AN ALGEBRAIC APPROACH TO DEGENERATE APPELL POLYNOMIALS AND THEIR HYBRID FORMS VIA DETERMINANTS

It is remarkable that studying degenerate versions of polynomials from algebraic point of view is not limited to only special polynomials but can also be extended to their hybrid polynomials.Indeed for the first time,a closed determinant expression for the degenerate Appell polynomials is derived.The determinant forms for the degenerate Bernoulli and Euler polynomials are also investigated.A new class of the degenerate Hermite-Appell polynomials is investigated and some novel identities for these polynomials are established.The degenerate Hermite-Bernoulli and degenerate Hermite-Euler polynomials are considered as special cases of the degenerate Hermite-Appell polynomials.Further,by using Mathematica,we draw graphs of degenerate Hermite-Bernoulli polynomials for different values of indices.The zeros of these polynomials are also explored and their distribution is presented.

ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PADE-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS

To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined. By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for Padé-type approximation are explicitly given.

The Study of Root Subspace Decomposition between Characteristic Polynomials and Minimum Polynomial

Let Abe the linear transformation on the linear space V in the field P, Vλibe the root subspace corresponding to the characteristic polynomial of the eigenvalue λi, and Wλibe the root subspace corresponding to the minimum polynomial of λi. Consider the problem of whether Vλiand Wλiare equal under the condition that the characteristic polynomial of Ahas the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that Vλi=Wλi. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”

Determinants of Modern Contraceptive Uptake among the Youth in Akuse Township in the Lower Manya Krobo Municipality, Ghana

Background: Access to safe and effective contraception is crucial for sexual and reproductive health to be at its best. This allows improved pregnancy results and the avoidance of unintended births. Teenagers’ views on using contraceptives are influenced by the information available at their disposal. The study assessed the influential factors affecting the utilization of modern contraceptives among the youth in Akuse in the Eastern Region of Ghana. Methods: The data for the study were gathered from 378 consented youth aged 15 - 24 years using a quantitative cross-sectional study design and a well-structured questionnaire. Person’s Chi-Square test was adopted to measure the association between the outcome variable and selected independent variables. Logistic regression models were utilized to measure the odds of the factors influencing modern contraceptive use among the respondents. Results: The results of the study suggested a high knowledge level of contraceptives;however, the prevalence of utilization was low. The level of education and age were factors found to influence the uptake of contraceptives. A bivariate analysis to examine the association between selected socio-demographic variables revealed that educational level (p = 0.044), religious affiliation (p = 0.002), and ethnicity (p = 0.016) were statistically associated with modern contraceptive use among the respondents. All other tested demographic variables including the age groups, gender, marital status, and residential status proved otherwise at the observed p-values greater than the 0.05 threshold. Respondents who indicated staying with partners were 6.79 times more likely to use contraceptives compared to their counterparts staying with a parent, after controlling for all other covariates. Conclusions: Based on the findings of this study, it was concluded that high contraceptive knowledge influences the choice of contraceptive preferred by the youth, and contraceptive use is also influenced by the advancement in age and educational level of the youth. It is however recommended that parents, guidance, and teachers be entreated to offer relevant and timely sexual information or education as these will most likely improve the uptake of modern contraceptives among the targeted population.

TOEPLITZ DETERMINANTS IN ONE AND HIGHER DIMENSIONS

In this study,we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk U.Furthermore,these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc inℂn.The obtained results provide the bounds of Toeplitz determinants in higher dimensions for various subclasses of normalized univalent functions.

Rabies by Dog Bites in Benin: Determinants of Good Knowledge of the Rabies Risk among Subjects Exposed in the Plateau Department in 2023

Rabies, a viral zoonosis, is one of the neglected tropical diseases (NTDs) that mainly affects poor and vulnerable populations. Human rabies is endemic in Benin. Although listed among the notifiable diseases, it nevertheless remains a neglected and under-reported disease. The objective of this study was to investigate the determinants of knowledge of rabies risk among subjects exposed to human rabies by dog bites in the Plateau department of Benin. Methods: This was a cross-sectional study with an analytical aim which involved a sample of 900 people aged at least 18 years old chosen randomly and residing in the Plateau department for at least six months. A pre-tested and validated digital questionnaire served as a data collection tool. Data analysis was done with R 3.4.1 software. The prevalence ratio was used as a measure of association. The confidence threshold was set at 5%. We carried out a multiple logistic regression using the likelihood ratio (top-down approach) in order to identify the main factors which influenced the good knowledge of human rabies in the population of Plateau. Results: In the study population, 80.89% of the subjects surveyed lived in a rural area. The sex ratio (Male/Female) was 1.11. The average age of the subjects was 40.24 ± 14.52 years. The prevalence of dog bites was 4.11% (95% CI = [3.00;5.63]) or 37 subjects bitten. After evaluation of knowledge of rabies risk in the study population, the main factors which positively influenced good knowledge of human rabies were age (p linear trend 0.01), residence of the subject in the municipality of Pobè (ORa = 8.34;95% CI = 0.83 - 1.23;p < 0.01), individual dwelling (ORa = 17.95;95% CI = 13.12 - 49.58;p < 0. 01), educational level of at least secondary (ORa = 5.53;CI 95 = 3.06 - 10.15;p = 0.01), the existence of a veterinary center (ORa = 3.66;CI 95 = 1.07 - 12.43 p = 0.01) and knowledge of a dog bites victim (ORa = 6.55;95% CI = 2.73 - 8.25;p < 0.01). Conclusion: This population study confirms that human rabies remains a public health problem with a need to raise awareness among populations on good prevention practices against human rabies after a dog bite in the Plateau department. This could help to improve strategies surveillance for the control and eradication of human rabies through the implementation of the One Health approach.

