Product Space Research Articles

Further research on separated degrees of M-fuzzifying convex spaces

In this paper, we redefine the concepts of join spaces and product spaces of M-fuzzifying convex spaces. Then we further investigate the S_i (i=0,1,2) separated degrees of an M-fuzzifying convex space in a logical viewpoint. Finally, we study the S_i (i=0,1,2) separated degrees of an M-fuzzifying convex space from the aspect of convergence structures.

Application of Polarized Projective Mapping combined with Ultra Flash Profiling to a complex − high fatigue product category: The case of Greek grape marc spirits

Grape marc spirits in general and Greek grape marc spirits specifically, is an alcoholic drinks category with very limited information on its sensory properties, despite its commercial importance. Polarized Projective Mapping (PPM), a method first introduced by Ares et al. in 2013, is promising but has not been applied in combination with Ultra Flash Profiling (UFP) to sets of many and complex-high fatigue products such as spirits. Our objective was to employ UFP combined to PPM in order to profile a set of 39 grape marc spirits from Greece and relate the results to several production information of the samples such as raw materials, origin region and distillation style, and to evaluate the combination of those methods in such a category. Projective Mapping was used for pole selection and five sessions of PPM followed for the evaluation of all samples. Unlike application in previous studies we had assessors carry out the UFP task first before each PPM session. ‘Global PPM’ and ‘Global UFP’ product spaces gave 39.7% and 43.1% explained variances in the first two axes respectively, when data from all sessions and assessors were analyzed in STATIS. The biggest impact for classification of the products was from the grape varieties used as raw material, even though distillation style had a serious impact as well. Clear product clusters were obtained and combination of those methods worked well for that category. Nevertheless, there is still a need for a global method of defining the “poles” in large sample sets.

Relational generalized iterated function systems

In this paper, we introduce a wider class of generalized iterated function systems, called relational generalized iterated function systems. More precisely, the classical contraction condition for functions defined on product spaces is weakened by means of an equivalence relation. In particular, if we consider the total equivalence relation, we recover the classical generalized iterated function systems. Our main result states that each compact subset of the underlying metric space generates, via a sequence of iterates, a fixed point of the associated fractal operator, called an attractor of the system. We also establish a structure result for the attractors and a theorem concerning the continuous dependence of the attractor on the associated compact set. Ultimately, we provide some examples which illustrate our main results.

The Path Innovation of Regional Economic Growth: Data Analysis Based on China

AbstractInnovative development to some extent overcomes the path-dependent characteristics of knowledge proximity, and the diversity and complexity of regional knowledge and technology foster regional economic growth. This study constructs a dynamic mechanism for maximizing the enterprise profit to explore the conditions enabling regions that can achieve knowledge accumulation and technological leapfrogging. Utilizing the fitness landscape methodology based on customs data from 30 provinces in China and employing genetic algorithms, we analyze two modes of enterprises with similar and different initial conditions under varying leapfrogging scenarios. By applying profit maximization rules within the fitness landscape of the product space, we examine how to promote regional industrial transformation and upgrade. The results indicate that a larger leap range in the production of products by enterprises, reflecting stronger innovative capabilities, significantly enhances the region’s ability for industrial transformation and upgrade. There is a strong positive relationship between the initial industrial foundation and regional economic development. When regions share a similar industrial base, the complexity of regional knowledge becomes more critical. Therefore, an increase in the foundational development of industries and technology-intensive products can enhance the potential for regional path innovation. Past research has primarily been based on the path dependence theory, emphasizing that the most crucial aspect for underdeveloped economies is the accumulation and development of foundational capabilities. However, our study contends that, for emerging economies, it is crucial to enhance the diversity and complexity of knowledge and technology. Particularly, it is essential to, based on a comprehensive understanding of their own strengths, open up new areas and new arenas in development, continually shaping new growth drivers and new strengths. This emerges as a significant factor in regional growth and leapfrog development.

Complex choices: Pole selection for polarized projective mapping applied to a complex product set

AbstractPolarized projective mapping (PPM) employs reference samples called poles, positioned in the sensory space, to which samples are compared. This structure enables comparisons across sessions. Selecting the right poles is therefore critical for accurate and reliable comparisons. The current study examined whether simple poles (single fruit juices) or complex poles (fruit juice blends) better represented the sensory space in PPM using a trained panel (n = 12). Thirteen commercial fruit juices, including single juices and blends, were assessed using projective mapping to understand product diversity and select three simple and three complex poles. PPM was then conducted using simple poles in one session and complex poles in another. Six of the thirteen juices used in PPM and three new juices were evaluated to validate the results. Findings indicated that while both simple and complex poles described the product space, simple poles were more able to capture the diversity of the sensory space.Practical ApplicationsThe study's insights into appropriate pole selection in PPM for complex product sets contribute to the technique's usability and robustness. The guidance on pole selection ensures selected poles anchor the product space, enabling meaningful comparisons across sessions. The finding that simple poles are able to describe complex product sets may assist in minimizing fatigue, a common challenge in evaluating complex products like wine, spirits, and cider. By addressing the unique challenges of complex products, this research enhances the useability of PPM as a valuable tool for comprehensively evaluating sensory characteristics in product categories prone to fatigue and limited consumption.

