Finite Value Research Articles

Uniqueness results of Differential Polynomials of L-functions sharing one finite value

Uniqueness results of Differential Polynomials of L-functions sharing one finite value

Determining sheath edge electric field around cylindrical pins of a DC-biased hairpin resonator probe

The sheath-edge electric field ( Es ) is an important parameter to patch the quasi-neutral pre-sheath and non-neutral sheath regions. The choice of Es significantly influences the theoretically estimated values of the sheath width, potential, and ion density distribution inside the sheath, as determined by the Poisson equation. The precise nature of Es has been a persistent subject of investigation, giving rise to the question of whether it should be zero or possess a finite value, as proposed by various authors. In this study, we determine the values of Es by solving Poisson’s equation as a boundary-value problem, utilizing experimentally determined values of sheath radius from a DC-biased hairpin probe. The obtained values of Es are found to be finite and closely align with the analytical expressions presented by Riemann (1991 J. Phys. D: Appl. Phys. 24 493) and Kaganovich (2002 Phys. Plasmas 9 4788). Additionally, the impact of electron-penetrating sheaths and interacting sheaths on the applicability of the hairpin probe in low-pressure plasmas is briefly discussed.

Subgrid moving contact line model for direct numerical simulations of bubble dynamics in pool boiling of pure fluids

This contact line vicinity model is conceived as a subgrid model for the DNS of bubble growth in boiling. The model is based on the hydrodynamic multiscale theory and is suitable for the partial wetting case. On the smallest length scale (distance from the contact line) ∼ 100 nm, the interface slope is controlled by the Voinov angle. It is the static apparent contact angle (ACA) that forms due to evaporation, similarly to previous models neglecting the contact line motion. The calculation of the Voinov angle is performed with the generalized lubrication approximation and includes several nanoscale effects like those of Kelvin and Marangoni, vapor recoil, hydrodynamic slip length and interfacial kinetic resistance. It provides the finite values of the heat flux, pressure and temperature at the contact line. The dynamic ACA is obtained with the Cox-Voinov formula. The microscopic length of the Cox-Voinov formula (Voinov length) is controlled mainly by the hydrodynamic slip. The integral heat flux passing through the contact line vicinity is almost independent of the nanoscale phenomena, with the exception of the interfacial kinetic resistance and is mostly defined by the dynamic ACA. Both the dynamic ACA and the integral heat flux are the main output parameters of the subgrid model, while the local superheating and the microscopic contact angle are the main input parameters. The model is suitable for the grid sizes > 1 µm.

A Force Function Formula for Solutions of Nonlinear Weakly Singular Volterra Integral Equations

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation(WSVIE), $u(x)=f(x)+\int_{0}^{x}\frac{t^{\mu-1}}{x^\mu}[u(t)]^\beta dt$. AL-Jawary and Shehan used Daftardar-Jafari Method(DJM) and solved the above integral equation for the investigation parameter $\mu>1$ using specific force functions with $\mu$ and $\beta$ values and obtained unique solutions. We have discovered a force function $f(x)=x^{k_1}-{\frac{x^{\gamma k_1}}{\gamma k_1+\mu}}$, that allows the introduction of noise terms phenomena discovered by Wazwaz; that cancel out the terms of the power series in the successive solution terms $u_m$, $m=0,1,2,...,n$: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by $x^{g(n)}$. Such that the integral solution can be written as $u(x)=u_0+\sum_{m=1}^{n}u_m$, where $n$ is finite. Simplifying the solution terms we get the unique solution $u(x)=x^{k_1}$, irrespective of $n-$value in the truncation point. We discovered a formula relation between the last solution term $u_n$ and the truncation point as $u_n=a_nx^{g(n)}$. Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter $\mu>1$. We extend the parameter range to include $\mu>1$ and $0<\mu\leq 1$ for our solution. In addition, for any chosen rational parameter $k_1$, the solution $u(x)=x^{k_1}$ is extrapolated to be valid for all integer parameter values $\beta\geq 2$, and positive rational parameter value $\mu>0$ and for any finite value of $n\geq2$.

Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence

Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number Reλ is infinitely large, thus raising doubts about published power-law exponents at finite values of Reλ, for either the second-order velocity structure function (δu)2¯ or the energy spectrum. Here, we focus on the transport equation of (δu)2¯ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as Reλ increases and finally disappears when Reλ is sufficiently large. An approach based on the dual scaling of (δu)2¯, i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields (δu)2¯∼r2/3 when Reλ is infinitely large. This approach also yields (δu)n¯∼rn/3 when Reλ is infinitely large. These results seem to be supported by the trend, as Reλ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations.

Slow passage through a transcritical bifurcation in piecewise linear differential systems: Canard explosion and enhanced delay

In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay zd(λ,ɛ) as a function of the bifurcation parameter λ and the singular parameter ɛ. We quantify the maximal delay by constructing a piecewise linear (PWL) transcritical minimal model and studying the dynamics near the slow-manifolds. Our findings encompass all potential maximum delay behaviours within the range of parameters, allowing us to identify: (i) the trivial scenario where the maximal delay tends to zero with the singular parameter; (ii) the singular scenario where zd(λ,ɛ) is not bounded, and also (iii) the transitional scenario where the maximal delay tends to a positive finite value as the singular parameter goes to zero. Moreover, building upon the concepts by Vidal and Françoise (2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter λ changes, the system presents a non-bounded canard explosion leading to an enhanced delay phenomenon at the critical value. Our understanding of the maximal delay zd(λ,ɛ) of a single normal form, allows us to determine both, the amplitude of the canard cycles and, in the enhanced delay case, the increase of the amplitude for each passage.

Bouncing Cosmology in f(R,G)$f(R,\mathcal {G})$ Gravity with Thermodynamic Analysis

AbstractThe evolution of the universe in a modified gravity theory that includes the terms Ricci scalar () and the Gauss‐Bonnet invariant () is studied. The function is obtained using the e‐folding number and reconstruction technique by assuming an appropriate parameterization of the scale factor . In this model, the various cosmological parameters are analyzed to explicate the bouncing scenario of the universe with the help of the contraction and expansion phases of the universe before and after the bouncing point of the model, respectively. A violation of the null energy condition is found. Additionally, the ghost condensate nature of the model in the neighborhood of the bouncing point () is seen. Furthermore, the deceleration parameter is not defined at the bouncing point, i.e., at and approaches a finite value in later times (). Finally, the evolution of the Hawking temperature and the validity of the second law of thermodynamics in the bouncing scenario of our model are discussed.

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

Novel consensus framework on discrete-time positive multi-agent systems

This paper investigates the consensus of discrete-time positive multi-agent systems. A framework on the consensus of positive multi-agent systems is constructed. By introducing a finite point and a self-feedback term, a novel consensus protocol is proposed to drive all states to common nonnegative finite values. An improved error model is established for positive multi-agent systems. Based on the consensus error model, the consensus of the agents is transformed into the stability of the error systems. Under the proposed consensus protocol and the new consensus error model, the consensus of agents is realized. A copositive Lyapunov function is chosen to derive the consensus of the multi-agent systems. Such a consensus approach is extended for time-delay multi-agent systems. All positivity and consensus conditions can be solved by virtue of linear programming. The main novelties of this paper lie in that: (i) A novel consensus error model is established, (ii) A finite value consensus is achieved, and (iii) Linear programming-based conditions are presented for the consensus design. Finally, two illustrated examples are given to verify the validity of the theoretical findings.

The Fresnel Approximation and Diffraction of Focused Waves

In this paper, diffraction of scalar waves by a screen with a circular aperture is explored, considering the incidence of either a collimated beam or a focused wave, a historical review of the development of the theory is presented, and the introduction of the Fresnel approximation is described. For diffraction by a focused wave, the general case is considered for both high numerical aperture and for finite values of the Fresnel number. One aim is to develop a theory based on the use of dimensionless optical coordinates that can help to determined the general behaviour and trends of different system parameters. An important phenomenon, the focal shift effect, is discussed as well. Explicit expressions are provided for focal shift and the peak intensity for different numerical apertures and Fresnel numbers. This is one application where the Rayleigh–Sommerfeld diffraction integrals provide inaccurate results.

