This paper explains why the critical line sits at the real part equal to one-half by treating it as an intrinsic boundary of a reparametrized complex plane (“z-space”), not a mere artifact of functional symmetry. In z-space the real part is defined by a geometric-series map that gives rise to a rulebook for admissible analytic operations. Within this setting we rederive the classical toolkit—the eta–zeta relation, Gamma reflection and duplication, theta–Mellin identity, functional equation, and the completed zeta—without importing analytic continuation from the usual s-variable. We show that access to the left half-plane occurs entirely through formulas written on the right, with boundary matching only along the line with the real part equal to one-half. A global Hadamard product confirms the consistency and fixed location of this boundary, and a holomorphic change of variables transports these conclusions into the classical setting.

Holomorphic Extensions of Generalized Hadamard Product

Artificial Intelligence (AI) has been dramatically applied to healthcare in various tasks to support clinicians in disease diagnosis and prognosis. It has been known that accurate diagnosis must be drawn from multiple evidence, namely clinical records, X-Ray images, IoT data, etc called the multi-modal data. Despite the existence of various approaches for multi-modal medical data fusion, the development of comprehensive systems capable of integrating data from multiple sources and modalities remains a considerable challenge. Besides, many machine learning models face difficulties in representation and computation due to the uncertainty and diversity of medical data. This study proposes a novel multi-modal fuzzy knowledge graph framework, called FKG-MM, which integrates multi-modal medical data from multiple sources, offering enhanced computational performance compared to unimodal data. In addition, the FKG-MM framework is based on the fuzzy knowledge graph model, one of the models that represent and compute effectively with medical data in tabular form. Through some experiment scenarios utilizing the well-known BRSET dataset on multi-modal diabetic retinopathy, it has been experimentally validated that the feature selection method, when combining image features with tabular medical data features, gives the highest reliability results among 5 methods including Feature Selection Method, Tensor Product, Hadamard Product, Filter Selection, and Wrapper Selection. In addition, the experiment also confirms that the accuracy of FKG-MM increases by 12–14% when combining image data with tabular medical data than the related methods diagnosing only on tabular data.

Abstract Gao and Xie conjectured that the inverse Kazhdan–Lusztig polynomial of any matroid is log‐concave. Although these polynomials are not necessarily real‐rooted, we conjecture that the Hadamard product of an inverse Kazhdan–Lusztig polynomial of degree with is real‐rooted. Using the theory of interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. As a consequence, the log‐concavity of inverse Kazhdan–Lusztig polynomials for paving matroids follows from Newton's inequalities.

Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .

A hybrid DNN model using novel integrated interface features for predicting protein-protein complexes binding affinity.

Spectral radius inequalities for generalized Hadamard products of exponentiated nonnegative matrices

Two subclasses of starlike and convex functions analytic in the unit open disk using q-derivative operator have been investigated in the present paper. The necessary and sufficient condition for the function belonging to these classes have been obtained. We further examine various properties, such as the Hadamard product and the quasi-Hadamard product. The coefficient estimates for the function belonging to these classes are also found. Mathematics Subject Classification 2020: 30C45, 30C50

This paper presents a comprehensive study of the Saigo-type fractional integral operator S0 α + ,β,γ applied to a broad class of generalized Q-functions Qσ κ, , ζ ξ , , η v, , r ϑ,τ ,ρ(t) characterized by multiple complex parameters. We establish explicit closed-form expressions for fractional integrals of weighted generalized Q-functions in terms of Hadamard products involving the original functions and Wright-type hypergeometric functions 2Ψ1. Detailed numerical examples with explicit parameter values demonstrate the computational feasibility and practical applicability of the derived formulas. The results contribute significantly to fractional calculus theory and provide valuable tools for solving fractional differential equations involving generalized special functions with memory effects.