Interpolation of generalized Ky Fan inequality for normal map on Eaton triple: Using gradients of G-increasing functions

In this paper, by using gradients of Gateaux differentiable G-increasing functions, we establish some interpolation results for the following G-majorization triangle inequality: (x+y)↓≺Gx↓+y↓ for vectors x,y of an Eaton triple with normal map (⋅)↓. In particular, we show refinements for the majorization Ky Fan's eigenvalue inequality λ(x+y)≺mλ(x)+λ(y) for x,y∈Hn, where Hn is the space of n×n Hermitian matrices and λ(⋅) is the eigenvalue operator on Hn. We derive analogous results for an inner product space with its standard norm, and for real skew-symmetric matrices with their singular values.

Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data

In the vast statistical literature, there are numerous probability distribution models that can model data from real-world phenomena. New probability models, nevertheless, are still required in order to represent data with various spread behaviors. It is a known fact that there is a great need for new models with limited support. In this study, a flexible probability model called the unit Maxwell-Boltzmann distribution, which can model data values in the unit interval, is derived by selecting the Maxwell-Boltzmann distribution as a base-line model. The important characteristics of the derived distribution in terms of statistics and mathematics are investigated in detail in this study. Furthermore, the inference problem for the mentioned distribution is addressed from the perspectives of maximum likelihood, method of moments, least squares, and maximum product space, and different estimators are obtained for the unknown parameter of the distribution. The derived distribution outperforms competitive models according to different fit tests and information criteria in the applications performed on four actual air pollutant concentration data sets, indicating that it is an effective model for modeling air pollutant concentration data.

Product Space Clustering with Graph Learning for Diversifying Industrial Production

During economic crises, diversifying industrial production emerges as a critical strategy to address societal challenges. The Product Space, a graph representing industrial knowledge proximity, acts as a valuable tool for recommending diversified product offerings. These recommendations rely on the edges of the graph to identify suitable products. They can be improved by grouping similar products together, which results in more precise suggestions. Unlike the topology, the textual data in nodes of the Product Space graph are typically unutilized in graph clustering methods. In this context, we propose a novel approach for economic graph learning that incorporates learning node data alongside network topology. By applying this method to the Product Space dataset, we demonstrate how recommendations have been improved by presenting real-life applications. Our research employing a graph neural network demonstrates superior performance compared to methods like Louvain and I-Louvain. Our contribution introduces a node data-based deep graph clustering graph neural network that significantly advances the macroeconomic literature and addresses the imperative of diversifying industrial production. We discuss both the advantages and limitations of deep graph learning models in economics, laying the groundwork for future research.

On the intermixed method for mixed variational inequality problems: another look and some corrections

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.

HGE: Embedding Temporal Knowledge Graphs in a Product Space of Heterogeneous Geometric Subspaces

Temporal knowledge graphs represent temporal facts (s,p,o,?) relating a subject s and an object o via a relation label p at time ?, where ? could be a time point or time interval. Temporal knowledge graphs may exhibit static temporal patterns at distinct points in time and dynamic temporal patterns between different timestamps. In order to learn a rich set of static and dynamic temporal patterns and apply them for inference, several embedding approaches have been suggested in the literature. However, as most of them resort to single underlying embedding spaces, their capability to model all kinds of temporal patterns was severely limited by having to adhere to the geometric property of their one embedding space. We lift this limitation by an embedding approach that maps temporal facts into a product space of several heterogeneous geometric subspaces with distinct geometric properties, i.e.\ Complex, Dual, and Split-complex spaces. In addition, we propose a temporal-geometric attention mechanism to integrate information from different geometric subspaces conveniently according to the captured relational and temporal information. Experimental results on standard temporal benchmark datasets favorably evaluate our approach against state-of-the-art models.

Targeted sampling of natural product space to identify bioactive natural product-like polyketide macrolides

Polyketide or polyketide-like macrolides (pMLs) continue to serve as a source of inspiration for drug discovery. However, their inherent structural and stereochemical complexity challenges efforts to explore related regions of chemical space more broadly. Here, we report a strategy termed the Targeted Sampling of Natural Product space (TSNaP) that is designed to identify and assess regions of chemical space bounded by this important class of molecules. Using TSNaP, a family of tetrahydrofuran-containing pMLs are computationally assembled from pML inspired building blocks to provide a large collection of natural product-like virtual pMLs. By scoring functional group and volumetric overlap against their natural counterparts, a collection of compounds are prioritized for targeted synthesis. Using a modular and stereoselective synthetic approach, a library of polyketide-like macrolides are prepared to sample these unpopulated regions of pML chemical space. Validation of this TSNaP approach by screening this library against a panel of whole-cell biological assays, reveals hit rates exceeding those typically encountered in small molecule libraries. This study suggests that the TSNaP approach may be more broadly useful for the design of improved chemical libraries for drug discovery.