Dissociation of J/ψ and Y Using Dissociation Energy Criteria in N-Dimensional Space

The analytical exact iteration method (AEIM) has been widely used to calculate N-dimensional radial Schrodinger equation with medium-modified form of Cornell potential and is generalized to the finite value of magnetic field (eB) with quasiparticle approach in hot quantum chromodynamics (QCD) medium. In N-dimensional space, the energy eigenvalues have been calculated for any states (n, l). These results have been used to study the properties of quarkonium states (i.e, the binding energy and mass spectra, dissociation temperature, and thermodynamical properties in the N-dimensional space). We have determined the binding energy of the ground states of quarkonium with magnetic field and dimensionality number. We have also determined the effects of magnetic field and dimensionality number on mass spectra for ground states of quarkonia. But the main result is quite noticeable for the values of dissociation temperature in terms of magnetic field and dimensionality number for ground states of quarkonia after using the criteria of dissociation energy. At last, we have also calculated the thermodynamical properties of QGP (i.e., pressure, energy density, and speed of sound) using the parameter eB with ideal equation of states (EoS). A preprint has previously been published (Solanki et al., 2023).

Three point amplitudes in matrix theory

We compute the three graviton amplitude in the Banks-Fischler-Shenker-Susskind matrix model for M-theory. Even though the three point amplitude is determined by super Poincare invariance in eleven dimensional M-theory, it requires a non-trivial computation in the matrix model. We consider a configuration where all three gravitons carry non-zero longitudinal momentum. To simplify the problem, we compactify one additional dimension and relate the amplitude to a supersymmetric index computation. We find agreement with the expected answer even at finite values of N.

Optimized coupling of photocatalysis and cavitation for phenol degradation: Use of an extended-kinetic approach

The coupling of hydrodynamic cavitation (HC) and photocatalysis (PC) predominantly exhibits a complementary nature, which is highlighted through a synergistic index. However, this calculation, based on pseudo-first order kinetics, fails to accurately represent the concentration data, often skewed by the formation of reaction intermediates. To address this, an ‘extended’ kinetic approach previously developed with two parameters, which accounts for the formation of intermediates, was adopted. The PC process was optimized for a simulated wastewater containing phenol of concentration (C0) of 100 ppm by applying a UV-A light source of intensity 175 ± 8 W/cm2 on an operating volume (VL) of 200 mL. Catalyst loading, solution pH and initial concentration (C0) were optimized. These optimal parameters were used to operate HC-PC (VL = 3500 mL) at a comparable illumination intensity across the 2 techniques’ reactors. It was observed that phenol conversion was observable only with halved C0 (= 50 ppm) for the HC-PC system, increasing catalyst loading from 0.5 to 1.0 g/L exhibited no increase in phenol conversion. The obtained results from experiments were interpreted with the help of the two-parameter model. For the PC system, the initial rate constant (k0) exhibits a similar trend to the final oxidation extent, however, it did not compare quantitatively. The second parameter – ‘y’ showed a high finite value, denoting the need for dosing external oxidants. The trends of the two-parameters with respect to different relevant parameters presented will help leverage this kinetic approach for AOPs, while optimizing and translating processes to larger scales of operation.

Mode-II stress intensity factors for special-shaped cylindrical shells in torsion

Special-shaped cylindrical shells in torsion are encountered in engineering applications, which will induce the Mode-II singular stress fields next to circumferential crack tips. The Mode-II stress intensity factor (SIF) represents the strength of these singular stress fields and is an important fracture parameter. The crack problems for the cylindrical shells are typical finite boundary value issues. It is very difficult to get the exact solutions by using the classical theory of elasticity. In present article, based on the concepts of the conservation law and the elementary strength theory, a hybrid methodology to determine the Mode-II SIFs for circumferential cracked special-shaped cylindrical shells in torsion is suggested. Due to the introduction of elementary mechanics, the corresponding errors are also inevitably carried into the Mode-II SIFs. However, the finite element analysis indicates that the error of the Mode-II SIFs calculated by present method is acceptable.

