Cosine Function Research Articles

First-order compartment model solutions – Exponential sums and beyond

First-order compartment models are common tools for modelling many biological processes, including pharmacokinetics. Given the compartments and the transfer rates, solutions for the time-dependent quantity (or concentration) curves can normally be described by a sum of exponentials. This paper investigates cases that go beyond simple sums of exponentials. With specific relations between the transfer rate constants, two exponential rate constants can be equal, in which case the normal solution cannot be used. The conditions for this to occur are discussed, and advice is provided on how to circumvent these cases. An example of an analytic solution is given for the rare case where an exact equality is the expected result. Furthermore, for models with at least three compartments, cases exist where the solution to a real-valued model involves complex-valued exponential rate constants. This leads to solutions with an oscillatory element in the solution for the tracer concentration, i.e., there are cases where the solution is not a simple sum of (real-valued) exponentials but also includes sine and cosine functions. Detailed solutions for three-compartment cases are given. As a tentative conclusion of the analysis, oscillatory solutions appear to be tied to cases with a cyclic element in the model itself.

New belief divergence measure based on cosine function in evidence theory and application to multisource information fusion

Multisource information fusion technology significantly benefits from using information across various sources for decision-making, particularly by leveraging evidence theory to manage uncertain information efficiently. Nonetheless, dealing with highly conflicting evidence presents a considerable challenge. To tackle this issue, this paper introduces a new belief divergence measure within the framework of evidence theory. The proposed measure, which incorporates the cosine function and pignistic probability transformation, is adept at quantifying the disparity between the evidences while maintaining key properties, such as boundedness, non-degeneracy and symmetry. Moreover, building upon the concepts of proposed belief divergence and belief entropy, this paper further proposes a new fusion method that employs a weighted evidence average prior to the application of Dempster’s rule. The performance of the proposed method is validated on several applications, and the results demonstrate its superior ability to absorb highly conflicting evidence compared with existing methods.

Dynamical analysis and circuit realization of a high complexity fourth-order double-wing chaotic system with transient chaos and its application in image encryption

This paper proposes a fourth-order double-wing chaotic system with high complexity. After conducting a dynamic analysis, it is found that the system exhibits transient chaos and a rare inverse period-doubling bifurcation phenomenon in the bifurcation diagram. The system also exhibits attractor coexistence, with periodic, quasi-periodic, indicating high sensitivity to initial values. These phenomena sufficiently demonstrate the rich dynamical characteristics of chaotic systems. By introducing an impulse function with a cosine function in the foundation of the proposed system, it is found that controllable wing number and staircase burst oscillations occur. Furthermore, the number of wings and oscillation periods vary with changes in parameters, which has significant implications in engineering applications. The circuit design and construction are carried out using the Multisim simulation software, and the digital circuit is realized by using a Field-Programmable Gate Array (FPGA). It is found that the simulation results and the actual implementation results are highly consistent with the phase portrait of the system, thus demonstrating the feasibility of the circuit. Finally, by combining the proposed system with a DNA encryption algorithm, a novel image encryption algorithm with multiple layers of encryption is designed, greatly enhancing the security of encrypted images. The security of this encryption algorithm is analyzed in terms of information entropy, key space, correlation, and resistance to attacks. It is found that the proposed encryption algorithm exhibits high confidentiality and resistance to attacks. The proposed system has significant reference value in secure communication when applied to image encryption.

FLSSnet: Few labeled samples segmentation network for coated fuel particle segmentation

Current image semantic segmentation networks based on convolutional neural networks require extensive training samples and pixel-level annotations to achieve satisfactory segmentation performance. This paper proposes a novel FLSSnet segmentation network based on an encoder-decoder structure. A modified MobileNetV3 backbone network, a contour-assisted training branch, and an improved semi-supervised training method with a cosine function to dynamically update the pseudo-label training weight are added to the network, which effectively reduces its dependency on labeled samples and improves its segmentation accuracy. Compared with other image segmentation networks, the proposed network does not need too many labeled samples for training to achieve the high segmentation accuracy required for industrial inspection. The effectiveness of the proposed network is validated on the coated fuel particle segmentation dataset and the MICCAI’s 2015 gland segmentation public dataset. It has great industrial application value in automating the thickness measurement of each layer of coated fuel particles and other fields.

Monotonicity Results of Ratios between Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine

In the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions.

