Approximate Bases and Interpolation in Non-Separable Banach Spaces: A Unified Analytical Framework

This article addresses the extension of classical spectral theory to the setting of non-separable Banach spaces (NSBS), a field that remains underdeveloped despite its increasing relevance in functional analysis and mathematical physics. While spectral theory in separable Banach spaces has been thoroughly investigated—particularly through the work of Fredholm, Riesz, and others—the transition to nonseparable spaces presents both technical and conceptual challenges. These arise mainly from the absence of countable dense subsets and orthonormal or Schauder bases, which are instrumental in formulating and proving classical theorems. The main objective of this work is to provide a rigorous and systematic framework for spectral theory in NSBS, focusing on the classification of spectral components, the generalisation of key theorems, and the topological and algebraic behaviour of operators acting on such spaces. We begin by recalling the foundations of spectral theory in the classical (separable) case and proceed to identify the structural modifications required in the non-separable context. A central part of the analysis is devoted to the residual and continuous spectra, whose properties exhibit significant deviations from their counterparts in separable settings. We propose new definitions, prove generalised versions of the spectral theorem, and characterise the structure of the spectrum under various topological assumptions. Particular attention is given to the stability of the spectrum under perturbations—both compact and non-compact—and to the conditions under which the classical spectral decomposition remains valid. Several illustrative examples are provided to highlight the influence of non-separability on spectral behaviour. The results presented here contribute to a deeper understanding of spectral phenomena in infinitedimensional analysis, particularly in cases where separability cannot be assumed. They offer theoretical tools for future work in operator theory, interpolation, partial differential equations, inverse problems, and the theory of operator algebras. This article thereby lays the groundwork for further mathematical developments in non-separable environments and suggests new directions for interdisciplinary applications.

We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikod\'{y}m compacta.

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

Antinormally ordering of phase operators and the algebra of weak limits

Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb–Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.

Massive structural interventions and rising density pressures have made urban public spaces increasingly perceived as stressful. From a human perspective, they frequently lack attractiveness. In response to this loss of significance, the method presented here introduces a diagnostic approach to identifying the causes of stress in urban public spaces, which have long remained a Black box in urban research. It provides planners with a tool to assess how dense urban areas are perceived and experienced by people. The diagnostic part of the approach involves identifying the causes of individual stress reactions through a systematic analysis of various measurable and observable characteristics of urban public space. As a key innovation, it integrates quantifiable hard factors with qualifiable soft factors within a unified analytical framework. The method organises the findings in a Stressor Matrix, facilitating a systematic evaluation of urban environments. The method’s core principles and components are presented based on a case study in the German city of Würzburg, one of five analysed case studies of this paper. Finally, it presents initial insights into six neuralgic stress-inducing factors, which are shopfront zones, public space elements, street space organisation, traffic, parking and acoustics.

Throughout this paper, a closed form expression of the multiple access interference (MAI) limited bit error rate (BER) is provided for the multiwavelength optical code-division multiple-access system when the system is working above the nominal transmission rate limit imposed by the passive encoding-decoding operation. This system is known in literature as the optical overlapped code division multiple access (OV-CDMA) system. A unified analytical framework is presented emphasizing the impact of optical hard limiter (OHL) on the BER performance of such a system. Results show that the performance of the OV-CDMA system may be highly improved when using OHL preprocessing at the receiver side.

<sec> <title>BACKGROUND</title> Virtual hospitals are defined as the delivery of hospital-level health care via digital services such as videoconferencing technologies, digital platforms, and remote monitoring. The purpose is to leverage virtual technologies to deliver accessible patient-centered care, improve the overall health of communities, and implement improvements driven by real-time access to patient data. Patients and their caregivers have increasingly favored these alternative and complementary service delivery models alongside traditional in-person care. However, this is still a complex issue, and the current literature points to a variety of challenges that need to be overcome to provide optimal models of care. </sec> <sec> <title>OBJECTIVE</title> This study aims to identify unique challenges to virtual hospitals providing multidisciplinary care for patients at home, and co-design recommendations to improve the patient experience. </sec> <sec> <title>METHODS</title> This research is a mixed methods exploratory case study of a virtual hospital in Sydney, New South Wales, Australia. The methods include (1) document analysis: this will be used to identify the current formal processes that govern the virtual hospital; (2) secondary analysis of data: a detailed investigation of existing data collected by the virtual hospital, for example, patient-reported experience measures, patient-reported outcome measures, and other available data; (3) observations of current practices; (4) semistructured interviews; (5) co-designed focus groups; (6) economic analysis; (7) comparative case study; and finally, (8) a triangulation analysis. To synthesize these findings within a unified analytical framework, all data will be subjected to open or axial coding following the Strauss framework, allowing the convergence of themes across codes and methods. </sec> <sec> <title>RESULTS</title> As of April 2025, observations and initial documentation analysis have begun. Multidisciplinary team meetings and clinician shadowing have commenced, along with analysis of Standards of Practice documents. </sec> <sec> <title>CONCLUSIONS</title> This protocol outlines a mixed methods case study on a new virtual hospital located in Sydney, New South Wales, Australia. We anticipate our results to provide a comprehensive understanding of patient experience through a range of quantitative and qualitative research activities. We will achieve this by identifying challenges for patients, carers, and health care workers, and documenting informal solutions discovered through our research activities. </sec> <sec> <title>INTERNATIONAL REGISTERED REPORT</title> DERR1-10.2196/72729 </sec>

