Real Part Research Articles

Optical Color Image Encryption Algorithm Based on Two-Dimensional Quantum Walking

The double random phase encoding (DRPE) image encryption method has garnered significant attention in color image processing and optical encryption thanks to its R, G, and B parallel encryption. However, DRPE-based color image encryption faces two challenges. Firstly, it disregards the correlation of R, G, and B, compromising the encrypted image’s robustness. Secondly, DRPE schemes relying on Discrete Fourier Transform (DFT) and Discrete Fractional Fourier Transform (DFRFT) are vulnerable to linear attacks, such as Known Plaintext Attack (KPA) and Chosen Plaintext Attack (CPA). Quantum walk is a powerful tool for modern cryptography, offering robust resistance to classical and quantum attacks. Therefore, this study presents an optical color image encryption algorithm that combines two-dimensional quantum walking (TDQW) with 24-bit plane permutation, dubbed OCT. This approach employs pseudo-random numbers generated by TDQW for phase modulation in DRPE and scrambles the encrypted image’s real and imaginary parts using the generalized Arnold transform. The 24-bit plane permutation helps reduce the R, G, and B correlation, while the generalized Arnold transform bolsters DRPE’s resistance to linear attacks. By incorporating TDQW, the key space is significantly expanded. The experimental results validate the effectiveness and security of the proposed method.

Nonlinear Wave Interaction of Nanorods Embedded in a Viscoelastic Medium

Abstract Purpose Nonlinear interactions between two acoustic waves in nanorods traveling at various wave numbers, group velocities, and frequencies are examined in this study. Methods The nonlinear equation of the nanorod in a viscoelastic medium is obtained using the theory of nonlocal elasticity. Furthermore, the multiple-scale expansion method is applied to study strongly dispersive, weakly nonlinear waves in a nonlocal viscoelastic medium. Using this expansion technique, we can derive the coupled nonlinear Schrödinger equations as the governing equations, which we solve as differential equations of some parameters by expanding the field quantities into an asymptotic series of the smallness parameter. Results We give the nonlinear plane wave solutions to these equations in several special cases. The plane wave solutions show how the wave amplitude affects the frequencies of nonlinear plane waves. Additionally, we show numerically how the real and imaginary parts of the group velocities and natural frequency of the system for a carbon nanotube in a viscoelastic medium are affected by the nonlocal, damping, and stiffness parameters.

Effects of Mn and Ru substitutions on the electrical properties and AC susceptibility of Tl(BaSr)CaCu2O7 superconductor

In this paper, the effects of M = Mn and Ru substitutions at the Tl site of Tl[Formula: see text]Mx(BaSr)CaCu2O7 (Tl-1212) on the electrical properties and AC susceptibility were investigated. The X-ray diffraction (XRD) and field emission scanning electron microscopy (FESEM) methods were used to examine the phase and microstructure, respectively. All samples showed diffraction patterns with Tl-1212 as the major phase and an increase in the Tl-1223 phase with increasing Mn and Ru substitutions. The microstructure showed partial melting and randomly oriented grains in all samples. The electrical resistance measurements showed metallic normal state behavior for all samples. The zero-resistance transition temperature [Formula: see text] was the highest for x = 0.4 where [Formula: see text] was 107[Formula: see text]K and 90[Formula: see text]K for Mn and Ru substitutions, respectively. The superconducting transition, [Formula: see text] obtained from AC susceptibility measurements increased with Mn and Ru substitutions up to x = 0.4. Using Bean’s model, the current density at the peak temperature of the imaginary part of the AC susceptibility, [Formula: see text]([Formula: see text] was determined to be between 16 and 22[Formula: see text]A[Formula: see text]⋅[Formula: see text]cm[Formula: see text] for Mn, and between 15 and 19[Formula: see text]A[Formula: see text]⋅[Formula: see text]cm[Formula: see text] for Ru-substituted samples. Mn substitution showed a higher transition temperature compared with Ru. This study showed that optimization of the average Cu valence was achieved with Mn and Ru substitutions for x = 0.4 at the Tl site. The ionic radii of the substituting and substituted elements are important factors in enhancing the transition temperature of Tl(BaSr)CaCu2O7 superconductor.

