Concept Of Decomposition Research Articles

Orthogonal Decompositions of Regular Graphs and Designing Tree-Hamming Codes

The paper introduces a concept of graph decomposition. That is orthogonal decompositions. Orthogonal decompositions of a graph H is a partitioning H into subgraphs of H such that any two subgraphs intersect in at most one edge. This decompositions are called G-orthogonal decompositions of H if and only if every subgraph in such decompositions is isomorphic to the graph G. Such decomposition appear in a lot of applications; statistics, information theory, in the theory of experimental design and many others. An approach of constructing orthogonal decompositions of regular graph is introduced here. Application of this approach for constructing tree -- orthogonal decompositions of complete bipartite graph is considered. Further, the use of orthogonal decompositions for designing hamming tree-codes is also discussed along with examples. The study shows that such codes have an efficient properties when are used to detect and correct the errors that may occur during a transmission of data through a network. Furthermore, we present a method for the recursive construction of orthogonal decompositions.

A dependent circular-linear model for multivariate biomechanical data: Ilizarov ring fixator study.

Biomechanical and orthopaedic studies frequently encounter complex datasets that encompass both circular and linear variables. In most cases (i) the circular and linear variables are considered in isolation with dependency between variables neglected and (ii) the cyclicity of the circular variables is disregarded resulting in erroneous decision making. Given the inherent characteristics of circular variables, it is imperative to adopt methods that integrate directional statistics to achieve precise modelling. This paper is motivated by the modelling of biomechanical data, that is, the fracture displacements, that is used as a measure in external fixator comparisons. We focus on a dataset, based on an Ilizarov ring fixator, comprising of six variables. A modelling framework applicable to the six-dimensional joint distribution of circular-linear data based on vine copulas is proposed. The pair-copula decomposition concept of vine copulas represents the dependence structure as a combination of circular-linear, circular-circular and linear-linear pairs modelled by their respective copulas. This framework allows us to assess the dependencies in the joint distribution as well as account for the cyclicity of the circular variables. Thus, a new approach for accurate modelling of mechanical behaviour for Ilizarov ring fixators and other data of this nature is imparted.

Exact steady states in the asymmetric simple exclusion process beyond one dimension

The asymmetric simple exclusion process (ASEP) is a paradigmatic nonequilibrium many-body system that describes the asymmetric random walk of particles with exclusion interactions in a lattice. Although the ASEP is recognized as an exactly solvable model, most of the exact results obtained so far are limited to one-dimensional systems. Here, we construct the exact steady states of the ASEP with closed and periodic boundary conditions in arbitrary dimensions. This is achieved through the concept of transition decomposition, which enables the treatment of the multidimensional ASEP as a composite of the one-dimensional ASEPs. Published by the American Physical Society 2024

Application of Singular Spectrum Analysis (SSA) Decomposition in Artificial Neural Network (ANN) Forecasting

Over time, various forecasting methods have been introduced. An example is the Hybrid model. This model can enhance the forecast accuracy compared to a single model. The Hybrid Singular Spectrum Analysis (SSA)-Artificial Neural Network (ANN) model combines the concepts of decomposition and forecasting. The Hybrid SSA-ANN forecasting works through two stages. Firstly, SSA decomposes the data into trend, seasonal, noise, and residue components. Secondly, the decomposed data is predicted using the ANN model, specifically the LSTM and GRU models. The Hybrid SSA-ANN model has been proven to improve forecasting accuracy. The Hybrid SSA-LSTM model improves the forecast accuracy by 78% compared to the single LSTM forecasting model. This can be seen from the respective RMSE values of 4.36 changing to 0.97 and MAPE values of 5.2% changing to 1.16%. Similarly, the Hybrid SSA-GRU model improves the forecast accuracy by 79% compared to the single GRU forecasting model. This can be observed from the respective RMSE values of 4.86 changing to 1.01 and MAPE values of 6.33% changing to 1.36%. In a case study using weekly data of crude oil's opening prices, the application of SSA decomposition can enhance the forecast accuracy by 78-79% in ANN forecasting

A coupling concept for Stokes-Darcy systems: The ICDD method

We present a coupling framework for Stokes-Darcy systems valid for arbitrary flow direction at low Reynolds numbers and for isotropic porous media. The proposed method is based on an overlapping domain decomposition concept to represent the transition region between the free-fluid and the porous-medium regimes. Matching conditions at the interfaces of the decomposition impose the continuity of velocity (on one interface) and pressure (on the other one) and the resulting algorithm can be easily implemented in a non-intrusive way. The numerical approximations of the fluid velocity and pressure obtained by the studied method converge to the corresponding counterparts computed by direct numerical simulation at the microscale, with convergence rates equal to suitable powers of the scale separation parameter ε in agreement with classical results in homogenization.

