Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution of P-Type $\ell_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.

This paper focuses on studying the mapping properties of singular integral operators on product spaces. The boundedness of such operators is established on Triebel-Lizorkin spaces whenever their kernel functions belong to Grafakos and Stefanov class. Our results generalize and improve some previously known results.

The Ali-Cesaro Stolz theorem, introduced in this paper, extends the classical Cesaro-Stolz theorem by addressing cases where the latter does not yield results or provides a "Does Not Existe (DNE) form. This novel theorem leverages the properties of limits and Z-transforms, offering a robust framework for analyzing sequences and series. The theorem is particularly effective in scenarios where traditional approaches fail, providing new insights and tools for mathematical analysis. This paper presents a detailed formulation of the Ali-Cesaro Stolz theorem, including proofs, applications, and a discussion on its implications and advantages over existing methods.

We introduce the notion of functional equiangular lines in finite rank modules over subrings of $\mathbb{R}$. We show that van Lint-Seidel relative and Gerzon universal bounds hold for this most general lines.

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann-Liouville fractional integral. Then the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied for derivation of its properties. In particular, we characterize its domain, null-space, and projector operator, establish the interrelations between its different realisations, and present a generalized fractional Taylor formula involving the generic fractional derivative. Then we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed form formula for its unique solution, and study its complete monotonicity.

Planar differential systems whose angular velocity is constant are called rigid differential systems. We characterize the existence and non-existence of limit cycles, and the finite and infinite singularities of the rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree in the Poincaré disc. Moreover, we classify the phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearitie that can exhibit one limit cycle.

We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if {τj} n j=1 is p-adic γ-equiangular lines in Qd p , then (1) |n| 2 ≤ |d| max{|n|, γ2 }. We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel [Indag. Math., 1966] relative bound for equiangular lines in the p-adic case. Keywords: Equiangular lines, p-adic Hilbert space.

The Fourier Continuous Derivative ($D_C$) offers a unique perspective on fractional differentiation grounded in the theory of Fourier series. This approach has the potential to address problems across various disciplines, including physics, engineering, and mathematics. The primary insight underpinning this approach is that a convex function defined on $\mathbb{Z}$ retains its convexity on $\mathbb{R}$. This paper delves into the Fourier Continuous Derivative, compares it with traditional fractional derivatives, and outlines its possible real-world applications, such as modeling viscoelastic materials, solving wave equations, and financial data analysis.

Wagyu beef is gaining worldwide popularity, primarily due to the fineness of its marbling. Cur-rently, the evaluation of this marbling is performed visually by graders. This method has several issues: varying evaluation standards among graders, reduced accuracy due to long working hours and external factors causing fatigue, and fluctuations in grading standards due to the grader's mood at the time. This paper proposes the use of fractal analysis for the grading evaluation of beef marbling to achieve automatic grading without the inconsistencies caused by human factors. In the experiments, cross-sectional images of the parts used for visual judgment were taken, and fractal analysis was performed on these images to evaluate them using fractal dimensions. The results confirmed a correlation between the marbling evaluation and the fractal dimensions, demonstrating that quantitative evaluation can be achieved, moving away from qualitative visual assessments.

Background: The key objective of this research was to identify the dominant parameters and determine the patient status of pediatric patients with congenital heart disease in the ICU. The data we collected from the Congenital Heart Disease (CHD) Surgery Unit (ICU) in the National Heart Foundation Hospital and Research Institute, Dhaka, Bangladesh. Data from 300 ICU patients was collected and analyzed. We choose patients who are less than 12 years old at the time of ICU admission and whose ICU stay is longer than 24 hours. (2) Methods: According to the doctor’s advice, the dependent and independent variables are selected. The relationship between these parameters in the ICU was selected by linear modeling, which is associated with mortality risk. These parameters are used in the linear regression model and the ARIMA model for prediction. The prediction model is used for pediatric congenital heart disease patients in the ICU. (3) Results: The relation between CVP and BP (95% CI=0.2113,0.353 r=0.2841249), PEEP, and FiO2 (95% CI=0.6992,0.770 r=0.7367744) is statistically significant (p value<0.001). pH and PO2 were statistically significant, but there were correlations (95% CI=0.161,0.308 r=0.2362575). The statistically significant (p < 0.001) parameters were selected for prediction. In this study, BP, FiO2, and PO2 parameters were used in the linear regression model and the ARIMA model for prediction. We predict the next-hour patient’s status. The ARIMA model was found to have less error than the linear regression model. (4) Conclusions: This study helps identify the important parameters that are measured in the ICU to monitor patient status and predict the future value of those parameters, with the ultimate goal of providing physicians with an early signal of future patient conditions.

In this paper, we analyze the generalized Ulam-Hyers stability of affine type AQ Functional Equation in various Banach Spaces using Direct and Fixed Methods

We address weighted composition operators between Bers-type spaces on generalized Hua-Cartan-Hartogs domains and provide necessary and sufficient conditions for their boundedness and compactness. We then apply our results to study the boundedness and the compactness of weighted composition operators between Bers-type spaces on five different domains: generalized Hua domains, Cartan-Hartogs domains, generalized Cartan-Hartogs domains, generalized Cartan-Hartogs domains over different Cartan domains and generalized ellipsoidal-type domains.

In this paper, several classes of rough Marcinkiewicz integral operators along surfaces of revolution on product spaces are investigated. We prove the Lp boundedness of these operators when their kernels functions belong to a class of functions related to a class of functions introduced by Grafakos-Stefanov. The results in this paper extend and improve several known results on Marcinkiewicz integrals.

During our digital transition, Docker containers have become essential for adaptable application deployment, although posing inherent security threats. This study utilizes Trivy for static analysis and Falco for dynamic analysis to identify and address vulnerabilities in Docker environments before deployment and in real-time. The IoT sector is now dealing with security issues that are made more severe by extensive connectivity and the prevalence of WiFi. This research promotes the use of AWS cloud security technologies such as IoT Core, CloudWatch, and GuardDuty to protect IoT devices from threats, with the goal of maintaining user privacy and ensuring safety in our interconnected digital world. This comprehensive strategy aims to enhance security measures in both Docker and IoT sectors, strengthening the security framework in our interconnected world.

Toll plaza optimization is essential for improving traffic flow and easing congestion on road networks. This study aims to suggest and assess potential optimization solutions for the Peshawar toll plaza, considering variables like traffic volume, toll-collecting effectiveness, and customer facilitation. Different resources are used for data collection, like the National Highway Authority of Pakistan. Physical data is collected from the established Toll Plaza of Peshawar, along with the development of each line. The goal of this study is to suggest and assess potential optimization solutions for the Peshawar toll plaza, considering variables like traffic volume, toll-collecting effectiveness, and customer happiness. The analysis of present traffic patterns, the detection of bottlenecks, and the creation of plans to ease congestion and enhance overall traffic flow are all part of the optimization process. The approaches used in this study include data analysis, simulation, and traffic modeling. PTV VISSIM is software for transportation planning, design, and analysis. The simulations consider various conditions, such as variable traffic volumes, toll collection methods, and lane configurations. When comparing the different optimization strategies, performance indicators like trip time, latency, and queue length are utilized to determine which approach is practical. Waiting time Vs. Existing each lane, etc, is explained. In the end, comparing queue lengths and throughput across different lane scenarios shows that maximizing throughput with existing lanes yields up to 45% improvement with E-tag lanes, 40% with existing E-tag lanes, 38% with E-tag plus additional toll and cash lanes, 32% with three cash lanes, and 28% with two E-tag lanes. Failure to regularly monitor and adjust the existing lane pattern scenario based on traffic patterns can lead to queues being 20% longer. Without proactive evaluation and optimization, there is a potential 63% opportunity for queue length improvement

Khosravi, Drnov\v{s}ek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.

