Mathematical Deduction Research Articles

High-efficient sample point transform algorithm for large-scale complex optimization

Decomposition algorithms and surrogate model methods are frequently employed to address large-scale, intricate optimization challenges. However, the iterative resolution phase inherent to decomposition algorithms can potentially alter the background vector, leading to the repetitive evaluation of samples across disparate iteration cycles. This phenomenon significantly diminishes the computational efficiency of optimization. Accordingly, a novel approach, designated the Sample Point Transformation Algorithm (SPTA), is put forth in this paper as a means of enhancing efficiency through a process of mathematical deduction. The mathematical deduction reveals that the difference between sample points in each iteration loop is a simple function related to the inter-group dependent variables. Consequently, the SPTA method achieves the comprehensive transformation of the sample set by establishing a surrogate model of the difference between the sample sets of two cycles with a limited number of sample points, as opposed to conducting a substantial number of repeated samplings. This SPTA is employed to substitute the most time-consuming step of direct calculation in the classical optimization process. To validate the calculation efficiency, a series of numerical examples were conducted, demonstrating an improvement of approximately 75 % while maintaining optimal accuracy. This illustrates the advantage of the SPTA in addressing large-scale and complex optimization problems.

A Closure Contact Model of Self-Affine Rough Surfaces Considering Small-, Meso-, and Large-Scale Stage Without Adhesive

Contact interface is essential for the dynamic response of the bolted structures. To accurately predict the dynamic characteristics of bolted joint structures, a fractal extension of the segmented scale model, i.e., the JK model, is proposed in this paper to comprehensively analyze the dynamic contact performance of engineering surfaces and revisit the multi-scale model based on the concept of asperities. The influence of asperity geometry, dimensionless material properties, and the elastic, elastoplastic, and full plastic mechanical models of a single asperity is established considering the asperity–substrate interaction. Then, a segmented scale contact model of rough surfaces is proposed based on the island distribution function in a strict sense. The mechanical contact process of determining rough surfaces is divided into small-scale, medium-scale, and large-scale stages. Moreover, cross-scale boundary conditions, i.e., al1′, al2′, and al3′, are provided through strict mathematical deduction. The results show that the real contact area and contact stiffness are positively correlated with fractal dimension and negatively correlated with fractal roughness. On a small scale, the contact damping decreases with an increase in load. In meso-scale and large-scale stages, the contact damping increases with the load. Finally, the reliability of the proposed model is verified by setting up three groups of modal vibration experiments.

A boundary-type algorithm based on the discrete ordinates for neutron transport equation

The spatial distribution of neutron flux in the core of a nuclear reactor plays a crucial role in ensuring nuclear safety. It can be determined by solving the neutron transport equation, where computational resource consumption is gradually receiving more attention, especially in problems with multiple reflective boundary conditions. It is common practice to assume initial boundary values in one angular direction to obtain a global solution, and then use reflective boundary conditions to solve for the other directions, iterating until convergence. However, this approach has the drawback of requiring the global solution, and each iteration necessitates storing the neutron flux values, consuming time and storage space. In order to address this issue, this paper proposes a precise and efficient boundary-type method, the Half Boundary Method (HBM). This method establishes relationships between adjacent node values through integration and mathematical deduction, ultimately obtaining the relationship among any node and the boundary values in any direction. As a result, iteration is only required for boundary values, reducing the iteration amount and thus saving time and storage space. Using the discrete ordinate (SN) method for angular discretization and HBM for spatial discretization, this paper solved the steady-state neutron transport equation for two-dimensional systems modeled with Cartesian geometry. The method is validated using several benchmark problems, including the 2D ISSA benchmark, the 2D mono-group k-eigenvalue problem and BWR rod bundle test problem. All of these problems demonstrate good results and effectively reduced computational time.

Wide-sense stationary discrete-time random processes: Analyses in the discrete-Fresnel and discrete-frequency domains

This article investigates the expectation and autocorrelation functions of random processes in the discrete-Fresnel and the discrete-frequency domains, originating from wide-sense stationary (WSS) random processes in the discrete-time domain. Through mathematical deduction, it is demonstrated that WSS random processes in the discrete-time domain exhibit the same expectation and autocorrelation functions as transformed to the discrete-Fresnel domain. In contrast, this property is not observed in the discrete-frequency domain. White, pink, and brown random processes are considered to illustrate these behaviors in the discrete-Fresnel and discrete-frequency domains. Furthermore, this study numerically compares data communication systems based on single-carrier with cyclic prefix, orthogonal chirp-division multiplexing, and orthogonal frequency-division multiplexing schemes, which are respectively designed in the discrete-time, discrete-Fresnel, and discrete-frequency domains. The numerical results show that orthogonal chirp-division multiplexing and single-carrier with cyclic prefix schemes achieve comparable communication performance under any additive noise modeled by a wide-sense stationary random process, regardless of the chosen linear equalization or the absence of it.

Secure impulsive synchronization for derivatively coupled networks against DoS attacks: A memory-compensated strategy

This work is concerned with the synchronization issue of complex networks suffering from Denial-of-Service (DoS) attacks. The derivative coupling has been proposed to model the nodes which are capable of perceiving state changing. An overarching handling procedure is initially introduced to convert derivatively coupled dynamic networks into a general form. To offset the information missing resulting from DoS attacks, an intricately crafted memory-based self-triggered impulsive controller (MBSTIC) is devised. The introduced scheme involves the meticulous collection of nodes’ historical state information. By jointly considering Lyapunov stability theorem and mathematical deduction approach, sufficient conditions for ensuring synchronous behavior for controlled networks are derived. A numerical example is given to demonstrate that the proposed method exhibits considerable performance when facing DoS attacks.

On the wavefunction cutoff factors of atomic hydrogen confined by an impenetrable spherical cavity

AbstractThe Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite‐Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi‐linear, and a large cavity region where the cutoff function is a step‐like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed.

Exploring α−ψ−ϕ contractive mapping: novel fixed point theorems in complete b-metric spaces

Background This paper explores α-ψ-ϕ contractive mappings, extending the field of self-map and fixed-point theorems. Methods We analyze α-ψ-ϕ contractive mappings using rigorous mathematical proofs and logical deductions. Results A key main result is established, supported by intuitive corollaries and practical examples, highlighting the applicability of our findings. Conclusions Our work provides a fresh perspective on contractive mappings, simplifying complex mathematical concepts and enriching the literature on fixed-point theorems.

Expression and prediction of pump and pumping station characteristics

Abstract According to the relevant literature of pumps and pumping stations and the expression and conversion method of pump efficiency and pump cavitation proposed by the author, through further careful mathematical deduction, the expression of pump and pump device characteristics is proposed. The proposed expressions for centrifugal pumps (and volute type mixed flow pumps) and axial flow pumps (and guide vane type mixed flow pumps) are applicable to both pumps and pump units; it is applicable not only to models, but also to similar prototype pumps and prototype pumping units of any diameter and parameters. The research and realization of performance expression of pumps and pump devices have great theoretical significance and practical application value for improving the design level of pumps and pumping stations and for accurately predicting the characteristics of pumps and pumping stations.

An unconditionally-stable well-posed relativistic particle pusher

Particle pushers widely used in Particle-in-Cell(PIC) simulations are commonly required to be unconditionally stable and meet the basic physical laws. In this work, basing on central difference, we propose an unconditionally stable and well posed particle pusher. By mathematical deduction, the high-dimensional non-linear problem for solving tensor-form relativistic Lorentz force law equation is transformed to a quartic scalar problem and a following linear problem. Some practical suggestions for programming and some numerical results are also given.

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

Quantifying residual oil in a household oil tank

On a chilly evening, a practical challenge was posed to 2nd level students taking an extra-curricular math class: Imagine a household cylindrical oil tank lying on its side. How could we determine how much heating oil remains in the tank? In this paper, we ask students to engage in mathematical deduction to devise methods for approximating a solution to this intriguing engineering problem. The only tools that we have access to are measuring tape, and a long clean stick, so resourcefulness becomes key. We establish a mathematical relationship between the parameters that we have access to measure and the volume of oil that remains in the tank. Expanding the scope, we encapsulate this solution within a Jupyter Notebook (Python), allowing for the calculation of oil in any cylindrical tank.

A Deep Learning Classification Scheme for PolSAR Image Based on Polarimetric Features

Polarimetric features extracted from polarimetric synthetic aperture radar (PolSAR) images contain abundant back-scattering information about objects. Utilizing this information for PolSAR image classification can improve accuracy and enhance object monitoring. In this paper, a deep learning classification method based on polarimetric channel power features for PolSAR is proposed. The distinctive characteristic of this method is that the polarimetric features input into the deep learning network are the power values of polarimetric channels and contain complete polarimetric information. The other two input data schemes are designed to compare the proposed method. The neural network can utilize the extracted polarimetric features to classify images, and the classification accuracy analysis is employed to compare the strengths and weaknesses of the power-based scheme. It is worth mentioning that the polarized characteristics of the data input scheme mentioned in this article have been derived through rigorous mathematical deduction, and each polarimetric feature has a clear physical meaning. By testing different data input schemes on the Gaofen-3 (GF-3) PolSAR image, the experimental results show that the method proposed in this article outperforms existing methods and can improve the accuracy of classification to a certain extent, validating the effectiveness of this method in large-scale area classification.

Dynamic characteristics of expectations of short-term interest rate and a generalized Vasicek model

This study's principal mathematical deduction exploits the importance of the specification of long-term equilibrium level in the mean-reversion short-term interest rate model—such as the CKLS (Chan, Karolyi, Longstaff, and Sanders) model—to describe the dynamic characteristics of future short-term interest rate expectations, especially long-horizon expectations. Therefore, we present a preferred model by introducing a stochastic long-run equilibrium level factor to extend the specifications of the Vasicek model's short-term interest rate dynamics. Using this new type of short-term interest rate as a driver we develop a two-factor affine arbitrage-free model of term structure, the generalized Vasicek mode. The empirical results show that the model not only has a good sample fitting ability to the Chinese government bond yield curve, but also has better performance in capturing the dynamic characteristics of short-term interest rates and short-term interest rate expectations. This study provides not only a promising avenue for future research on improving interest rate modeling techniques and their practical applications in the financial industry but also a new literature base for accurately identifying public expectations and explaining the underlying mechanisms of expectation changes.

A Generalized Microwave Absorption Criterion of Two‐Port Networks for Electromagnetic Interference Shielding

AbstractHerein, the Z11 argument tangent is proposed as a novel criterion for assessing energy loss (absorption) of the reciprocal two‐port networks composed of the shielding material. With a simple mathematical deduction, it can be found that the relationship between the network absorption and the Z11 argument tangent is inverted. The absorption decreases with increasing the Z11 argument tangent, and the Z11 argument is approximately equal to the power factor angle. Additionally, a series of samples are presented to verify their absorptions and the Z11 argument tangents. The numerical simulations on other works are also presented to calculate the Z11 argument tangent and validate the relationship between the Z11 argument tangent and electromagnetic absorption from another side. The experiment and simulation results agree well with that of the mathematical deduction. The findings can be applied in most passive shielding materials and provide a promising approach to guiding and developing absorption‐dominated electromagnetic shielding materials for next‐generation communication applications.

Single Isotopologue for In-Sample Calibration and Absolute Quantitation by LC-MS/MS.

Targeted mass spectrometry (MS)-based absolute quantitative analysis has been increasingly used in biomarker discovery. The ability to accurately measure the masses by MS enabled the use of isotope-incorporated surrogates having virtually identical physiochemical properties with the target analytes as calibrators. Such a unique capacity allowed for accurate in-sample calibration. Current in-sample calibration uses multiple isotopologues or structural analogues for both the surrogate and the internal standard. Here, we simplified this common practice by using endogenous light peptides as the internal standards and used a mathematical deduction of "heavy matching light, HML" to directly quantify an endogenous analyte. This method provides all necessary assay performance parameters in the authentic matrix, including the lower limit of quantitation (LLOQ) and intercept of the calibration curve, by using only a single isotopologue of the analyte. This method can be applied to the quantitation of proteins, peptides, and small molecules. Using this method, we quantified the efficiency of heart tissue digestion and recovery using sodium deoxycholate as a detergent and two spiked exogenous proteins as mimics of heart proteins. The results demonstrated the robustness of the assay.