Anomaly detection in PV systems using constrained low-rank and sparse decomposition

PV (photovoltaic) systems, also known as solar panel systems, play an essential role in the mitigation of greenhouse gas emissions and the promotion of renewable energy. Through the conversion of sunlight into usable energy, electricity is generated without emitting greenhouse gases and producing pollutants. Notwithstanding the evolutionary significance of PV systems, the occurrence of defects and anomalies in PV systems may result in diminished power output, consequently impeding the efficiency of the systems and potentially resulting in hazards in certain circumstances. Therefore, early detection of faults and anomalies in PV systems is imperative to guarantee the reliability, efficiency, and safety of the systems. In this article, we develop a signal decomposition for the purpose of anomaly detection in PV systems. The proposed methodology is grounded on the concept of low-rank and sparse decomposition, with consideration given to the signs of the decomposed low-rank and sparse components, as well as the smooth variations within and between periods in the mean signals. Through the implementation of Monte Carlo simulations, we showcase the efficacy of our proposed methodology in identifying anomalies of varying durations and magnitudes in PV systems. A case study is employed to validate the proposed methodology in detecting anomalies in real PV systems.

A Study on Degree Based Topological Indices of Harary Subdivision Graphs With Application

Combinatorial design theory and graph decompositions play a critical role in the exploration of combinatorial design theory and are essential in mathematical sciences. The process of graph decomposition involves partitioning the set of edges in a graph G. An n-sun graph, characterized by a cycle with an edge connecting each vertex to a terminating vertex of degree one, is introduced in this study. The concept of n-sun decomposition is applied to certain even-order graphs. The indices covered in this study include the general connectivity index of the harary graphs, Zagreb indices, symmetric division degree indices and randic indices.

SVD enclosure of a class of interval matrices

This paper develops the concept of the singular value decomposition (SVD) of a class of interval matrices. The methodology relies on tighter outer estimations of eigenvalues and their corresponding eigenvectors of an interval matrix. The interval enclosure of every eigenvalue of an interval matrix is determined using an iterative procedure, and an algorithm is proposed for determining the corresponding eigenvector enclosure. Using these concepts, the SVD enclosure of an interval matrix is obtained. Numerical examples are provided to demonstrate the principles of the proposed methodologies at different stages.

Оценка добавленной стоимости во внешней торговле: современные подходы

The increase in globalization processes, which accelerated the fragmentation of production chains in the global economy, updated the task of clarifying the contribution of each of the element of the chain to the final result. With the consolidation of foreign economic cooperation, on the one hand, new opportunities have opened up for the inclusion in the international division of labor for more and more countries and companies, and on the other hand, risk threats of destroying or breaking relationships in these global value chains have intensified. This updated the task of specifying the real volumes of export-import turnover, cleared of double counted elements. In the early 2010s almost simultaneously there have been launched 3 basic concepts of export decomposition in the categories of value-added trade, prepared by UNCTAD, OECD and the EU. The system-functional analysis of the proposed approaches made it possible, at the same time, to reveal a number of keen questions that asked for clarification, in particular, the underestimation of intangible assets in creating value-added in exports within the framework of a specific global value chain, the measurement of value-added embedded in re-export (re-import) turnover of the country, etc. These aspects received due coverage in foreign and domestic economic literature. However, the latest developments by foreign specialists of the methodology of decomposing exports into the internal and external components of net production and double counted elements in the domestic scientific turnover are still being introduced with some delay. In this regard, the article set the task of conducting a comparative analysis of the main new results achieved, focusing on the methods introduced by A. Nagengast – R. Stehrer (2014) and A. Borin – M. Mancini (2019). At the same time, the main attention was concentrated on identifying breakthrough methodological "increments." It was revealed that new results are in demand not only in the context of macroeconomic analysis of transformational processes in the global economy, but also to an increasing extent to adjust foreign trade policy and develop ways to mitigate the risks of deterioration of bilateral foreign trade relations.

The Properties of Topological Manifolds of Simplicial Polynomials

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.

Psychophysical neutrality and its descendants: a brief primer for dual-aspect monism

A key ingredient of the metaphysical doctrine of dual-aspect monism is a psychophysically neutral domain, of which mental and physical aspects arise as epistemic descendants that manifest themselves by decomposition. This primer first introduces some elementary notions to define the basic concepts needed to understand the approach, such as those of states, state spaces, observables, partitions and correlations. Using these notions, the concepts of decomposition and manifestation are explained, and a differentiated view of the mereological distinction of wholes and parts is outlined. Next, a number of historical and contemporary accounts of psychophysical neutrality with philosophical (Plato, Spinoza, Schelling, Kant), scientific (Bohm, Pauli, Jung, Connes, Gibson), and artistic (sculpture, music) flavor are given as illustrative examples. Finally, correlations between the psychophysically neutral, the mental, and the physical domain of reality are discussed, in which these correlations are substantiated by meaning.