Objective: To employ Latent Class Analysis (LCA) to investigate dietary intake patterns among Sudanese children aged 0 to 2 years and to examine the association of these patterns with sociodemographic factors. Methods: This study leveraged the Sudan Multiple Indicator Cluster Survey (MICS) 2014 data to uncover dietary intake patterns among 7,362 children using latent class analysis (LCA). We investigated class memberships concerning demographic and socioeconomic factors. The model's adequacy was determined using several fit indices, including BIC, AIC, entropy, CAIC, and SABIC, providing a holistic evaluation of the model's accuracy in capturing dietary behaviors. Results: Three latent classes were identified: Class 1 (55%) with an average nutrition composition, Class 2 (28%) with limited nutrition composition, and Class 3 (17%) with good nutrition composition. Significant associations were found between latent class membership and sociodemographic factors, particularly mother's education level and household wealth. The three-class solution provided the best balance between model fit and class distinction. Conclusions: The LCA revealed distinct dietary intake patterns and underscored the influence of sociodemographic factors on child nutrition. The findings suggest that targeted nutritional interventions should be developed according to the specific needs of different latent classes. The study also highlights the utility of LCA as a robust statistical and machine learning tool in public health research, capable of informing tailored interventions and policies for improving child nutrition. Implications: The study emphasizes the importance of maternal education and socio-economic status in shaping dietary behaviors of children in Sudan. It implies the need for policies that address educational disparities, food security, and economic development as part of comprehensive nutritional interventions.

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established, when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model.

This paper aims to demonstrate that, beyond the small world of Riemann-Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to the different paths taken. Notions of time and scale sequences are introduced and general associated basic derivatives, namely right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to get both known and new fractional order derivatives. Several fractional derivatives are considered, mainly: Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Let $\{\tau_n\}_{n=1}^\infty$ and $\{\omega_m\}_{m=1}^\infty$ be two modular Parseval frames for a Hilbert C*-module $\mathcal{E}$. Then for every $x \in \mathcal{E}\setminus\{0\}$, we show that \begin{align}\label{UE} \|\theta_\tau x \|_0 \|\theta_\omega x \|_0 \geq \frac{1}{\sup_{n, m \in \mathbb{N}} \|\langle \tau_n, \omega_m\rangle \|^2}. \end{align} We call Inequality (\ref{UE}) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (\ref{UE}) is the noncommutative analogue of breakthrough Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, Inequality (\ref{UE}) extends Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

This study explores the properties and behavior of bi-close-to-convex functions, introducing new subclasses defined by their relationship with fractional differential operator and bi-univalent functions. Using the Faber polynomial technique, we derive upper bounds for the n^{th} coefficient of functions in these classes. We also investigate the erratic behavior of initial coefficients in bi-close-to-convex functions, as characterized by the (λ,q)-fractional differintegral operator. Furthermore, we address Fekete-Szegö problems and present notable findings from our investigation, contributing to the continued growth and refinement of geometric function theory, yielding new insights and practical uses.

Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ denote the class of normalized analytic functions $f$ in the open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z$. In the second section of this article we find the sharp upper bounds for the initial coefficients $a_{3}$, $a_{4}$ and $a_{5}$ and the sharp upper bound for module of the Hankel determinant $|H_{2,3}(f)|$ for the functions from the class $\mathcal{S}_{\cos}^{\ast}$. The first result of the next section deals with the sharp upper bounds of the logarithmic coefficients $\gamma_{3}$ and $\gamma_{4}$ and we found in addition the sharp upper bound for $\left|H_{2,2}\left(F_{f}/2\right)\right|$. For obtaining these results we used the very useful and appropriate Lemma 2.4 of N.E. Cho et al. [Filomat 34(6) (2020), 2061--2072], and the technique for finding the maximum value of a three variable function on a closed cuboid. All the maximum found values were checked by using MAPLE\texttrademark{} computare software, and we also found the extremal functions in each cases. All of our present results are the best ones and give sharp versions of those recently published in [Hacet. J. Math. Stat. 52, 596--618, 2023].

Principal Component Analysis (PCA) is a method that identifies common directions within multivariate data and presents the data in as few dimensions as possible. One of the advantages of PCA is its objectivity, as the same results can be obtained regardless of who performs the analysis. However, PCA is not a robust method and is sensitive to noise. Consequently, the directions identified by PCA may deviate slightly. If we can teach PCA to account for this deviation and correct it, the results should become more comprehensible. The methods for doing this and an issue with this are presented.

The commercialization of historic streets constitutes a pivotal aspect of urban cultural heritage, and the comprehension of their visual perception serves as a valuable asset to urban planning and cultural conservation efforts. However, current research concerning the disparities in visual perception among diverse demographics within historic streets, as well as their spatiotemporal dynamics, remains insufficient. This study aims to utilize mobile eye-tracking technology to delve deeply into the visual perception characteristics of diverse demographics within commercialized historic streets, while also analyzing their spatiotemporal dynamics. We explore the disparities in visual perception of historic streets among groups with varying degrees of architectural knowledge and professional literacy. Through meticulous observation using mobile eye-tracking technology, we elucidate the visual perception patterns of different demographic characteristics through statistical analysis and spatiotemporal modeling. The findings reveal significant discrepancies in visual attention and perception among various groups. Participants with a higher understanding of architecture tend to focus more on architectural details and structures within the streetscape compared to those with a lower understanding, who exhibit greater interest in elements such as shops and signage. Moreover, during the process of perceiving the streetscape, participants’ eye movements transition from surrounding landscapes and historical edifices upon initial entry to the district, gradually shifting towards the commercial activities and distinctive architecture within the district. This study delves deeply into the visual perception characteristics and spatiotemporal dynamics of commercialized historic streets, providing invaluable insights for urban planning and cultural preservation endeavors. By revealing the impact of different demographic characteristics on the visual perception of historic streets, it offers new insights for urban planning and cultural heritage conservation.

The notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.

Graph visualization has evolved significantly over time, offering various methods to effectively represent complex data structures. This study introduces a novel force-directed algorithm for 3D data visualization, specifically tailored for commonly used datasets in linked data visualization. Leveraging the power of graph theory and methods, the algorithm aims to address the challenge of creating visually appealing and easily interpretable visualizations, which often necessitate advanced user interaction. By utilizing 3D space and the Unity Engine, the research successfully visualizes data and develops interactive visualizations, overcoming limitations of basic force-directed implementations. The main contribution of presented research is in the force-directed algorithm with springs and controlled placement as a visualization technique that combines the use of springs and attractive forces to stabilize a graph in a 3D environment. The combination of these two techniques also helps prevent the formation of a single large cluster within the graph, enhancing the overall visualization quality.

We investigate a singularly perturbed q-difference differential Cauchy problem with polynomial coefficients in complex time and space with quadratic nonlinearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter with values in some Banach space of formal power series in space with analytic coefficients on shrinking domains in time. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in the parameter for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as q tends to 1. In particular the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all its arguments that solves a nonlinear partial differential Cauchy problem.

