular for any σ‐ideal I on the Baire space such that is strongly proper assuming every set of reals is ∞‐Borel and there is no ω1‐sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.

Self-adjoint and Markovian extensions of infinite quantum graphs.

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, graph ends, and self‐adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of finite volume for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self‐adjointness of the Gaffney Laplacian — the underlying metric graph does not have finite volume ends. If, however, finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.

Myths and misconceptions of data-driven methods: Applications to process safety analysis

With a rapid realization of process digitization, data-driven methods are being increasingly adopted in process safety analysis. However, the use of data-driven methods contains a varied degree of myths and misconceptions resulting from the application of a method or the data representation that does not follow a proper scientific notion. These myths and misconceptions cause significant errors in terms of results and their interpretability. Hence, the purpose of this study is set to analyze the most common myths and misconceptions of data-driven methods observed in the recent literature. In the current work, we have analyzed 500 public domain articles from 1990 to 2020, published in 10 renowned safety journals. The analysis attempts to address the following questions: (i) What are the key data in process safety analysis? (ii) What are the sources of data? (iii) What does the data-driven method mean? (iv) What are the common myths and misconceptions of data-driven methods? and (v) How frequently such myths and misconceptions are occurring? After analyzing the 500 articles, it is observed that most of the myths are related to improper data representation, missing appropriate assumptions, and blanket use of methods without a detailed understanding of their limitations. The authors believe this work will help peers to avoid the myths studied here, use data-driven methods with scientific rigor, and present findings in a meaningful way.

Thermodynamics of Reduced State of the Field.

Recent years have seen the flourishing of research devoted to quantum effects on mesoscopic and macroscopic scales. In this context, in Entropy 2019, 21, 705, a formalism aiming at describing macroscopic quantum fields, dubbed Reduced State of the Field (RSF), was envisaged. While, in the original work, a proper notion of entropy for macroscopic fields, together with their dynamical equations, was derived, here, we expand thermodynamic analysis of the RSF, discussing the notion of heat, solving dynamical equations in various regimes of interest, and showing the thermodynamic implications of these solutions.

A Functional Programming Language with Versions

While modern software development heavily uses versioned packages, programming languages rarely support the concept of versions in their semantics, which makes software updates more bulky and unsafe. This paper proposes a programming language that intrinsically supports versions. The main goals are to design core language features to support multiple versions in one program and establish a proper notion of type safety with those features. The proposed core calculus, called Lambda VL, has versioned values, each containing different values under different versions. We show the construction of the type system as an extension of coeffect calculus by mapping versions to computational resources. The type system guarantees the existence of a valid combination of versions for a program. The calculus enables programming languages to use multiple versions of a package within a program. It will serve as a basis for designing advanced language features like module systems and semantic versioning.

Normality, Controllability and Properness in Optimal Control

In this paper we show that, for optimal control problems involving equality and inequality constraints on the control function, the notions of normality and properness (or the Mangasarian–Fromovitz constraint qualification) of a trajectory relative to the set of constraints are equivalent. This is in contrast with some differences recently obtained between the theories of mathematical programming and optimal control, and it provides an important insight in the derivation of first and second order necessary optimality conditions for infinite dimensional problems.

Emergent fracton dynamics in a nonplanar dimer model

We study the late time relaxation dynamics of a pure $U(1)$ lattice gauge theory in the form of a dimer model on a bilayer geometry. To this end, we first develop a proper notion of hydrodynamic transport in such a system by constructing a global conservation law that can be attributed to the presence of topological solitons. The correlation functions of local objects charged under this conservation law can then be used to study the universal properties of the dynamics at late times, applicable to both quantum and classical systems. Performing the time evolution via classically simulable automata circuits unveils a rich phenomenology of the system's non-equilibrium properties: For a large class of relevant initial states, local charges are effectively restricted to move along one-dimensional 'tubes' within the quasi-two-dimensional system, displaying fracton-like mobility constraints. The time scale on which these tubes are stable diverges with increasing systems size, yielding a novel mechanism for non-ergodic behavior in the thermodynamic limit. We further explore the role of geometry by studying the system in a quasi-one-dimensional limit, where the Hilbert space is strongly fragmented due to the emergence of an extensive number of conserved quantities. This provides an instance of a recently introduced concept of 'statistically localized integrals of motion', whose universal anomalous hydrodynamics we determine by a mapping to a problem of classical tracer diffusion. We conclude by discussing how our approach might generalize to study transport in other lattice gauge theories.