Enhanced Network Compression Through Tensor Decompositions and Pruning.

Network compression techniques that combine tensor decompositions and pruning have shown promise in leveraging the advantages of both strategies. In this work, we propose enhanced Network cOmpRession through TensOr decompositions and pruNing (NORTON), a novel method for network compression. NORTON introduces the concept of filter decomposition, enabling a more detailed decomposition of the network while preserving the weight's multidimensional properties. Our method incorporates a novel structured pruning approach, effectively integrating the decomposed model. Through extensive experiments on various architectures, benchmark datasets, and representative vision tasks, we demonstrate the usefulness of our method. NORTON achieves superior results compared to state-of-the-art (SOTA) techniques in terms of complexity and accuracy. Our code is also available for research purposes.

A study on automated improvement of securities trading strategies using machine learning optimization algorithms

Abstract Automation in securities trading offers advantages over human subjective trading, such as immunity to subjective emotional factors, high efficiency, and the ability to monitor multiple stocks simultaneously, making it a cutting-edge development path in the securities trading industry. In this paper, we first apply the concept of time-frequency decomposition, gradually moving from the first-order moments of securities prices to the higher-order moments. We then combine this with the EMD time-frequency decomposition method to analyze the securities price sequence and extract the characteristics of the securities price fluctuations. Finally, we use the differential long- and short-term memory network to construct an automatic optimization trading system. We compare the system’s performance with traditional technical analysis indexes, as well as the annualized returns of PPO and A2C models on various securities, to verify its performance under unilateral rising, oscillating rising, and plummeting quotes. Finally, we conducted a live test on 1000 GEM stocks. The system in this paper outperforms all traditional technical indicators, with an average annualized return of 71.85% at the lowest and 127.27% at the highest among 5 securities, demonstrating excellent performance. In the three quotes of Ningde Times, Aier Dental, and Goldfish that are rising one way, rising and falling over time, and rising again, the annualized returns of this paper’s system are 77.13%, 67.16%, and 12.66%, which are higher than those of the PPO and A2C models.

Parallel Prediction Method of Knowledge Proficiency Based on Bloom’s Cognitive Theory

Knowledge proficiency refers to the extent to which students master knowledge and reflects their cognitive status. To accurately assess knowledge proficiency, various pedagogical theories have emerged. Bloom’s cognitive theory, proposed in 1956 as one of the classic theories, follows the cognitive progression from foundational to advanced levels, categorizing cognition into multiple tiers including “knowing”, “understanding”, and “application”, thereby constructing a hierarchical cognitive structure. This theory is predominantly employed to frame the design of teaching objectives and guide the implementation of teaching activities. Additionally, due to the large number of students in real-world online education systems, the time required to calculate knowledge proficiency is significantly high and unacceptable. To ensure the applicability of this method in large-scale systems, there is a substantial demand for the design of a parallel prediction model to assess knowledge proficiency. The research in this paper is grounded in Bloom’s Cognitive theory, and a Bloom Cognitive Diagnosis Parallel Model (BloomCDM) for calculating knowledge proficiency is designed based on this theory. The model is founded on the concept of matrix decomposition. In the theoretical modeling phase, hierarchical and inter-hierarchical assumptions are initially established, leading to the abstraction of the mathematical model. Subsequently, subject features are mapped onto the three-tier cognitive space of “knowing”, “understanding”, and “applying” to derive the posterior distribution of the target parameters. Upon determining the objective function of the model, both student and topic characteristic parameters are computed to ascertain students’ knowledge proficiency. During the modeling process, in order to formalize the mathematical expressions of “understanding” and “application”, the notions of “knowledge group” and “higher-order knowledge group” are introduced, along with a parallel method for identifying the structure of higher-order knowledge groups. Finally, the experiments in this paper validate that the model can accurately diagnose students’ knowledge proficiency, affirming the scientific and meaningful integration of Bloom’s cognitive hierarchy in knowledge proficiency assessment.

Analysis of Student Learning Difficulties in Solving Calculus II Course Case Studies

Calculus II is a compulsory course in the Mathematics Education Study Programme of PGRI Adi Buana University Surabaya which is programmed by third semester students. This course learns about the basic concepts of integrals and their applications, so students are required to be able to understand and analyse problems appropriately. This study aims to determine the difficulties experienced by students in solving Calculus II problems as well as to find out the causes. The research subjects were five students of Mathematics Education of Universitas PGRI Adi Buana Surabaya in the academic year 2022/2023. This research uses a qualitative approach using test and interview methods. The results obtained from this study are in the form of types of student difficulties in solving problems, namely difficulty in determining the final result, difficulty in understanding the concept of function decomposition, difficulty in applying concepts to problems, difficulty in calculating, difficulty in starting the first step of the calculation. Furthermore, for the causes of these difficulties, including, lack of accuracy in understanding the problems given, lack of mastery of concepts, errors in the calculation process, and lack of understanding of the concept of initial identification of integral forms.

