Algebraic Means Research Articles

Comparative Analysis of Rainfall Estimation Methods Using Observation Data and Satellite Data in Coastal Areas of Kalimantan Barat Province

This study analyzes the accuracy of satellite rainfall data compared to observed data at several stations in West Kalimantan from 1998 to 2019. Three methods—Normal Ratio, Algebraic Mean, and Inverse Square Distance—were employed to address missing rainfall data. Among these, the Inverse Square Distance method provided the highest coefficient of determination (R²), with values nearing one at Ketapang, Tebas, and Singkawang stations. Evaluation using RMSE and Relative Bias revealed variations in data accuracy across stations. Satellite data showed high accuracy at Sukadana and Ketapang but lower accuracy at Singkawang, highlighting the need for location-specific calibration. Graphical analyses showed consistent annual rainfall patterns between satellite and observed data despite differences in magnitude during specific years. Comparative studies of three data sources—observational, satellite, and BMKG—demonstrated similar trends, with satellite data exhibiting fewer fluctuations and higher stability. Satellite rainfall data is a reliable alternative for estimating rainfall patterns in areas with limited observational data. However, further adjustments are required to enhance its accuracy in specific locations. This study underscores the importance of refining satellite data models for better hydrological management and prediction accuracy.

The Emotional Meaning of Pure Music

Music can have extrinsic and/or intrinsic meaning. The former is relevant in the case of program music. The latter conforms to pure (absolute) music, i.e. music that can be understood without reference to extrinsic sources. Taking the intrinsic content of music as basic, we must ask about its nature. I propose to identify it with aesthetic emotion. As tonal music is organized by series of chords relative to the context of a tonal scale, the question is how music forms can be mapped onto aesthetic emotions. I propose to ground the analysis on a two-dimensional space of emotions, where one dimension refers to arousal (activity) and the other dimension refers to valence. Relating valence with consonance and arousal with entropic uncertainty leads to an account which directly relates structural and probabilistic aspects of tonal music with its affective content. The present bare-bone semantics of pure music proposes an explicit modeling of the affective response based on an algebraic meaning conception.

Pre-training enhanced spatio-temporal graph neural network for predicting influent water quality and flow rate of wastewater treatment plant: Improvement of forecast accuracy and analysis of related factors

Efficient management of wastewater treatment plants (WWTPs) necessitates accurate forecasting of influent water quality parameters (WQPs) and flow rate (Q) to reduce energy consumption and mitigate carbon emissions. The time series of WQPs and Q are highly non-linear and influenced by various factors such as temperature (T) and precipitation (Precip). Conventional models often struggle to account for long-term temporal patterns and overlook the complex interactions of parameters within the data, leading to inaccuracies in detecting WQPs and Q. This work introduced the Pre-training enhanced Spatio-Temporal Graph Neural Network (PT-STGNN), a novel methodology for accurately forecasting of influent COD, ammonia nitrogen (NH3-N), total phosphorus (TP), total nitrogen (TN), pH and Q in WWTPs. PT-STGNN utilizes influent data of the WWTP, air quality data and meteorological data from the service area as inputs to enhance prediction accuracy. The model employs unsupervised Transformer blocks for pre-training, with efficient masking strategies to effectively capture long-term historical patterns and contextual information, thereby significantly boosting forecasting accuracy. Furthermore, PT-STGNN integrates a unique graph structure learning mechanism to identify dependencies between parameters, further improving the model's forecasting accuracy and interpretability. Compared with the state-of-the-art models, PT-STGNN demonstrated superior predictive performance, particularly for a longer-term prediction (i.e., 12 h), with MAE, RMSE and MAPE at 12-h prediction horizon of 2.737 ± 0.040, 4.209 ± 0.060 and 13.648 ± 0.151 %, respectively, for the algebraic mean of each parameter. From the results of graph structure learning, it is observed that there are strong dependencies between NH3-N and TN, TP and Q, as well as Precip, etc. This study innovatively applies STGNN, not only offering a novel approach for predicting influent WQPs and Q in WWTPs, but also advances our understanding of the interrelationships among various parameters, significantly enhancing the model's interpretability.

Circle fit optimization for resonator quality factor measurements: Point redistribution for maximal accuracy

The control of material loss mechanisms is playing an increasingly important role for improving coherence times of superconducting quantum devices. Such material losses can be characterized through the measurement of planar superconducting resonators, which reflect losses through the resonance's quality factor Ql. The resonance quality factor consists of both internal (material) losses as well as coupling losses when resonance photons escape back into the measurement circuit. The combined losses are then described as Ql−1=Re{Qc−1}+Qi−1, where Qc and Qi reflect the coupling and internal quality factors of the resonator, respectively. To separate the relative contributions of Qi and Qc to Ql, diameter-correcting circle fits use algebraic or geometric means to fit the resonance signal on the complex plane. However, such circle fits can produce varied results, so to address this issue, we use a combination of simulation and experiment to determine the reliability of a fitting algorithm across a wide range of quality factor values from Qi≪Qc to Qc≪Qi. In addition, we develop a measurement protocol that can not only reduce fitting errors by factors ≳2 but also mitigates the influence of the measurement background on the fit results. This technique can be generalized for other resonance systems beyond superconducting resonators. Published by the American Physical Society 2024

Validating an algebraic approach to characterizing resonator networks

Resonator networks are ubiquitous in natural and engineered systems, such as solid-state materials, electrical circuits, quantum processors, and even neural tissue. To understand and manipulate these networks it is essential to characterize their building blocks, which include the mechanical analogs of mass, elasticity, damping, and coupling of each resonator element. While these mechanical parameters are typically obtained from response spectra using least-squares fitting, this approach requires a priori knowledge of all parameters and is susceptible to large error due to convergence to local minima. Here we validate an alternative algebraic means to characterize resonator networks with no or minimal a priori knowledge. Our approach recasts the equations of motion of the network into a linear homogeneous algebraic equation and solves the equation with a set of discrete measured network response vectors. For validation, we employ our approach on noisy simulated data from a single resonator and a coupled resonator pair, and we characterize the accuracy of the recovered parameters using high-dimension factorial simulations. Generally, we find that the error is inversely proportional to the signal-to-noise ratio, that measurements at two frequencies are sufficient to recover all parameters, and that sampling near the resonant peaks is optimal. Our simple, powerful tool will enable future efforts to ascertain network properties and control resonator networks in diverse physical domains.

Monadic Expressions and Their Derivatives

There are several well-known ways to compute derivatives of regular expressions due to Brzozowski, Antimirov or Lombardy and Sakarovitch. We propose another one which abstracts the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we first introduce a new derivation technique based on the graded module monad and then show an application of this technique to generalize the parsing of expressions with capture groups and back references. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

A two‐phase method of moments model for high‐impact polystyrene phase inversion conversion and molecular properties

AbstractA challenge for high‐impact polystyrene (HIPS) production and design is the more accurate and low computational cost prediction of the phase separation and phase inversion conversion, the first conversion determining the onset of the two‐phase system and the second conversion defining the pathway toward a complex, for example, salami‐like morphology. In this work, a two‐phase deterministic method of moments (MoM) model running in several minutes is therefore developed in the intermediate styrene conversion range (up to around 30%), considering for simplicity only diffusional limitations on termination on an average basis. It is showcased that the phase separation conversion can be taken as 2 m%, and the phase inversion conversion should be calculated by algebraic means. Interestingly, the latter conversion varies with the initial reaction conditions either on a time basis (e.g., variation of initial radical concentration) or both a time and conversion basis (e.g., relative contribution of rubber and St partitioning coefficient). A comparison with the commonly used pseudo‐homogeneous MoM model reveals that by accounting for mass transfer in a more representative two‐phase model, the amount of monomer consumed is slightly reduced, the grafting efficiency decreases with increasing conversion instead of increasing, and the styrene composition in the graft copolymer decreases. The current work additionally puts forward two‐phase MoM data to facilitate future benchmarking with other (stochastic) modeling approaches and enables kinetic insights into heterogeneous grafting polymerization, further promoting the production of multiphase industrial polymer products.

Atom counting based on Voronoi averaged STEM intensities using a crosstalk correction scheme

If quantitative scanning transmission electron microscopy is used for very precise thickness measurements with atomic resolution, it is commonly referred to as »atom counting«. Due to scattering and the finite probe extent the signal recorded in one atomic column is dependent not only on its own height but also on the height of its neighbours. Especially for thicker specimens this crosstalk effect can have significant impact on the measured intensity. If it is not appropriately accounted for in the evaluation, it can result in a deterioration of accuracy that impedes the possibility of actual atom counting. However, as the number of possible neighbour configurations can be excessively large, a comprehensive consideration of all in the evaluation reference is neigh impossible. This work proposes a method that allows for the a-posteriori reduction of crosstalk during the evaluation by algebraic means. Based on a parametric model, which is described in detail in the article, the crosstalk is expressed by an invertible matrix. Applying the inverted matrix to the measurement yields crosstalk corrected intensity values with very little computational effort. These can subsequently be evaluated by direct comparison to simple reference data. The working principle of the method is presented on the example of crystalline gold. The crosstalk parametrisation is found by fitting a model to sets of specifically created multislice simulations. The parameters are given for both aberration corrected and uncorrected STEM. Subsequently the abilities and potential of the technique are assessed in simulative studies on multiple model systems including gold nanoparticles. Overall a significant and robust improvement of the attainable precision can be demonstrated making the proposed method a promising tool for reference-based atom counting.

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

Abstract Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

Polynomial solutions of generalized quartic anharmonic oscillators

This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schrödinger operator in terms of the generators of a nilpotent group, which we call the quartic group. Energy eigenvalues are then seen to depend on the values of the two Casimir operators of the group. This dependence exhibits a scaling law which follows from the scaling properties of the group generators. Demanding that the potential gives rise to polynomial solutions in a particular Lie algebra element puts constraints on the four potential parameters, leaving only two of them free. For potentials satisfying such constraints, at least one of the energy eigenvalues and the corresponding eigenfunctions can be obtained in closed analytic form by pure algebraic means. With our approach, we extend the class of quasi-exactly solvable quartic oscillators which have been obtained in the literature by means of the more common textrm{sl}(2,{mathbb {R}}) algebraization. Finally, we show how solutions of the generalized quartic oscillator problem give rise to solutions for a charged particle moving in particular non-constant electromagnetic fields.

The invariant solution of Smoluchowski coagulation equation with homogeneous kernels based on one parameter group transformation

The governing equation of self-preserving size distribution can be obtained by similarity transformation from the Smoluchowski coagulation equation. To deal with the variable upper limit of convolution, the one parameter group transformation is introduced to get the invariant form of the governing equation. However, whether the equation has an invariant solution depending on asymptotic property of the coagulation kernel at the lower-end boundary. For homogeneous kernel, it can be discussed with three classes, type I has no similarity solution, type II has similarity solution which should be analyzed, and type III has similarity solution. Under the existence of similarity solution, an improved iDNS algorithm is developed with fourth order Runge–Kutta method to get the invariant solution of Smoluchowski coagulation equation more efficiently. Based on the invariant solution, a new definition of entropy in statistical physics is proposed to distinguish the convergence of Smoluchowski coagulation equation mathematically. The new entropy is a simple function of algebraic mean volume and has a maximum. At long time, the entropy production approaches to zero, which means the convergence of solution for Smoluchowski coagulation equation mathematically, and the Cercignani conjecture is almost true for Smoluchowksi coagulatin equation.

Approaching the Standard of Knowledge Representation in the Web by Algebraic Means

Approaching the Standard of Knowledge Representation in the Web by Algebraic Means

Monadic Expressions and their Derivatives

We propose another interpretation of well-known derivatives computations from regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we introduce a new derivation technique based on the graded module monad. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

Efficiency at optimal performance: A unified perspective based on coupled autonomous thermal machines.

We show that coupled autonomous thermal machines, in the presence of three heat reservoirs and following a global linear-irreversible description, can have efficiency at maximum power (EMP) which is analogous in form to the EMP of models with two (hot and cold) reservoirs. In particular, the temperature dependence of EMP in the coupled model is via only the ratio of hot and cold temperatures if the intermediate reservoir temperature is expressed as an algebraic mean of these temperatures. Many popular expressions of EMP in the literature can be recovered by making a choice of some standard mean. Further, the universal properties of EMP near equilibrium can be explained in terms of the properties of symmetric means. For the case of broken time-reversal symmetry, a universal second-order coefficient of 6/49 is predicted in the series expansion of EMP, analogous to the 1/8 coefficient in the time-reversal symmetric case.

Towards an approach to teachers’ professional development: How to work with algebraic thinking in the early years

This article presents the first intervention cycle of a design-based research program on teacher professional development. The study aims to understand how the formative process helped teachers to understand what algebraic thinking means and how to work with it in the early years. Data analysis was based on three principles of design: teacher’s role and actions, professional learning tasks for teachers, and discursive interactions among participants. The results suggest that these design principles contributed to teachers’ understanding of the meaning of algebraic thinking and how to promote it in elementary students.

Limit state according to the condition of loss of stability of equilibrium forms

Objective. The purpose of the study is to determine the group of the limiting state according to the condition of loss of stability of the equilibrium form of structures. Method. The study is based on the provisions of the theory of stability of equilibrium states of building structures; branching theory of solutions of nonlinear equations; perturbation method; methods of catastrophe theory.Result. The results of the analysis of the post-critical behavior of structures based on the solution of the problem in higher approximations and from the fundamental provisions of the theory of catastrophes are generalized. It is proved that the study of the stability of equilibrium forms of structures using algebraic means and geometric images of the theory of catastrophes makes it possible to unambiguously determine the type of critical bifurcation points, predict the nature of the behavior of the structure, and determine the limit state group to which the state reached by the structure should be attributed. Conclusion. It seems necessary to rename the ordinal numbers of the types of critical points of bifurcations so that they coincide with the numbers of the groups of limit states corresponding to them.

Extended study on the application of the sextic potential in the frame of X(3)-sextic

The main aim of the present paper is to extensively study the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are explicitly found, by algebraic means, the so-called quasi-exact solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands at the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, as well as the shape evolution within an isotopic chain. The numerical results are given for 39 nuclei, namely, 98–108Ru, 100–102Mo, 116–130Xe, 180–196Pt, 172Os, 146–150Nd, 132–134Ce, 152–154Gd, 154–156Dy, 150–152Sm, 190Hg and 222Ra. Across this study, it seems that the higher QES order improves our results by decreasing the root mean square, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly at the critical point.

Solution of Smoluchowski coagulation equation for Brownian motion with TEMOM

The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole, and it can be decomposed into the following two functions by similarity transformation: one is a function of time (the particle k-th moments), and the other is a function of dimensionless volume (self-preserving size distribution). In this paper, a simple iterative direct numerical simulation (iDNS) is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment, which has been solved with the Taylor-series expansion method of moment (TEMOM) in our previous work. The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature, and then the method is extended to the field of Brownian agglomeration over the entire size range. The results show that the difference between the lognormal function and the self-preserving size distribution is significant. Moreover, the thermodynamic constraint of the algebraic mean volume is also investigated. In short, the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship; thus, a complete method to solve the Smoluchowski coagulation equation asymptotically is established.

Arithmetic Framework to Optimize Packet Forwarding among End Devices in Generic Edge Computing Environments.

Multi-access edge computing implementations are ever increasing in both the number of deployments and the areas of application. In this context, the easiness in the operations of packet forwarding between two end devices being part of a particular edge computing infrastructure may allow for a more efficient performance. In this paper, an arithmetic framework based in a layered approach has been proposed in order to optimize the packet forwarding actions, such as routing and switching, in generic edge computing environments by taking advantage of the properties of integer division and modular arithmetic, thus simplifying the search of the proper next hop to reach the desired destination into simple arithmetic operations, as opposed to having to look into the routing or switching tables. In this sense, the different type of communications within a generic edge computing environment are first studied, and afterwards, three diverse case scenarios have been described according to the arithmetic framework proposed, where all of them have been further verified by using arithmetic means with the help of applying theorems, as well as algebraic means, with the help of searching for behavioral equivalences.

On the Way to the Standard of Knowledge Representation in the Web by Algebraic Means

On the Way to the Standard of Knowledge Representation in the Web by Algebraic Means