This paper defines a novel semigroup construction, called a relatively commutative semigroup. A semigroup is called relatively commutative if there exist $k,r,t,s\in \mathbb{Z^{+}}$ and $x,y\in S$ such that $(ab)^{k}=b^{r}ax$ and $(ab)^{t}=yba^{s}$ for all $a,b\in S$. The manuscript also provides significant results and examples exploring relatively commutative semigroups' structural characteristics. Additionally, the study addressing the relationships between various commutative semigroups and the relatively commutative semigroup demonstrates that every quasicommutative, $\mathcal{H}$-commutative, and $(m,n)$-commutative semigroup, as well as every normal semigroup and the center of a semigroup, is a relatively commutative semigroup. It further offers examples that illustrate that the reverse of the aforementioned relationships is not always applicable. Furthermore, it establishes that all semigroups with two elements are, in fact, either a left relatively commutative semigroup, a right relatively commutative semigroup, or a relatively commutative semigroup. Moreover, the paper confirms that while every commutative semigroup is also a relatively commutative semigroup, the reverse is valid when the relatively commutative semigroup has order 2. Besides, the paper delineates the necessary conditions for a one-sided ideal as an ideal within a relatively commutative semigroup, and demonstrates that if a relatively commutative semigroup is regular, it is reduced.

In this study, we define a new generalization of the hybrid $k$-Pell sequence consisting of hybrid numbers with generalized hybrid $k$-Pell numbers as coefficients. We establish some algebraic properties and also the Binet formula, generating function, and exponential generating function related to this new sequence. In addition, some identities are provided as sum identities, and Catalan, Cassini, and d'Ocagne's identities. The particular cases are studied, namely, the hybrid numbers with hybrid $k$-Pell numbers as coefficients, the hybrid numbers with hybrid $k$-Pell--Lucas numbers as coefficients, and the hybrid numbers with hybrid Modified $k$-Pell numbers as coefficients.

This article establishes new general two- and three-dimensional integral inequalities. The first result involves four functions: two main functions defined on the positive real line and two auxiliary functions defined on the unit interval. As a significant contribution, the upper bound obtained is quite simple; it is expressed only as the product of the unweighted integral norms of these functions. The main ingredient of the proof is an original change of variables methodology. The article also presents a three-dimensional extension of this result. This higher-dimensional version uses a similar structure but with nine functions: three main functions defined on the positive real line and six auxiliary functions defined on the unit interval. It retains the simplicity and sharpness of the upper bound. Both results open up new directions for applications in analysis. This claim is supported by various examples, including some based on power, logarithmic, trigonometric, and exponential functions, as well as some secondary but still general integral inequalities.

This paper introduces an axiomatic approach in the theory of energy dissipation in Hilbert envelopes on motion waveforms emanating from various vibrating dynamical systems. A Hilbert envelope is a curve tangent to peak points on a motion waveform. The basic approach is to compare non-modulated vs. modulated waveforms in measuring energy loss during the vibratory motion $m(t)$ at time $t$ of a moving object such as a walker, runner, biker or the action of any spring system recorded in a video. Modulation of $m(t)$ is achieved by using Mersenne primes to adjust the frequency $\omega$ in the Fourier transform $m(t)e^{\pm j2\pi \omega t}$ on motion waveform $m(t)$, where the frequency $\omega$ is a Mersenne prime. Expenditure of energy $E_{m(t)}$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform at time $t$. Energy dissipation is measured in terms of the difference between modulated and non-modulated $m(t)$.

Some Cauchy-type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule, and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials, and some trigonometric functions. Further, we obtain some exponential, logarithmic, and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.

A mathematical model based on a discrete newborn set is proposed to describe the evolution of a sex-age-structured population, taking into account the temporary pair of sexes, infinite ranges of reproductive age of sexes, and maternal care of offspring. Pair formation is modeled by a weighted harmonic mean type function. The model is based on the concept of density of families composed of mothers with their newborns. All individuals are divided into the pre-reproductive and reproductive age groups. Individuals of the pre-reproductive class are divided into the newborn and teenager groups. Newborns are under maternal care while the teenagers can live without maternal care but cannot mate. Females of the reproductive age group are divided into singles and those who care for their offspring. The model is composed of a coupled system of integro-partial differential equations. Sufficient conditions for the existence of a one-parameter class of separable solutions of this model are found in the case of stationary vital rates.

In this paper, an improved two-stage and three-stage optional randomized response (ORR) models for quantitative variables that make the use of additive scrambling was proposed. These two-stage and three-stage models achieve efficient estimation of the mean and sensitivity level simultaneously in the single sample by using two questions. It is found that the proposed models perform better than the existing ORR models in terms of estimating sensitive attribute and sensitivity level simultaneously. It is found that the proposed three stage ORR model provides better estimates than the two-stage and one-stage ORR models and offers more privacy to the respondents with suitable choice of design parameters. The properties of the proposed models are demonstrated with the help of a numerical study.

The concept of soft derivative, introduced by Molodtsov in 1999, is one of the fundamental concepts in soft analysis. The handled paper defines partial soft derivative and studies some of its basic properties, such as the relation between partial soft derivative and boundedness, some basic partial soft derivative rules, e.g., sum rule, constant multiple rule, and difference rule, the relation between soft derivative and partial soft derivative, the relation between classical partial derivative and partial soft derivative, and the geometric interpretation of partial soft derivative. Moreover, it exemplifies the theoretical part of the study and provides figures for the geometric interpretation. Finally, this study discusses the need for further research.

Throughout this article, a two-phase sampling (TPS) technique is employed to estimate the population mean of the sensitive variable. The current article endeavours to develop a chain ratio type estimator for the estimation of sensitive variable(s) in the presence of non-response and measurement error simultaneously by utilizing ORRT models under a two-phase sampling technique. The significant aspects associated with the suggested estimator characterized by bias and mean squared error have been evaluated. Besides this, the utterance for the minimum mean squared error for the optimal values has also been identified. The supremacy of the proposed estimator has been compared with the modified existing estimators under the TPS scheme by using two sensitive auxiliary variables. To clarify the theoretical findings, a simulation study along with a hypothetical generated population and a real population which is based on abortion rates from Statistical Abstract of the United States: 2011 are also addressed in this study.

In this paper, we prove an Ambarzumyan-type theorem for a Conformable fractional diffusion operator, i.e. we show that $q(x)$ and $p(x)$ functions are zero if the eigenvalues are the same as the eigenvalues of zero potentials.