In recent decades, with the ever-growing scale of video data, near-duplicate videos continue to emerge. Data quality issue caused by near-duplicate videos is becoming more and more prominent, which has affected the application of normal videos. Although current studies on near-duplicate video detection can be helpful to uncover data quality issues for videos, they still lack a process of automatic merging for the video data represented by high-dimensional features, which are difficult to automatically clean the near-duplicate videos to improve data quality for video datasets. At present, there are few studies on near-duplicate video data cleaning. The existing studies have the sensitive problems of video data orderliness and clustering initial center under a condition that prior distribution is unknown, which seriously affect the accuracy of near-duplicate video data cleaning. To address the above issues, an automatic near-duplicate video data cleaning method based on a consistent feature hash ring is proposed in this paper. First, a residual network with convolutional block attention modules, a long short-term memory deep network, and an attention model are integrated to construct an RCLA deep network with the multi-headed attention mechanism to extract spatiotemporal features of video data. Then, a consistent feature hash ring is constructed, which can effectively alleviate the sensitivity of video data orderliness while providing a condition of near-duplicate video merging. To reduce the sensitivity of the initial cluster centers to results of near-duplicate video cleansing, an optimized feature distance-means clustering algorithm is constructed by utilizing a mountain peak function on a consistent feature hash ring, which can implement automatic cleaning of near-duplicate video data. Finally, experiments are conducted based on a commonly used dataset named CC_WEB_VIDEO and a coal mining video dataset. Compared with some existing works, simulation results demonstrate that the performance of the proposed method.

Suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A, using the Hausdorff outer measure, is measurable in the sense of Carathèodory; we want to find an extension of the expected value, w.r.t the Hausdorff measure, that's unique, finite and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of ℝᴬ. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 < Hʰ(Fₖ***) < +∞; and (fₖ*) is a sequence of functions where {(x,fₖ*(x))∶x∈ dom(Fₖ***)}=Fₖ***. Thus, the extended expected value of or E**[f,Fₖ***] is: ∀(ε>0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique, finite, and "natural" (defined on §2.3 & §2.4) for all f in a prevalent subset of ℝᴬ. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.

This paper presents an analysis of the number of zeros in the binary representation of natural numbers. The primary method of analysis involves the use of the concept of the fractional part of a number, which naturally emerges in the determination of binary representation. This idea is grounded in the fundamental property of the Riemann zeta function, constructed using the fractional part of a number. Understanding that the ratio between the fractional and integer parts of a number, analogous to the Riemann zeta function, reflects the profound laws of numbers becomes the key insight of this work. The findings suggest a new perspective on the interrelation between elementary properties of numbers and more complex mathematical concepts, potentially opening new directions in number theory and analysis.

Let $(\{f_n\}_{n=1}^\infty, \{\tau_n\}_{n=1}^\infty)$ and $(\{g_n\}_{n=1}^\infty, \{\omega_n\}_{n=1}^\infty)$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f) \cap\mathcal{D}(\theta_g))\setminus\{0\}$, we show that \begin{align}\label{UB} \|\theta_f x\|_0\|\theta_g x\|_0 \geq \frac{1}{\left(\displaystyle\sup_{n,m \in \mathbb{N} }|f_n(\omega_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(\tau_n)|\right)^p}, \end{align} where \begin{align*} & \theta_f: \mathcal{D}(\theta_f) \ni x \mapsto \theta_fx \coloneqq \{f_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}), \quad \theta_g: \mathcal{D}(\theta_g) \ni x \mapsto \theta_gx \coloneqq \{g_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle \textit{[arXiv:2308.00312v1]} and unbounded uncertainty principle \textit{[arXiv:2312.00366v1]} for Banach spaces.

The emergence of Internet of Things (IoT) devices is revolutionizing healthcare. This dynamic healthcare environment relies on Internet of Healthcare Things (IoHT) and Internet of Medical Things (IoMT) technologies. Economic, social, and technological opportunities from the IoHT are transforming healthcare systems. This article covers the history, present condition, and future prospects of IoHT, as well as its topologies, platforms, services, applications, market trends, and solution types. It examines security issues, requirements, and a threat taxonomy, studying smart healthcare privacy and security. It explores future research and the challenges IoHT-based solutions face. International health rules and regulations based on the IoT and their social and economic effects on long-term development are also examined. Integrating IoMT into healthcare infrastructures enhances scalability, efficiency, reliability, and accuracy. Through communication networks and data-processing algorithms, IoMT creates smart hardware and software platforms that enhance decision-making. IoMT devices' low resources raise privacy and security concerns. Security and privacy issues are organised by healthcare IoMT adoption in this study. We give an extensive attack taxonomy against the IoMT infrastructure, encompassing devices, security mechanisms, and their pros and cons. Furthermore, it describes the privacy and security requirements for any unique security solutions against IoMT's multiple attack vectors. This article explains the complicated difficulties, security concerns, and future research pathways needed to safeguard and ensure the long-term expansion of these game-changing technologies in the healthcare ecosystem.

As data driven models gain importance in driving decisions and processes, recently it becomes increasingly important to visualize the data with both speed and accuracy. A massive amount of data is presently generated in the educational sphere from various learning platforms, tools, and institutions. Visual analytics of educational big data has capability to improve student learning, develop strategies for personalized learning, and improve faculty productivity. How-ever, there are limited advancements in the education domain for data-driven decision making leveraging the recent advancements in the field of machine learning. Some of the recent tools such as Tableau, Power BI, Microsoft Azure suite, Sisense etc. leverage artificial intelligence and machine learning techniques to visualize data and generate insights out of it, however their ap-plicability in educational advances is limited. This paper focuses on leveraging machine learn-ing and visualization techniques to demonstrate their utility through a practical implementation using k-12 state assessment data compiled from the institutional websites of the State of Texas and Louisiana. Effective modeling and predictive analytics are the focus of the sample use case presented in this research. Our approach demonstrates the applicability of web technology in conjunction with machine learning to provide a cost-effective and timely solution to visualize and analyze big educational data. Additionally, ad-hoc visualization provides contextual anal-ysis within areas of concern for Education Agencies (EA).

In recent years, online review system attracts attention to assess seller-customer relations in the world of e-commerce. To address the quality concern of online review, especially incentivized ones, this study evaluates credibility and consistency based on reviews’ volume, length, and content to distinguish the impact of incentives on customer review behavior, improving review quality, and purchase decision-making. Software product reviews collected from software review websites, including Capterra, Software Advice, and GetApp undergo Exploratory Data Analysis (EDA) to reveal critical features such as cost, support, usability, and product features. The indirect impact of companies’ size, direct impact of users’ experience, and different impacts of changing situations during years on the incentive reviews’ volume are major findings of sentiment analysis. A/B testing results show a minimal to no impact of such reviews on purchasing decisions, highlighting discrepancies in credibility and consistency in volume, length, and content. Employing advanced techniques like Sentence-BERT (SBERT) and TF-IDF, the study explores semantic differences in reviews to improve recommendation systems for a more customized shopping experience. This approach seeks to establish a framework that discerns between review types and their effect on customer behavior. The findings contribute to developing more sophisticated and consistent e-commerce solutions, emphasizing the importance of authentic and reliable online reviews in influencing consumer choices.

In this paper, we study the constrained minimization problem for an energy functional which is related to the following Kirchhoff type equation \begin{equation*} -\Big(\eta+b\big(\int_{\R^{3}}|\nabla u|^{2}dx\big)^{s}\Big)\Delta u+V(x)u=\mu u +\lambda|u|^{p}u,\end{equation*} where $b$ is a positive constant, parameters $\eta\geq0, \lambda>0$, exponents $s>0$, $0

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.

Electromobility has gained significant attention in the auto- motive industry, with car manufacturers striving to improve the charging experience for their users. In this study, we compare multiple realistic charging strategies in order to quantify the benefits arising from better planning of charging sessions. We base our evaluation on electricity price data from three regions: Sweden (SE3 region), Germany (DE region), and Denmark (DK2 region). This data is obtained from the ENTSO-E Transparency Platform, covering the two full years 2021 and 2022. We enhance this dataset with commercial price forecasts from the company StormGeo and benchmark it against a naive forecasting method. The primary aim of our work is to compare how the strategies of varying levels of complexity perform across the three different countries in terms of minimizing energy costs for electric car users.