Unintended Pregnancy Determinants among Antenatal Clinic Attendees: A University of Port Harcourt Teaching Hospital Experience

An unintended pregnancy is a pregnancy that is either mistimed or unplanned. The objectives of this study were to determine the prevalence of unintended pregnancy as well as to document the determinant factors among pregnant women attending antenatal clinics at the University of Port Harcourt Teaching Hospital (UPTH), Port Harcourt, Rivers State, Nigeria. It was a descriptive, cross-sectional study of 215 women attending the antenatal clinic of UPTH from July to August 2022. Information on socio-demographic characteristics, reproductive history, desirability of the current pregnancy at the time of conception, knowledge and use of contraceptive methods were collected using a pretested questionnaire and analyzed using SPSS version 26.0. Chi-square test was used for test of associations with the level of significance set at P < 0.05. The prevalence rate of unintended pregnancy from the study was 16.28% (≈16%). The contraceptive awareness was very high (209, 97.21%), however, 101 (46.98%) participants had never used any form of contraceptives. Univariate analysis using Chi-square test showed a statistically significant association between age and unintended pregnancy (P = 0.042), level of education and unwanted pregnancy (P = 0.033) as well as parity and unintended pregnancy (P = 0.019). The prevalence of unintended pregnancy among women attending antenatal clinics was high, possibly due to low contraceptive usage. More efforts should be geared towards ensuring access to comprehensive contraceptive care and contraceptive methods, this will enhance uptake and reduce the rate of unintended pregnancy.

Determinants of Early Survival of Breast Cancer Patients in Yaoundé-Cameroon

Introduction: Breast cancer is one of the leading causes of death worldwide. We carried out this study with the aim of evaluating the determinants of early survival of women with breast cancer in two hospitals in the city of Yaoundé. Methodology: This was an analytical cross-sectional study with retrospective and prospective data collection of breast cancer patients during 6 years in two Hospitals of Yaoundé from January 2017 to December 2022. We consulted the files in search of epidemiological, clinical, paraclinical, therapeutic and survival variables. We completed the survival data directly from the patients or their relatives after their consent. We analyzed the data using SPSS version 23.0 software. Survival analysis was done using the Kaplan-Meier method and survival curves were compared using the Log Rank test. Factors influencing survival were evaluated using the Cox model. The significance threshold (P value) was set at 0.05 at 95% confidence interval. The study was approved by the ethics committees. Results: We included 500 patients whose ages varied between 22 and 83 years with a mean age of 47.19 ± 11.61 years. The most represented age group was 30 to 45 years old (45.8%). Less than half (41.6%) were postmenopausal. The most frequent reason for consultation was a breast lump (79.9%). The most common clinical stage at presentation was stage-3 (47.6%). Infiltrating ductal carcinoma was the most represented histological type (84.7%). The most represented histological grade was grade 2 (40.2%). Immunohistochemistry was performed in 34.20% of cases. The most represented molecular subtype was triple negative (41.8%) followed by Luminal A (30%). Concerning treatment, 17.2% did not receive any, 45% had surgery, 79.4% had chemotherapy, 34.2% hormone therapy, and 14.6% radiotherapy. The survival of patients with breast cancer at 1, 2, 3, 4 and 5 years was respectively 90.6%;83.1%;74.2%;69.8% and 59.2%. The median survival was not reached;however, the first quartile (Q1) was 36 months (3 years). Independent factors associated with reduced survival were breast ulceration (aHR = 3.23;p = 0.002), bilateral tumor location (aHR = 9.2;p < 0.001) and clinical stage 3 (aHR = 1.72;p = 0.010) while patients classified ACR3 on imaging (aHR = 0.19;p = 0.005) had improved survival. Conclusion: Breast cancer survival from 1 to 5 years decrease from 90 to 59%. Mortality was highest in the first 40 months. Independent factors associated with reduced survival were breast ulceration, bilateral tumor location and clinical stage 3 while patients classified ACR3 on imaging had improved survival.