Abstract The differential subordinations theory is also known as the admissible functions method and it was initiated by S.S. Miller and P.T. Mocanu in two publications in 1978 and 1981. The same two reputed researchers have introduced the notion of differential superordination as a dual notion of differential subordination in 2003. Generalizing the differential subordinations, J.A. Antonino and S. Romaguera introduced the notion of strong differential subordination in 1994. G.I. Oros has introduced the notion of strong differential superordination as a dual notion of strong differential subordination in 2009. Using the differential subordinations and superordinations, strong differential subordinations and superordinations methods it was easier to prove some classical results from this domain, and also many extensions or generalizations of these, at the same time new results being obtained. A useful tool in the study of different types of operators is offered by the theory of classical differential subordinations and superordinations and the theory of strong differential subordinations and superordinations. Some remarkable results in the differential and integral operators theory were successfully proved and new results were also obtained using those theories. In this article we establish several properties for the operator $$ IR_{\lambda ,l}^{m,n}$$ I R λ , l m , n defined as the Hadamard product of Ruscheweyh operator and multiplier transformation by using the classic and strong theories of differential subordinations and superordinations and get differential sandwich-type theorems. Interesting corollaries follow for particular functions used as best subordinant and best dominant.

Skin cancer poses a significant threat to life, necessitating early detection. Skin lesion segmentation, a critical step in diagnosis, remains challenging due to variations in lesion size and edge blurring. Despite recent advancements in computational efficiency, edge detection accuracy remains a bottleneck. In this paper, we propose a lightweight UNet with multi-module synergy and dual-domain attention for precise skin lesion segmentation to address these issues. Our model combines the Swin Transformer (Swin-T) block, Multi-Axis External Weighting (MEWB), Group multi-axis Hadamard Product Attention (GHPA), and Group Aggregation Bridge (GAB) within a lightweight framework. Swin-T reduces complexity through parallel processing, MEWB incorporates frequency domain information for comprehensive feature capture, GHPA extracts pathological information from diverse perspectives, and GAB enhances multi-scale information extraction. On the ISIC2017 and ISIC2018 datasets, our model achieves mIoU and DSC scores of 81.22% and 89.64%, and 81.65% and 89.90%, respectively. These results demonstrate improved segmentation accuracy with low parameter count and computational cost, aiding physicians in diagnosis and treatment. https://github.com/SolitudeWolf/ESMDL-UNet.git.

Static-dynamic class-level perception consistency in video semantic segmentation.

Membrane rafts are cellular portals to external stimuli that trigger signaling cascades for sophisticated yet remarkable biochemical activities. Visualization of the topographic evolution of membrane rafts remains unreported on live cells due to the nanosized and dynamic nature. Here, an imaging strategy involving atomic force microscopy and Hadamard product is developed to unveil membrane-raft features. Michigan Cancer Foundation-7 (MCF-7) cells were subjected to fibrinogen or manganese(II) (Mn2+)/resveratrol, both of which are ligands of integrin αVβ3 embedded within membrane rafts; the former promotes metastasis, and the latter enables apoptosis. MCF-7 cellular membranes responded to the two stimulants markedly different. The size, height, spatiotemporal trajectory, and persistent time of ligand-activated nanodomains are revealed. This approach opens up a visualized platform toward the understanding of activation-associated signaling cascades.

Modeling and analyzing tunnel driving behavior provides insights into driving behavior characteristics and state identification. Previous studies have primarily extracted single or multiple driving behavior features, neglecting their overall time-varied patterns. This study aimed to develop a driving behavior spectrum that considers the coupling effect of driving behavior time series patterns, drivers’ physiological characteristics, and multidimensional environment factors encompassing acoustic, lighting, traffic volume, and road segment type, and to establish a driving state identification model in tunnels. First, a real vehicle test was conducted to collect data on driving behavior, drivers’ physiology, and tunnel environment, from which 13 variables were extracted. A fuzzy comprehensive evaluation method was then applied to assess the complexity of the tunnel environment. Second, the driving behavior spectrum was created for each driver by introducing a single feature recurrence matrix spectrum radius (SRMSR). Then, the hidden Markov model and the criteria importance through intercriteria correlation weighting method were employed to evaluate and classify the driving states. Finally, the composite feature recurrence matrix spectrum radius (CRMSR) based on SRMSR was derived using the Hadamard product and employed as an input variable for a Light Gradient Boosting Machine driving state identification model. The results indicated that the proposed CRMSR was effective in identifying tunnel driving states, enhancing model accuracy as an input. In addition, the proposed method can pinpoint the critical tunnel zones requiring enhanced safety design based on the identification of driving states. It can be used to monitor and identify risky driving states, providing a data foundation for early warning systems and aiding in tunnel design to enhance overall safety.

