The aim of this paper is to compare single versus multiple gall stone disease in patients undergoing laparoscopic cholecystectomy. A total of one hundred and thirty patients (males- 52, females- 78) within age group 20-65 years presented with cholelithiasis were divided into 2 groups. Group 1 comprised of single stone patients and group II had multiple stones. Each group had 65 patients. Parameters compared were symptoms and operative findings between both groups, Analysis of results was carried with chi0 square test. Most common clinical symptoms found to be fever in both groups (87%- group 1, 90%- group 2). Other symptoms were dyspepsia seen in 25% and 30%, nausea/ vomiting in 45% and 38% in group 1 and 2 patients respectively. Females were commonly affected and maximum case were observed in age group 40-50 years. Multiple gall stones patients had more symptoms and prevalence of conversion to open surgery was more in these patients. Group 1 patients showed gall bladder distention in 56 and group 2 in 62 patients, gall bladder contraction was seen in 6 patients in group 1 and 3 in group 2, adhesions were exhibited by 11 patients in group 1 and 20 in group 2. 4 patients in group 2 converted to open surgery. Gangrenous gall bladder was seen in 1 in group 1 and 5 in group 2.

The aim of this paper is to compare different treatment modalities in management of plantar fasciitis. Forty- five patients with plantar fasciitis (males- 2- females- 20) were randomized into 3 groups. Group 1 patients received conventional treatment, group 2 patients received heat treatment with silicone heel pad and group 3 patients received active plantar fascia stretching with sham calf stretching. Patient was evaluated using foot function index (FFI) and the foot and ankle disability index (FADI). In group 1, foot function pre-intervention index score and post-intervention score was 68.2 and 43.5, in group 2 was 64.8 and 6.9 and in group 3 was 70.5 and 1.6 respectively. A significant difference was observed (P< 0.05). In group 1, pre-intervention foot and ankle disability score and post-intervention score was 61.5 and 76.4, in group 2 was 66.5 and 95.2 and in group 3 was 61.3 and 97.6 respectively. A significant difference was observed (P< 0.05). Active plantar fascia stretching with sham calf stretching exercise resulted in most significant improvement in FFI and FADI score followed by treatment with heat and silicone heel pad and conventional treatment.

In this paper, the \(q\)-derivative operator and the principle of subordination were employed to define a subclass \(\mathcal{B}_q(\tau,\lambda,\phi)\) of analytic and bi-univalent functions in the open unit disk \(\mathcal{U}\). For functions \(f(z)\in\mathcal{B}_q(\tau,\lambda,\phi)\), we obtained early coefficient bounds and some Fekete-Szegö estimates for real and complex parameters.

In this paper, we give characterizations of separation criteria for bitopological spaces via \(ij\)-continuity. We show that if a bitopological space is a separation axiom space, then that separation axiom space exhibits both topological and heredity properties. For instance, let \((X, \tau_{1}, \tau_{2})\) be a \(T_{0}\) space then, the property of \(T_{0}\) is topological and hereditary. Similarly, when \((X, \tau_{1}, \tau_{2})\) is a \(T_{1}\) space then the property of \(T_{1}\) is topological and hereditary. Next, we show that separation axiom \(T_{0}\) implies separation axiom \(T_{1}\) which also implies separation axiom \(T_{2}\) and the converse is true.

The strategic value of the Navy’s submarine capabilities is greatly influenced by the ability of its supporting infrastructure to be able to maintain the confidentiality of submarine operations. Based on these conditions, this study aims to identify and analyze the operational readiness of submarines to increase the deterrence of the state defense system at sea. This study uses Interpretive Structural Modeling (ISM) and MIC-MAC diagrams. The ISM method is used to analyze the effect on the criteria. MICMAC diagrams are used as criteria for determining the related variables. Based on the results of the ISM and MICMAC diagram analysis, it was found that fifteen elements were divided into \(4\) (four) quadrants. Quadrant I (Autonomous) consists of six elements, such as \((C_1)\), \((C_2)\), \((C_3)\), \((C_4)\), \((C_{13)}\), \((C_{15})\). Quadrant II (Dependent) consists of six elements, such as \((C_5)\), \((C_6)\), \((C_7)\), \((C_{11})\), \((C_{12})\), \((C_{14})\). Quadrant IV (Independent) consists of three elements, such as \((C_8)\), \((C_9)\), \((C_{10})\). This research is expected to provide benefits to both academics and practitioners in the field of submarines and defense strategy.

A q-rung orthopair fuzzy matrix (q-ROFM), an extension of the Pythagorean fuzzy matrix (PFM) and intuitionistic fuzzy matrix (IFM), is very helpful in representing vague information that occurs in real-world circumstances. In this paper we define some algebraic operations, such as max-min, min-max, complement, algebraic sum, algebraic product, scalar multiplication \((nA)\), and exponentiation \((A^n)\). We also investigate the algebraic properties of these operations. Furthermore, we define two operators, namely the necessity and possibility to convert q-ROFMs into an ordinary fuzzy matrix, and discuss some of their basic algebraic properties. Finally, we define a new operation(@) on q-ROFMs and discuss distributive laws in the case where the operations of \(\oplus_{q}, \otimes_{q}, \wedge_{q}\) and \(\vee_{q}\) are combined each other.

There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there is an infinite number of possible definitions of fractional derivatives, all are correct as differential operators each of them must be properly defined its algebra. We introduce a generalized version of fractional derivative that extends the existing ones in the literature. To those extensions it is associated a differentiable operator and a differential ring and applications that shows the advantages of the generalization. We also review the different definitions of fractional derivatives and it is shown how the generalized version contains the previous ones as a particular cases.

Let \(G = (X, Y; E)\) be a bipartite graph with two vertex partition subsets \(X\) and \(Y\). \(G\) is said to be balanced if \(|X| = |Y|\) and \(G\) is said to be bipancyclic if it contains cycles of every even length from \(4\) to \(|V(G)|\). In this note, we present spectral conditions for the bipancyclic bipartite graphs.

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify some particular sums such as the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive formulae for simplifying general repeated sums as well as a variant containing binomial coefficients. Additionally, we study the \(m\)-th difference of a sequence and show how sequences whose \(m\)-th difference is constant can be related to binomial coefficients.

In this paper, we present the notion of generalized \(F\)-expansive mapping in complete rectangular metric spaces and study various fixed point theorems for such mappings. The findings of this paper, generalize and improve many existing results in the literature.