Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

In this paper, we introduced many new notions and new contraction named as S-weakly contraction after that we obtained the p-common best proximity point theorems for different types of contractions in the setting of complete metric spaces by using weak P_p-property and proved the uniqueness of these points. Also we presented some examples to prove the validity of our results.

In this article some important observations have been reported on recent works related to weakly tripotent rings and locally invo-regular rings. Our findings give additional results as well as correct some recent results on weakly tripotent rings and locally invo-regular rings appeared in Rendiconti Sem. Mat. Univ. Pol. Torino (2021) and Azerbaijan Journal of Mathematics (2021) respectively. In addition we exhibit that if the Jacobson radical J(A) of a ring is strongly involution t-clean then the characteristic of J(A) need not be four. This finding improves an important result appeared in Eur. J. Pure Appl. Math (2022).

In this note we consolidate and give a brief description of several recently published results in ring theory having disproof. These results have been published during 2016 to 2021 in the so called non-predatory reputed mathematical journals indexed in the well known database like Scopus. We have considered results on rings in which each element is a sum of two idempotents appeared in Canad. Math. Bull. (2016), weakly tripotent rings appeared in Bull. Korean Math. Soc. (2018) and Rendiconti Sem. Mat. Univ. Pol. Torino (2021), involution t-clean rings appeared in Eur. J. Pure Appl. Math (2022) and strongly 2-nil clean rings with units of order two appeared in Eur. J. Pure Appl. Math (2023).

In this paper we provide some important and significant observations on invo-regular rings appeared in Ann. Univ. Mariae Curie-Sklodowska Sect. A Mathematica (2018). In addition we exhibit that strongly invo-regular rings, invo-regular rings and quasi invo-regular rings all coincide with the well known and well characterized tripotent rings and all these rings further coincide with the well known unit regular rings.

In 1895, Korteweg and de Vries (KdV), derived their celebrated equation describing the motion of waves of long wavelength in shallow water. In doing so they made a number of quite reasonable assumptions, incompressibility of the water and irrotational fluid. The resulting equation, the celebrated KdV equation, has been shown to be a very reasonable description of real water waves. However there are other phenomena which have an impact on the shape of the wave, that of vorticity and viscosity. This paper examines how a constant vorticity affects the shape of waves in electrohydrodynamics. For constant vorticity, the vertical component of the velocity obeys a Laplace equation and also has the usual lower boundary condition. In making the vertical component of the velocity take central stage, the Burns condition can be thus bypassed.

The aim of this paper is to obtain Karush-Kuhn-Tucker optimality conditions for weakly efficient solutions to vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient solutions to the constrained vector optimization problem are presented. As well as some examples. The results presented in this paper generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces and real Banach spaces to Hadamard manifolds, respectively.

We investigate properties of strongly useful topologies, i.e. topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.

The task of weakly supervised point cloud semantic segmentation has received widespread attention and also has been widely used in autonomous driving, robotics, and modern industries. Due to the high dimensionality and large volume of point cloud data, many technical difficulties appeal to weakly supervised semantic processing, especially segmentation. However, weakly-supervised schemes usually provide only partial labelling information of the underlying point cloud data and thus need to effectively extract local features, geometric information, and contextual relationships only using these limited labelled data for supervised learning. To improve weakly-supervised semantic segmentation, we propose a novel segmentation network through the boundary-based feature aggregation based on a K-NN algorithm with down-sampling operation, and also introduce the smoothness loss and Siamese loss for effective segmentation. The experiments on public datasets demonstrate that our presented segmentation method exceeds most of the existing fully supervised and weakly-supervised methods in terms of mIoU. Specifically, our network has high segmentation accuracy on the labels of objects with similar geometrical structures, such as ceiling, wall, floor, chair and table, reaching 91.2%, 98.8%, 83.3%, 75.3% and 80.2%, respectively. Extensive experiments also illustrate its robustness, effectiveness, and generalization of the proposed weakly supervised segmentation network.