We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of $\mathcal{L}^p$ spaces for $1< p<\infty$. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.

The special affine Fourier transform (SAFT) is a useful and powerful analyzing tool in signal processing, optics and communications. In this paper, we mainly discuss the randomized nonuniform sampling and reconstruction for random signals bandlimited in the SAFT domain. First, we show that the nonuniform sampling is identical to the uniform sampling after a pre-filter in the sense of second order statistic characters. Then, we propose an approximate reconstruction based on sinc interpolation for the nonuniform sampling of random signals bandlimited in the SAFT domain. Finally, we give the mean square error estimate for the proposed approximate recovery approach.

Let $\mathcal{X}$ be a Banach space. Let $\{\tau_j\}_{j=1}^n, \{\omega_k\}_{k=1}^m\subseteq \mathcal{X}$ and $\{f_j\}_{j=1}^n$, $\{g_k\}_{k=1}^m\subseteq \mathcal{X}^*$ satisfy $ |f_j(\tau_j)|\geq 1$ for all $ 1\leq j \leq n$, $|g_k(\omega_k)|\geq 1 $ for all $1\leq k \leq m$. If $x \in \mathcal{X}\setminus \{0\}$ is such that $x=\theta_\tau\theta_f x=\theta_\omega\theta_g x$, then we show that \begin{align}\label{FKDB} \|\theta_fx\|_0\|\theta_gx\|_0\geq \frac{\bigg[1-(\|\theta_fx\|_0-1)\max\limits_{1\leq j,r \leq n,j\neq r}|f_j(\tau_r)|\bigg]^+\bigg[1-(\|\theta_g x\|_0-1)\max\limits_{1\leq k,s \leq m,k\neq s}|g_k(\omega_s)|\bigg]^+}{\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|f_j(\omega_k)|\right)\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|g_k(\tau_j)|\right)}. \end{align} We call Inequality (\ref{FKDB}) as \textbf{Functional Kuppinger-Durisi-B\"{o}lcskei Uncertainty Principle}. Inequality (\ref{FKDB}) improves the uncertainty principle obtained by Kuppinger, Durisi and B\"{o}lcskei \textit{[IEEE Trans. Inform. Theory (2012)]} (which improved the Donoho-Stark-Elad-Bruckstein uncertainty principle \textit{[SIAM J. Appl. Math. (1989), IEEE Trans. Inform. Theory (2002)]}). We also derive functional form of the uncertainity principle obtained by Studer, Kuppinger, Pope and B\"{o}lcskei \textit{[EEE Trans. Inform. Theory, (2012)]}.

In this paper, a positive answer to the open question of Fabino et al. is given, and the concept of I\c sik type set valued contractions is introduced and fixed point theorem for such contractions is established. Examples to support main theorems are given, and an application to integral inclusion is given.

The current research examines the many characteristics of the $\ell_{p}(\widetilde{F}(\widetilde{r},\widetilde{s}))$ $(1\leq p<\infty)$ and $\ell_{\infty}(\widetilde{F}(\widetilde{r},\widetilde{s}))$ spaces which are the generalized forms of those by Candan in 2024 using Fibonacci numbers and two non-zero real numbers in accordance with a predetermined rule, we have made an effort to go through all the characteristics and features which the author of earlier versions thought are the most valuable. This manuscript contains all the information required to describe the matrix class $(\ell_{1},\ell_{p}(\widetilde{F}(\widetilde{r},\widetilde{s})))$ $(1\leq p<\infty)$ which are going to be given in detail in the following sections of the manuscript. We are going to offer estimates for the norms of the bounded linear operators $L_{A}$ formed by those matrix transformations utilizing the Hausdorff measure of non-compactness and identify prerequisites to derive the relevant subclasses of compact matrix operators. When the findings of the current study are compared to those found in the literature, it can be said that the newly found ones are more inclusive and comprehensive.

The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to the time variable are provided: one of Gevrey nature, and another of mixed type Gevrey and q-Gevrey. This asymptotic phenomena is observed due to the modification of the norm established on the space of coefficients of the formal solution. The techniques used are based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis-Sibuya theorem.

In this work, we introduce a new concepts for $\alpha$-type $F$-Suzuki contraction and $\alpha$-type $F$-weak-Suzuki contraction in the context of $b$-metric spaces. Compared to the $\alpha$-type $F$-contraction and $F$-Suzuki contraction mappings, these contractions are essentially weaker. For these type of contraction mappings, sufficient conditions are established for the fixed point's existence and uniqueness in $b$-metric spaces. As a result, the findings encompass several generalizations. To show the usability of our obtained results, we provide a supportive example and an application to a non-linear differential equation with fractional order.

In this study we introduce the new classes $\mathcal{M}_{\alpha}(\sin)$ and $\mathcal{M}_{\alpha}(\cos)$ of $\alpha$-convex functions associated with sine and cosine functions. Also, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of Zalcman functional for the class $\mathcal{M}_{\alpha}(\cos)$ for the case $n=3$, showing that the Zalcman conjecture holds for this value. Moreover, the problem of the Fekete-Szeg\H{o} functional estimate for these classes is studied.

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo type fractional stochastic delay differential systems (FSDDSs) with Poisson jumps by utilizing delayed perturbation of Mittag-Leffler function. Moreover, by using Burkholder-Davis-Gundy's inequality, Doob's martingale inequality and Holder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

We consider a Cauchy problem for differential equations in a Hilbert space $X$. The problem is stated in a time interval $I$, which can be finite or infinite. Weuse a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we state and prove convergence criteria for both a general fixed point problem and thecorresponding Cauchy problem. These criteria provide necessary and sufficient conditions on a sequence $\{u_n\}$ which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi univalent function defined in open unit disk connected with a linear q-convolution operator, which are associated with the quasi-subordination. We find coefficients estimate |h_2 |,|h_3 | for functions in these subclasses. Several known and new consequences of these results are also pointed out.

In this paper, two types of the general fractional derivatives of distributed order as well as a corresponding fractional integral of distributed type are defined and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of the one-parametric Sonin kernels with a power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann-Liouville and the Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

In this study, we focus on an abstract form of summation equations, which can be regarded as a particular class of nonlinear difference equations. We investigate sufficient conditions for the existence of discrete almost automorphic solutions by appealing to the fixed point theory and establish a linkage between the existence of discrete almost automorphic solutions and the existence of bounded solutions for the nonlinear difference equations in the spotlight. Consequentially, we propose a discrete counterpart of the Bohr-Neugebauer theorem for the qualitative theory of difference equations.

In this study, we deal with a nonlinear system of neutral Volterra integro-differential equations (NVIDEs) with infinite delay. We study stability, uniform stability and instability of zero solution of the system of NVIDEs with infinite delay. We obtain three new results with regard to these concepts via two new Lyapunov– Krasovskii functionals (LKFs), respectively. As the main contributions of this work, we improve and generalize some earlier results in the relevant database from the linear case to more general nonlinear cases.

The Internet of Things (IoT) aims to connect all objects and facilitate data interchange, with data accuracy being paramount. While IoT has seen extensive adoption in conventional sectors, its application in the petroleum industry, especially with downhole equipment, poses challenges due to the extreme high-temperature and high-pressure conditions, making accurate measurements elusive. This Paper we aim to enhance the accuracy of downhole pressure measurements in slim-hole drilling parameter monitoring, mitigate the temperature influences on strain gauges, and surmount the intrinsic limitations of the traditional BP neural network, such as sluggish convergence speeds, high sensitivity to initial weights and biases, unstable learning rates, and a tendency to fall into local minima. This study introduces a temperature compensation model using a chaotic-initiated adaptive whale optimization algorithm (C-I-WOA) to optimize a BP neural network, which is henceforth referred to as the C-I-WOA-BP model. Initially, we harnessed chaotic mapping techniques to refine the initialization process of the whale optimization algorithm (WOA) population, thereby bolstering the heterogeneity of the algorithm. Subsequently, we integrated an adaptive weight recalibration mechanism, refined the encompassing methodology of the WOA, and bolstered its capabilities for expansive global scouting and meticulous local refinement in the solution arena. Subsequently, the WOA was used to optimize the weight coefficients of the BP neural network. Finally, we conducted a detailed comparative analysis by juxtaposing the performance metrics of the BP, C-WOA-BP, I-WOA-BP, and C-I-WOA-BP networks. The experimental results indicate that the C-I-WOA-BP model demonstrates faster convergence speed, superior global search capabilities, and enhanced local exploitation potential compared to the other models. Additionally, it provides a more stable and accurate reduction in temperature-induced errors in drilling pressure parameter measurements.

We examine a linear q-difference differential equation which is singular in complex time t at the origin. Its coefficients are polynomial in time and bounded holomorphic on horizontal strips in one complex space variable. The equation under study represents a q-analog of a singular partial differential equation, recently investigated by the author, which comprises Fuchsian operators and entails a forcing term that combines polynomial and logarithmic type functions in time. A sectorial holomorphic solution to the equation is constructed as a double complete Laplace transform in both time t and its complex logarithm log t and Fourier inverse integral in space. For a particular choice of the forcing term, this solution turns out to solve some specific nonlinear q-difference differential equation with polynomial coefficients in some positive rational power of t. Asymptotic expansions of the solution relatively to time t are investigated. A Gevrey type expansion is exhibited in a logarithmic scale. Furthermore, a formal asymptotic expansion in power scale is displayed, revealing a new fine structure involving remainders with both Gevrey and q-Gevrey type growth.

In this paper, we explore a new norm on L^p-space called r-norm, and hence producing a new normed space called r-normed space. Measure theoretic and functional analytic properties are covered such as the relation between r-norm and the classical p-norm. An application of r-normed space in probability theory and statsitics is also presented, notably, its role as a rigorous framework in describing the coefficient of determination.

By framing the problem of determining the state of Schrodinger’s cat as a problem of searching for Hidden Markov Model (HMM), this work shows that Schrodinger’s cat is dead with stationary probabilities P(Cat = Dead) = 1 and P(Cat = Alive) = 0 consistently in various observation-based modelling of the HMM.

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L_2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

As a new approach, for a nonzero normed algebra $A$, we will define some different classes of algebra fuzzy norms on $A$ generated by homomorphisms and continuous homomorphisms. Also as a source of examples and counterexamples in the field of fuzzy normed algebras, separate continuity of the elements of each class are investigated.

We consider a classial case of irrational integrals containing a square root of a quadratic polynomial. It is well known that they can be expressed in terms of elementary functions by one of three Euler’s substitutions. It is less known that the Euler substittutions have a beautiful geometric interpretation. In the framework of this interpretation one can see that the number 3 is not the most suitable. We show that it is natural to introduce the fourth Euler substitution. By the way, it is not clear who was the first to attribute these three substitutions to Euler. In his original treatise Leonhard Euler uses two substitutions which are sufficient to cover all cases.

For a nonzero normed linear space X‎, ‎we will define some different classes of fuzzy norms on X generated by linear and bounded linear functionals‎. ‎Also separate continuity of the elements of each class are investigated‎. ‎The aim of this research is to introduce a source of examples and counterexamples in the field of fuzzy normed spaces‎.

Given a fuzzy normed space $ \left( X,N \right) $‎, ‎we will introduce the notion of fuzzy near best approximation within a relative distance $ \rho \geq 0 $‎. ‎Some basic properties are characterized and also many examples for illustration are presented‎.

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be unbounded continuous 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f) \cap\mathcal{D}(\theta_g))\setminus\{0\}$, we show that \begin{align}\label{UB} \mu(\operatorname{supp}(\theta_f x))\nu(\operatorname{supp}(\theta_g x)) \geq \frac{1}{\left(\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|\right)\left(\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|\right)}. \end{align} where \begin{align*} &\theta_f:\mathcal{D}(\theta_f) \ni x \mapsto \theta_fx \in \mathcal{L}^1(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{D}(\theta_g) \ni x \mapsto \theta_gx \in \mathcal{L}^1(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} We call Inequality (\ref{UB}) as \textbf{Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Along with recent \textbf{Functional Continuous Uncertainty Principle} (derived in [arXiv:2308.00312v1 [math.FA], 1 August 2023]), Inequality (\ref{UB}) also improves Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

This study explores the potential of data science software solutions like Customer Relationship Management Software (CRM) for increasing the revenue generation of businesses. We focused on those businesses in Accommodation and Food Service sector across the European Union (EU). The investigation is contextualized within the rising trend of data-driven decision making, examining the potential correlation between data science application and business revenue. Employing a comprehensive evaluation of Eurostat datasets from 2014 to 2021, we used both univariate and multivariate analyses, we assessed e-commerce sales data across countries, focusing on the usage of big data and CRM tools. Big data utilization showed a clear, positive relationship with enhanced e-commerce sales. However, CRM tools exhibited a dualistic impact: while their use in marketing showed no significant effect on sales, their application in non-marketing functions had a negative correlation. These findings underscore the potential role of CRM and data science solutions in enhancing business performance in the EU's Accommodation and Food Service industry.

In this paper, we establish certain Lp bounds for several classes of rough Marcinkiewicz integrals over surfaces of revolution on product spaces. By using these bounds and using an extrapolation argument, we obtain the Lp boundedness of these Marcinkiewicz integrals under very weak conditions on the kernel functions. Several previous results on Marcinkiewicz operators are essentially extended or improved. Our results represent natural extensions and improvements of several known results on Marcinkiewicz integrals over symmetric spaces.

We address weighted composition operators ψCφ from α-Bloch spaces to Bers-type spaces of bounded holomorphic functions on Y, where Y is a generalized Hua domain of the first kind, and obtain some necessary and sufficient conditions for the boundedness and compactness of those operators.

Motivated by results on the location of zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström-Kakeya Theorem, and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients, and also on their moduli. The results presented naturally apply to certain classes of lacunary polynomials.

This study takes a two-echelon agricultural products supply chain (APSC) composed of an agricultural product processor (processor) and an agricultural product supermarket (supermarket) as the research object. In order to ensure the stable supply of high-quality agricultural products, this paper considers both fairness concern and corporate social responsibility (CSR), and establishes four Stackelberg game models with or without considering fairness concern and CSR, and obtains the influence mechanism of fairness concern and CSR on bilateral quality efforts in APSC through comparative analysis and numerical analysis. Research shows that the supermarket’s fairness concern can take the role of "profit distribution mechanism", but it is not conducive to improving bilateral quality efforts resulting in the inability to improve market demand, and it is not conducive to improving the utility of the supply chain system and consumer surplus; both members of APSC sharing CSR can not only improve the quality of agricultural products and promote the expansion of market demand, but also can achieve Pareto improvement of both profit of all members and supply chain system. Besides, The CSR shared by both members has a greater impact on bilateral quality efforts decision than the supermarket’s fairness concern, and thus the positive effect of CSR can overcome the negative effect of supermarket’s fairness concern.

Let $(Omega, mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in \Omega}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $\Delta \{(\alpha, \alpha):\alpha \in \Omega\}$ is measurable in the measure space $\Omega\times \Omega$, then we show that \begin{align}\label{CRBA} \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}\langle \tau_\alpha, \tau_\beta\rangle \geq \frac{-(\mu\times\mu)(\Delta)}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [Ann. of Math., 1947]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, Commun. Korean Math. Soc., 2023]. We also derive Banach space version of Inequality (1).