Let C n be a complex correlation matrix of size n × n . It is claimed that ( per ( C n ) ) 2 ≥ per ( C n ∘ C n ¯ ) , where C n ∘ C n ¯ is the Hadamard product of C n and C ¯ n (the complex conjugation to each entry of C n ), which is known as the Chollet permanental conjecture, raised in 1982. It is clear from the definitions that if all off-diagonal entries lie in the unit interval [ 0 , 1 ] , then the conjecture holds true. However, in general, the conjecture was merely proved to be valid for n = 2, 3 in the year 1987 and n = 4 in the year 2023. Let r be a positive real number and a := 2 / ( n − 2 ) ( ( r 4 − 2 ) n + 2 ) with n ≥ 5 . It is shown in this paper that if all off-diagonal entries of C n belong to the annulus T 0 ( a , ra ) centred at the origin with inner radius a and outer radius ra for any r ≥ 2 , then ( per ( C n ) ) 2 ≥ per ( C n ∘ C n ¯ ) . The positions of the annuli that serve the Chollet's permanental conjecture can be varied infinitely depending on r and can be translated to the annuli T 0 ( a s + 1 , ar s + 1 ) for any real number s ≥ 0 . The correlation matrices on these annuli are also shown to satisfy an inequality. The investigation of the conjecture for n = 5 is further studied, and it turns out that any matrix permuting similar to a ridge matrix always satisfy the conjecture.

Abstract In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps $$T:\mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}[x_1,\dots ,x_n]$$ T : R [ x 1 , ⋯ , x n ] → R [ x 1 , ⋯ , x n ] such that $$Tx^\alpha = t_\alpha x^\alpha $$ T x α = t α x α for all $$\alpha \in \mathbb {N}_0^n$$ α ∈ N 0 n with $$t_\alpha \in \mathbb {R}$$ t α ∈ R and $$Tp\ge 0$$ T p ≥ 0 on $$\mathbb {R}^n$$ R n for all $$p\in \mathbb {R}[x_1,\dots ,x_n]$$ p ∈ R [ x 1 , ⋯ , x n ] with $$p\ge 0$$ p ≥ 0 on $$\mathbb {R}^n$$ R n . We discuss representations of T , give characterizations of diagonal positivity preservers, and compare these to previous (partial) results in the literature. On the side we get a characterization of linear maps preserving moment sequences and a new proof of Schur’s product formula. The tool of diagonal positivity preservers simplifies several other existing proofs in the literature. We give a full characterization of generators A of diagonal positivity preservers, i.e., $$e^{tA}$$ e tA is a diagonal positivity preserver for all $$t\ge 0$$ t ≥ 0 . We give the connection of these generators to infinitely divisible moment sequences.

Cancer is a highly heterogeneous disease characterized by complex molecular changes. Subtypes identified through multi-omics data hold significant promise for improving prognosis and facilitating personalized precision treatment. Recent multi-omics integration methods have mostly focused on capturing complementary information from different data types, often overlooking potential interactions between omics data. Here we develop a novel method named interactive multi-kernel learning (iMKL), which incorporates omics-omics interactions alongside heterogeneous data types under the unsupervised multi-kernel learning framework, to improve subtype identification. Using the sample-similarity kernel for each dataset, we propose a joint Hadamard product strategy to capture higher-order interactive effects from different omics data types. We applied iMKL to two renal cell carcinoma (RCC) datasets—clear renal cell carcinoma (ccRCC) and type II papillary renal cell carcinoma (type II pRCC)—both including miRNA expression, mRNA expression, and DNA methylation data. Stability analysis through random sampling of patients or features demonstrated that iMKL exhibits strong robustness and accuracy in identifying patient subtypes. The identified subtypes revealed dramatic differences in patient survival, with both ccRCC and type II pRCC classified into three distinct subtypes. The findings in the real application highlight potential biomarkers associated with adverse patient outcomes and demonstrate substantial advancement in cancer subtype identification. The iMKL method effectively identifies tumor molecular subtypes that are strongly associated with clinical features and survival rates, providing valuable insights for accurate cancer subtyping, clinical decision-making, and the realization of personalized treatment strategies.

