Search Results (14)

This paper develops a practical computational framework for the Bayesian Cournot model with bilateral incomplete cost information, where each player is uncertain about the opponent’s marginal cost, drawn from a continuous compact interval $[c_{*}, c^{*}]$ with $0<c_{*}<c^{*}<\infty$. The infinite dimensionality of the functional strategy spaces (mappings from types to production quantities) renders analytical closed-form solutions infeasible in this continuous-type setting. To overcome this challenge, we restrict the strategy spaces to finite-dimensional differentiable sub-manifolds—specifically, one-parameter families of oscillatory functions (cosine, sine, and mixed forms). After suitable affine Q-rescaling to map the oscillatory range into the production interval $[0, Q]$, and with parameter ranges satisfying $\alpha , \beta >(\pi /2)/c_{*}$, these curves ensure near-exhaustivity: the joint production map $(\alpha , \beta )\mapsto (x_{\alpha }\left(s\right), y_{\beta }\left(t\right))$ covers ${[0, Q]}^2$ densely for every fixed cost pair $(s, t)$, thereby recovering (up to density and closure) the full ex-post payoff space. We introduce the ex-post payoff mapping $\Phi (s, t, x, y)=(e_s(x, y)\left(t\right), f_t(x, y)\left(s\right))$, which collects every realizable payoff pair once nature draws the types and players select their strategies. The image of $\Phi$ defines the general payoff space of the game, and its non-dominated points constitute the general ex-post Pareto frontier—all efficient realized outcomes across type-strategy realizations, without dependence on private probability measures over types. Using multi-objective genetic algorithms, we numerically approximate this frontier (and selected collusive compromises) within the restricted but representative sub-manifolds. The resulting frontiers are computationally accessible, robust to parameter variations, and validated through hypervolume convergence, sensitivity analysis, and comparisons with NSGA-II, PSO and scalarization methods. The findings are significant because they provide decision-makers in oligopolistic markets (e.g., electric vehicles) with viable, implementable production policies that explore efficient trade-offs under genuine cost uncertainty, without requiring explicit forecasts of the opponent’s type distribution—a limitation of traditional expected-utility approaches. By focusing on ex-post efficiency, the method reveals belief-independent compromise solutions that may guide tacit coordination or collusive outcomes in real-world strategic settings. Full article

This paper introduces a novel class of generalized contractions, termed $(\alpha ,\eta ,(Q,h),\mathcal{L})$-contraction mapping, within the context of triple controlled metric-type spaces, extending the framework of fixed point theory in controlled structures. The proposed mapping is defined using $\alpha$-admissible and $\eta$-subadmissible functions, in conjunction with a control pair $(Q,h)$ of upper class of type I, and incorporates Wardowski’s function $\mathcal{L}$-contraction condition. Under suitable hypotheses, we establish both the existence and uniqueness of fixed points for this class of mappings. Several corollaries are derived as special cases of the main result. Moreover, we provide a nontrivial application by analyzing the solvability of a nonlinear equation involving powers of the sine function, thereby illustrating the utility of the developed theory. Full article

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image of the unit disc under the q-sine function exhibits a visually appealing resemblance to a figure-eight shape that is symmetric about the real axis. Additionally, we investigate some important geometrical problems like necessary and sufficient conditions, coefficient bounds, Fekete-Szegö inequality, and partial sum results for the functions belonging to this newly defined subclass. Full article

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided. Full article

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. Full article

This study introduces a subclass $S_{qs}^{*}$ of starlike functions associated with the q-analogue of the sine function defined in symmetric unit disk. This article comprises the investigation of sharp coefficient bounds, and the upper bound of the third-order Hankel determinant for this class. It also includes the findings of Zalcman and generalized Zalcman conjectures for functions of this class. Full article

Utilizing $\left(p,q\right)$-numbers and $\left(p,q\right)$-concepts, in 2016, Duran et al. considered $\left(p,q\right)$-Genocchi numbers and polynomials, $\left(p,q\right)$-Bernoulli numbers and polynomials and $\left(p,q\right)$-Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced $(p,q)$-special polynomials and numbers and have described some of their properties and applications. In this paper, using the $(p,q)$-cosine polynomials and $(p,q)$-sine polynomials, we consider a novel kinds of $(p,q)$-extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the $\left(p,q\right)$-integral representations and $\left(p,q\right)$-derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. Full article

