Algebraic Functions Research Articles

LiDICS: a light dynamics interface for multi-domain simulation based on the FMI-standard

The assessment of adaptive façades faces several challenges, including the representation of adjustable material and geometric properties within building performance simulation. This paper introduces a methodical and model-based interface dedicated to this issue. Adjustable material and geometric façade properties can be captured and continuously adapted within the building performance simulation using this model, without the need to reinitialise the simulation as required with matrix-based methods. The method comprises two steps: model training and export. Model training transforms ray-tracing-based sampling outcomes into an algebraic function. Model export generates a Functional Mock-up Unit. The LiDICS method is applied to switchable membrane cushion constructions with complementary printing patterns. The model training achieves a high accuracy ( r 2 = 99.4 % ). Subsequently, an inter-model validation is conducted using annual simulation from the RADIANCE 3-phase method, likewise resulting in high accuracy ( r 2 = 98.5 % ). Lastly, a brief analysis is conducted to compare four operation strategies regarding coupled comfort and energy assessment.

Anisotropic quark stars in $f(R,L_m,T)$ gravity

Abstract We investigate the impact of $f(R,L_m,T)$ gravity on the internal structure of compact stars, expecting this theory to manifest prominently in the high-density cores of such stars. In this study, we begin by considering the algebraic function $f(R,L_m,T) = R + \alpha T L_m$, where $\alpha$ represents the matter-geometry coupling constant. We specifically choose the matter Lagrangian density $L_m= -\rho$ to explore compact stars with anisotropic pressure. To this end, we employ the MIT bag model as an equation of state. We then numerically solve the hydrostatic equilibrium equations to obtain mass-radius relations for quark stars, examining static stability criteria, the adiabatic index, and the speed of sound. Finally, we use recent astrophysical data to constrain the coupling parameter $\alpha$, which may lead to either larger or smaller masses for quark stars compared to their counterparts in general relativity.

Analysis and Implementation of Quantum Gates (X, Z, Y, Hadamard CNOT and CCNOT) by using matrices for Quantum Computing

The constraints of present supercomputers in addressing exponentially intricate problems are apparent, as these jobs sometimes necessitate hundreds or even thousands of years due to the exponential increase in computational requirements. Quantum computing, which utilizes concepts from quantum physics and operates with qubits, offers a promising solution by significantly decreasing problem-solving durations to only seconds or minutes. Using IBM's quantum computing platform, this work investigates the construction and analysis of fundamental quantum gates, including X, Z, Y, Hadamard, CNOT, and CCNOT. This work presents a comprehensive analysis of these gates using matrix representations, demonstrating their operational mechanics and algebraic functions, and fundamental role of matrix theory in quantum gate operations and its contribution to the future development of quantum computing.

On the automorphisms of the Drinfeld modular groups

Let [Formula: see text] be the ring of elements in an algebraic function field [Formula: see text] over [Formula: see text] which are integral outside a fixed place [Formula: see text]. In contrast to the classical modular group [Formula: see text] and the Bianchi groups, the Drinfeld modular group [Formula: see text] is not finitely generated and its automorphism group [Formula: see text] is uncountable. Except for the simplest case [Formula: see text] not much is known about the generators of [Formula: see text] or even its structure. We find a set of generators of [Formula: see text] for a new case. On the way, we show that every automorphism of [Formula: see text] acts on both the cusps and the elliptic points of [Formula: see text]. Generalizing a result of Reiner for [Formula: see text] we describe for each cusp an uncountable subgroup of [Formula: see text] whose action on [Formula: see text] is essentially defined on the stabilizer of that cusp. In the case where [Formula: see text] (the degree of [Formula: see text]) is [Formula: see text], the elliptic points are related to the isolated vertices of the quotient graph [Formula: see text] of the Bruhat–Tits tree. We construct an infinite group of automorphisms of [Formula: see text] which fully permutes the isolated vertices with cyclic stabilizer.

A New Integral Transform and Applications: HK-Transform

In this paper, we propose a new integral transform called the Transform and employ it to analytically solve the Volterra integral equations (VIE.) of the first kind. To achieve this First we derive the transform of basic algebraic and transcendental mathematical functions. The fundamental characteristics of the transform, which can be used to solve a variety of functional equations, including, we then discuss the fundamental properties, such as ordinary differential equations and integral equations, then we discover the exact solution for general first class V.I.E. To illustrate the applicability of the transform, numerical problems are examined and handled carefully step-by-step. The results show that the proposed Transform new integral transform produces accurate solutions for first-kind VIE. without requiring time-consuming computations.

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Rigorous numerical integration of algebraic functions

Rigorous numerical integration of algebraic functions

Two-temperature thermochemical nonequilibrium model based on the coarse-grained treatment of molecular vibrational states.

