Weak Version Research Articles

On some weak versions of graded-Noetherian modules and rings

On some weak versions of graded-Noetherian modules and rings

Partial Algebras and Implications of (Weak) Matrix Properties

Matrix properties are a type of property of categories which includes the ones of being Mal’tsev, arithmetical, majority, unital, strongly unital, and subtractive. Recently, an algorithm has been developed to determine implications M⇒lex∗N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{M}\\Rightarrow _{\ extrm{lex}_*}\ extrm{N}$$\\end{document} between them. We show here that this algorithm reduces to constructing a partial term corresponding to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{N}$$\\end{document} from a partial term corresponding to M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{M}$$\\end{document}. Moreover, we prove that this is further equivalent to the corresponding implication between the weak versions of these properties, i.e., the one where only strong monomorphisms are considered instead of all monomorphisms.

Small amendment arguments: how they work and what they do and do not show

AbstractThe small improvement argument has been said to establish that the standard weak preference or value relation can be incomplete. We first show that the argument is one of three possible ‘small amendment arguments’, each of which would yield the same conclusion. Generalizing the analysis thus, we subsequently present a strong and a weak version of small amendment arguments and derive the exact rationality conditions under which they reveal incompleteness. The results show that the arguments (in any of their variants) need not reveal a problem for the possibility of rational choice. In fact, it can be argued that they only reveal such a problem if the underlying relation is complete rather than incomplete.

One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has aradical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremumbeing finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2 (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case.

A Fisher-type evolution equation with applications in image processing

The aim of this paper is to present some applications of the Neumann Laplacian in image processing, along with the necessary mathematical background. We prove weak and strong versions of the maximum principle for weak solutions of elliptic and parabolic problems and apply them to a Fisher K.P.P.-type equation. The original contribution lies in the application of this equation in image processing, where various diffusion-like effects can be achieved. Additionally, a review of the basics of linear and nonlinear PDEs with Neumann boundary conditions is provided, along with updated bibliography and recent qualitative results. There are also some new theoretical results developed in this work.

On relationships between vector optimization problems and vector variational inequalities using directional convexificators

In this work, we formulate the vector variational inequalities of Stampacchia and Minty types in terms of directional convexificators, and relate them to a vector optimization problem. Our approach consists of using a suitable directional generalized convexity, given in terms of directional convexificators, to help us figure out the necessary and sufficient conditions for a point to be an efficient solution to the vector optimization problem. We also investigate the weak versions of the vector variational inequalities and provide several results for determining weak efficient solutions. An example illustrating both our findings and the limits of some earlier research is provided.

Categoricity and multidimensional diagrams

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.

Affectivity in its Relation to Personal Identity

AbstractMy aim is to propose affectivity as a criterion for personal identity. My proposal is to be taken in its weak version: affectivity as only one of the criteria for personal identity. I start by arguing for affectivity being a better candidate as a criterion for personal identity than thinking. Next, I focus on synchronic vs. diachronic and on ontic vs. epistemic distinctions (my proposal will concern diachronic ontic personal identity) and consider the realm of affectivity in its temporal dimension. In my argument, I rely on the genuineness of affectivity and I exploit chiefly long- or everlasting feelings (or emotions) and those which involve the feeling subject more rather than less. I conclude by stating that genuine long-lasting feelings (or emotions) are constitutive of diachronic ontic personal identity.

Weak-form latent space dynamics identification

Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique.We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers’, radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers’ simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 16% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers’ yields a 140X speedup compared to the corresponding full order model.Python code to reproduce the results in this work is available at (https://github.com/MathBioCU/PyWSINDy_ODE) and (https://github.com/MathBioCU/PyWLaSDI).

Similarities and differences between specification and non-uniform specification

Abstract Pavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Breaking through the doors of perception, consciousness, and existence: to what extent does psychedelic phenomenology ontologically depend on external factors?

AbstractAs reliable facilitators of characteristically unique altered states of consciousness that are notoriously difficult to comprehend, psychedelics naturally attract a multidisciplinary scope for their research, with direct relevance to neuroscience, pharmacology, psychology, as well as philosophy of mind, metaphysics, and epistemology — disciplines united by the common inquiry into the nature of conscious experience. Accordingly, the central focus of this paper concerns the metaphysical debate between internalism and externalism about whether psychedelic phenomenology — in particular, its ‘breakthrough’ level — could be said to directly ontologically depend on external factors, as well as the interpretive implications arising from each of those positions. The main contributions of the author of this paper include the provision of a coherent account of the essential phenomenal characteristics of psychedelic experiences, the integration of relatively fragmented philosophical debates on psychedelic phenomenology into the traditional internalist-externalist framework in metaphysics, as well as the exploration of the extent to which the externalist position on psychedelic phenomenology — traditionally associated with non-naturalism — is compatible with naturalistic explanatory frameworks. In Part 1, psychedelic phenomenology will be outlined first, after which the central inquiries of this paper will be contextualised and qualified. In Part 2, the internalist position on psychedelic phenomenology — both its strong and weak versions — will be discussed. In Part 3, the externalist position will be explored, which will be followed by a final evalutation between the three positions presented in this paper. The paper will conclude in favour of the weak internalist position, and that whilst the externalist position remains significantly more speculative than both the strong and the weak internalist positions, the externalist position cannot be rejected altogether until further progress in consciousness research is made.

Matrix norm based hybrid Shapley and iterative methods for the solution of stochastic matrix games

In this paper, we present four alternative solution methods to Shapley iteration for the solution of stochastic matrix games. We first combine the extended matrix norm method for stochastic matrix games with Shapley iteration and then state and prove the weak and strong hybrid versions of Shapley iterations. Then, we present the semi-extended matrix norm and iterative semi-extended matrix norm methods, which are analytic-solution-free methods, for finding the approximate solution of stochastic matrix games without determining the strategy sets. We illustrate comparisons between the Shapley iteration, weak and strong hybrid Shapley iterations, semi-extended matrix norm method, and iterative semi-extended matrix norm method with several examples. The results reveal that the strong and weak hybrid Shapley iterations improve the Shapley iteration and decrease the number of iterations, and the strong hybrid Shapley iteration outperforms all the other proposed methods. Finally, we compare these methods and present their performance analyses for large-scale stochastic matrix games as well.

A long-term, regional-level analysis of Zipf's and Gibrat's laws in the United States

We test the validity of Zipf's and Gibrat's laws for city size distributions at the regional level from 1900 to 2010 by considering US states. Zipf's law is satisfied for a majority of states, but for the United States as a whole it only held during the first half of the twentieth century. The null hypothesis of a power law is not rejected at the national level or for most states (the maximum number of rejections in one year is 13 states out of 48). There is evidence supporting a weak version of Gibrat's law in the long-term; mean growth is independent of initial population for most city sizes over the entire United States and in 27 states, while the variance of growth is size-dependent.

The Metaphysical Foundations of the Principle of Indifference

Abstract The arguments in favor of the Principle of Indifference fail to explain its fruitfulness in science. Using the recent metaphysical concept of Grounding, I devise an explanation that can justify a weak version of the principle and discuss an instance of its application in Quantum mechanics.

Modeling the interplay of oscillatory synchronization and aggregation via cell–cell adhesion

We present a model of systems of cells with intracellular oscillators (‘clocks’). This is motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity. Cells undergo random motion and adhere to each other. The adhesion strength between neighbors depends on their clock phases in addition to a constant baseline strength. The oscillators are linked via Kuramoto-type local interactions. The model is an advection-diffusion partial differential equation with nonlocal advection terms. We demonstrate that synchronized states correspond to Dirac-delta measure solutions of a weak version of the equation. To analyze the complex interplay of aggregation and synchronization, we then perform a linear stability analysis of the incoherent, spatially uniform state. This lets us classify possibly emerging patterns depending on model parameters. Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: There is aggregation into separate synchronized clusters with or without global synchrony; global synchronization without aggregation; or unexpectedly a ‘phase wave’ pattern characterized by spatial gradients of clock phases. A 2D Lattice-Gas Cellular Automaton model confirms and illustrates these results.

The achievement gap thesis reconsidered: artificial intelligence, automation, and meaningful work

AbstractJohn Danaher and Sven Nyholm have argued that automation, especially of the sort powered by artificial intelligence, poses a threat to meaningful work by diminishing the chances for meaning-conferring workplace achievement, what they call “achievement gaps”. In this paper, I argue that Danaher and Nyholm’s achievement gap thesis suffers from an ambiguity. The weak version of the thesis holds that automation may result in the appearance of achievement gaps, whereas the strong version holds that automation may result on balance loss in possibilities for workplace achievements, i.e., in the appearance of an overall gappier work landscape. Against the strong version of the achievement gap thesis, I characterize situations where automation may result in boons to meaning-conferring workplace achievements: the appearance of what I call achievement spread and achievement swaps. Nevertheless, Danaher and Nyholm are right to worry about some uses of automation whereby human workers become subservient to AI. But these situations are better framed, I argue, as autonomy gaps rather than achievement gaps.

Does environmental regulation pressure induce the green innovation of enterprises? Quasi-natural experiment of China's air pollution prevention and control action plan

ABSTRACT Air Pollution Prevention and Control Action Plan (APPCAP) is one of the most influential command-and-control environmental regulations (CMC) in China. Whether it can promote the green innovation performance of enterprises remains unclear. Based on the ‘Green Patent List’ issued by the World Intellectual Property Organization (WIPO) and the Chinese listed companies’ data, this paper applies the quasi-natural experiment methods of the difference-in-differences model and difference-in-difference-in-differences model to identify the impact of environmental regulation pressure on the green innovation of enterprises. The study finds that implementing the APPCAP promotes the enterprises’ green innovation performance in quantity and quality. Heterogeneity analysis shows that the green innovation effect of APPCAP is more obvious for those samples in the eastern region, with features of capital-intensive, labour-intensive, and low market concentration. Furthermore, our findings further support the weak version of the Porter hypothesis (PH), Whose path is that the implementation of APPCAP induces enterprises’ green innovation by promoting their innovation investment. These findings provide policy implications for the coordination of environmental regulation and green transformation in pursuit of the goal of carbon peaking and carbon neutrality goals.

The impact of environmental regulation on innovation and international competitiveness

The purpose of this paper is to analyze the impact of environmental regulation on innovation and international competitiveness. We test the weak, narrow, and strong versions of Porter's hypotheses by looking at the impact of environmental regulation on exports both directly and indirectly through innovation and by introducing the role of pollution intensity in moderating the impact of stringent regulation on innovation and international competitiveness. Green policies are measured with the OECD Environmental Stringency Policy Index, distinguishing between market, non-market instruments, and technology support policies. Differently from previous papers, we adopt the technology gap approach to trade, which is suitable for relating environmental regulation to trade competitiveness and we apply the simultaneous-equation system econometric model with a moderating factor represented by pollution intensity. The results support the weak and strong versions of Porter’s hypotheses and find that the positive impact of regulation on innovation and exports increases with a country’s pollution intensity, suggesting that green policies, if properly coordinated, can represent a win–win strategy, fostering, at the same time, sustainability and international competitiveness.

Expected utility, independence, and continuity

In this paper, we provide two novel expected utility theorems by suitably adjusting the independence and continuity axioms. Our first theorem characterizes expected utility preferences using weak versions of the independence axiom (with varying mixture weights) and a new weak continuity axiom. Our second theorem characterizes these preferences using weaker versions of the independence axiom (with mixture weights fixed at 1/2) and a strong topological continuity axiom. We provide useful examples to illustrate the tightness of these characterizations.

Birkhoff-James orthogonality and the simultaneous zero inclusion property

A normed space X has the simultaneous zero inclusion (S0I) property if, for every invertible bounded linear operator T on X, 0 lies in the spatial numerical range of T if and only if 0 lies in the spatial numerical range of T−1. It is shown that the only absolute symmetric norms on C2 with the S0I property are scalar multiples of the Euclidean norm. However, the S0I property does not characterize inner products in a slightly wider class of norms—for example, there is a family of absolute polygonal norms on C2 that have the S0I property. There are two keys: (1) The S0I property is a weak version of symmetric Birkhoff-James orthogonality. (2) Complex spaces of dimension two are constructed, based on Radon curves, in which Birkhoff-James orthogonality is symmetric.