Ensuring precise segmentation of colorectal polyps holds critical importance in the early diagnosis and treatment of colorectal cancer. Nevertheless, existing deep learning-based segmentation methods are fully-supervised, requiring extensive precise manual pixel-level annotation data, which leads to high annotation costs. Additionally, it remains challenging to train large-scale segmentation models when confronted with limited colonoscopy data. To address these issues, we introduce the general segmentation foundation model—Segment Anything Model (SAM) into the field of medical image segmentation. Fine-tuning the foundation model is an effective approach to tackle sample scarcity. However, current SAM fine-tuning techniques still rely on precise annotations. To overcome this limitation, we propose WSPoly-SAM, a novel weakly-supervised approach for colonoscopy polyp segmentation. WSPoly-SAM utilizes weak annotations to guide SAM in generating segmentation masks, which are then treated as pseudo-labels to guide the fine-tuning of SAM, thereby reducing the dependence on precise annotations data. To improve the reliability and accuracy of pseudo-labels, we have designed a series of enhancement strategies to improve the quality of pseudo-labels and mitigate the negative impact of low-quality pseudo-labels. Experimental results on five medical image datasets demonstrate that WSPoly-SAM outperforms current fully-supervised mainstream polyp segmentation networks on the Kvasir-SEG, ColonDB, CVC-300, and ETIS datasets. Furthermore, by using different amounts of training data in weakly-supervised and fully-supervised experiments, it is found that weakly-supervised fine-tuning can save 70% to 73% of annotation time costs compared to fully-supervised fine-tuning. This study provides a new perspective on the combination of weakly-supervised learning and SAM models, significantly reducing annotation time and offering insights for further development in the field of colonoscopy polyp segmentation.

We define the notions of strong and strict recurrency for actions of countable ordered groups on $\sigma$-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

(1) Background: Evidence-based policymaking requires data about the local population's socioeconomic status (SES) at detailed geographical level, however, such information is often not available, or is too expensive to acquire. Researchers have proposed solutions to estimate SES indicators by analyzing Google Street View images, however, these methods are also resource-intensive, since they require large volumes of manually labeled training data. (2) Methods: We propose a methodology for automatically computing surrogate variables of SES indicators using street images of parked cars and deep multiple instance learning. Our approach does not require any manually created labels, apart from data already available by statistical authorities, while the entire pipeline for image acquisition, parked car detection, car classification, and surrogate variable computation is fully automated. The proposed surrogate variables are then used in linear regression models to estimate the target SES indicators. (3) Results: We implement and evaluate a model based on the proposed surrogate variable at 30 municipalities of varying SES in Greece. Our model has $R^2=0.76$ and a correlation coefficient of $0.874$ with the true unemployment rate, while it achieves a mean absolute percentage error of $0.089$ and mean absolute error of $1.87$ on a held-out test set. Similar results are also obtained for other socioeconomic indicators, related to education level and occupational prestige. (4) Conclusions: The proposed methodology can be used to estimate SES indicators at the local level automatically, using images of parked cars detected via Google Street View, without the need for any manual labeling effort.

Background: Ovarian, cervical, and endometrial cancers stand as fatal health killers for women's mental and physical health, especially affecting female reproductive systems. Exploiting weakly supervised learning and explainable AI techniques are crucial for fast, accurate, and robust automatic marker detection, efficient prevention, and primary treatment of gynecological tumors. Methods: With respect to the PRIMSA methodology, weakly supervised learning schemes and deep learning-based schemes are investigated in the cross-subject fields of clinical diagnostic imaging and explainable AI technology. Related methods, opening research problems, and challenging subjects are explored in the screening and treatment of gynecological tumors and performing cancer image diagnosis in the latest study. Results: Keynote approaches combining ultrasound medicine, explainable AI, and clinical imaging technology are summarized. In combination with learning-based methods with medical imaging, prospective insights are put forward for refreshed concepts on treatments in the area of gynecological oncology, with a feasible range of applications for their corresponding AI techniques. Conclusions: Explainable AI and deep learning-based approaches are capable of performing accurate classification between benign and malignant tumors, yielding a pathway to care for women's health, improving the survival rate of patients, pacing with disease prediction, and matching the strategic goal of "Healthy China 2030".

The generalized dynamic model of the rotor system, presented in the paper, is the first model that takes into account the interconnected oscillations of the “rotor - weakly conductive fluid – foundation” system under the action of such parameters as fluid and rotor motion, linear eccentricity, friction forces, foundation vibration and nonlinear characteristics of rolling bearings, as well as the action of a magnetic field on the fluid. Consistent equations of motion for the system “rotor - weakly conductive fluid – foundation” were derived and solved analytically. Forced and natural oscillations of the system were analyzed, and the distinctive features of the rotor system dynamics were revealed. The values of frequencies and amplitudes, which are one of the main factors determining the dynamic behavior of the system, were obtained and studied.