A Pressure Projection Scheme With Near‐Spectral Accuracy for Nonhydrostatic Flow in Domains With Open Boundaries

AbstractWe describe a pressure projection scheme for the simulation of incompressible flow in cubic domains with open boundaries based on fast Fourier transforms. The scheme is implemented in flow_solve, a numerical code designed for process studies of rotating, density‐stratified flow. The main algorithmic features of the open‐boundary code are the near‐spectral accuracy of the discrete differentiation and a dynamic two‐dimensional domain decomposition that scales efficiently to large numbers of processors. The simulated flows are not required to be periodic or to satisfy symmetry conditions at the open boundaries owing to the use of mixed series expansions combining cosine and singular Bernoulli polynomial basis functions. These expansions facilitate the imposition of inhomogeneous boundary conditions and allow the code to be used for offline, one‐way nesting within an arbitrarily embedded subdomain of a larger scale simulation. The projection scheme is designed to exploit a simple and powerful numerical engine: inversion of Poisson's equation with homogeneous Neumann boundary conditions using fast cosine transforms. Here, we describe the mathematical transformations used to accommodate the imposition of space‐ and time‐varying boundary conditions. The utility of the approach for process studies and for nesting within submesoscale‐resolving ocean models is demonstrated with simulations of wind‐driven near‐inertial waves in the upper ocean.

Geodesic Basis Function Neural Network.

In the learning of existing radial basis function neural networks-based methods, it is difficult to propagate errors back. This leads to an inconsistency between the learning and recognition task. This article proposes a geodesic basis function neural network with subclass extension learning (GBFNN-ScE). The geodesic basis function (GBF), which is defined here for the first time, uses the geodetic distance in the manifold as a measure to obtain the response of the sample with respect to the local center. To learn network parameters by back-propagating errors for the purpose of correct classification, a specific GBF based on a pruned gamma encoding cosine function is constructed. This function has a concise and explicit expression on the hyperspherical manifold, which is conducive to the realization of error back propagation. In the preprocessing layer, a sample unitization method with no loss of information, nonnegative unit hyperspherical crown (NUHC) mapping, is proposed. The sample can be mapped to the support set of the GBF. To alleviate the problem that one-hot encoding is not effective enough in the differential expression of data labels within a class, a subclass extension (ScE) learning strategy is proposed. The ScE learning strategy can help the learned network be more robust. For the working of GBFNN-ScE, the original sample is projected onto the support set of GBF through the NUHC mapping. Then the mapped samples are sent to the nonlinear computation units composed of GBFs in the hidden layer. Finally, the response obtained in the hidden layer is weighted by the learned weight to obtain the network output. This article theoretically proves that the separability of the data with ScE learning is stronger. The experimental results show that the proposed GBFNN-ScE has a better performance in recognition tasks than existing methods. The ablation experiments show that the ideas of the GBFNN-ScE contribute to the algorithm performance.

Robust constrained cosine arctangent based adaptive filtering for feedback control in hearing aids

The striking problem of acoustic feedback remains a persistent challenge in hearing aids (HAs), imposing limitations on attainable amplification and significantly deteriorating sound quality, often manifesting as disruptive howling artifacts. With the objective of feedback mitigation, the feedback path is estimated using an adaptive filter. The constrained least mean square (CLMS) technique is commonly employed in adaptive filtering applications where it is necessary to meet specific linear constraints. However, algorithm's robustness is compromised when impulsive or non-Gaussian noise interference occurs. In lieu of this, a novel algorithm known as the robust constrained cosine arctangent (CCAT) adaptive filtering is introduced to enhance the convergence behavior. This has been formulated with the inclusion of linear constraints to the suggested cosine function integrated with the arctangent technique. This algorithm aims to address the issue of feedback cancellation in hearing aids along a set of constraints. To achieve a minimal steady-state error and expedite the convergence of the method, a novel policy that adjusts the step factor has been suggested. CCAT-VSS is a modified version of CCAT that includes a variable step size (VSS). The simulation findings demonstrate that the suggested method exhibits superior performance for misalignment, added stable gain and certain voice quality assessments.

Applying the Sine-Cosine Optimization Algorithm to the Parametric Estimation Problem in Three-Phase Induction Motors

The steady-state analysis of electrical machines requires a detailed characterization of their equivalent electrical circuit, which adequately represents the transformation and interaction between electrical and mechanical energy. This research aims to characterize the equivalent circuit of three-phase induction motors by minimizing the mean square error between the measured and calculated torque variables. These torques are obtained from data provided by the manufacturer, including starting, peak, and full-load torques. A metaheuristic optimization technique is applied to solve the resulting nonlinear programming model based on the interactions between the sine and cosine functions. The numerical results obtained with this algorithm demonstrate its efficiency in terms of response quality, reaching objective function values of less than \(1\times10^{-8}\) with regard to the measured and calculated variables. Simulation results in two test systems allow concluding that the parametric estimation problem in three-phase induction motors is a multimodal optimization problem. This implies a potentially infinite set of solutions that minimize the root mean square error and adequately represent the behavior of the motor's output torque under various probable operating conditions.

Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model

Abstract Exploring nonlinear equations and systems with predetermined coefficient values constrains the depth of understanding of the dynamics inherent in various applications and phenomena represented by such equations. On the contrary, exploration of nonlinear models with free coefficients offers avenues for improved development and ongoing refinement. In light of this, this study aimed to reassess the Wu–Zhang (WZ) system and Sawada–Kotera (SK) model by introducing arbitrary coefficients. Our goal is to identify the constraints necessary to ensure the existence of soliton solutions. Through the application of two distinct approaches, namely, the sine–cosine function method and tanh–coth expansion method, we systematically examine the conditions that facilitate the emergence of solitons within the WZ system and SK model. The insights gained from this analysis are supported by the presentation of 2D and 3D plots, providing a visual depiction of the propagation characteristics exhibited by the obtained solutions. The findings of the current work on conditions for the existence of soliton solutions for both generalized Wu–Zhang and generalized Sawada–Kotera models are novel and presented here for the first time.

Superlattice Delineated Fermi Surface Nesting and Electron-Phonon Coupling in CaC6

The superconductivity of CaC6 as a function of pressure and Ca isotopic composition was revisited using DFT calculations on a 2c–double hexagonal superlattice. The introduction of superlattices was motivated by previous synchrotron absorption and Raman spectroscopy results on other superconductors that showed evidence of superlattice vibrations at low (THz) frequencies. For CaC6, superlattices have previously been invoked to explain the ARPES data. A superlattice along the hexagonal c-axis direction is also illustrative of atomic orbital symmetry and periodicity, including bonding and antibonding s-orbital character implied by cosine-modulated electronic bands. Inspection of the cosine band revealed that the cosine function has a small (meV) energy difference between the bonding and antibonding regions, relative to a midpoint non-bonding energy. Fermi surface nesting was apparent in an appropriately folded Fermi surface using a superlattice construct. Nesting relationships identified phonon vectors for the conservation of energy and for phase coherency between coupled electrons at opposite sides of the Fermi surface. A detailed analysis of this Fermi surface nesting provided accurate estimates of the superconducting gaps for CaC6 with the change in applied pressure. The recognition of superlattices within a rhombohedral or hexagonal representation provides consistent mechanistic insight on superconductivity and electron−phonon coupling in CaC6.

Explicit solutions for the ground displacement of a sliding model simulating both the frictional and rotational effects under idealized acceleration pulses

AbstractFor the first time, the present work derives explicit equations predicting the downward ground displacement of a sliding model simulating both the frictional and rotational effects under idealized acceleration pulses, in the form of simple formulas. Explicit equations allow not only accurate predictions in all cases, but also analysis of the solutions and derivation of expressions for limit cases. Half and full cycles of (i) rectangular, (ii) triangular, (iii) trapezoidal, and (iv) sinusoidal pulses and slopes both under static stability and instability are considered. For this purpose, for pulse cases (i)–(iii) recently proposed implicit analytical solutions are used, while for case (iv), first the analytical equations predicting the sliding displacement and velocity of a sinusoidal pulse in terms of time are obtained and then the time duration of motion is estimated, by using the Bhaskara approximation of the sine and cosine functions. Then, from these, solutions for the particular limit cases corresponding to the “conventional” sliding‐block model (Case A) and the post‐failure run‐off movement without any applied pulse (Case B) are derived. The results for Case A provide a useful tabulation of sliding‐block solutions, some of which are not reported in the literature. The results for Case B provide novel predictions of the time duration of motion in the case of post‐failure movement. The general solutions are analyzed graphically and the deviation from the solutions of Cases A and B is illustrated. Finally, the explicit solutions are compared to solutions of actual accelerograms.

A novel underwater weak signal detection method based on parameter optimized VMD and 3D chaotic system

Based on chaos theory, combined with improved variational mode decomposition, a novel method for underwater weak signal detection and frequency extraction is proposed. The aim of this research is to address the challenge of detecting underwater signals and extracting their frequencies under ultra-low Signal-to-Noise Ratio (SNR). A novel three-dimensional (3D) chaotic system with infinitely shrinking attractors is proposed, which incorporates a cosine function to achieve chaos control and enhancement. The nonlinear characteristics of the system, such as Lyapunov exponent and bifurcation, are analyzed in detail. Then, we design a frequency array and perform scale transformation on the chaotic system to construct a detection model that covering all frequencies. The model can transition from critical chaos to large period or intermittent chaos in the presence of external periodic signals by adjusting the parameters of the cosine function. Considering the relationship between the generation of intermittent chaos and the frequency of target signal, a frequency extraction algorithm based on chaos sparrow search algorithm optimization variational mode decomposition (CSSA-VMD) and Hilbert transform is designed. The algorithm can estimate the real frequency of underwater signals by performing noise reduction and envelope analysis on intermittent chaotic signals. The experimental results demonstrate that the proposed system can detect weak signals in complex marine backgrounds and extract their frequency features, with a detection SNR of -48 dB and a frequency error of less than 0.01%. Importantly, the circuit of the detection system has been implemented, verifying its feasibility for detecting and tracking underwater signals.

Generalized cosine-Gaussian breathing solitons with transverse cross modulation in nonlocal nonlinear media

Based on the nonlocal nonlinear Schrödinger equation, the propagation characteristics of transverse cross-modulated cosine-Gaussian (TCMCG) breathing solitons in strongly nonlocal media are studied. A series of mathematical expressions are provided to describe the propagation dynamics characteristics of the TCMCG breathing solitons, including the soliton width, soliton width change rate, soliton width extreme values, change rates of soliton width extreme values, phase wavefront curvature and phase wavefront curvature extreme values, etc. The research results indicate that by modulating TCMCG breathing solitons, generalized high-order spatial solitons can be formed with periodic changes in light intensity modes but unchanged in beam width. The cosine function parameter has a significant impact on the propagation dynamics characteristics, which appears in the initial expression and plays a cross modulation role in the transverse spatial light intensity. The effect of the initial incident power on soliton width extreme values and phase wavefront curvature extreme values is also discussed.

Broadband vibration mitigation using a two-dimensional acoustic black hole phononic crystala).

Acoustic black holes (ABHs) are known as efficient structural dampers. Periodic lattices are identified as an efficient way to forbidden wave propagation in targeted frequency bandgaps (BGs). The paper demonstrates the possibility to merge the ABH effect with Bragg BGs. The geometrical layout leading to this double effect consists of a plate of periodically modulated thickness by a combination of cosine functions of the spatial coordinates constituting an ABH-like cell, coated with a thin damping layer. The resulting metamaterial allows the realization of solid, stiff, and nonresonant panels over a wide frequency range, including low frequencies, without increasing the mass. First, the band structure is analyzed in the conservative case (without damping layer) using a plane wave expansion model following Kirchhoff's assumptions. The results show the existence of low-frequency BGs that can be controlled by only three geometric parameters, which are defined on the type of lattice chosen (square or hexagonal). Next, a finite size panel is designed for the hexagonal lattice. Experimental characterization of the demonstrator with and without viscoelastic coating shows very attractive broadband vibration mitigation performances due to the fact that the dissipation produced by the ABH effect does not deter filtering effects produced in the BG.

Uniform rate inflation on the brane

We propose a model of uniform rate inflation on the brane. The potential is given by a hyperbolic cosine function plus a negative cosmological constant. The equation of motion is solved analytically without using slow-roll approximation. The result is that the inflaton field is rolling at a constant speed. The prediction for cosmological perturbations depends on the field value at the end of inflation. The experimental constraints could be satisfied in the parameter space.

Client-centered detached modular housing: natural language processing-enabled design recommender system

Abstract Designing modular housing is a complex task that necessitates a thorough understanding of the diverse needs of clients in terms of both design aesthetics and floor plan layout. Furthermore, adhering to design for manufacture and assembly (DfMA) principles adds to the complexity, as these are essential modular home requirements. Traditional construction methods frequently fail to meet the specific needs of both clients and DfMA, potentially resulting in suboptimal design solutions. Incorporating client requirements during the design phase necessitates the use of an effective system and framework to reduce changes in subsequent project stages. Existing literature lacks a suitable approach, particularly in the context of modular housing. To address this gap, this paper introduces an artificial intelligence–building information modeling recommender system (RS) for detached modular housing design. The system processes client requirements entered as text utilizing the Word2vec algorithm with the GloVe dataset, refined through transfer learning using surveyed client data of housing needs. The system recommends three distinct modular building design alternatives sourced from a building information modeling models database using cosine and Euclidean similarity functions. A sensitivity analysis ensures that client needs are considered fairly, increasing the robustness of the RS. By incorporating natural language processing, this system transforms the construction industry by making initial designs more client-centric compared with traditional methods. Furthermore, it promotes improved collaboration among clients, design, and construction teams, reducing modifications to design in later stages of construction.

A Modified Sine Cosine Algorithm for Numerical Optimization

Sine cosine algorithm (SCA) is a recently developed meta-heuristic method based on the characteristics of sine and cosine functions. However, SCA algorithm may suffer premature convergence in solving complex optimization problems. To mitigate this limitation, a modified SCA (MSCA) is developed to achieve an effective balance between exploration and exploitation. Specifically, the conversion parameter can be configured adaptively by using the individual fitness information in the evolution process to guide the individual search behavior. After that, an inertia weight is embedded in the position updating equation to further improve the search accuracy and accelerate the convergence speed. The performance of MSCA algorithm is investigated on 13 benchmark functions and the chaotic time series prediction problem. Experimental results demonstrate that MSCA algorithm is more effective than the other optimization methods.

Efficient Optimization of a Support Vector Regression Model with Natural Logarithm of the Hyperbolic Cosine Loss Function for Broader Noise Distribution

While traditional support vector regression (SVR) models rely on loss functions tailored to specific noise distributions, this research explores an alternative approach: ε-ln SVR, which uses a loss function based on the natural logarithm of the hyperbolic cosine function (lncosh). This function exhibits optimality for a broader family of noise distributions known as power-raised hyperbolic secants (PHSs). We derive the dual formulation of the ε-ln SVR model, which reveals a nonsmooth, nonlinear convex optimization problem. To efficiently overcome these complexities, we propose a novel sequential minimal optimization (SMO)-like algorithm with an innovative working set selection (WSS) procedure. This procedure exploits second-order (SO)-like information by minimizing an upper bound on the second-order Taylor polynomial approximation of consecutive loss function values. Experimental results on benchmark datasets demonstrate the effectiveness of both the ε-ln SVR model with its lncosh loss and the proposed SMO-like algorithm with its computationally efficient WSS procedure. This study provides a promising tool for scenarios with different noise distributions, extending beyond the commonly assumed Gaussian to the broader PHS family.

Classification feature selection and dimensionality reduction based on logical binary sine-cosine function arithmetic optimization algorithm

Arithmetic optimization algorithm (AOA) is a meta-heuristic algorithm inspired by mathematical operations. AOA has been diffusely used for optimization issues on continuous domains, but few scholars have studied discrete optimization problems. In this paper, we proposed Binary AOA (BAOA) based on two strategies to handle the feature selection problem. The first strategy used S-shaped and V-shaped shift functions to map continuous variables to discrete variables. The second strategy was to combine four logical operations (AND, OR, XOR, XNOR) on the basis of the transfer function, and constructed a parameter model based on the sine and cosine function. An enhanced logic binary sine–cosine function arithmetic optimization algorithm (LBSCAOA) was proposed to realize the position update of variables. Its purpose was to improve the algorithm's global search capabilities and local exploitation capabilities. In the simulation experiments, 20 datasets were selected to testify the capability of the proposed algorithm. Since KNN had the advantages of easy understanding and low training time complexity, this classifier was selected for evaluation. The performance of the improved algorithm was comprehensively evaluated by comparing the average classification accuracy, the average number of selected features, the average fitness value and the average running time. Simulation results showed LBSCAOA with V-Shaped “V4” stood out among many improved algorithms. On the other hand, LBSCAOA with V-Shaped “V4” was used as a representative to compare with other typical feature selection algorithms to verify its competitivenes.