The Peak Stability Analysis through Hysteresis Phenomenon on Heterogeneous Networks

The macroscopic fundamental diagram (MFD) is a nonuniversal changing process over network traffic status which indicates different shapes in different networks. Hysteresis is observed in the MFD of some urban networks. It is a unique phenomenon when the network remains at low stability level and usually appears around the congestion period. This paper analyzed network peak stability through focusing on hysteresis. The formation mechanism of hysteresis is deduced from the mathematical method based on previous research studies. The precondition of hysteresis and the changing process of network state can be figured by mathematical deduction. It indicates that hysteresis only occurs conditionally in the period of macroscopic congestion and is not a universal phenomenon. Heterogeneity is an important factor leading to network instability. The hysteresis patterns of different peaks in MFD are different due to the variation of network flow. Real data are collected from Atlanta’s urban network to verify the analysis of hysteresis. To discuss the changing process of hysteresis in different peaks, a three-stage division is proposed and time series is presented as a third dimension in MFD. It is worth noticing that the existence and form of hysteresis in morning and evening peaks are different. Although there is a higher peak flow in the morning peak, the stability of the evening peak performs better when hysteresis occurs in the network. The different fluctuations in the morning and evening peaks are exhibited through the 3D version of MFD. The otherness of hysteresis in different peaks is explained through a 3D coordinate system with cross-compared corresponding indexes.

Social Influence in Signed Networks

Social influence has been widely discussed in various disciplines due to its important sociological significance. However, the dynamics of social influence in signed networks have nonetheless received fairly little attention. In this article, we propose a generalized Pólya urn model that considers the effect of negative relationships and is capable of comprehending the specific mechanisms of homophily and xenophobia in the dynamics. Based on the mathematical deduction, we find that the signed network guides social influence in a trend toward equality. Simulation shows that a higher effect or larger proportion of negative relationships may break the self-reinforcement and make the market more equal. The collective dynamics in the signed network are more predictable but generate path dependence. We also find that a balanced structure has no impact on the average market share but is helpful in removing path dependence and promoting system stability.

Abstract B014: The number analysis of digital data in Nature’s ecosystem: The unified general dynamics theory of “Unity of One” and its fractional distribution, action and interaction principle, and algorithms for computerized translational informatics