The divination function of China's Yijing has led to its circulation for thousands of years. In our exploration of Yijing's characteristics using big data, we have discovered variations in results between the coin toss method and the ancient yarrow-stalk method of divination, known as "the number for the great expansion method of divination(大衍之数)". The yarrow-stalk method serves as the fundamental method of divination in Yijing and continues to hold significance in studying the essential characteristics of Yijing. Despite the complexity of yarrow calculations, advancements in computer technology and big data have simplified its application. By employing the yarrow-stalk method, we simulated changes in the 64 hexagrams, calculated probabilities and proportions of hexagram alterations, and derived fundamental characteristics and patterns of hexagrams. Additionally, we constructed the spatial representation of lines and hexagrams. Through a binary system rearrangement, we created a 64x64 matrix illustrating hexagram transformations. Subsequently, we generated 100 million random hexagrams and analyzed line and hexagram changes accordingly. Our findings indicate the following:(1) Big data analysis reveals evident asymmetry in the hexagrams obtained through the yarrow-stalk method, with a triangular fractal characteristic forming the background.(2) Each of the 64 hexagrams exhibits a distinct probability distribution when transforming into other hexagrams, which can be categorized into five types.(3)The occurrence probabilities of Laoyang, Laoyin, Shaoyang, and Shaoyin are 18.61%, 6.387%, 31.38%, and 43.62% respectively. The probabilities of Yin and Yang occurrences are nearly equal, each close to 50%. However, the probability of Laoyang is approximately three times higher than that of Laoyin.(4) Visualized and analyzed the characteristics of hexagram changes greater than 100000 times using 3D statistical maps and Sankey diagram.These results demonstrate that the yarrow-stalk method effectively unveils the characteristics and underlying patterns of the 64 hexagrams. This study provides a novel approach for scientifically exploring the internal laws governing the 64 hexagrams in Yijing.

The paper examines rational Darboux-Crum transforms (RDCTs) of the Jacobi-reference (JRef) potential on the line, with an emphasis on the transformations using only normalizable seed solutions. The Cooper-Ginocchio-Khare (CGK) potential constitutes the simplest example of this rich family of exactly solvable one-dimensional (1D) potentials. It was explicitly confirmed that that an arbitrary Darboux-Crum transformation (DCT) of the given Sturm-Liouville equation (SLE) can be represented as a chain of sequential Darboux deformations of the associated Liouville potentials. If the rational SLE (RSLE) under consideration is exactly solvable and the transformation function (TF) used at each step is nodeless then each RSLE constructed in such a way was proven being exactly solvable itself. The exact solvability of RDCTs of the JRef canonical SLE (CSLE) was also confirmed for the DCTs using pairs of juxtaposed eigenfunctions (an extension of the renowned Adler theorem to the RCSLEs of our interest). By re-expressing the reference polynomial fraction (RefPF) of the transformed CSLE in terms of the Krein determinant (KD), instead of Crum Wronskian (CW), it was shown that the eigenfunctions under consideration are expressible in terms of polynomial solutions of a generalized Heun equation with positions of the most poles dependent on the exponent parameters. The outlined technique was extended to two isospectral SUSY partners of the JRef potential (as well as to its third sibling with the inserted new lowest-energy level) which, by analogy with the CGK potential, are quantized by polynomial solutions of the 2-Heun equation with all four poles at some fixed positions on the real axis.

This study investigated the factors affecting consumers’ acceptance to use cashless payment services in Malaysia through the construction of the Cashless Society Acceptance (CSA) model. The Technology Readiness Index 2.0 and Unified Theory of Acceptance and Use of Technology 2 were employed to create the CSA model. A total of 434 questionnaires were collected from Malaysian consumers, aged 18 years and above. The results show that Perceived Usefulness, Perceived Ease of Use, and Discomfort have the most significant influence on the consumers’ acceptance of cashless payment services with R-square of 71.4%.

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be continuous 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align*} \mu(\operatorname{supp}(\theta_f x)) \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}. \end{align*} where \begin{align*} &\theta_f: \mathcal{X} \ni x \mapsto \theta_fx \in \mathcal{L}^1(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{X} \ni x \mapsto \theta_gx \in \mathcal{L}^1(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} This solves a problem asked by K. M. Krishna in the paper 'Functional Continuous Uncertainty Principle' \textit{[arXiv:2308.00312v1]}.

In this paper, we classify the function space of all real-valued functions on R denoted as F(R,R) into 28 distinct blocks. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between four well-known real-valued functions in real analysis: the Cantor function, Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored.

We say that a system {zmF(z)}m=0∞ is a Beurling system if F is an outer function. Beurling’s approximation theorem asserts that if F is an outer function from H2(D) then the system {zmF(z)}m=0∞ is complete in the space H2(D). We prove that a Beurling system with F∈Hp(D),1≤p&lt;∞ is an M−bases in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p&lt;∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different methods of summation we characterize outer functions F for which the expansion converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.

Chinese divination of I Ching has a history of thousands of years. The six lines changes in 64 hexagrams have exceeded one billion scenarios, and the inherent laws among them have not been well revealed to this day. By using big data's technology and coin toss method, this paper simulates the change of 64 hexagrams, and explores the probability and proportion of each hexagram during the change from the perspective of quantification, as well as the maximum and minimum conversion rate of hexagram change. This paper summarizes the basic characteristics and the basic law of hexagram change, and accordingly constructs the spatial form of hexagram change, and reveals the hidden secrets of the ancient Book of changes (I Ching). To achieve this goal, we randomly toss three coins 600 million times to generate 100 million hexagrams. According to the basic rules of hexagram divination, we calculate the hexagram changes. The results showed that: (1) Changes of things are mostly simple changes. The probability of 1 billion randomly generated hexagrams from one to three dynamic lines is close to 80%. (2) About 17% of the 64 hexagrams have no dynamic lines, which means that a significant proportion of the 64 hexagrams are invariant. (3) Small probability changing hexagram certainty and large probability changing hexagram uncertainty. (4) After generating one billion hexagrams at random, the topographic map with the probability of changing hexagrams has axial symmetry and fractal geometric characteristics, and the fractal characteristics are mainly manifested in that the changes on both sides of the symmetry are presented based on the triangle background. These results reflect the obvious characteristics and internal regularity of the changes in the 64 hexagrams of the I Ching. This article provides new ideas for scientific exploration of the internal laws of the 64 hexagrams in the I Ching.

In a p-uniformly convex and uniformly smooth Banach space, let the pair of variational inequality and fixed point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of symmetric structure mutually. Under mild registrations, we prove weak and strong convergence of the suggested algorithms to a common solution of this pair of VIFPPs, respectively. Lastly, an illustrative example is furnished to verify the applicability and implementability of our proposed approaches.

In this paper, we design two inertial-type subgradient extragradient algorithms with line-search process for solving the pseudomonotone variational inequality problems (VIPs) and common fixed-point problem (CFPP) of finite Bregman relatively nonexpansive mapping and a Bregman relatively demicontractive mapping in p-uniformly convex and uniformly smooth Banach spaces, which are more general than Hilbert spaces. Under mild conditions, we derive weak and strong convergence of the suggested algorithms to a common solution of the VIPs and CFPP, respectively. Additionally, an illustrated example is furnished to back up the feasibility and implementability of our proposed methods.

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $\{\tau_\alpha\}_{\alpha\in \Omega}$, $\{\omega_\beta\}_{\beta \in \Delta}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align}\label{UE} \log (\mu(\Omega)\nu(\Delta))\geq S_\tau(h)+S_\omega (h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{\alpha \in \Omega, \beta \in \Delta}|\langle\tau_\alpha , \omega_\beta\rangle|}{2}\right) , \quad \forall h \in \mathcal{H}_\tau \cap \mathcal{H}_\omega, \end{align} where \begin{align*} &\mathcal{H}_\tau := \{h_1 \in \mathcal{H}: \langle h_1 , \tau_\alpha \rangle \neq 0, \alpha \in \Omega\}, \quad \mathcal{H}_\omega := \{h_2 \in \mathcal{H}: \langle h_2, \omega_\beta \rangle \neq 0, \beta \in \Delta\},\\ &S_\tau(h):= -\displaystyle\int\limits_{\Omega}\left|\left \langle \frac{h}{\|h\|}, \tau_\alpha\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, \tau_\alpha\right\rangle \right|^2\,d\mu(\alpha), \quad \forall h \in \mathcal{H}_\tau, \\ & S_\omega (h):= -\displaystyle\int\limits_{\Delta}\left|\left \langle \frac{h}{\|h\|}, \omega_\beta\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, \omega_\beta\right\rangle \right|^2\,d\nu(\beta), \quad \forall h \in \mathcal{H}_\omega. \end{align*} We call Inequality (\ref{UE}) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (\ref{UE}) improves the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}\label{CUE} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{p} \nu(\operatorname{supp}(\theta_g x))^\frac{1}{q} \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))^\frac{1}{p} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{q}\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}. \end{align} where \begin{align*} &\theta_f: \mathcal{X} \ni x \mapsto \theta_fx \in \mathcal{L}^p(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{X} \ni x \mapsto \theta_gx \in \mathcal{L}^p(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K} \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (\ref{CUE}) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle obtained by M. Krishna in \textit{[arXiv:2304.03324v1, 2023]}. It also answers a question asked by Prof. Philip B. Stark to the author.

Let $\{f_j\}_{j=1}^n$ and $\{g_k\}_{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then we show that \begin{align}\label{UE} \log (nm)\geq S_f (x)+S_g (x)\geq -p \log \left(\displaystyle\sup_{y \in \mathcal{X}_f\cap \mathcal{X}_g, \|y\|=1}\left(\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(y)g_k(y)|\right)\right), \quad \forall x \in \mathcal{X}_f\cap \mathcal{X}_g, \end{align} where \begin{align*} &\mathcal{X}_f\coloneqq \{z\in \mathcal{X}: f_j(z)\neq 0, 1\leq j \leq n\}, \quad \mathcal{X}_g\coloneqq \{w\in \mathcal{X}: g_k(w)\neq 0, 1\leq k \leq m\},\\ &S_f (x)\coloneqq -\sum_{j=1}^{n}\left|f_j\left(\frac{x}{\|x\|}\right)\right|^p\log \left|f_j\left(\frac{x}{\|x\|}\right)\right|^p, \quad S_g (x)\coloneqq -\sum_{k=1}^{m}\left|g_k\left(\frac{x}{\|x\|}\right)\right|^p\log \left|g_k\left(\frac{x}{\|x\|}\right)\right|^p, \quad \forall x \in \mathcal{X}_g. \end{align*} We call Inequality (1) as \textbf{Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (1) reduces to the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality (1).

We develop a quantitative microscopic theory of decaying Turbulence by studying the dimensional reduction of the Navier-Stokes loop equation for the velocity circulation. We have found an infinite dimensional manifold of solutions of the Navier-Stokes loop equation\cite{M93, M23PR} for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This family of solutions corresponds to a fractal curve in complex space $\mathbb C^d$, described by an algebraic equation between consecutive positions plus a nonlinear periodicity condition. We derive the constrained SDE for the evolution of the fractal curve at a fixed moment of physical time as a function of an auxiliary stochastic time. We expect this stochastic process to cover our fixed manifold of the solutions of the decaying Turbulence. The energy density of the fluid decays as $\mathcal E_0/t$, where $\mathcal E_0$ is an initial dissipation rate. Presumably, we have found a new phase of extreme Turbulence yet to be observed in real or numerical experiments.

Botnet Attacks are one of the many types of ways to attack users in their day to day browsing, often of different trends and techniques to take your information without your control. Most firewalls can amply protect you against these trends and techniques, but scarily enough, recent articles and research have shown that they are starting to bypass said firewalls through layers of attacks, confusing your best line of defense. In this Report, we will be looking into exactly what are Botnets and the different subsets, what we can do to combat Botnet Attacks, where do they come from, their origin, the analysis of their kit and what we can put together from information collected. Through this we will be able to reach a conclusion of what we can use to possibly mitigate the effect they have on the internet and stop them from enacting their purpose.

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{\omega_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. If $ x \in \mathcal{X}\setminus\{0\}$ is such that $\theta_fx$ is $\varepsilon$-supported on $M\subseteq \{1,\dots, n\}$ w.r.t. p-norm and $\theta_gx$ is $\delta$-supported on $N\subseteq \{1,\dots, n\}$ w.r.t. p-norm, then we show that \begin{align}\label{ME} &o(M)^\frac{1}{p}o(N)^\frac{1}{q}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|f_j(\omega_k) |}\max \{1-\varepsilon-\delta, 0\},\\ &o(M)^\frac{1}{q}o(N)^\frac{1}{p}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(\tau_j) |}\max \{1-\varepsilon-\delta, 0\},\label{ME2} \end{align} where \begin{align*} \theta_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad \theta_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^n \in \ell^p([n]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequalities (\ref{ME}) and (\ref{ME2}) as \textbf{Functional Donoho-Stark Approximate Support Uncertainty Principle}. Inequalities (\ref{ME}) and (\ref{ME2}) improve the finite approximate support uncertainty principle obtained by Donoho and Stark \textit{[SIAM J. Appl. Math., 1989]}.

A family of singularly perturbed q-difference-differential equations under the action of a small complex perturbation parameter is studied. The action of the formal monodromy around the origin is present in the equation, which suggests the construction of holomorphic solutions holding logarithmic terms in both, the formal and the analytic level. We provide both solutions and describe the asymptotic behavior relating them by means of q-gevrey asymptotic expansions of some positive order, with respect to the perturbation parameter. On the way, the development of a space product of Banach spaces in the Borel plane is needed to provide a fixed point for a coupled system of equations.

3D measurement is a high-throughput method that can record a large amount of information. In this study, we have developed new methods that could be used for morphological analysis of plants from the information contained in 3D data. Specifically, we investigated characteristics that can be measured by scale (dimension) and/or visual assessment by humans. The characteristics that can be measured on a scale-related dimension were tested based on the bounding box, convex hull, column solid, and voxel. Furthermore, for characteristics that can be evaluated by visual assessment, we propose a new method using normal vectors and local curvature (LC) data. For these examinations, we used our highly accurate all-around 3D plant modelling system. The correlation coefficients between manual measurements and the scale-related methods were all above 0.9. In particular, the differences in LC calculated from the normal vector data allowed us to visualize and quantify the concavity and convexity of leaves. Furthermore, we also found a difference in the time point at which leaf blistering began to develop among the cultivars. The precise 3D model made it possible to perform quantitative measurements of lettuce size and morphological characteristics. In addition, the newly proposed LC-based analysis method made it possible to quantify the characteristics that rely on visual assessment.

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{\omega_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} o(M)^\frac{1}{q}o(N)^\frac{1}{p}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(\tau_j) |}, \end{align*} where $q$ is the conjugate index of $p$. Then for all $x \in \mathcal{X}$, we show that \begin{align}\label{FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{1}{1-o(M)^\frac{1}{q}o(N)^\frac{1}{p}\displaystyle\max_{1\leq j,k\leq n}|g_k(\tau_j)|}\right)\left[\left(\sum_{j\in M^c}|f_j(x)|^p\right)^\frac{1}{p}+\left(\sum_{k\in N^c}|g_k(x) |^p\right)^\frac{1}{p}\right]. \end{align} We call Inequality (1) as \textbf{Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}.