This article proposes a new lifetime-generated family of distributions called the sine-exponentiated Weibull-H (SEW-H) family, which is derived from two well-established families of distributions of entirely different nature: the sine-G (S-G) and the exponentiated Weibull-H (EW-H) families. Three new special models of this family include the sine-exponentiated Weibull exponential (SEWE${}_x$), the sine-exponentiated Weibull Rayleigh (SEWR) and sine-exponentiated Weibull Burr X (SEWBX) distributions. The useful expansions of the probability density function (pdf) and cumulative distribution function (cdf) are derived. Statistical properties are obtained, including quantiles ($Q_U$), moments ($M_O$), incomplete $M_O$ ($I{M}_O$), and order statistics ($O_S$) are computed. Six numerous methods of estimation are produced to estimate the parameters: maximum likelihood ($M_L$), least-square ($L_S$), a maximum product of spacing ($M{P}_R{S}_P$), weighted $L_S$ ($W{L}_S$), Cramér–von Mises ($C_{R}VM$), and Anderson–Darling ($A_D$). The performance of the estimation approaches is investigated using Monte Carlo simulations. The total factor productivity (TFP) of the United Kingdom food chain is an indication of the efficiency and competitiveness of the food sector in the United Kingdom. TFP growth suggests that the industry is becoming more efficient. If TFP of the food chain in the United Kingdom grows more rapidly than in other nations, it suggests that the sector is becoming more competitive. TFP, also known as multi-factor productivity in economic theory, estimates the fraction of output that cannot be explained by traditionally measured inputs of labor and capital employed in production. In this paper, we use five real datasets to show the relevance and flexibility of the suggested family. The first dataset represents the United Kingdom food chain from 2000 to 2019, whereas the second dataset represents the food and drink wholesaling in the United Kingdom from 2000 to 2019 as one factor of FTP; the third dataset contains the tensile strength of single carbon fibers (in GPa); the fourth dataset is often called the breaking stress of carbon fiber dataset; the fifth dataset represents the TFP growth of agricultural production for thirty-seven African countries from 2001–2010. The new suggested distribution is very flexible and it outperforms many known distributions. Full article

The flow behavior inside a miniature centrifugal pump, under a periodic pulse flow rate, was studied by means of numerical simulation. For a given incoming periodic pulse flow with a sine wave, the performance of the centrifugal pump was investigated in the section with increasing flow and the section with decreasing flow, and the special points of the flow rate and the periodic flow were identified. Further, the energy gradient method and the Q-criterion were adopted to analyze the internal vertical structure and flow stability. It was found that the regions with large variations in velocity and total pressure were mainly located at the leading edge of the suction surface and the middle area of the pressure surface of the blades. Irregular pressure fluctuation frequency under the periodic pulse flow was shown; this was mainly concentrated in the low-frequency zones close to the impeller’s rotational frequency. In addition, for the same flow rate in the periodic pulse flow, the pressure frequency fluctuation for the increasing flow rate section was higher than that observed for the decreasing flow rate section. It was found that the most unstable sections appeared in the first half-period of the flow rate variation (large flow rate), according to the distributions of the Q criteria of the vortex and the energy gradient function K. In this section, motions of strong vortices led to large gradients of the mechanical energy. Full article

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class. Full article

In this paper, we introduce q-cosine and q-sine Euler polynomials and determine identities for these polynomials. From these polynomials, we obtain some special properties using a power series of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. We investigate the approximate roots of q-cosine Euler polynomials that help us understand these polynomials. Moreover, we display the approximate roots movements of q-cosine Euler polynomials in a complex plane using the Newton method. Full article

This paper constructs and introduces $(p,q)$ -cosine and $(p,q)$ -sine Bernoulli polynomials using $(p,q)$ -analogues of ${(x+a)}^n$ . Based on these polynomials, we discover basic properties and identities. Moreover, we determine special properties using $(p,q)$ -trigonometric functions and verify various symmetric properties. Finally, we check the symmetric structure of the approximate roots based on symmetric polynomials. Full article

In this paper, we define cosine Bernoulli polynomials and sine Bernoulli polynomials related to the q-number. Furthermore, we intend to find the properties of these polynomials and check the structure of the roots. Through numerical experimentation, we look for various assumptions about the polynomials above. Full article

Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of $\mathcal{S}$ -matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the $\mathcal{S}$ -matrix in terms of abstract functional integral over a primary scalar field, the $\mathcal{S}$ form of a grand canonical partition function is found. By expression of $\mathcal{S}$ -matrix in terms of the partition function, representation for $\mathcal{S}$ in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the $\mathcal{S}$ -matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of $\mathcal{S}$ (more precisely, of $-ln\mathcal{S}$ ) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory ${\phi }^4$ (with suggestions for ${\phi }^6$ ) and nonpolynomial sine-Gordon theory. A new definition of the $\mathcal{S}$ -matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the $\mathcal{S}$ -matrices of the theory. For simplicity of numerical calculation, the $D=1$ case is considered, and propagator for free theory G is in the form of Gaussian function typically in the Virton–Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process. Full article