Although the high-fidelity state-to-state (StS) model accurately describes high-temperature thermochemical nonequilibrium flows, its practical application is hindered by the prohibitively high computational cost. In this paper, we develop a reduced-order model that leverages the widelyused two-temperature (2T) framework and a coarse-grained treatment of molecular vibrational states to achieve accuracy comparable to the StS model while ensuring computational efficiency. We observe that the multigroup coarse-grained model (CGM), lumping vibrational energy levels into several groups, yields results close to the StS model for the high-temperature postshock oxygen flows, even using only two groups. However, theone-group CGM (CGM-1G), equivalent to the 2T model but using the StS kinetics, fails to approximate the StS results. Analysis of microscopic group properties reveals that the failure of the CGM-1G stems from the inability to capture the non-Boltzmann effects of mid-to-high vibrational levels, overestimating apparent dissociation rates and vibrational energy loss in the dissociation-dominated region. We then propose an analytical distribution function of vibrational groups by incorporating Treanor-like terms and an additional linear term (addressing the dissociation depletion of high-lying levels). Building upon this algebraic group distribution function and reconstructing vibrational levels within each group using the vibrational temperature, we develop a new 2T model called CG2T, which demonstrates accuracy much closer (than the CGM-1G) to the StS results for the postshock oxygen flows with varying degrees of thermochemical nonequilibrium. Moreover, a fullyconnected neural network is pretrained to substitute the module for the mass and vibrational energy source terms to enhance computational efficiency, achieving about 30-fold speedup in the CG2T model without sacrificing accuracy.

Speed‐sensorless control for SPMSM via novel exponential super‐twisting ADRC

AbstractAlthough the conventional super‐twisting reaching law (CSTRL) maintains the desired tracking performance for disturbed systems, its implementation is challenging due to its dependence on accurate system dynamics. To mitigate this dependence and reduce chattering of CSTRL, a novel exponential super‐twisting reaching law (NESTRL) is proposed, leveraging the dynamic adaptability of the exponential reaching law and eliminating chattering. By using NESTRL, the proposed controller can achieve a fast response without chattering. Additionally, a comprehensive procedure for finite convergence time analysis is provided. The convergence conditions of the system's state variables are determined using a Lyapunov function, and the stability of system is proven using an algebraic Lyapunov function. Finally, simulations and experimental results demonstrate that the proposed approach is valid and effective, showing improvements compared to the traditional super‐twisting active disturbance rejection control and linear active disturbance rejection control in sensorless control.

Exploring cosmological evolution and constraints in [formula omitted] teleparallel gravity

This study explores the extension of teleparallel gravity within the framework of general relativity, introducing an algebraic function f(T) dependent on the torsion scalar T. Motivated by the teleparallel formulation, we investigate cosmological implications, employing the simplest parametrization of the dark energy equation of state. Our chosen f(T) function, f(T)=α(−T)n, undergoes stringent constraints using recent observational data (H(z), SNeIa, BAO, and CMB). The model aligns well with cosmic dynamics, exhibiting quintessence behavior. The evolution of the deceleration parameter, the behavior of dark energy components, and the Om(z) diagnostic further reveal intriguing cosmological phenomena, emphasizing the model’s compatibility with quintessence scenarios.

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

PENGEMBANGAN E-MODUL BERBANTUAN FLIPBOOK PADA MATERI APLIKASI TURUNAN FUNGSI ALJABAR KELAS XI SMAN 8 LUBUKLINGGAU

The aim of this research is to determine the validity, practicality and effectiveness of the Flipbook-Assisted E-Module on the Algebra Function Derivative Application Material for Class XI IPA 2 SMA Negeri 8 Lubuklinggau. The research used is RD using the ADDIE model. Data collection techniques are interviews, observation, questionnaires and tests. The research subjects were students of class XI Science 2 at SMA Negeri 8 Lubuklinggau.It is-assisted moduleflipbook what was developed was declared valid, this was based on a language validator of 0.86, a media validator of 0.94 and a material validator of 0.95. The results of the validity test recapitulation show a high score with an average of 0.91. The teacher's practicality result is 4.85, testone to one evaluation was 4.52, small group practicality was 4.31, it was concluded thatIt is-assisted moduleflipbook in the application material for algebraic function derivatives, it shows an average score of 4.56 and is said to be practical. Effectiveness based on scoreN-gain amounting to 0.71 in the high category. therefore it is concluded thatIt is-The flipbook-assisted module on the application of algebraic function derivatives developed is valid, practical and effective.

Transfer Efficiency and Organization in Turbulent Transport over Alpine Tundra

The exchange of momentum, heat and trace gases between atmosphere and surface is mainly controlled by turbulent fluxes. Turbulent mixing is usually parametrized using Monin–Obukhov similarity theory (MOST), which was derived for steady turbulence over homogeneous and flat surfaces, but is nevertheless routinely applied to unsteady turbulence over non-homogeneous surfaces. We study four years of eddy-covariance measurements at a highly heterogeneous alpine valley site in Finse, Norway, to gain insights into the validity of MOST, the turbulent transport mechanisms and the contributing coherent structures. The site exhibits a bimodal topography-following flux footprint, with the two dominant wind sectors characterized by organized and strongly negative momentum flux, but different anisotropy and contributions of submeso-scale motions, leading to a failure of eddy-diffusivity closures and different transfer efficiencies for different scalars. The quadrant analysis of the momentum flux reveals that under stable conditions sweeps transport more momentum than the more frequently occurring ejections, while the opposite is observed under unstable stratification. From quadrant analysis, we derive the ratio of the amount of disorganized to organized structures, that we refer to as organization ratio (OR). We find an invertible relation between transfer efficiency and corresponding organization ratio with an algebraic sigmoid function. The organization ratio further explains the scatter around scaling functions used in MOST and thus indicates that coherent structures modify MOST. Our results highlight the critical role of coherent structures in turbulent transport in heterogeneous tundra environments and may help to find new parametrizations for numerical weather prediction or climate models.

Implementation of Problem Based Learning Approach Culturally Responsive Teaching to Enhance Engagement and Learning Outcomes in Algebraic Function Limit Material

This research aims to evaluate the engagement and learning outcomes of students through the Problem Based Learning model with a Culturally Responsive Teaching approach, particularly in the topic of algebraic function limit system. Based on observations and interviews conducted at Muhammadiyah Mlati High School, it was found that students in the eleventh grade of Social Sciences were less active in participating in mathematics classes, which are part of the mandatory curriculum based on the 2013 Curriculum. This study utilized the Classroom Action Research method conducted in three cycles. Each cycle included planning, implementation, observation, and reflection stages. The observation results indicated an increase in learning activity with a score of 56.25% in the first cycle, 72.13% in the second cycle, and 91.38% in the third cycle. This increase in learning activity correlates with the improvement in students' learning outcomes, where the N-gain value in the first cycle was 0.48 (medium category), 0.51 in the second cycle (medium category), and 0.56 in the third cycle (medium category), indicating that the success indicator (≥75.00) has been achieved. Thus, this research is considered successful and concluded at the end of the third cycle.

Algebraic solutions of linear differential equations: An arithmetic approach

Given a linear differential equation with coefficients in Q ( x ) \mathbb {Q}(x) , an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck–Katz p p -curvature conjecture.

Generalized techniques for constructing the generating functions of some algebraic and rational functions

We construct generating functions for general forms of certain classes of algebraic and rational functions using only basic tools of Mathematics. This approach unifies, extends and generalizes several techniques of constructing generating functions.

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

ABSTRACT In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

Unsteady flow of nanofluid over a sheet of variable thickness with nonlinear kinematics

The transport of heat in an unsteady flow of nanofluid has been discussed in this paper. The flow is maintained over a non-flat, variably porous, and moving sheet in the present simulation. Moreover, the sheet has both constant temperature and nanoparticle concentration at its surface. Unsteadiness in all field variables (i.e. the flow, temperature, and nanoparticles concentration) has been produced by the time-dependent quantities, specified at the surface of the sheet (i.e. kinematics of sheet). The well-known Buongiorno model of nanofluid has been undertaken in the present studies. Furthermore, the simulation includes the particular cases of uniform and linear (polynomial & algebraic functions) stretching/shrinking and injection/suction velocities. Besides that observations of the same nature have been recorded for the non-flat sheet. The boundary layer form of the governing PDEs has been utilized to evaluate the exact status of field variables in the flow domain. Appropriate conditions are imposed by considering the geometry of the flow problem. The problem at hand is simplified by introducing a new set of functions for the field variables because of boundary inputs. After this treatment, the system of boundary value ODEs, which contains several dimensionless numbers (parameters), has appeared, and the problem is solved numerically for various values of parameters, involved in it. More precisely, effects of unsteadiness, wall roughness, surface deformation parameters, Prandtl, Schmidt, thermophoretic, and Brownian motion numbers have been seen on profiles of four field variables. Besides that, the numerical results are also obtained for the skin friction coefficient, Nusselt, and Sherwood numbers. The present modelled problem and its solution are compared with classical models of that particular nature and with their solutions.

Robust clustering algorithm: The use of soft trimming approach

The presence of noise or outliers in data sets may heavily affect the performance of clustering algorithms and lead to unsatisfactory results. The majority of conventional clustering algorithms are sensitive to noise and outliers. Robust clustering algorithms often overcome difficulties associated with noise and outliers and find true cluster structures. We introduce a soft trimming approach for the hard clustering problem where its objective is modeled as a sum of the cluster function and a function represented as a composition of the algebraic and distance functions. We utilize the composite function to estimate the degree of the significance of each data point in clustering. A robust clustering algorithm based on the new model and a procedure for generating starting cluster centers is developed. We demonstrate the performance of the proposed algorithm using some synthetic and real-world data sets containing noise and outliers. We also compare its performance with that of some well-known clustering techniques. Results show that the new algorithm is robust to noise and outliers and finds true cluster structures.