Identifying Key Products to Trigger New Exports: An Explainable Machine Learning Approach

Abstract Tree-based machine learning algorithms provide the most precise assessment of the feasibility for a country to export a target product given its export basket. However, the high number of parameters involved prevents a straightforward interpretation of the results and, in turn, the explainability of policy indications. In this paper, we propose a procedure to statistically validate the importance of the products used in the feasibility assessment. In this way, we are able to identify which products, called explainers, significantly increase the probability to export a target product in the near future. The explainers naturally identify a low dimensional representation, the Feature Importance Product Space, that enhances the interpretability of the recommendations and provides out-of-sample forecasts of the export baskets of countries. Interestingly, we detect a positive correlation between the complexity of a product and the complexity of its explainers.

Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter

This article concerns the asymptotic behavior of solutions for a class of nonclassical diffusion equation with time-dependent perturbation coefficient and degenerate memory. We prove the existence and uniqueness of time-dependent global attractors in the family of time-dependent product spaces, by applying the operator decomposition technique and the contractive function method. Then we study the asymptotic structure of time-dependent global attractors as \(t\to \infty\). It is worth noting that the memory kernel function satisfies general assumption, and the nonlinearity \(f\) satisfies a polynomial growth of arbitrary order. For more information see https://ejde.math.txstate.edu/Volumes/2024/22/abstr.html

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.

On an analogue of a property of singular M-matrices for the Lyapunov and Stein operators

A well-known result for a singular irreducible M-matrix A is that the only nonnegative vector that belongs to the range space of A is the zero vector. In this paper, we prove an analogue of this result for the Lyapunov and Stein transformations, which act on the inner product space of real symmetric matrices.

BANKSY unifies cell typing and tissue domain segmentation for scalable spatial omics data analysis.

Spatial omics data are clustered to define both cell types and tissue domains. We present Building Aggregates with a Neighborhood Kernel and Spatial Yardstick (BANKSY), an algorithm that unifies these two spatial clustering problems by embedding cells in a product space of their own and the local neighborhood transcriptome, representing cell state and microenvironment, respectively. BANKSY's spatial feature augmentation strategy improved performance on both tasks when tested on diverse RNA (imaging, sequencing) and protein (imaging) datasets. BANKSY revealed unexpected niche-dependent cell states in the mouse brain and outperformed competing methods on domain segmentation and cell typing benchmarks. BANKSY can also be used for quality control of spatial transcriptomics data and for spatially aware batch effect correction. Importantly, it is substantially faster and more scalable than existing methods, enabling the processing of millions of cell datasets. In summary, BANKSY provides an accurate, biologically motivated, scalable and versatile framework for analyzing spatially resolved omics data.

General single/multiple integral inequalities and their applications to stability of time-delay systems

This paper is concerned with the stability of time delay systems. A new type of multiple integral inner product and a new class of multiple integral inner product space are firstly introduced. Based on them, general single/multiple integral inequalities are developed, and some previous integral inequalities including single and multiple integral inequalities can be regarded as their special cases. Secondly, suitable Lyapunov–Krasovskii functionals (LKF) are constructed. On the basis of the proposed integral inequalities, the derivative of the LKF can be gauged more accurate than those in the literature. As a result, some improved stability criteria of time-delay systems are formulated in terms of linear matrix inequalities. Through two numerical examples, it is shown that these criteria can provide larger delay upper bounds than some existing ones, showing the effectiveness and superiority of the proposed method.

Shape-Aware Synthon Search (SASS) for Virtual Screening of Synthon-Based Chemical Spaces.

Virtual screening of large-scale chemical libraries has become increasingly useful for identifying high-quality candidates for drug discovery. While it is possible to exhaustively screen chemical spaces that number on the order of billions, indirect combinatorial approaches are needed to efficiently navigate larger, synthon-based virtual spaces. We describe Shape-Aware Synthon Search (SASS), a synthon-based virtual screening method that carries out shape similarity searches in the synthon space instead of the enumerated product space. SASS can replicate results from exhaustive searches in ultralarge, combinatorial spaces with high recall on a variety of query molecules while only scoring a small subspace of possible enumerated products, thereby significantly accelerating large-scale, shape-based virtual screening.

Representative Functions, Variational Convergence and Almost Convexity

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Integration or fragmentation: the arrow of China's lithium product development.

Lithium is an indispensable resource for the next generation of clean technology. Promoting the development of lithium industry has become a global consensus, with China being no exception. The development process involves not only the growth and degeneration of lithium products but also the path-dependency issues arising from resources and technology. This study, based on the perspective of product space structure, constructs China's lithium product network to study its pattern evolution and predict the development direction. Then, according to the current situation and pattern evolution trends, potential advantageous lithium products and advantage-degraded lithium products are identified, and tactics for expanding chains and breaking chains are formulated for them, presenting strategies for the integrated and fragmentated development of China's lithium products. This aims to steer China's lithium products towards a more orderly and closely interconnected direction, representing the arrow of development for China's lithium products.