This review paper provides a comprehensive overview of machine vision pose measurement algorithms. The paper focuses on the state-of-the-art algorithms and their applications. The paper is structured as follows: The introduction in Section 1 provides a brief overview of the field of machine vision pose measurement. Section 2 describes the commonly used algorithms for machine vision pose measurement. Section 3 discusses the factors that affect the accuracy and reliability of machine vision pose measurement algorithms. Section 4 presents the applications of machine vision pose measurement in various fields. The paper provides specific examples of how machine vision pose measurement is used in each of these fields. Finally, Section 5 summarizes the paper and provides future research directions. The paper highlights the need for more robust and accurate algorithms that can handle varying lighting conditions and occlusion. It also suggests that the integration of machine learning techniques may improve the performance of machine vision pose measurement algorithms. Overall, this review paper provides a comprehensive overview of machine vision pose measurement algorithms, their applications, and the factors that affect their accuracy and reliability. It provides a valuable resource for researchers and practitioners working in the field of computer vision.

In the present paper we introduce three new classes of bi-univalent functions connected with Gregory coefficients. For functions in each of these three bi-univalent function classes we have derived the estimates of the Taylor--Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ and Fekete-Szeg\H{o} functional problems for functions belonging to these new subclasses. We defined three subclasses of the class of the bi-univalent functions $\Sigma$, namely $\mathfrak{HG}_{\Sigma}$, $\mathfrak{GM}_{\Sigma}(\mu)$ and $\mathfrak{G}_{\Sigma}(\lambda)$ by using the subordinations with the function whose coefficients are Gregory's numbers. First, we proved that these classes are not empty, i.e. contains other functions than the identity one. Using the well-known Carath\'eodory Lemma for the functions with real positive parts in the open unit disk, together with an estimation due to P. Zaprawa (see https://doi.org/10.1155/2014/357480) and another one of Libera and Zlotkiewicz, we gave upper bounds for the above mentioned initial coefficients and for the Fekete-Szeg\H{o} functionals. The main results are followed by some particular cases, and the novelty of the definitions and the proofs could involve further studies for such type of similarly defined subclasses.

In this paper, we introduce a way of representing a given mapping as the sum of odd and even mappings. Then, by using this representation, we investigate the stability of various forms for the general nonic functional equation.

Badminton tactics refer to the techniques and strategies employed by players to win a match. Analyzing these tactics can help players improve their performance and outsmart their opponents. To study the tactics of top players, we use a gaming tree to analyze matches between two of the most powerful badminton players in history: Lin and Lee. By employing the Nash Equilibrium, we can discover the most beneficial strategies for both players, which reflect their most powerful techniques. Additionally, with the help of this gaming tree, we can precisely predict how players will implement their tactics. Empirical experimental results demonstrate that our proposed method not only evaluates and identifies each player's weaknesses and strengths but also has powerful capabilities to predict their tactics.

We consider the second order differential equation y'' + ω^(2)ρ(x)y = 0 where ω is a positive parameter. The principal concern here is to find conditions on the function ρ^(−1/2)(x), which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for ω sufficiently large. In particular, if c_(νk)(α) denotes the k-th positive zero of the general Bessel (cylinder) function C_(ν)(x; α) = J_(ν)(x) cos α−Y_(ν)(x) sin α of order ν, and if |ν| < 1/2, we prove that (−1)^(m)∆^(m+2)c_(νk)(α) > 0 (m = 0, 1, 2, ...; k = 1, 2, ...), where ∆a_(k) = a_(k+1) − a_(k). This type of inequalities was conjectured by Lorch and Szego in 1963. We also show that the differences of the zeros of various orthogonal polynomials with higher degrees possess the sign-regularity.

We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of a suitable Abel-type transform of the function itself. Thus, the computation of N Legendre coefficients can be performed efficiently in O(NlogN) operations by means of a single Fast Fourier Transform of the Abel-type transform of f(x). We also show how the symmetries associated with the Abel-type transform can be exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, numerical tests are presented to exemplify and confirm the theoretical results.

This paper aims to unify q-derivative/q-integral and h-derivative/h-integral in a single definition, this will be called q−h-derivative/q−h-integral. These notions are further extended on finite interval [a,b] in the form of left and right q−h-derivatives and q−h-integrals. Some inequalities are studied for q−h-integrals which are directly connected with well known results. In diverse fields of science and engineering the theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified by using the concept of q−h-derivative/q−h-integral.

We study the existence of positive solutions for a system of Hadamard fractional differential equations on an infinite interval with nonnegative nonlinearities, subject to coupled nonlocal boundary conditions which contain Hadamard fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main results, we use the Guo-Krasnosel'skii fixed point theorem and the Leggett-Williams fixed point theorem.

The paper's main goal is to introduce a novel type of interpolative convex contraction and build up some fresh findings for it using the interpolative convex contractions' progressive approach. We have established certain conclusions using orbitally S-complete and Suzuki type contractions in F-metric spaces. My research aims to examine the fixed point method's efficacy in solving fractional differential equation with boundary conditions.

In this work, new refinements of some numerical radius inequalities are proved. Namely, new improvements and refinements purify the recent inequalities of some famous inequalities concerning the numerical radius of Hilbert space operators. The proven inequalities in this work are not only an improvement over old inequalities, but rather they are stronger than them. Several examples that support the validity of our results are established as well.

We prove Lp estimates of a class of generalized Marcinkiewicz integral operators with mixed homogeneity on product domains. By using these estimates along with an extrapolation argument we obtain the boundedness of our operators under very weak conditions on the kernel functions. Our results in this paper improve and extend several known results on both generalized Marcinkiewicz integrals and parabolic Marcinkiewicz integrals on product domains.

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier-Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability, the remainder is in terms of an entropy, not a metric. Recently, a stability result for H¹ was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an L^p norm. Afterwards, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H¹ convergence is identified in this paper. Moreover, an explicit H¹ bound via a moment assumption is shown. Also, the L^p stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

The main goal of this article is the characterization of Herz-type Besov-Morrey spaces K˙p,qαN˙μ,rs and Triebel-Lizorkin-Morrey spaces K˙p,qαE˙μ,rs via ball means of differences, atomic, molecular and wavelet decompositions. There are provided necessary facts for implying these results, which can help to explore such spaces on different domains in further researches. Proposed spaces allow to extend the class of studying functions, containing Leray projection, heat semi-group and other operators, which can be helpful in exploring weak, strong and mild solutions of nonlinear partial differential equations.

In the literature, frames generated by unitary representations of groups (known as group-frames) are studied only for Hilbert spaces. We make first study of frames for Banach spaces generated by isometric invertible representations of discrete groups on Banach spaces. These frames are characterized using left regular, right regular, Gram-matrices and group-matrices on classical sequence spaces. A sufficiently large collection of functional-vector pairs using the double commutant of the representation is identified which generate group-frames for Banach spaces. Subsequently, we study Schauder frames generated by time-frequency shift operators on finite dimensional Banach spaces. We derive Moyal formula, fundamental identity of Gabor analysis, Wexler-Raz criterion and Ron-Shen duality in functional form.

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{\omega_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}\label{UE} \|\theta_f x\|_0^\frac{1}{p}\|\theta_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(\omega_k)|}\quad \text{and} \quad \|\theta_g x\|_0^\frac{1}{p}\|\theta_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(\tau_j)|}. \end{align} where \begin{align*} \theta_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad \theta_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (\ref{UE}) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (\ref{UE}) improves Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.