Added mass of oscillating bodies in stratified fluids

The concept of added mass is generalized to stratified fluids, accounting for the presence of internal waves. Once the added mass of a moving body is known, so is the hydrodynamic force exerted on it by the fluid, and the energy imparted by it to the fluid. As a function of frequency, added mass is complex. Its real part is associated with inertia and its imaginary part, only present in the frequency range of propagating waves, with wave damping. Owing to causality, these two parts satisfy Kramers–Kronig relations. The added masses of an elliptic cylinder of horizontal axis, typical of two-dimensional bodies, and a spheroid of vertical axis, typical of three-dimensional bodies, are deduced from their dipole strengths, themselves deduced from their representations as single layers. The wave power is shown to be a maximum, for fixed oscillation amplitude, at approximately $0.8$ times the buoyancy frequency. In the temporal domain, added mass appears as a new memory force taking the form of a convolution integral. The kernel of this integral combines algebraically decaying oscillations at the buoyancy frequency on the one hand; and an exponentially damped oscillation for the horizontal motion of the spheroid, implying short-term memory, an aperiodic algebraic decay for its vertical motion, implying long-term memory, and a constant for the motion of the cylinder, implying everlasting memory, on the other hand. A limitation of the study is its restriction to translational motion.

Structural, morphological, and dielectric properties of poly(vinylidene fluoride)/(Bi0.5Ba0.25Sr0.25) (La0.5Ti0.5) O3 composites

AbstractFerroelectric flexible polymer composite films of poly(vinylidene fluoride) (PVDF) with different weight percentages (10.0%, 20.0%, 30.0%, 40.0%, and 50.0%) of (Bi0.5Ba0.25Sr0.25) (La0.5Ti0.5) O3 were synthesized by solvent casting method. Analysis using the x‐ray diffractometer and Fourier transform infrared spectroscopy tools revealed the presence of the different phases (α, β) of PVDF. Scanning electron microscope morphology illustrates the homogenous dispersion of ceramics particles in the matrix. The differential scanning calorimetry curve (under 25–400°C) gives the value of melting temperature (Tm=169.98°C) and percentage crystallinity is about 29.45% for 50 wt.% composite. The findings demonstrated that the composite has good thermal stability up to 169.98°C. Dielectric analysis shows ferroelectric and dielectric behavior and also distinguishes α and β phases of PVDF. It was found that the composite film containing 50 wt.% (Bi0.5Ba0.25Sr0.25) (La0.5Ti0.5) O3 had a dielectric constant (εr) of 38 (at 100 Hz) and dielectric loss of 0.059, which are the maximum among the samples. The relative permittivity rises with an increase in filler content due to electrostatics and interfacial polarization between PVDF (CH2CF2 dipole) and ceramic. The real and imaginary part of impedance spectroscopy with frequency (Cole–Cole plots) for composite (10.0–50.0 wt.%) at 25°C are also investigated. The current study provides a fresh perspective on the fact that PVDF‐(Bi0.5Ba0.25Sr0.25) (La0.5Ti0.5) O3 composite system has improved thermal stability and dielectric characteristics.

Probing the four-fermion operators via the transverse double spin asymmetry at the Electron-Ion Collider

The chirality-flipping operators of light fermions are currently poorly constrained by experimental analyses due to the lack of interference with standard model (SM) amplitudes in traditional observables. In this work, we propose to investigate the semileptonic scalar and tensor four-fermion operators of electron and quarks through the transverse double spin asymmetry (DSA) at Electron-Ion Collider, where both the electron and proton beams could be highly transversely polarized. Due to the chirality-flipping nature of these operators, we demonstrate that their interference with the SM results in an unsuppressed contribution to the DSA, and could lead to nontrivial azimuthal cos2ϕ and sin2ϕ distributions that are linearly dependent on their Wilson coefficients. This new method has the potential to significantly improve the current constraints on these scalar and tensor four-fermion operators without relying on theoretical assumptions about other types of new physics effects, particularly for the tensor-type operator of the u quark. Additionally, our findings indicate that both the real and imaginary parts of these operators can be simultaneously constrained and offer a new opportunity for probing potential CP-violation effects. However, it is important to note that these results would be sensitive to the quark transversity distributions, which are currently poorly constrained by the experimental data, but could be significantly improved at the upcoming Electron-Ion Collider. Therefore, our work opens up a new avenue to utilize this new spin asymmetry for exploring the new physics effects from the scalar/tensor four-fermion operators. Published by the American Physical Society 2024

Enhanced emergent electromagnetic inductance in Tb5Sb3 due to highly disordered helimagnetism

In helimagnetic metals, ac current-driven spin motions can generate emergent electric fields acting on conduction electrons, leading to emergent electromagnetic induction (EEMI). Recent experiments reveal the EEMI signal generally shows a strongly current-nonlinear response. In this study, we investigate the EEMI of Tb5Sb3, a short-period helimagnet. Using small angle neutron scattering we show that Tb5Sb3 hosts highly disordered helimagnetism with a distribution of spin-helix periodicity. The current-nonlinear dynamics of the disordered spin helix in Tb5Sb3 indeed shows up as the nonlinear electrical resistivity (real part of ac resistivity), and even more clearly as a nonlinear and huge EEMI (imaginary part of ac resistivity) response. The magnitude of the EEMI reaches as large as several tens of μH for Tb5Sb3 devices on the scale of several tens of μm, originating to noncollinear spin textures possibly even without long-range helimagnetic order.

Area–time–energy efficient architecture of CBNS‐based fast Fourier transform

AbstractA new approach for implementing the fast Fourier transform (FFT) using complex binary number system (CBNS) is presented in this paper. The advantage of using CBNS is that instead of processing the real and imaginary parts of a complex number separately, we can process both as a single entity. A total of four FFT architectures have been developed, which are 16‐, 64‐, 256‐, and 1024‐point FFT architectures. The proposed FFT is designed using Verilog‐HDL and implemented using a Virtex‐7 field‐programmable gate array (FPGA). The maximum clock rate achieved by the proposed FPGA‐based design is 376.1 MHz. The FPGA resource utilization of the FFT is less than the state‐of‐the‐art FFT designs. An application‐specific integrated circuit (ASIC) for the FFT is also designed. The normalized energy per FFT of the proposed ASIC is 0.14 nJ, and the normalized area per FFT of the ASIC is 1.08 mm2 which is less than other existing designs.

Dynamic Monitoring of Biomolecular Hydrodynamic Dimensions by Magnetization Motion on Quartz Crystal Microbalance.

Hydrodynamic dimension (HD) is the primary indicator of the size of bioconjugated particles and biomolecules. It is an important parameter in the study of solid-liquid two-phase dynamics. HD dynamic monitoring is crucial for precise and customized medical research as it enables the investigation of the continuous changes in the physicochemical characteristics of biomolecules in response to external stimuli. However, current HD measurements based on Brownian motion, such as dynamic light scattering (DLS), are inadequate for meeting the polydisperse sample demands of dynamic monitoring. In this paper, we propose MMQCM method samples of various types and HD dynamic monitoring. An alternating magnetic field of frequency ωm excites biomolecule-magnetic bead particles (bioMBs) to generate magnetization motion, and the quartz crystal microbalance (QCM) senses this motion to provide HD dynamic monitoring. Specifically, the magnetization motion is modulated onto the thickness-shear oscillation of the QCM at the frequency ωq. By analysis of the frequency spectrum of the QCM output signal, the ratio of the magnitudes of the real and imaginary parts of the components at frequency ωq ± 2ωm is extracted to characterize the particle size. Using the MMQCM approach, we successfully evaluated the size of bioMBs with different biomolecule concentrations. The 30 min HD dynamic monitoring was implemented. An increase of ∼10 nm in size was observed upon biomolecular structural stretching. Subsequently, the size of bioMBs gradually reduced due to the continuous dissociation of biomolecules, with a total reduction of 20∼40 nm. This HD dynamic monitoring demonstrates that the release of biomolecules can be regulated by controlling the duration of magnetic stimulation, providing valuable insights and guidance for controlled drug release in personalized precision medicine.

Fermion self-energy and damping rate in a hot magnetized plasma

We derive a general expression for the fermion self-energy in a hot magnetized plasma by using the Landau-level representation. In the one-loop approximation, the Dirac structure of the self-energy is characterized by five different functions that depend on the Landau-level index n and the longitudinal momentum pz. We derive general expressions for all five functions and obtain closed-form expressions for their imaginary parts. The latter receive contributions from three types of on shell processes, which are interpreted in terms of Landau-level transitions, accompanied by a single photon (gluon) emission or absorption. By making use of the imaginary parts of the self-energy functions, we also derive the Landau-level dependent fermion damping rates Γn(pz) and study them numerically in a wide range of model parameters. We also demonstrate that the two-spin degeneracy of the Landau levels is lifted by the one-loop self-energy corrections. While the spin splitting of the damping rates is small, it may be important for some spin and chiral effects. We argue that the general method and the numerical results for the rates can have interesting applications in heavy-ion physics, astrophysics, and cosmology, where strongly magnetized QED or QCD plasmas are ubiquitous. Published by the American Physical Society 2024

Probing the Bardeen–Kiselev black hole with the cosmological constant caused by Einstein equations coupled with nonlinear electrodynamics using quasinormal modes and greybody bounds

In this work, we investigate a static and spherically symmetric Bardeen–Kiselev black hole (BH) with the cosmological constant, which is a solution of the Einstein-non-linear Maxwell field equations. We compute the quasinormal frequencies for the Bardeen–Kiselev BH with the cosmological constant due to electromagnetic and gravitational perturbations. By varying the BH parameters, we discuss the behavior of both real and imaginary parts of the BH quasinormal frequencies and compare these frequencies with the Reissner–Nordström–de Sitter BH surrounded by quintessence (RN-dSQ). Interestingly, it is shown that the responses of the Bardeen–Kiselev BH with the cosmological constant and the RN-dSQ under electromagnetic perturbations are different when the charge parameter q, the state parameter w and the normalization factor c are varied; however, for the gravitational perturbations, the responses of the Bardeen–Kiselev BH with the cosmological constant and the RN-dSQ are different only when the charge parameter q is varied. Therefore, compared with the gravitational perturbations, the electromagnetic perturbations can be used to understand nonlinear and linear electromagnetic fields in curved spacetime separately. Another interesting observation is that, due to the presence of Kiselev quintessence, the electromagnetic perturbations around the Bardeen–Kiselev BH with the cosmological constant damps faster and oscillates slowly; for the gravitational perturbations, the quasinormal mode decays slowly and oscillates slowly. We also study the reflection and transmission coefficients along with the absorption cross section in the Bardeen–Kiselev BH with the cosmological constant; it is shown that the transmission coefficients will increase due to the presence of Kiselev quintessence.

The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements [formula omitted] and [formula omitted]

The non-first-order-factorizable contributions1 to the unpolarized and polarized massive operator matrix elements to three-loop order, AQg(3) and ΔAQg(3), are calculated in the single-mass case. For the F12-related master integrals of the problem, we use a semi-analytic method based on series expansions and utilize the first-order differential equations for the master integrals which does not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to O(ε5) in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable x∈]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x∈]0,1] at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-x region. We also derive expansions in the region of small and large values of x. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.

Towards a systematic study of non-thermal leptogenesis from inflaton decays

This paper investigates non-thermal leptogenesis from inflaton decays in the minimal extension of the canonical type-I seesaw model, where a complex singlet scalar ϕ is introduced to generate the Majorana masses of right-handed neutrinos (RHNs) and to play the role of inflaton. First, we systematically study non-thermal leptogenesis with the least model dependence. We give a general classification of the parameter space and find four characteristic limits by carefully examining the interplay between inflaton decay into RHNs and the decay of RHNs into the standard-model particles. Three of the four limits are truly non-thermal, with a final efficiency larger than that of thermal leptogenesis. Two analytic estimates for these three limits are provided with working conditions to examine the validity. In particular, we find that the strongly non-thermal RHNs scenario occupies a large parameter space, including the oscillation-preferred K range, and works well for a relatively-low reheating temperature TRH ≥ 103 GeV, extending the lower bound on the RHN mass to 2 × 107 GeV. The lepton flavor effects are discussed. Second, we demonstrate that such a unified picture for inflation, neutrino masses, and baryon number asymmetry can be realized by either a Coleman-Weinberg potential (for the real part of ϕ) or a natural inflation potential (for the imaginary part of ϕ). The allowed parameter ranges for successful inflation and non-thermal leptogenesis are much more constrained than those without inflationary observations. We find that non-thermal leptogenesis from inflaton decay offers a testable framework for the early Universe. It can be further tested with upcoming cosmological and neutrino data. The model-independent investigation of non-thermal leptogenesis should be useful in exploring this direction.

Unveiling the enhanced optical limiting of polyaniline – Fullerene composite investigated by z-scan

The reverse saturable absorption (RSA) and optical limiting (OL) mechanisms of polyaniline (PANI) and fullerene doped PANI (PANI-C60) solutions were investigated by open-aperture z-scan studies, using a nanosecond pulsed Nd:YAG laser operating at 532 nm wavelength as the excitation source. The nonlinear absorption coefficient (β), OL threshold, and imaginary part of nonlinear susceptibility (χi(3)) of PANI and PANI-C60 solutions were determined for three different laser input pulse energies and are found to be in the order of 10−9 cm/W, 106 W/cm2, and 10−10 esu respectively. The results showed enhanced OL efficiency for the PANI-C60 solution, making it a promising candidate for applications in optoelectronics and optical limiting. The transmission electron microscope (TEM) images revealed the uniform distribution of C60 of particle size in the range of 5–10 nm in the PANI matrix. The composite formation of PANI-C60 was confirmed by Fourier transform infrared spectroscopy (FTIR). The presence of C60 and its charge transfer complex (CTC) formation with PANI are responsible for the enhanced OL efficiency. The linear refractive index values were measured using UV–Visible absorption spectra and compared with the theoretical values. The investigation of the OL mechanism and the calculation of its associated NLO parameters for PANI and PANI-C60 solutions are reported for the first time.

A comprehensive theoretical analysis on structural, electronic, optical, and mechanical properties of Sr2VRuO6 compound

In this paper, using the FP-LAPW technique as implemented in the Wien2k code, we have studied the structural, electronic, optical, and mechanical properties of strontium-based compound Sr2VRuO6. In this study, we explore the properties of Sr2VRuO6 using various approximations. We present the total energy versus energy, employing the Wu-Cohen-Generalized Gradients Approximation (WC-GGA) for both nonmagnetic and ferromagnetic states. To determine the stability of this material in cubic structure, the tolerance factor (t) and octahedral factor (μ) have been calculated. The lattice parameters of Sr2VRuO6 were determined, and subsequent calculations yielded the compressive modulus, Young's modulus, shear modulus, and Poisson's ratio in both two and three dimensions. Analyzing the band structure of Sr2VRuO6 within the mBJ-GGA approximation the half metallic character is observed with an indirect bandgap of 2.34 eV. In addition, the total and partial density of states, as well as charge density maps for Sr2VRuO6 have been calculated. Furthermore, we investigated the real and imaginary parts of the dielectric function as functions of energy for Sr2VRuO6. Additionally, the study delved into the variation of the refractive index, absorption coefficient, reflectivity, and energy loss with respect to energy for Sr2VRuO6.

Effect of oxygen defects on ferroelectric oxides: Correlation between linear and nonlinear dielectric responses

Oxygen defects are introduced into PbZr0.5Ti0.5O3 films and their impact on ferroelectric behavior, linear dielectric response (LDR), and nonlinear dielectric response (NDR) is studied. Apart from a notable decrease in both polarization and LDR permittivity, the frequency spectra of LDR exhibit a distinctive loss peak. The peak position varies with temperature and oxygen defect concentration. NDR parameters, including the Rayleigh coefficient (α1), the slope (α2) correlating the imaginary part of permittivity with the ac electric field, and the slope (k) between the real and the imaginary parts of permittivity, demonstrate diverse behaviors in response to temperature and oxygen defect concentration. The similarity in the frequency spectra of tanδ and 1/k reveals the correlation between LDR and NDR induced by the presence of oxygen defects. These observations are attributed to the behavior of the single and composite defects associated with oxygen vacancies.