Plant diseases pose a severe threat to global agricultural production, significantly challenging crop yield, quality, and food security. Therefore, accurate and efficient disease detection is crucial. Current detection methods have clear limitations: CNN-based methods struggle to model long-range dependencies effectively and have weak generalization abilities. Transformer-based methods, while adept at long-range feature modeling, face issues with large parameter sizes and inefficient calculations due to the quadratic complexity of the self-attention mechanism in relation to image size. To address these challenges, this paper proposes the MamSwinNet model. Its core innovation lies in: using the Efficient Token Refinement module with an overlapping space reduction method, relying on depthwise separable convolutions designed with “stride + 3” convolution kernels to expand the image block overlap area and fully preserve boundary spatial structure. This generates high-quality tokens and converts them into a fixed number of latent tokens, reducing computational complexity while maximizing the retention of key features. It integrates the Spatial Global Selective Perception (SGSP) module and the Channel Coordinate Global Optimal Scanning (CCGOS) module. The SGSP module uses a dual-branch structure (the spatial modeling branch introduces 2D-SSM to scan four directions for capturing long-range dependencies, and the residual compensation branch supplements features to prevent loss; the two branches are combined using Hadamard product to enhance spatial detail modeling). The CCGOS module combines channel and spatial attention by embedding positional information through global average pooling in the height and width dimensions, using the Mamba block for channel-selective scanning and generating an attention map, enabling precise association of key channel features like color with spatial distribution. Experimental results show that the model achieves F1 scores of 79.47%, 99.52%, and 99.38% on the PlantDoc, PlantVillage, and Cotton datasets, respectively. The model has only 12.97M parameters (52.9% less than the Swin-T model) and a computational cost as low as 2.71GMac, significantly improving computational efficiency. This study provides an efficient and reliable intelligent solution for large-scale crop disease detection.

This study examines the necessary requirements for some analytic function subclasses, especially those associated with the generalized Mittag-Leffler function, to be classified as univalent function subclasses that are determined by particular geometric constraints. The core methodology revolves around the application of the Hadamard (or convolution) product involving a normalized Mittag-Leffler function Mκ,χ(ζ), leading to the definition of a new linear operator Sχ,ϑκℏ(ζ). We investigate inclusion results in the recently defined subclasses Ξ˜(ϖ,ϱ),L^(ϖ,ϱ),K^(ϖ,ϱ) and F^(ϖ,ϱ), which generalize the classical classes of starlike, convex, and close-to-convex functions. This is achieved by utilizing recent developments in the theory of univalent functions. In addition, we examine the behavior of functions from the class Rθ(E,V) under the action of the convolution operator Wχ,ϑκh(ζ), establishing sufficient criteria for the resulting images to lie within the subclasses of analytic function. Also, certain mapping properties related to these subclasses are analyzed. In addition, the geometric features of an integral operator connected to the Mittag-Leffler function are examined. A few particular cases of our main findings are also mentioned and examined and the paper ends with the conclusions regarding the obtained results.

Time series is common across disciplines, however the analysis of time series is not trivial due to inter- and intra-relationships between ordered data sequences. This imposes limitation upon the interpretation and importance estimate of the features within a time series. In the case of multivariate time series, these features are the individual time series and the time steps, which are intertwined. There exist many time series analyses, such as Autocorrelation and Granger Causality, which are based on statistic or econometric approaches. However analyses that can inform the importance of features within a time series are uncommon, especially with methods that utilise embedded methods of neural network (NN). We approach this problem by expanding upon our previous work, Pairwise Importance Estimate Extension (PIEE). We made adaptations toward the existing method to make it compatible with time series. This led to the formulation of aggregated Hadamard product, which can produce an importance estimate for each time point within a multivariate time series. This subsequently allows each time series within a multivariate time series to be interpreted as well. Within this work, we conducted an empirical study with univariate and multivariate time series, where we compared interpretation and importance estimate of features from existing embedded NN approaches, an explainable AI (xAI) approach, and our adapted PIEE approach. We verified interpretation and importance estimate via ground truth or existing domain knowledge when it is available. Otherwise, we conducted an ablation study by retraining the model with Leave-One-Out and Singleton feature subsets to see their contribution towards model performance. Our adapted PIEE method was able to produce various feature importance heatmaps and rankings inline with the ground truth, the existing domain knowledge or the ablation study.