Logical Entropy of Information Sources.

In this paper, we present the concept of the logical entropy of order m, logical mutual information, and the logical entropy for information sources. We found upper and lower bounds for the logical entropy of a random variable by using convex functions. We show that the logical entropy of the joint distributions X1 and X2 is always less than the sum of the logical entropy of the variables X1 and X2. We define the logical Shannon entropy and logical metric permutation entropy to an information system and examine the properties of this kind of entropy. Finally, we examine the amount of the logical metric entropy and permutation logical entropy for maps.

Hosoya Polynomial for Subdivided Caterpillar Graphs.

BACKGROUND: Computing Hosoya polynomial for a graph associated with a chemical compound plays a vital role in the field of chemistry. From Hosoya polynomial, it is easy to compute the Weiner index(Weiner number) and Hyper Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected by the physicochemical properties of chemical compounds. Caterpillar trees are used in chemical graph theory to represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices. METHODS: The Hosoya polynomial of a graph G is defined as H(G;x) = Σd(G)K=0 d(G.k)xk. In order to compute the Hosoya polynomial, we need to find its coefficient d(G.k) which is the number of pairs of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance , 2 ≤ m ≤ (n + 1)k in the form of sets. Then finding the cardinality of these sets and adding them up will give us the value of coefficient d(G.m) . Finally, using the values of coefficients in the definition, we get the Hosoya polynomial of uniform subdivision of caterpillar graph. RESULT: In this work, we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph. CONCLUSION: Caterpillar trees are among the important and general classes of trees. Thorn rods and thorn stars are the important subclasses of caterpillar trees. The idea of the present research article is to provide a road map to those researchers who are interested in studying the Hosoya polynomial for different trees.

A gender-based study of symptoms and risk factors associated with mortality in COVID-19 patients in a tertiary care hospital, Lahore.

OBJECTIVE: To determine the symptoms and risk factors associated with gender-specific mortality among coronavirus disease-2019 patients. METHODS: The retrospective, descriptive, cross-sectional study was conducted at the coronavirus disease-2019 ward of the Jinnah Hospital, Lahore, Pakistan, and comprised record of confirmed cases of coronavirus disease-2019 diagnosed on the basis of characteristic clinical symptoms, radiological findings and polymerase chain reaction positivity from May 1 to August 31, 2020. Clinical symptoms, comorbidities and outcomes were extracted from the medical records. Data was analysed using SPSS 23. RESULTS: Of 337 cases, 132(39.2%) died. Among the deceased, 84(64%) were males with median age 61.5 (interquartile range: 22) and 48(36%) were females with median age 54.5 (interquartile range: 25).There were more female non-survivors who suffered from kidney disease 10(66.7%) than male non-survivors 5(33.3%) (p<0.05). Ischaemic heart disease was more common among males than females (p=1.62). CONCLUSIONS: The mortality rate in males was higher compared to females. The symptoms and risk factors associated with mortality varied between the genders.

n-polynomial exponential type p-convex function with some related inequalities and their applications.

In this paper, the idea and its algebraic properties of n-polynomial exponential type p-convex function have been investigated. Authors prove new trapezium type inequality for this new class of functions. We also obtain some refinements of the trapezium type inequality for functions whose first derivative in absolute value at certain power are n-polynomial exponential type p-convex. At the end, some new bounds for special means of different positive real numbers are provided as well. These new results yield us some generalizations of the prior results. Our idea and technique may stimulate further research in different areas of pure and applied sciences.

A Patient-Specific Single Sensor IoT-Based Wearable Fall Prediction and Detection System.

Falls in older adults are a major cause of morbidity and mortality and are a key class of preventable injuries. This paper presents a patient-specific (PS) fall prediction and detection prototype system that utilizes a single tri-axial accelerometer attached to the patient's thigh to distinguish between activities of daily living (ADL) and fall events. The proposed system consists of two modes of operation: 1) fast mode for fall predication (FMFP) predicting a fall event (300-700 msec) before occurring and 2) slow mode for fall detection (SMFD) with a 1-sec latency for detecting a fall event. The nonlinear support vector machine classifier (NLSVM)-based FMFP algorithm extracts seven discriminating features for the pre-fall case to identify a fall risk event and alarm the patient. The proposed SMFD algorithm utilizes a Three-cascaded 1-sec sliding frames classification architecture with a linear regression-based offline training to identify a single and optimal threshold for each patient. Fall incidence will trigger an alarming notice to the concern healthcare providers via the Internet. Experiments are performed with 20 different subjects (age above 65 years) and a total number of 100 associated falls and ADL recordings indoors and outdoors. The accuracy of the proposed algorithms is furthermore validated via MobiFall Dataset. FMFP achieves sensitivity and specificity of 97.8% and 99.1%, respectively, while SMFD achieves sensitivity and specificity of 98.6% and 99.3%, respectively, for a total number of 600 measured falls and ADL cases from 77 subjects.

Role of key players in paradigm shifts of prostate cancer bone metastasis.

The decreased bone mineral density and compromised bone strength predispose individuals to skeletal osteoporosis. Both prostate cancer and bone metastasis caused by cancer invasion have remained a great challenge to researchers. With the advancement in the fields of biochemistry and biomechanics, the molecular mechanisms that make prostate cancer metastasize to bone have recently been identified, and they provide new molecular targets for drug development. Many biochemical by-products have been identified to help in understanding the interaction between the bone and the tumor. Enhanced clinical management of patients with bone metastases was reported during the past decade; however, the anticipated risk and the response to the therapy are still challenging to assess. In this review, the key players that play a dominant role in secondary osteoporosis are addressed. An attempt is made to provide the readers with a clear understanding of the communication pathways between each of the cell types involved in this vicious cycle. Furthermore, the role of Wnts, sclerostin, RANKL, PTHrP, and their respective clinical studies are addressed in this study.

New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity.

The inequality of Popoviciu, which was improved by Vasic and Stankovic (Math. Balk. 6:281-288, 1976), is generalized by using new identities involving new Green's functions. New generalizations of an improved Popoviciu inequality are obtained by using generalized Montgomery identity along with new Green's functions. As an application, we formulate the monotonicity of linear functionals constructed from the generalized identities, utilizing the recent theory of inequalities for n-convex functions at a point. New upper bounds of Grüss and Ostrowski type are also computed.