Relativistic massive and massless scalar fields bound to the Bumblebee gravity's black hole

In this letter, we examine massive and massless scalar quasibound states bound in the static spherically symmetric black hole solution of the Einstein-Bumblebee space-time. We start with constructing the governing relativistic fields' dynamical equation, i.e. the Klein-Gordon equation, component by component. With the help of the ansatz of separation of variables, we find the exact solution of the angular part in terms of Spherical Harmonics while the radial exact solution is discovered in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that reveals a complex-valued energy levels expression for the case of massive scalar, where the real part is the scalar particle's energy and the imaginary part represents the quasibound state's decay. And for a massless scalar, a pure imaginary energy levels is obtained. The quasibound states, thus, describe decaying relativistic states bound in the black hole's gravitational potential well. At the end, we investigate the Hawking radiation of the static Bumblebee black hole's horizon by deriving and investigating the radiation distribution function.

An Analytical Approach for Naturalistic Cooperative and Competitive EEG-Hyperscanning Data: A Proof-of-Concept Study.

Investigating the neural mechanisms underlying both cooperative and competitive joint actions may have a wide impact in many social contexts of human daily life. An effective pipeline of analysis for hyperscanning data recorded in a naturalistic context with a cooperative and competitive motor task has been missing. We propose an analytical pipeline for this type of joint action data, which was validated on electroencephalographic (EEG) signals recorded in a proof-of-concept study on two dyads playing cooperative and competitive table tennis. Functional connectivity maps were reconstructed using the corrected imaginary part of the phase locking value (ciPLV), an algorithm suitable in case of EEG signals recorded during turn-based competitive joint actions. Hyperbrain, within-, and between-brain functional connectivity maps were calculated in three frequency bands (i.e., theta, alpha, and beta) relevant during complex motor task execution and were characterized with graph theoretical measures and a clustering approach. The results of the proof-of-concept study are in line with recent findings on the main features of the functional networks sustaining cooperation and competition, hence demonstrating that the proposed pipeline is promising tool for the analysis of joint action EEG data recorded during cooperation and competition using a turn-based motor task.

Structural, Dielectric, Electrical, and Magnetic Characteristics of Bi0.8Ba0.1Er0.1Fe0.96Cr0.02Mn0.02O3 Nanoparticles

Bi0.8Ba0.1Er0.1Fe0.96Cr0.02Mn0.02O3 (BBEFCMO) multiferroic ceramic was synthesized through the sol-gel route. The impact of incorporating various dopants into both A and B sites of the BiFeO3 was investigated, and structural, Raman, dielectric, electric, and magnetic properties were studied. X-ray diffraction analysis and Raman spectroscopy revealed a rhombohedral structure with the R3c space group for the doped material (BBEFCMO). Dielectric properties were examined across a frequency range of 102–106 Hz. The present multiferroic material exhibits a colossal dielectric constant and minimal dielectric loss tangent, making it suitable for applications in energy storage. Furthermore, the Cole-Cole type of relaxation was deduced from the imaginary part of the modulus for both grain and boundary-grain contributions. Overall, this study indicates that substituting ions in both A and B sites of BiFeO3 significantly enhances its multiferroic properties, as evidenced by dielectric and magnetic measurements.

Global Receptive Field Designed Complex-Valued Convolutional Neural Network Equalizer for Optical Fiber Communications

In this paper, an improved complex-valued convolutional neural network (CvCNN) structure to be placed at the received side is proposed for nonlinearity compensation in a coherent optical system. This complex-valued global convolutional kernel-assisted convolutional neural network equalizer (CvGNN) has been verified in terms of Q-factor performance and complexity compared to seven other related nonlinear equalizers based on both the 64 QAM experimental platform and the QPSK numerical platform. The global convolution operation of the proposed CvGNN is more suitable for the calculation process of perturbation coefficients, and the global receptive field can also be more effective at extracting effective information from perturbation feature maps. The introduction of CvCNN can directly focus on the complex-valued perturbation feature maps themselves without separately processing the real and imaginary parts, which is more in line with the waveform-dependent physical characteristics of optical signals. Based on the experimental platform, compared with the real-valued neural network with small convolutional kernel (RvCNNC), the proposed CvGNNC improves the Q-factor by ∼2.95 dB at the optimal transmission power, while reducing the time complexity by ∼44.7%.