IMPLEMENTING COMPUTATIONAL THINKING IN ELEMENTARY SCHOOL ICT EDUCATION AND EVALUATION USING THE KIRKPATRICK METHOD

The lack of understanding of computational concepts and problem-solving among students who still consider computers merely as tools for gaming and entertainment, along with their limited exposure to computational Thinking, constrains their ability to address problems systematically. Although the 2013 Curriculum emphasizes critical Thinking, basic data analysis, problem-solving, creativity, and innovation, aligned with the computational thinking method, it remains a distinct challenge for students and teachers to integrate it into their teaching. The objective of computational thinking education is to shift the paradigm and introduce students to computational thinking skills, particularly in ICT Education. Testing this method employs the Kirkpatrick method, which demonstrates a significant improvement in students' ability to solve problems with a more systematic and computational approach. Students can comprehend the concepts of decomposition, pattern recognition, abstraction, and algorithm development. This improvement is reflected in their performance on various computer-based problem-solving tasks. The benefits of implementing computational Thinking include students applying computational thinking principles not only in computer-based learning but also in their daily lives, gaining more confidence in facing technology-based challenges.

Syzygies, constant rank, and beyond

We study linear PDE with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of primary decomposition is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.

Adaptive task decomposition physics-informed neural networks

As a new approach for solving partial differential equations, physics-informed neural networks (PINNs) have received extensive attention in recent years and have made breakthroughs in various fields. Serving as a neural network-based universal solver, they possess advantages such as high continuity and being meshless. However, an increasing number of studies have pointed out that PINNs may fail to converge to accurate results when dealing with complex tasks. In this paper, we highlight the differences between training PINNs and regular neural network tasks, summarizing three key characteristics of training PINNs. Based on these findings, we propose a strategy of task decomposition and progressive learning, which successfully explains the effectiveness of existing methods for time-dependent problems and reveals their limitations. For time-dependent and time-independent problems, we partition the tasks into physically complete and progressively challenging sub-tasks. The division is based on the resolution requirements for time-independent problems and the coverage of the time domain for time-dependent problems. Through task parameters and the decomposition of the loss terms, we adaptively control the progressive increase in task complexity during the training process. The proposed method overcomes the limitations of existing approaches, incurs no additional computational cost, and is applicable to both time-dependent and time-independent problems, demonstrating higher generality. Furthermore, building upon the concepts of task decomposition and progressive learning, we introduce a stable and efficient minibatch training method for the first time. In a series of numerical experiments on baseline PDE problems, our approaches outperformed other state-of-the-art algorithms.

Parameter estimation of hybrid pulse streams based on sub-Nyquist sampling and Shift-invariant subspace projection

Finite Rate of Innovation (FRI) sampling theory offers an efficient approach for the sub-Nyquist measurement of the short pulse stream signal, but it can only address parameter measurement problems under waveform consistency. To overcome the measurement challenge of the hybrid pulse streams, this paper proposes a subspace projection-based FRI parameter estimation method under sub-Nyquist sampling conditions. Our method is based on the concept of subspace decomposition, which involves finely dividing different pulse signals within their respective Shift-invariant (SI) subspaces. We use the generating function of the corresponding space as a decoupling object to achieve parameter estimation through FRI measurement methods. The sampling process is efficient and simple, requiring only single-channel sampling to measure the characteristic parameters of all hybrid pulses. We not only provide theoretical guarantees for the implementation of this approach but also conduct numerous numerical simulations and hardware verification experiments. Our experimental results demonstrate that our method achieves higher parameter measurement accuracy compared to classical methods in a simpler manner.

Векторная декомпозиция сейсмических данных методами волнового обращения во времени

The paper analyzes the stage of decomposition of the initial seismic data in the methods of wave reversal in time when constructing seismic attributes. Within the framework of the formal approach of mapping the data of one space into the data of a space of a higher dimension, a classification of existing approaches in seismic exploration is given. Identification of the decomposition stage in the seismic data processing workflow makes it possible to highlight the differences in existing approaches to building seismic attributes and predict the future direction of seismic data processing. The concept of vector decomposition, originally used in the RTH method, is introduced. The variety of depth seismic attributes obtained in the RTH method based on vector decomposition allows solving a wide range of problems in the exploration and development of hydrocarbon deposits at a new qualitative level. The RTH method includes, as a special case, the PSDM, AVO, AI methods and is an alternative to the MVA, FWI methods, as well as the method of a velocity model bilding based on fast beam migration algorithms. A close connection between the technique of wavefront time reversal in seismic exploration and analogous time reversal in optics and acoustics is noted. Examples of seismic data processing using vector decompositio to identify zones of natural fracturing in shale oil are given.