New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (RST) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general RST recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical RST can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of rough naturals are also studied and variants are characterised in the penultimate section.

World Health Organization defines palliative care as the active total care of patients whose disease is not responding to curative treatment. Palliative care for the terminally ill is based on a multidimensional approach to provide whole-person comfort care while maintaining optimal function; dental care plays an important role in this multidisciplinary approach. The aim of the present study is to review significance of dentist's role to determine whether mouth care was effectively assessed and implemented in the palliative care setting. The oral problems experienced by the hospice head and neck patient clearly affect the quality of his or her remaining life. Dentist plays an essential role in palliative care by the maintenance of oral hygiene; dental examination may identify and cure opportunistic infections and dental disease like caries, periodontal disease, oral mucosal problems or prosthetic requirement. Oral care may reduce not only the microbial load of the mouth but the risk for pain and oral infection as well. This multidisciplinary approach to palliative care, including a dentist, may reduce the oral debilities that influence the patient's ability to speak, eat or swallow. This review highlighted that without effective assessment of the mouth, the appropriate implementation of care will not be delivered. Palliative dental care has been fundamental in management of patients with active, progressive, far-advanced disease in which the oral cavity has been compromised either by the disease directly or by its treatment; the focus of care is quality of life.

In the context of general rough sets, the act of combining two things to form another is not straightforward. The situation is similar for other theories that concern uncertainty and vagueness. Such acts can be endowed with additional meaning that go beyond structural conjunction and disjunction as in the theory of * -norms and associated implications over L-fuzzy sets. In the present research, algebraic models of acts of combining things in generalized rough sets over lattices with approximation operators (called rough convenience lattices) is invented. The investigation is strongly motivated by the desire to model skeptical or pessimistic, and optimistic or possibilistic aggregation in human reasoning, and the choice of operations is constrained by the perspective. Fundamental results on the weak negations and implications afforded by the minimal models are proved. In addition, the model is suitable for the study of discriminatory/toxic behavior in human reasoning, and of ML algorithms learning such behavior.

Background:Calendula officinalis (C. officinalis), commonly known as pot marigold, is a medicinal herb with excellent antimicrobial, wound healing, and anti-inflammatory activity.Aim:To evaluate the efficacy of C. officinalis in reducing dental plaque and gingival inflammation.Materials and Methods:Two hundred and forty patients within the age group of 20-40 years were enrolled in this study with their informed consent. Patients having gingivitis (probing depth (PD) ≤3 mm), with a complaint of bleeding gums were included in this study. Patients with periodontitis PD ≥ 4 mm, desquamative gingivitis, acute necrotizing ulcerative gingivitis (ANUG), smokers under antibiotic coverage, and any other history of systemic diseases or conditions, including pregnancy, were excluded from the study. The subjects were randomly assigned into two groups – test group (n = 120) and control group (n = 120). All the test group patients were advised to dilute 2 ml of tincture of calendula with 6 ml of distilled water and rinse their mouths once in the morning and once in the evening for six months. Similarly, the control group patients were advised to use 8 ml distilled water (placebo) as control mouthwash and rinse mouth twice daily for six months. Clinical parameters like the plaque index (PI), gingival index (GI), sulcus bleeding index (SBI), and oral hygiene index-simplified (OHI-S) were recorded at baseline (first visit), third month (second visit), and sixth month (third visit) by the same operator, to rule out variable results. During the second visit, after recording the clinical parameters, each patient was subjected to undergo a thorough scaling procedure. Patients were instructed to carry out regular routine oral hygiene maintenance without any reinforcement in it.Results:In the absence of scaling (that is, between the first and second visit), the test group showed a statistically significant reduction in the scores of PI, GI, SBI (except OHI-S) (P < 0.05), whereas, the control group showed no reduction in scores when the baseline scores were compared with the third month scores. Also, when scaling was performed during the third month (second visit), there was statistically significant reduction in the scores of all parameters, when the third month scores were compared with the sixth month scores in both groups (P < 0.05), but the test group showed a significantly greater reduction in the PI, GI, SBI, and OHI-S scores compared to those of the control group.Conclusion:Within the limits of this study, it can be concluded that calendula mouthwash is effective in reducing dental plaque and gingivitis adjunctive to scaling.

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A teacher's knowledge base consists of knowledge of mathematics content, knowledge of student epistemology, and pedagogical knowledge. It has severe implications on the understanding of student's knowledge of content, and the learning context in general. The necessity to formalize the different content knowledge in approximate senses is recognized in the education research literature. A related problem is that of coherent formalizability. Responsive or smart AI-based software systems do not concern themselves with meaning, and trained ones are replete with their own issues. In the present research, many issues in modeling teachers' understanding of content are identified, and a two-tier rough set-based model is proposed by the present author. The main advantage of the proposed approach is in its ability to coherently handle vagueness, granularity and multi-modality. An extended example to equational reasoning is used to demonstrate these.

We develop two algebraic semantics for bitten rough set theory ([19]) over similarity spaces and their abstract granular versions. Connections with choice based generalized rough semantics developed in [15] by the present author and general cover based rough set theories are also considered.