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Euler’s Number in Right Triangles: Euler’s Number in Right Triangles
No full textThis paper reveals an explicit and intriguing formula that yields the precise value of Euler’s number (e) from the side lengths of a right angled triangle
Metallic Means in Primitive Pythagorean Triples : Metallic Ratios substantiated in Pythagorean Triangles and other Right Angled Triangles
Full text linkThis paper elaborates the precise correlation between metallic ratios and the different families of primitive Pythagorean triples. This paper also discusses the explicit formulae those provide the mathematical relationships between different metallic numbers. Main purpose of this paper is to highlight the geometric substantiation of metallic means in Pythagorean triples and other right triangles, and the trigonometric expression of the metallic ratios
Exact Solutions and Stability Analysis of Pulse-Front Pairs in Coupled Complex Ginzburg–Landau Equations
Full text linkThis work introduces new exact solutions demonstrating how localized pulses and fronts can coexist in coupled complexGinzburg–Landau systems. Using a novel analytical method, we establish conditions for the stability and phase-lockingof these structures, revealing relationships between amplitude, wave-number, and dispersion effects. In practical opticalsetups like dual-core fibers, these solutions can produce stable wave patterns that transfer energy efficiently. Ourapproach addresses existing difficulties in analyzing complex dissipative systems and enhances understanding of theirwave interactions
Application of Metaheuristics for Facility Location Optimization
Full text linkFacility location problems are a fundamental component of supply chain optimization, particularly in agriculture, where collection centers must be strategically positioned to minimize transportation costs and improve accessibility for producers. Traditional mathematical programming techniques are suitable for small-scale problems but become computationally expensive as the problem size increases. To overcome these limitations, this study applies metaheuristic approaches, specifically Genetic Algorithm (GA) and Particle Swarm Optimization (PSO), to determine the optimal siting of agricultural collection centers in Elbasan, Albania. The case study considers thirteen administrative areas with annual production volumes used as demand weights, while distances are calculated using geographic coordinates. The proposed algorithms aim to minimize the weighted travel distance between farmers and assigned collection facilities. Results show that both GA and PSO successfully identify near-optimal solutions with significantly reduced total transportation costs compared to single-facility baselines.
Solving cubic and quartic equations by means of Vieta's formulas
Full text linkIn this paper, we will prove that with the use of Vieta's formulas, it is possible to apply a unified method in solving equations of the third and fourth degree
Controlling Chaos in the Lozi Map Using Hidden Variables: A Novel Approach for Stability Enhancement
Full text linkIn this paper, we provide a novel 3D system that demonstrates the existence of a hidden attractor. Although equilibrium points are present in the system, this hidden attractor cannot be found by examining equilibrium points or their surroundings, in contrast to self-excited attractors. Differential equation theory explains this behavior by showing that the system has intricate dynamics that are not directly dependent on equilibrium points. We added a hidden variable to the system to make it more complex and improve its stability by adding the following equation to the original system: zn+1=wzn+r3xn, where the hidden variable interacts with the original system through control parameters r1, r2, and r3, increasing chaos or improving the system stability. where it appeared us more complex dynamic behaviors
Ordering Unicyclic Graphs with a Fixed Girth by p-Sombor Indices
Full text linkThe p-Sombor index of a graphs G is deffned as, SOp(G) = X xy∈E(G) (d p (x) + d p (y)) 1 p , where d(x) represents the degree of vertex x in graph G. Our focus centers on exploring the p-Sombor index of unicyclic graphs, speciffcally addressing graphs with a predetermined girth. We determine the ffrst four smallest p-Sombor index and identifying the corresponding graphs that achieve these extremes
Solving cubic equation using Cardano’s method
Full text linkA cubic equation is solvable by radicals. This means that the solutions of the cubic equation can be obtained usingfour basic arithmetic operations which include addition, subtraction, multiplication, and division, and taking the squareand cube roots
A new generalization on metric space and its metrizability.
Full text linkIn this paper, our particular scope is to give a new generalization for the metric function and after that a proof of the metrizability of generalized φ-metric space. This new approach is influenced by the Chittenden’s metrization theorem
Significant Effect Of Amended Soil On Microbial Flora Of Soil And Plant Growth In Comparison With Natural Low Nutritive Soil
Full text linkSoil chemical properties, such as Carbon and Nitrogen levels, are crucial for fulfilling the basic needs of plants. The presence of secondary and tertiary nutrients plays a vital role in constructing soil structure, influencing the survival of normal microbial flora, seeds, and later plant growth. The use of chemical fertilizers addresses nutritional deficiencies, but reports indicate that only 40% is utilized by plants, leading to soil issues like salinity and drought logging. When soil lacks essential nutrients, external amendments are necessary, impacting plant growth and crop yield by enhancing normal microbial flora i.e. Plant growth-promoting rhizobacteria (PGPRs). Recognizing the significance of both chemical nutrients and physical properties for healthy plant growth, a soil amendment called "Fertisol soil" was developed. This involved adding basic and secondary nutrients to the soil. The amended soil underwent analytical characterization and was compared with natural soil through chemical analysis and microbial growth assessments. Comparative studies using the amended soil showed improved plant growth and flowering capacity compared to control plants in terms of height and flowering. This highlights the positive impact of balanced nutrient amendments on soil health and overall plant performance