Rationale of a Cross Sectional Analytic Study on Determinants of Recurrent Preeclampsia at University Clinics of Kinshasa (Democratic Republic of Congo) and at Victor Dupouy Hospital Center (France)

Research Background: Pre-eclampsia is one of main causes of materno-foetal mortality and morbidity worldwide, with a prevalence of 3% - 7%. Although considered a primiparous condition, it can nevertheless recur. Several factors appear to be associated with risk of recurrence of pre-eclampsia, such as the term of delivery of previous pregnancy, severity of disease, the existence of co-morbidities and the inter-genital space. Purpose: The aim of our study will be to analyse and identify in a population of pregnant women with a history of preeclampsia risk factors associated with occurrence of recurrent preeclampsia at University clinics of Kinshasa (Democratic Republic of Congo) and at Victor Dupouy Hospital Center (France). Methods: In this study, pregnant women with an history of preeclampsia who will give birth between November 2018 and October 2024 at University Clinics of Kinshasa (UCK) and Victor Dupouy Hospital Center (VDHC) will be included. This will be a cross-sectional analytical study, data from previous and subsequent pregnancies will be studied. Expected Result: The prevalence of recurrent preeclampsia in the study population will be determined. And we will highlight the factors that will determine the recurrence of preeclampsia by analysing the risk factors. Conclusion: Knowledge of the factors associated with recurrent preeclampsia could be an excellent tool for predicting and preventing the disease.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Solving Invariant Problem of Cauchy Means Based on Wronskian Determinant

This paper studied the invariance of the Cauchy mean with respect to the arithmetic mean when the denominator functions satisfy certain conditions. The partial derivatives of Cauchy’s mean on the diagonal are obtained by using the method of Wronskian determinant in the process of solving. Then the invariant equation is solved by using the obtained partial derivatives. Finally, the solutions of invariant equations when the denominator functions satisfy the same simple harmonic oscillator equation or the denominator functions are power functions that have been obtained.

Disparities in Prevalence and Determinants of Hypertension amongst Bamiléké Adults Residing in Two Different Agroecological Zones of Cameroon

Background: Cardiovascular diseases such as hypertension (HTN) are one of the main causes of death in Cameroon. This study aimed at assessing prevalence disparities and determinants of hypertension amongst Bamiléké adults residing in two different agroecological zones of Cameroon. Methods: A cross-sectional and descriptive survey was conducted among Bamiléké population living in the Highlands zone (Western region) and in the Monomodal Rainforest zone (Littoral region) of Cameroon from August 2016 to August 2017. Participants (962) were aged at least 20 years old. Data on sociodemographic, hemodynamic, anthropometric, and biochemical parameters and lifestyle of the participants were collected. Results: Results obtained revealed that 34.2% were hypertensive and those residing in the highland zone were more affected than those living in the monomodal rainforest zone (44.5% vs 22.9%). The different subtypes of HTN (Isolated systolic hypertension (14.1%), isolated diastolic hypertension (7.2%) and Systo-diastolic hypertension (23.3%)) were also more prevalent in the Highlands Zone. The most prevalent stage of HTN was pre-HTN (31.5%). However, people living in the monomodal rainforest zone were more affected by pre-HTN compared to Bamiléké living in the highland zone (33.6% vs. 29.6%). Results also showed that high consumption (≥ 3 times/week) of carbohydrate- and fat-rich foods, ageing, obesity, and marital status were associated with high blood pressure in both agroecological zones. Besides, secondary education (OR = 0.68;95% CI: 0.42 - 0.99) in the Highlands Zone and high (≥3 times/week) vegetable consumption (OR = 0.66;95% CI: 0.44 - 0.98) in the Monomodal Rainforest Zone had a protective effect on elevated blood pressure of population. Conclusion: There is a disparity in the prevalence of hypertension and some of its determinants among Bamiléké adults residing in different agroecological zones. This work highlights the need to advocate for local and ethno-cultural health policies to prevent, diagnose and manage hypertension.

Improving Video Watermarking through Galois Field GF(2^(4)) Multiplication Tables with Diverse Irreducible Polynomials and Adaptive Techniques

Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.