Abstract Nature’s ecosystem is governed by the physical and chemical principle of the mass-action law (MAL). The conventional classical specific aimed observation/statistics-based scientific R&D is considered a “Bottom-Up” approach. Here, this author introduces a new mathematical approach that established the MAL-unified general dynamics theory-based “Top-Down” algorithms of scientific translational, computerized simulation for Econo-green R&D. The new math approach evoked the general system analysis with sequential, pattern, combinatorial analysis, and mathematical induction and deduction, which is distinct from the aim-specific derivation of Michaelis-Menten, Henderson-Hasselbalch, Hill, Scatchard, and Langmuir theoretical equations. In short, the unified MAL-based biodynamics, pharmacodynamics, and bio-informatics (BD/PD/BI) include three general equations/algorithms for Unity of One Theory (UOT): (i) For action, the median-effect equation (MEE), also called the doctrine of the median (DOM), where the median-effect dose (Dm) is the universal reference of action potency and the dynamic-order (m) common-link in which m=1, >1, <1 indicates hyperbolic, sigmoidal and flat sigmoidal, respectively for dose-effect curves (DEC); (ii) For interaction, the combination-index equation (CIE), in which CI=1, <1, and >1 quantitatively indicates additive-effect, synergism, and antagonism, respectively; (iii) For the consequence of interaction, the dose-reduction index equation (DRIE), in which DRI=1, >1 and <1, digitally defines no-dose-reduction, favorable, and not favorable dose-reduction, respectively. The linearization of DECs with the MEE plot (MEP) led to the minimum two dose-data points theory (MTDPT), which provides efficient, cost-effective, Econo-green scientific R&D in biomedical sciences such as in molecules, cells, tissues, organs, whole body, and clinical protocol design and data analysis/simulation in the general R&D, in which 2 becomes 4, 3 become 5, and 4 becomes 6 dose-data points. The automatically added default two data points in the algorithm are dose zero and Dm-dose in all dynamic studies, designs, and dynamic simulations. The most important feature of the MAL-dynamics and informatics theory is that all mathematical terms of MEE, CIE, and DRIE are dimensionless relativity ratios, thus ensuring universal applicability regardless of units, physical states, and mechanisms. As of September 11, 2023, the MAL-BD/PD/CI/BI theory, algorithm, and computer software have been cited in over 1,550 scientific journals and 1,188 patents, indicating broad applications in scientific R&D and patent discovery. These citations encompass biomedical, agricultural, environmental, marine, physical, and food sciences. The effectors can be nanoparticles, molecules, chemicals, biologicals, radiation, UV, microwave, thermal, and photo effects, or their mix. Relevant details are available at HTTPS://www.combosyn.com; DOI:10.1016/j.jbc.2023.103419, and dx.doi.org/10.1124/jpet.122.185320. Citation Format: Ting-Chao Chou. The number analysis of digital data in Nature’s ecosystem: The unified general dynamics theory of “Unity of One” and its fractional distribution, action and interaction principle, and algorithms for computerized translational informatics [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Translating Cancer Evolution and Data Science: The Next Frontier; 2023 Dec 3-6; Boston, Massachusetts. Philadelphia (PA): AACR; Cancer Res 2024;84(3 Suppl_2):Abstract nr B014.

Development of novel optimal operating maps for combined cooling, heating, and power systems

Following Thermal Load (FTL)/Following Electricity Load (FEL) strategy plays a critical role in combined cooling, heating, and power systems (CCHPs) to balance multi-energy supply and load, develop advanced optimal strategy, and optimizing system design and configuration. However, this study found that the conventional FEL/FTL is misemployed as the optimal operating foundation by overlooking the thermal energy wastage during the power generation process. Additionally, this study highlights that current operating schemes relying on FEL/FTL may result in substantial global energy waste due to the disregard for factors such as power generation type, power grid efficiency, optimal cooling supply scheme, and correct operating boundary. In this study, mathematical deduction and case study are performed to demonstrate that the conventional deployment of FEL/FTL is misapplied as optimal solutions. Further, an accurate and comprehensive methodology to derive optimal operating guidelines for diverse CCHP configurations is proposed. The optimal operating maps are derived by comparing primary fuel consumptions under FEL, FTL, and power grid supply schemes. Optimal cooling supply schemes are derived to determine the overall optimal operating strategy. The key findings indicate that the proposed methods for supplying electricity, cooling, and heating offer significant advantages over conventional approaches. Further, the case studies emphasize the potential for primary energy savings of up to 26.6% in hybrid CCHPs. The proposed methodologies serve as valuable references for revising existing research results and pave the way to develop optimal operating solutions for interactive CCHPs.

Dynamics analysis of a predator-prey model with Allee effect and harvesting effort

<p>In the paper, a predator-prey model with the Allee effect and harvesting effort was proposed to explore the interaction mechanism between prey and predator. Under the framework of mathematical theory deduction, some conditions for the occurrence of transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations were derived with harvesting effort and the Allee effect as key parameters. Under the framework of bifurcation dynamics numerical simulation, the evolution process of specific bifurcation dynamics behavior was gradually visualized to reveal the influence mechanism of the Allee effect and harvesting effort. The research results indicated that the Allee effect and harvesting effort not only seriously affected the bifurcation dynamics essential characteristics of the model (1.3), but also could promote the formation of constant steady state and periodic oscillation persistent survival mode of prey and predator. Furthermore, it is worth noting that appropriate harvesting effort was beneficial for the formation of a sustainable survival cycle between prey and predator. In summary, we hoped that the research findings could contribute to the comprehensive promotion of bifurcation dynamics studies in the predator-prey model.</p>