Behavior plays a critical role in health and disease. Although molecular omics have advanced the understanding of biological mechanisms, they alone cannot explain the role of behavior in health, and traditional behavioral measurement technologies have struggled to capture the dynamic, multidimensional, and heterogeneous nature of real-world behaviors. With the development of digital technologies such as wearable devices, smartphone sensors, and ecological momentary assessment, behavioromics has emerged as a new research paradigm with a holistic systems perspective, integrating continuous, multimodal behavioral data with molecular, physiological, and environmental information. By organizing behavioral measurement, modeling, and interpretation within a unified analytical framework, behavioromics reveals dynamic behavioral patterns, supports mechanism-informed inference, and identifies actionable targets for early intervention. Despite ongoing challenges in data heterogeneity, causal interpretation, and ethical governance, behavioromics holds promise for reshaping disease prediction, early detection, and precision intervention, and for advancing proactive health through earlier, behavior-centered prevention.

This review critically examines the ballistic performance and impact resistance of composite fabrics, with a particular focus on their role in next-generation protective systems. While previous work has focused on general ballistic materials, relatively few studies have provided a systematic and integrated synthesis that links material selection, structural design parameters, and recent nanotechnology improvements. This work addresses this gap by offering a comprehensive classification of composite fabric configurations, considering key factors such as fiber types, fabric architecture, thickness, stacking sequence, and layer orientation. The influence of these parameters on deformation behavior, failure mechanisms, and energy absorption efficiency is discussed in detail. It also evaluates standardized testing methods, current reference benchmarks, and limitations of current testing protocols. Recent advances, including high-strength fibers, hybrid architectures, and multifunctional material interactions, are analyzed to highlight emerging trends and unresolved challenges. By organizingrecent progress within a unified analytical framework, this review offers researchers, engineers, and practitioners actionable insights for the design and optimization of advanced ballistic impact materials. Furthermore, it outlines key research gaps and future directions, particularly in the areas of multifunctional integration, lightweight design, and improved predictive modeling for high-strain-rate impact conditions.

This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework.

HRMARS - Purpose: Industry 4.0 technologies, originating in Germany in 2011, have expanded rapidly across global manufacturing. However, high investment costs, rapid technological evolution, and a shortage of skilled professionals have constrained their adoption in many traditional sectors. The garment manufacturing industry, dominated by small and medium-sized enterprises and characterized by low automation, outdated equipment, and manual operations (Huynh, 2024), faces particular challenges. This study examines the current research status of Industry 4.0 technologies adoption in the traditional garment manufacturing sector and identifies future development directions. Design/methodology/approach: A systematic literature review combined with bibliometric analysis was employed. Relevant studies were retrieved from the Web of Science (WOS) database using defined keywords, and visualization was performed with VOSviewer 1.6.19 to map research hotspots and trends. Findings: The study provides a comprehensive overview of Industry 4.0 technologies adoption in garment manufacturing, highlighting current research focuses and emerging directions for future exploration. Research limitations/implications: Only five core Industry 4.0 technologies?Internet of Things, Cloud Computing, Big Data Analytics, Artificial Intelligence, and Smart Manufacturing?were analyzed. Future research could broaden this scope to capture a more holistic technological landscape. Practical implications: Industry 4.0 technologies adoption can transform traditional garment production, enhance competitiveness, and support sustainable development. Originality/value: By integrating five foundational Industry 4.0 technologies into a unified analytical framework, this study establishes a theoretical foundation for future empirical research on digital transformation within traditional garment manufacturing.

The Mundell-Fleming model of international macroeconomics originated in the early 1960s and has been extended during the ensuing quarter century. This paper develops an exposition that integrates the various facets of the model and incorporates its extensions into a unified analytical framework. Attention is given to (1) the distinction between short-run and long-run effects of policies, (2) the implications of debt and tax financing of government expenditures, and (3) the role of the exchange rate regime in this regard. By identifying the key mechanisms operating in the model, the exposition clarifies the model's limitations and facilitates comparison with other, more current approaches.

In this letter, we present a reconfigurable intelligent surface (RIS)-assisted space shift keying (SSK) modulation and reflection phase modulation (RPM) scheme, where RIS embeds its own information in the reflection phase shift of the reflected radio frequency (RF) signal. More specifically, the RIS simultaneously performs two tasks: i) reflecting the impinging RF signal with a discrete phase shift and ii) embedding the information bits in the phase shift. We perform joint detection for the RPM and SSK symbols using a maximum likelihood detector, and a unified analytical framework is presented for theoretical analysis of the average bit error rate (ABER) and ergodic capacity of the proposed scheme. All numerical results are thoroughly verified through Monte Carlo simulation. Numerical results reveal that the proposed scheme outperforms the conventional SSK scheme.