Determinants of Tobacco Cultivation and the Associated Impacts of Adoption in Hurungwe District, Zimbabwe

Tobacco is an essential cash crop in Zimbabwe and a strategic livelihood option for hundreds of thousands of rural households. However, the crop is linked to negative environmental, economic, and social impacts. The existing studies on tobacco cultivation in Zimbabwe present contradictory findings on the determinants and impacts of adoption, leaving unanswered questions about the crop’s sustainability impact in the country. This article investigates the determinants of smallholder farmers’ decisions to grow tobacco and the associated impacts of adoption. Random and purposive sampling were used to select 273 household surveys, including tobacco and non-tobacco smallholder farmers, and 56 expert interviews to answer the research questions. We employed regression models alongside expert interviews and document analysis to identify the determinants influencing the decision-making process of smallholder farmers in Zimbabwe regarding tobacco cultivation. Additionally, our investigation aimed to elucidate the perceived impacts associated with the adoption of this agricultural practice. The regression analysis indicated that the farmer’s age, education level, farming experience, family size, household income, and perceived high farm profitability are significant drivers of tobacco adoption. We also discovered divergent and convergent perceptions of the critical impacts of tobacco cultivation. The study highlights the need for proactive multi-stakeholder collaboration and sustainable financial arrangements to address the negative impacts of tobacco production. As the primary stakeholder responsible for regulating and promoting agricultural activities, the Zimbabwean government should provide meaningful financial support, increase access to credit, and ensure better market facilities for alternative crops to reduce the over-dependence on tobacco.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Impact of Social Determinants of Health on Self-Perceived Resilience: An Exploratory Study of Two Cohorts of Adults with Congenital Heart Disease

Social determinants of health(SDOH)affect quality of life.We investigated SDOH impacts on self-perceived resilience among people with adult congenital heart disease(ACHD).Secondary analysis of data from two com-plementary studies:a survey study conducted May 2021–June 2022 and a qualitative study conducted June 2020–August 2021.Resilience was assessed through CD-RISC10 score(range 0–40,higher scores reflect greater self-perceived resilience)and interview responses.Sociodemographic and SDOH(education,employment,living situa-tion,monetary stability,financial dependency,area deprivation index)data were collected by healthcare record review and self-report.We used linear regression with robust standard errors to analyze survey data and performed a thematic analysis of interview data.Survey participants(N=127)mean age was 42±14 years;51%were female,87%white.ACHD was moderate(75%)or complex(25%);41%functional class C or D.Resilience(mean 30±7)varied by monetary stability:compared to people with difficulty paying bills,resilience was 15.0 points higher(95%CI:6.9–23.1,p<0.001)for people reporting having enough money and 14.2 points higher(95%CI:5.9–22.4,p=0.001)for those reporting just enough money.Interview participants’(N=25)mean age was 32 years(range 22–44);52%were female,72%white.ACHD was moderate(56%)or complex(44%);76%functional class C or D.Participants discussed factors affecting resilience aligned with each of the major SDOH,prominently,economic stability and healthcare access and quality.Financial stability may be important for supporting self-perceived resi-lience in ACHD.This knowledge can inform the development of resilience interventions for this population.

Sensitivity Analysis of Electromagnetic Scattering from Dielectric Targets with Polynomial Chaos Expansion and Method of Moments

In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.

Three-dimensional pseudo-dynamic reliability analysis of seismic shield tunnel faces combined with sparse polynomial chaos expansion

To address the seismic face stability challenges encountered in urban and subsea tunnel construction,an efficient probabilistic analysis framework for shield tunnel faces under seismic conditions is proposed.Based on the upper-bound theory of limit analysis,an improved three-dimensional discrete deterministic mechanism,accounting for the heterogeneous nature of soil media,is formulated to evaluate seismic face stability.The metamodel of failure probabilistic assessments for seismic tunnel faces is constructed by integrating the sparse polynomial chaos expansion method(SPCE)with the modified pseudo-dynamic approach(MPD).The improved deterministic model is validated by comparing with published literature and numerical simulations results,and the SPCE-MPD metamodel is examined with the traditional MCS method.Based on the SPCE-MPD metamodels,the seismic effects on face failure probability and reliability index are presented and the global sensitivity analysis(GSA)is involved to reflect the influence order of seismic action parameters.Finally,the proposed approach is tested to be effective by a engineering case of the Chengdu outer ring tunnel.The results show that higher uncertainty of seismic response on face stability should be noticed in areas with intense earthquakes and variation of seismic wave velocity has the most profound influence on tunnel face stability.

A coupled Legendre-Laguerre polynomial method with analytical integration for the Rayleigh waves in a quasicrystal layered half-space with an imperfect interface

The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space.However,it fails to obtain the correct stress at the interfaces in a layered half-space,especially when there are significant differences in material properties.Therefore,a coupled Legendre-Laguerre polynomial method with analytical integration is proposed.The Rayleigh waves in a one-dimensional(1D)hexagonal quasicrystal(QC)layered half-space with an imperfect interface are investigated.The correctness is validated by comparison with available results.Its computation efficiency is analyzed.The dispersion curves of the phase velocity,displacement distributions,and stress distributions are illustrated.The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated.Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space.It can save over 99%of the computation time.This method can be expanded to investigate waves in various layered half-spaces,including earth-layered media and surface acoustic wave(SAW)devices.

Generalized polynomial chaos expansion by reanalysis using static condensation based on substructuring

This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.