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method based on the weakly singular boundary integral equation, the quadrature techniques, dedicated for the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take the full advantage of the multi-patch technique. The generated collo-cation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by the simple examples with analytical solutions. The feature of the present multi-patch WSIGABEM is investigated by the comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner and the arch bridge are verified to be effective.

Based on the engineering background of Daliuta mining area, the fracture distribution and water movement within the weakly cemented overlay strata are studied through liquid-solid coupling physical simulation and COMSOL numerical simulation test. 3D fracture network of overburden strata in mining is plotted based on the result of the physical simulation test. The results showed that the ‘fractured zone’ of the weakly cemented overburden strata in the western area develops more evidently than that in the eastern area ---- wide-opened fractures form at both ends of the working area and this leads to a wide distribution of fractures in the overlying strata.After the initial caving of the roof, height of the penetrating fractures increases rapidly and the separated strata space becomes the main channel for water flow. Under the action of water flow and rock strata movement, mining-induced fractures in overlying rock display cyclic change of expansion, penetration and closure. When the working face is fully mined, the penetrating fractures in the overlying strata and mining-induced fractures up in the working face become the main passages for water flow. The experimental result provides a reference for underground reservoirs layout and maintenance and water-conserved mining in western China.

In this work, we introduce the concepts of r-fuzzy soft δ-open (resp. semi-open and α-open) sets on fuzzy soft topological space, the relations of these sets with each other are established. Next, we introduce the concepts of fuzzy soft δ-continuous (resp. β-continuous, semi-continuous, pre-continuous and α-continuous) functions, and some properties of these functions along with their mutual relationships are investigated. Also, a decomposition of fuzzy soft semi-continuity and a decomposition of fuzzy soft α-continuity are given. Finally, as a weaker form of fuzzy soft continuity by Aygunoglu et al. (2014), the concepts of fuzzy soft almost (resp. weakly) continuous functions are introduced, and some properties are specified. Moreover, we explore the notion of continuity in a very general setting namely fuzzy soft (L, M, N, O)-continuous functions, and a historical justification are obtained.

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α,f), where α=(K,γ,α) is a morphism of Σ-completions from F=(C,F,η) to G=(D,G,ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥαG,0(f):Schα(f)B. We then prove that there exists an endofunctor, ΥαG,0, of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G=(D,G,ρ), there exists a functor ΥG from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of ΥG, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥαG,0.

In this manuscript, we first introduce some properties of r-fuzzy soft α-open sets in fuzzy soft topological spaces based on the manuscript Aygunoglu et al. (Hacet. J. Math. Stat. 2014, 43, 193-208). In addition, we define the concepts of fuzzy soft α-closure (α-interior) operators, and investigate some properties of them. Furthermore, the concept of r-fuzzy soft α-connected sets is defined and characterized with help of fuzzy soft α-closure operators. Thereafter, we introduce and study the concepts of fuzzy soft almost (weakly) α-continuous mappings, which are weaker forms of fuzzy soft α-continuous mappings, and we discuss some properties of fuzzy soft α-continuity. Moreover, we establish that fuzzy soft α-continuity → fuzzy soft almost α-continuity → fuzzy soft weakly α-continuity, but the converse may not be true. It is also we show that the composition is fuzzy soft almost α-continuous mapping if, is fuzzy soft α-continuous mapping and is fuzzy soft almost continuous mapping. Finally, some new types of compactness via r-fuzzy soft α-open sets are defined, and the relationships between them are examined.

In the present study. At first, some properties of r-fuzzy soft β-closed sets are introduced on fuzzy soft topological spaces based on the paper Aygunoglu et al. (Hacet. J. Math. Stat. 2014, 43, 193-208). Also, we define the closure and interior operators with respect to the classes of r-fuzzy soft β-closed and r-fuzzy soft β-open sets, and investigate some of their properties. In addition, the concept of r-fuzzy soft β-connected sets is introduced and characterized with help of fuzzy soft β-closure operators. Thereafter, some properties of a fuzzy soft β-continuity are studied. Furthermore, we introduce and study the concepts of fuzzy soft almost (weakly) β-continuous functions, which are weaker forms of a fuzzy soft β-continuity. Also, the relationships between these classes of functions are specified with the help of some illustrative examples. Finally, we explore new types of fuzzy soft functions called, fuzzy soft β-irresolute (strongly β-irresolute, β-irresolute open, β-irresolute closed and β-irresolute homeomorphism) functions, and discuss some properties of them. Also, we show that fuzzy soft strongly β-irresolute → fuzzy soft β-irresolute → fuzzy soft β-continuity, but the converse may not be true.