Observables from classical black hole scattering in Scalar-Tensor theory of gravity from worldline quantum field theory

In this paper, we compute the two observables, impulse and waveform, in a black hole scattering event for the Scalar-Tensor theory of gravity with a generic scalar potential using the techniques of Worldline Quantum Field Theory. We mainly investigate the corrections to the above mentioned observables due to the extra scalar degree of freedom. For the computation of impulse, we consider the most general scenario by making the scalar field massive and then show that each computed diagram has a smooth massless limit. We compute the waveform for scalar and graviton up to 2PM, taking the scalar as massless. Furthermore, we discuss if the scalar has mass and how the radiation integrals get more involved than the massless case. We also arrive at some analytical results using stationary phase approximation. Interestingly, we also show that the λ4φ4 interaction vertex does not contribute to the radiation by showing that the integral has no non-zero finite value.

Schrödinger equation as a confluent Heun equation

This article deals with two classes of quasi-exactly solvable (QES) trigonometric potentials for which the one-dimensional Schrödinger equation reduces to a confluent Heun equation (CHE) where the independent variable takes only finite values. Power series for the CHE are used to get polynomial and nonpolynomial eigenfunctions. Polynomials occur only for special sets of parameters and characterize the quasi-exact solvability. Nonpolynomial solutions occur for all admissible values of the parameters (even for values which give polynomials), and are bounded and convergent in the entire range of the independent variable. Moreover, throughout the article we examine other QES trigonometric and hyperbolic potentials. In all cases, for a polynomial solution there is a convergent nonpolynomial solution.

Electric and elastic characteristics of the orthorhombic paramagnet after the quantum quench

Quantum quench (the sudden change) of the magnetic field in the orthorhombic and monoclinic paramagnets yields the renormalization of the spin quadrupole moment of the system. In the closed system, the renormalized spin quadrupole moment oscillates with the frequency, determined by the final value of the magnetic field. In the open system, these oscillations are damped in the steady-state regime. The renormalization of the electric permittivity, piezoelectric modulus, and elastic modulus of the system are dependent on both initial and finite values of the magnetic field via the spin quadrupole susceptibility demonstrating features at the critical values of the magnetic field, at which crossover of the energy levels of the paramagnet takes place.

Exploring Challenging Properties of Liquid Metallic Systems through Machine Learning: Liquid La and Li4Pb Systems.

In this machine learning (ML) study, we delved into the unique properties of liquid lanthanum and the Li4Pb alloy, revealing some unexpected features and also firmly establishing some of the debated characteristics. Leveraging interatomic potentials derived from ab initio calculations, our investigation achieved a level of precision comparable to first-principles methods while at the same time entering the hydrodynamic regime. We compared the structure factors and pair distribution functions to experimental data and unearthed distinctive collective excitations with intriguing features. Liquid lanthanum unveiled two transverse collective excitation branches, each closely tied to specific peaks in the velocity autocorrelation function spectrum. Furthermore, the analysis of the generalized specific heat ratio in the hydrodynamic regime investigated with the ML molecular dynamics simulations uncovered a peculiar behavior, impossible to discern with only ab initio simulations. Liquid Li4Pb, on the other hand, challenged existing claims by showcasing a rich array of branches in its longitudinal dispersion relation, including a high-frequency LiLi mode with a nonhydrodynamic optical character that maintains a finite value as q → 0. Additionally, we conducted an in-depth analysis of various transport coefficients, expanding our understanding of these liquid metallic systems. In summary, our ML approach yielded precise results, offering new and captivating insights into the structural and dynamic aspects of these materials.

Mechanical and thermal characterization of additively manufactured novel multilayer polymer composite through experiments and finite element simulation

This study presents the investigation of mechanical and thermal properties of additively manufactured novel multilayer composites (MLC). Wood and carbon-reinforced poly-lactic acid were used as the base materials for the 3D printing of the MLC. The tensile, compressive, and flexural strengths of the developed MLC were 51.47, 70.90, and 73.3 MPa, respectively. The finite element simulation values of tensile and compressive strength of the MLC were 51.01 and 70.02 MPa depicting negligible variations of 0.90% and 1.24%. The glass transition temperatures for the WRP, CFRP, and MLC specimens were 54.85 °C, 57.29 °C, and 56.78 °C, respectively.

Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple T-Values

In this paper, we will study finite multiple T-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author’s previous conjecture that the finite Euler sum space of weight, w, is isomorphic to a quotient Euler sum space of weight, w. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important Q-linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the Q-span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers.