We approximate the trigonometric function sine and cosine on the interval . This analysis provides two formulas to approximate sine and cosine. At first, we try to derive the formula which involves a square root, and then we derive another formula that does not require any use of a square root. Nevertheless, after deriving the procedure which requires no square root, we further try to increase its accuracy and then derive another formula that approximates trigonometric functions more accurately on the interval [0, pi/2]. So, this analysis provides mainly two types of procedures. One uses square roots, whereas the other does not. We also focus on ensuring the accuracy of these trigonometric functions in the interval [0,pi/2]. This accuracy analysis is portrayed using the graph. This graph shows the difference between the values generated by the functions defined here and the actual value of these functions. So, these graphs also indicate the error of these functions on that interval. Finally, we compare our approximation with the approximation formula of the 7th-century Indian Mathematician Bhaskara I.

Let $\mathcal{\mathcal{S}}_{\cos}^{\ast}$ denote the class of normalized analytic functions $f$ in the open unit disk $\mathbb{D}$ satisfying the subordination $\dfrac{zf^{\prime}(z)}{f(z)}\prec\cos z$. In the second section of this article we find the sharp upper bounds for the initial coefficients $a_{3}$, $a_{4}$ and $a_{5}$ and the sharp upper bound for module of the Hankel determinant $|H_{2,3}(f)|$ for the functions from the class $\mathcal{S}_{\cos}^{\ast}$. The first result of the next section deals with the sharp upper bounds of the logarithmic coefficients $\gamma_{3}$ and $\gamma_{4}$ and we found in addition the sharp upper bound for $\left|H_{2,2}\left(F_{f}/2\right)\right|$. For obtaining these results we used the very useful and appropriate Lemma 2.4 of N.E. Cho et al. [Filomat 34(6) (2020), 2061--2072], and the technique for finding the maximum value of a three variable function on a closed cuboid. All the maximum found values were checked by using MAPLE\texttrademark{} computare software, and we also found the extremal functions in each cases. All of our present results are the best ones and give sharp versions of those recently published in [Hacet. J. Math. Stat. 52, 596--618, 2023].

We developed a novel hybrid approach for solving global optimization, computer science, bio-medical and engineering real life applications that is based on the coupling of the Whale Optimizer and Sine Cosine Algorithms via a surrogate model. We relate the whale optimizer algorithm to balance between the exploitation and the exploration process in the proposed method. There exist confirmed techniques for searching approximate best optimal solutions, but our algorithm will further guarantee that such numerical and statistical solutions satisfy physical bounds of the standard and real life functions. Our experiments with the benchmark, bio-medical, computer science and engineering real life problems have illustrated the advantages of using a newly hybrid approach based on mixing Whale Optimizer and Sine Cosine algorithms. It holds considerable potential for reducing execution time for solving standard and real life problems and at the same time improving the quality of the solution.

In this study we introduce the new classes $\mathcal{M}_{\alpha}(\sin)$ and $\mathcal{M}_{\alpha}(\cos)$ of $\alpha$-convex functions associated with sine and cosine functions. Also, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of Zalcman functional for the class $\mathcal{M}_{\alpha}(\cos)$ for the case $n=3$, showing that the Zalcman conjecture holds for this value. Moreover, the problem of the Fekete-Szeg\H{o} functional estimate for these classes is studied.

The work analyses the way in which symptoms and life themes manifest in middle-aged adults diagnosed with major recurrent depression. Specifically, the relationships between symptoms, life themes, and life themes - symptoms have been analyzed. For this purpose, Spearman correlation, and the methods of Latent Semantic Indexing (LSI) were used. Seven symptoms and twenty-six life themes were identified in the patients analyzed as well as similarities of symptoms/life themes (at patient level), the ranking of the importance of symptoms on each life theme (at the level of the group of patients), the rankings of the similarities of life themes in relation to different symptoms or various groups of symptoms (at the level of the group of patients), and dysfunctional cycles of symptoms and life themes (at patient level). The findings only refer to the patients analyzed. Although our findings cannot be generalized, there is a possibility that some of them may also be encountered in other patients. However, the design of the work can be used to initiate other similar studies.

Teaching kinematic rotations is a daunting task for even some of the most advanced mathematical minds. However, envisioning and explaining the three-dimensional rotations can be highly simplified by changing the paradigm. This paradigm change allows for a high school student with a understanding of geometry to be able to not only develop the matrix but also explain the rotations at a collegiate level. The proposed method includes the assumption of a point (P) within the initial three-dimensional frame with axes (x ̂_i, y ̂_i, z ̂_i). It then utilizes a two-dimensional rotation view (2DRV) to measure how the coordinates of point P translate after the initial axis is rotated instead of using the established Euler's formula. The equations are used in matrix notation to develop a direction cosine matrix (DCM) for future equations. This method provides a high school student with an elementary and repeatable process to compose and explain kinematic rotations.

This paper proposes a novel facial expression recognition network called the Lightweight Facial Network with Spatial Bias (LFNSB). The LFNSB model balances model complexity and recognition accuracy. It has two key components: a lightweight feature extraction network (LFN) and a Spatial Bias (SB) module for aggregating global information. The LFN introduces combined channel operations and depthwise convolution techniques, effectively reducing the number of parameters while enhancing feature representation capability. The Spatial Bias module enables the model to focus on local facial features while also capturing the dependencies between different facial regions. Additionally, a novel loss function called Cosine-Harmony Loss is designed. This function optimizes the relative positions of feature vectors in high-dimensional space, resulting in better feature separation and clustering. Experimental results on the AffectNet and RAF-DB datasets show that the proposed LFNSB model performs excellently in facial expression recognition tasks. It achieves high recognition accuracy while significantly reducing the number of parameters, thus substantially lowering model complexity.

Malignant growth is the most widely recognized repulsive infections winning around the world, and the patients with disease are saved just when the malignant growth is distinguished at the beginning phase. Each kind of disease is interesting, with its own arrangement of development properties and hereditary changes. This paper presents the lung knob division and disease characterization by proposing an enhancement calculation. The general technique of the created approach includes four stages, such as pre-processing, division, highlight extraction, and the order. From the outset, the CT picture of the lung is taken care of to the division. When the division is done, the highlights are extricated through morphological and measurable and surface highlights like LOOP and LGP. At long last, the extricated highlights are given to the order step. Here, the characterization is done dependent on the Deep Belief Network (DBN) which is prepared by utilizing the proposed Chicken-Sine Cosine Algorithm (CSCA) which distinguish the lung tumor, giving two classes in particular, knob or non-knob. The presentation assessment of lung knob division and malignant growth grouping dependent on CSCA is figured utilizing three measurements to be specific, precision, affectability, and the explicitness.

In this paper, we examine two strategies for boosting the performance of ensembles of Siamese networks (SNNs) for image classification using two loss functions (Triplet and Binary Cross Entropy) and two methods for building the dissimilarity spaces (FULLY and DEEPER). With FULLY, the distance between a pattern and a prototype is calculated by comparing two images using the fully connected layer of the Siamese network. With DEEPER, each pattern is described using a deeper layer combined with dimensionality reduction. The basic design of the SNNs takes advantage of supervised k-means clustering for building the dissimilarity spaces that train a set of support vector machines, which are then combined by sum rule for a final decision. The robustness and versatility of this approach are demonstrated on several cross-domain image data sets, including a portrait data set, two bioimage and two animal vocalization data sets. Results show that the strategies employed in this work to increase the performance of dissimilarity image classification using SNN is closing the gap with standalone CNNs. Moreover, when our best system is combined with an ensemble of CNNs, the resulting performance is superior to an ensemble of CNNs, demonstrating that our new strategy is extracting additional information.

In this paper, we proposed the fusion of two projection based face recognition algorithms: local binary Patterns in DCT domain and singular value decomposition (SVD) characterized by its simplicity and efficiently. Standard databases ORL are used to test the experimental results which prove that proposed system achieves more accurate face recognition as compared to individual method.

In the paper, the author finds bounds of the sine and cosine along a circle on the complex plane in terms of two double inequalities for the norms of the sine and cosine along a circle on the complex plane.

The superconductivity of CaC6 as a function of pressure and of abundant Ca isotopes is revisited using DFT calculations on a 2c – double hexagonal superlattice. The introduction of superlattices is motivated by previous synchrotron absorption and Raman spectroscopy results on other superconductors that show evidence of superlattice vibrations at low (THz) frequencies. A 2c superlattice for CaC6 is 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 reveals that the cosine function has a small (meV) energy difference between bonding and antibonding regions relative to a midpoint non-bonding energy. Fermi surface nesting is apparent in an appropriately folded Fermi surface using a superlattice construct. Nesting relationships identify phonon vectors for conservation of energy and for phase coherency between coupled electrons at opposite sides of the Fermi surface. Detailed analysis of this Fermi surface nesting provides accurate estimates of the superconducting gaps for CaC6 with change in applied pressure. Recognition of superlattices within a rhombohedral or hexagonal representation provides consistent mechanistic insight on superconductivity and electron−phonon coupling in CaC6.

This study aims to apply a Recommendation System with Content Based Filtering method with Jaccard Similarity and Cosine Similarity algorithms on the E-Learning Platform. Recommendation systems deal with how to provide personalized recommendations to users efficiently. The Content Based Filtering method with Jaccard Similarity and Cosine Similarity algorithms can be used to calculate the similarity value between E-Courses on the E-Learning Platform. Implementation for Recommendation System using Google Colaboratory with Python programming language. In the application of the Recommendation System dataset, Coursera Free Dataset consists of 975 instances. The recommendation results use the Jaccard Similarity algorithm with an average similarity value of 0.3 while the value of Cosine Similarity with the average similarity value is 0.6 where the similarity value of Cosine Similarity is higher. Based on the results of the Mean Absolute Error in the low recommendation system, the average MAE value for all iterations of Jaccard Similarity algorithm is 0.013 and for the Cosine Similarity algorithm the average MAE value for all iterations is 0.014. This shows that the Recommendation System with Jaccard Similarity and Cosine Similarity algorithms can be used on the E-Learning Platform to provide efficiency solutions for personalized recommendations.

The goal of this paper is to propose a dual version of the direct cosine simplex algorithm (DDCA) for general linear problems. Unlike the two-phase and the big-M methods, our technique does not involve artificial variables. Our technique solves the dual Klee-Minty problem in two iterations and solves the dual Clausen’s problem in four iterations. The utility of the proposed method is evident from the extensive computational results on test problems adapted from NETLIB. Preliminary results indicate that this dual direct cosine simplex algorithm (DDCA) reduces the number of iterations of two-phase method.

In the paper, the author discusses and computes bounds of the sine and cosine along straight lines on the complex plane.

High Efficiency Video Coding (HEVC) uses an 8-point filter and a 7-point filter, which are based on the discrete cosine transform (DCT), for the 1/2-pixel and 1/4-pixel interpolations, respectively. In this paper, discrete sine transform (DST)-based interpolation filters (IF) are proposed. The first proposed DST-based IFs (DST-IFs) use 8-point and 7-point filters for the 1/2-pixel and 1/4-pixel interpolations, respectively. The final proposed DST-IFs use 12-point and 11-point filters for the 1/2-pixel and 1/4-pixel interpolations, respectively. These DST-IF methods are proposed to improve the motion-compensated prediction in HEVC. The 8-point and 7-point DST-IF methods showed average BD-rate reductions of 0.7% and 0.3% in the random access (RA) and low delay B (LDB) configurations, respectively. The 12-point and 11-point DST-IF methods showed average BD-rate reductions of 1.4% and 1.2% in the RA and LDB configurations for the Luma component, respectively.

Software defect prediction is an important means to guarantee software quality. Because there are no sufficient historical data within a project to train the classifier, cross-project defect prediction (CPDP) has been recognized as a fundamental approach. However, traditional defect prediction methods using feature attributes to represent samples, which can not avoid negative transferring, may result in poor performance model in CPDP. This paper proposes a multi-source cross-project defect prediction method based on dissimilarity space ( DM-CPDP). This method first uses the density-based clustering method to construct the prototype set with the cluster center of samples in the target set. Then, the arc-cosine kernel is used to form the dissimilarity space, and in this space the training set is obtained with the earth mover’s distance (EMD) method. For the unlabeled samples converted from the target set, the KNN algorithm is used to label those samples. Finally, we use TrAdaBoost method to establish the prediction model. The experimental results show that our approach has better performance than other traditional CPDP methods.

This article introduces ten similarity measures that utilize cosine and cotangent functions to compare (m,n)-rung orthopair fuzzy sets, this a super class of q-rung orthopair fuzzy sets. Further, we applied our established weighted similarity measures to medical diagnosis, pattern recognition,and building material problems and also presented the comparison between established measures with the existing cosine and cotangent measures of q-rung orthopair, pythagorean and intuitionistic fuzzy sets. Lastly, we considered a numerical example that demonstrates the effective and scientific application of these similarity measures in plant leaf disease classification. Furthermore, we present graphical interpretations to demonstrate the effectiveness of our established measures.

In this paper, we present a framework that automatically labels Latent Dirichlet Allocation (LDA) generated topics using sentiment and aspect terms from COVID-19 tweets to help the end-users by minimizing the cognitive overhead of identifying key topics labels. Social media platforms especially Twitter are considered as one of the most influential sources of information for providing public opinion related to a critical situation like the COVID-19 pandemic. LDA is a popular topic modelling algorithm that extracts hidden themes of documents without assigning a specific label. Thus automatic labelling of LDA-generated topics from COVID-19 tweets is a great challenge instead of following the manual labelling approach to get an overview of wider public opinion. To overcome this problem, in this paper, we propose a framework named \texttt{SATLabel} that effectively identifies significant topic labels using \textit{top unigrams features of sentiment terms and aspect terms clusters from LDA generated topics} of COVID-19 related tweets to uncover various issues related to the COVID-19 pandemic. The experimental results show that our methodology is more effective, simpler, and traces better topic labels compare to the manual topic labelling approach.

Speech enhancement technology seeks to improve the quality and intelligibility of speech signals degraded by noise, particularly in telephone communications. Recent advancements have focused on leveraging deep neural networks (DNN), especially U-Net architectures, for effective denoising. In this study, we evaluate the performance of a 6-level skip-connected U-Net constructed using either conventional convolution activation blocks (CCAB) or innovative global local former blocks (GLFB) across different processing domains: temporal waveform, short-time Fourier transform (STFT), and short-time discrete cosine transform (STDCT). Our results indicate that the U-Nets can receive better signal-to-noise ratio (SNR) and perceptual evaluation of speech quality (PESQ) when applied in the STFT and STDCT domains, with comparable short-time objective intelligibility (STOI) scores across all domains. Notably, the GLFB-based U-Net outperforms its CCAB counterpart in metrics such as CSIG, CBAK, COVL, and PESQ, while maintaining fewer learnable parameters. Furthermore, we propose domain-specific composite loss functions, considering the acoustic and perceptual characteristics of the spectral domain, to enhance the perceptual quality of denoised speech. Our findings provide valuable insights that can guide the optimization of DNN designs for causal speech denoising.

To address the large hysteresis and susceptibility to instability in high-speed deposition fusion moulding (FDM) 3D printing’s temperature control, this paper proposes an improved ant-lion optimization algorithm (OCALO) to optimize the PID parameters.The improvement aims to solve the problem that the traditional ant-lion optimization algorithm (ALO) reduces the population diversity and tends to fall into the local optimal solution when it is close to the global optimal solution.The generation of the initial solution is enhanced by the introduction of a new Tent-Logistic-Cotangent composite chaotic mapping. This integration into Elite Opposition-Based Learning aids in optimizing population diversity. Concurrently, the incorporation of cosine factors of a specified parameter into the elitist formula of the traditional ALO algorithm is aimed at reducing the tendency of ants to randomly gravitate towards ant-lions with lower fitness values. This adjustment helps in balancing the exploration-exploitation trade-off in the algorithmic process. Compared with two existing classical algorithms and three improved ALO algorithms, the algorithm improves the convergence speed, global search ability and the ability to jump out of the local optimal solution. The improved algorithm was combined with PID control to design an OCALO-PID temperature controller and simulated on a high-speed 3D printing temperature model identified by modelling. The results show that the method improves the transient and steady-state performance of reactor temperature control with good control accuracy and robustness. Finally, the proposed algorithm is applied to a physical experimental platform to verify the feasibility of the algorithm.

Jellyfish search (JS) algorithm impersonates the foraging behavior of jellyfish in the ocean. It is a new developed meta-heuristic algorithm that solves complex and real world optimization problems. The global explore capability and robustness of JS are strong, but JS still has great development space in solving complex optimization problems with high dimensions and multiple local optima. Therefore, an enhanced jellyfish search (EJS) algorithm is developed in this study, and three improvements are made: (i) By adding sine and cosine learning factors, the jellyfish can learn from both random individual and best individual during Type B motion in swarm to enhance the optimization capability and convergence speed; (ii) Adding local escape operator can skip local optimal trap and boost the exploitation ability of JS; (iii) Opposition-based learning operator and quasi-opposition learning operator can increase and strengthen the population distribution more diversified, and better individuals are selected from present and new opposition-solution to participates in the next iteration, which can boost the solution’s quality, meanwhile convergence speed is fasted and its precision is increased. In addition, the performance contrast of the developed EJS and some previous outstanding and advanced methods are evaluated on CEC2017, CEC2019 test suite and six real engineering example of case. It is demonstrated that EJS algorithm escaped the trap of local optimum, enhanced the solution’s quality and the calculation speed. What’s more, the practical engineering applications of EJS algorithm also verify its superiority and effectiveness in solving both constrained and unconstrained optimization problems, and it stretched one’s mind for solving such optimization problems.

In today’s world face detection is the most important task. Due to the chromosomes disorder sometimes a human face suffers from different abnormalities. For example, one eye is bigger than the other, cliff face, different chin-length, variation of nose length, length or width of lips are different, etc. For computer vision currently this is a challenging task to detect normal and abnormal face and facial parts from an input image. In this research paper a method is proposed that can detect normal or abnormal faces from a frontal input image. This method used Fast Fourier Transformation (FFT) and Discrete Cosine Transformation of frequency domain and spatial domain analysis to detect those faces.

The terrain of Changbai Mountain has great influence on the distribution of atmospheric flows and the occurrence and development of precipitation. This study quantitatively analyzes the regional characteristic of topographic perturbations and the relationship between terrain, wind and precipitation in Changbai Mountain region, by using a spectral analysis of the two-dimensional discrete cosine transform. Three domains with relatively heavy summer precipitation are selected as the study region. The results indicate that the overall terrain of Changbai Mountain region exhibits anisotropic characteristics. The terrain spectra of domain B are less than those of domain A and C across the whole wavelength (λ) bands, indicating that the large-scale topographic perturbations of domain B are relatively weak. The largest topographic spectral peak of domain C shows the most pronounced undulation of terrain among the three domains. The dominant wavelengths of terrain height variance for domain A and C, both close to the respective maximum wavelengths, indicate more prominent large-scale topographic perturbations. For domain A, the variation of the precipitation spectra is consistent with that of the wind spectra at the wavelength bands of λ < 390 km, showing a high correlation between wind field and the occurrence of rainfall. The inverse relationship at larger wavelengths indicates that multiple factors contribute to the occurrence of rainfall. For domain B, there is consistency in the fluctuations of terrain spectra, precipitation spectra, and wind spectra at the wavelength bands of λ < 278.3 km, implying that the smaller-scale terrain has an important effect on the occurrence of summer precipitation. For domain C, the variations of terrain spectra, precipitation spectra, and wind spectra are almost consistent across the whole wavelength bands, indicating that the large-scale terrain and minor terrain both play a crucial role in atmospheric uplift and the occurrence and development of summer rainfall.

Several depth image based rendering (DIBR) watermarking methods have been proposed, but they have various drawbacks, such as non-blindness, low imperceptibility, and vulnerability to signal or geometric distortion. This paper proposes a template based DIBR watermarking method that overcomes the drawbacks of previous methods. The proposed method exploits two properties to resist DIBR attacks: the pixel is only moved horizontally by DIBR, and the smaller block is not distorted by DIBR. The one dimensional (1D) discrete cosine transform (DCT) and curvelet domains are adopted to utilize these two properties. A template is inserted in the curvelet domain to restore the synchronization error caused by geometric distortion. A watermark is inserted in the 1D DCT domain to insert and detect a message from the DIBR image. Experimental results of the proposed method show high imperceptibility and robustness to various attacks, such as signal and geometric distortions. The proposed method is also robust to DIBR distortion and DIBR configuration adjustment, such as depth image preprocessing and baseline distance adjustment.

The transmission of light is one of the key optical processes in the terrestrial environment (the atmosphere and underlying surfaces). The dependence of light transmittance on the illumination/observation conditions and optical properties of the atmosphere–underlying system can be studied using the integro-differential radiative transfer equation. However, for numerous applications a set of analytical equations is needed to describe the transmitted light intensity and flux. In this paper, we describe various analytical techniques to study light transmittance through light scattering and absorbing media. A physical significance and improved mathematical accuracy of approximations are provided using the analytical models for the diffusion exponent, average cosine of the light field, spherical and plane albedos. The accuracy of various approximations is studied using exact radiative transfer calculations with various scattering phase functions, single-scattering albedos, observational conditions, and optical depths.

The concept of neutrosophic number is a significant mathematical tool to deal with real scientific problems because it can tackle indeterminate and incomplete information which exists generally in real problems. In this article, we use neutrosophic numbers (a + bI), where a and bI denote determinate component and indeterminate component respectively. We explore the situations in which the input information is needed to express in terms of neutrosophic numbers. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine unknown criteria weights. We define neutrosophic number harmonic mean operators and proved their basic properties. Then, we develop two novel MCGDM strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility and effectiveness of the proposed two strategies. Sensitivity analysis with variation of “I” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “I”. The efficiency of the developed strategies is ascertained by comparing the obtained results from the proposed strategies with the existing strategies in the literature.

A particular challenge in the conceptual design of direct-from orbit delivery systems is the seemingly well-known kinematics retain fallacies whose efficacies are reduced over great distances. Rotation about the local wing of an aerospace vehicle is almost never the pitch angle, yet modern application of kinematics often assumes such (with accompanying angular error). The same assertion is usually true about the nature of roll and yaw angles. Expressing motion in coordinates of rotating reference frames necessitates transformation between reference frames, and one such transformation is embodied in the Direction Cosine Matrices (DCM) formed by a sequence of three successive frame rotations. One of two ubiquitous sequences of three rotations used to construct the DCM involves first a rotation around the inertial z-axis, then intermediate y-axis, then finally about the body’s x-axis; a sequence commonly called the “aerospace sequence” or the a “3-2-1 rotation sequence”. The second ubiquitous sequence is the so-called “orbital sequence” or “3-1-3 rotation sequence” with rotations about the inertial z axis first, then about the intermediate x-axis, then finally about the body z-axis. This manuscript evaluates which sequence is the most advantageous for an object starting in space and then travels through the atmosphere to a target on the Earth’s surface. Six degrees of freedom of vehicle motion were simulated starting in orbit with a given thrust and commanded maneuver. The simulation performs all twelve possible rotation sequences (transforming inertial coordinates to body coordinates) with comparison by computational burden and error representing rotations about body axes (roll, pitch, and yaw respectively). Simulation precision is validated using the quaternion normalization condition indicating near machine precision (0.9\times{10}^{-15}) and reveals the so-called 132 rotation is the most accurate with an average error of 0.14° and a computational time of 0.013 seconds: resulting in a 97.95% increase in accuracy over the so-called 321 rotation and a 99.84% increase over the so-called 313 rotation.