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The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\\geq 2$ . Apart from results concerning the asymptotic behaviour of $p_{d}(n)$ , little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth powers. The second part of the paper is experimental and contains questions and conjectures concerning the arithmetic behaviour of the sequence $(p_{d}(n))_{n\\in \\mathbb {N}}$ , based on computations of $p_{d}(n)$ for $n\\leq 10^5$ for $d=2$ and $n\\leq 10^{6}$ for $d=3, 4, 5$ .

This work emphasizes the computational and analytical analysis of integral-differential equations, with a particular application in modeling avoidance learning processes. Firstly, we suggest an approach to determine a unique solution to the given model by employing methods from functional analysis and fixed-point theory. We obtain numerical solutions using the approach of Picard iteration and evaluate their stability in the context of minor perturbations. In addition, we explore the practical application of these techniques by providing two examples that highlight the thorough analysis of behavioral responses using numerical approximations. In the end, we examine the efficacy of our suggested ordinary differential equations (ODEs) for studying the avoidance learning behavior of animals. Furthermore, we investigate the convergence and error analysis of the proposed ODEs using multiple numerical techniques. This integration of theoretical and practical analysis enhances the domain of applied mathematics by providing important insights for behavioral science research.

This study undertakes a comprehensive analysis of second-order Ordinary Differential Equations (ODEs) to examine animal avoidance behaviors, specifically emphasizing analytical and computational aspects. By using the Picard–Lindelöf and fixed-point theorems, we prove the existence of unique solutions and examine their stability according to the Ulam-Hyers criterion. We also investigate the effect of external forces and the system’s sensitivity to initial conditions. This investigation applies Euler and Runge–Kutta fourth-order (RK4) methods to a mass-spring-damper system for numerical approximation. A detailed analysis of the numerical approaches, including a rigorous evaluation of both absolute and relative errors, demonstrates the efficacy of these techniques compared to the exact solutions. This robust examination enhances the theoretical foundations and practical use of such ODEs in understanding complex behavioral patterns, showcasing the connection between theoretical understanding and real-world applications.

In this study the scattering behaviour of Shear Horizontal (SH) waves by a circular lining embedded within a layered piezoelectric material was comprehensively investigated. The methods including the wave function expansion approach, the repetitive superposition of the mirror-image technique, and the complex-variable function method were employed. The boundaries of the layered structure were assumed to satisfy stress-free and electrically insulating conditions. Based on the boundary conditions imposed on the circular lining, a system of definite-integral equations for the solution was constructed. And then the numerical computations were carried out to precisely determine the dynamic stress and electric-field strength of the piezoelectric material in the vicinity of the lining. The impacts of key parameters such as the incident wave number, the thickness of the lining, and the thickness of the layers on the lining were meticulously explored. The results revealed that upon the incidence of SH waves into the interior of the structure, the stress concentrations induced by the electromechanical coupling effect at the inner and outer boundaries of the lining exhibited opposite trends. Specifically, the dynamic stress concentration coefficient at the inner boundary of the lining was notably higher than that at the outer boundary, whereas the pattern of the electric-field-strength concentration coefficient was reversed. Moreover, under high-frequency wave incidence the stress variation of the defect region was found to be more sensitive. Significantly, it is demonstrated that the damage resulting from the electromechanical coupling effect can be effectively mitigated by appropriately adjusting the thickness of the lining and the thickness of the layers.

Owing to enhanced thermal characteristics of nanomaterials, multidisciplinary applications of such particles have been utilized in the industrial and engineering processes, chemical systems, solar energy, extrusion processes, nuclear systems etc. The aim of current work is to suggests the thermal performances of thixotropic nanofluid with interaction of magnetic force. The suspension of microorganisms in thixotropic nanofluid is assumed. The investigation is further supported with the triple diffusion flow. The motivations for considering the triple diffusion phenomenon are associated to attaining more thermal applications. The flow pattern is subject to novel stagnation point flow. The convective thermal constraints are incorporated. The modeled problem is numerically evaluated by using shooting technique. Different consequences of physical parameters involving the problem are graphically attributed. The insight analysis is presented for proposed problem with different engineering applications. It is claimed that induced magnetic field enhanced due to magnetic parameter while declining results are observed for thixotropic parameter. The heat transfer enhances due to variation of Dufour number. Furthermore, low profile of nanoparticles concentration has been observed for thixotropic parameter and nano-Lewis number.

Owing to advanced thermal features and stable properties, scientists have presented many novel applications of nanomaterials in the energy sectors, heat control devices, cooling phenomenon and many biomedical applications. The suspension between nanomaterials with microorganisms is important in biotechnology and food sciences. With such motivations, the aim of current research is to examine the bioconvective thermal phenomenon due to Reiner–Philippoff nanofluid under the consideration of multiple slip effects. The assessment of heat transfer is further predicted with temperature dependent thermal conductivity. The radiative phenomenon and chemical reaction is also incorporated. The stretched surface with permeability of porous space is assumed to be source of flow. With defined flow constraints, the mathematical model is developed. For solution methodology, the numerical simulations are worked out via shooting technique. The physical aspects of parameters are discussed. It is claimed that suggested results claim applications in the petroleum sciences, thermal systems, heat transfer devices etc. It has been claimed that the velocity profile increases due to Bingham parameter and Philippoff constant. Lower heat and mass transfer impact is observed due to Philippoff parameter.

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains.

Even though the cycloidal rotor concept has been around for almost a century, it is still not as popular as it should be. Most often it is used to propel unmanned aerial vehicles or sea-going ships, or it is applied as a river- or sea-energy converter. Despite the possibility of directing the flow by changing the inclination angle of blades and the possibility of working in both directions, there are no scientific studies on the use of the concept in HVAC (heat, ventilation and air conditioning). One of the most important elements characterizing the operation of the cycloidal rotor is the cycloidal function describing the change in the angles of the blades during rotation. To properly design a cycloidal rotor for a preferred application, an analysis of the rotor geometrical parameters must be performed and analyzed. This was performed on a four-blade rotor equipped with CLARK Y blades. Using Ansys CFX software, a CFD model of a fan operating with various cycloidal functions was created. The results were compared with the experimental data with the use of the LDA technique. Different velocity profiles were obtained despite the use of cycloidal functions with similar waveforms and small angular differences. This is due to the considerable sensitivity of the cycloidal regulation system to differences in the geometrical sizes that describe it.

An algorithm and a MATLAB implementation for computing the Kummer function U ( a , b , x ) and its derivative is given in this paper. The algorithm is efficient and accurate. Numerical tests show that the MATLAB algorithm allows the computation of the function with ∼ 1 0 − 14 relative accuracy in the parameter region ( a , b , x ) ∈ (0,500) × (0,500) × (0,1000) in double-precision floating point arithmetic.

We propose fast numerical algorithms to improve the accuracy of singular vectors for a real matrix. Recently, Ogita and Aishima proposed an iterative refinement algorithm for singular value decomposition that is constructed with highly accurate matrix multiplications carried out six times per iteration. The algorithm runs for the problem that has no multiple and clustered singular values. In this study, we show that the same algorithm can be run with highly accurate matrix multiplications carried out five times. Also, we proposed four algorithms constructed with highly accurate matrix multiplications, two algorithms with the multiplications carried out four times, and the other two with the multiplications carried out five times. These algorithms adopt the idea of a mixed-precision iterative refinement method for linear systems. Numerical experiments demonstrate speed-up and quadratic convergence of the proposed algorithms. As a result, the fastest algorithm is 1.7 and 1.4 times faster than the Ogita-Aishima algorithm per iteration on a CPU and GPU, respectively.