Cities are the main sites of production, employment, and capital accumulation in advanced economies, which places urban economic performance at the centre of economic policy and urban governance. This work develops an integrated grey-based multi-criteria approach for assessing that performance and applies it to the sixteen G7 cities covered by the Global Power City Index (GPCI). Criterion weights are obtained with Grey RANCOM (G-RANCOM), which converts the ordinal rankings of a five-member expert panel into interval weights, and the cities are ranked with Grey MUNRA (G-MUNRA), which aggregates linear, vector, and non-linear normalization. Each performance entry is an interval bounded by the minimum and the maximum annual score observed in the GPCI Economy function over 2021–2025. Market size, economic vitality, and business environment emerge as the most influential criteria, and New York, London, and Tokyo occupy the highest positions, while Osaka, Milan, and Fukuoka occupy the last three. Robustness is tested via scenario analyses on the model parameters and through a global analysis of 100,000 replications in which all of them vary jointly. New York holds the first position in 74% of the replications and the three lowest positions are unchanged in 87%, whereas cities in adjacent middle positions are not separated reliably. Rankings produced by five established grey approaches, by crisp and fuzzy counterparts, and by the published GPCI Economy rankings agree with the reported ordering, with Spearman correlations between 0.92 and 1.00. The framework offers urban policymakers a transparent benchmarking tool for evidence-based competitiveness strategies.
The price index of daily commodities is an important indicator of inflation in a country. Adequate modeling of the price index is useful for efficient economic decision-making. Since the exact modeling of the price index is impossible, probabilistic modeling is a suitable alternative for this. The change in the price index of various commodities at different time points is usually dependent, and hence joint modeling is a suitable solution. In this paper, a new family of distributions is proposed for joint modeling of the price index of various Saudi commodities at two different time points and is named as Darwish Bivariate Family of Distributions (DBFDs). Some necessary properties of the family are presented, and some dependence measures are also computed. Conditional distributions for the proposed family are investigated alongside the method for random data generation from any member of the proposed family. The parameter estimation for the proposed family is discussed in general. A specific member of the family, namely the Darwish Bivariate Log-logistic (DBLL) distribution, is studied in detail. Some important properties of the DBLL distribution are presented. The DBLL distribution is used to model the price index of various Saudi commodities. It is found that the proposed DBLL distribution provides a better fit to model the price index in comparison with the other models used in the study.
This study aims to propose a class of binary Darboux transformations for an integrable nonlinear Schrödinger system with two symmetric matrix potentials. The associated Lax pairs of AKNS type ensure the existence of these binary Darboux transformations, whose explicit single-step applications yield soliton solutions. Notably, the M-matrix appearing in the formulation of the binary Darboux transformations must be constructed within an extended framework, allowing for cases where eigenvalues coincide with adjoint eigenvalues, thereby encompassing generalized Darboux transformations as well.
We study the half-integer family F(n,p)=3F2−2n+12,1,2p−12;2n+52,2p+12;1. Although this family belongs to the Karlsson–Minton class of 3F2(1) series with negative integral parameter differences, the most directly relevant symmetric formula of Shpot and Srivastava becomes singular at the free numerator parameter a=1(here a denotes the first numerator parameter of the Shpot–Srivastava reduction formula, the value of which controls the Beta factors in their evaluation). We show that this apparent singularity is removable and identify its finite value with the independently derived closed form. First, Euler’s integral representation and a recurrence for Jm,n(x)=∫01v2m(1−xv2)−ndv yield a closed form for 2F1−2n+12,1;2n+52;x as a polynomial in (1−x) plus arctanh(x)/x. Rainville’s integral formula then reduces F(n,p) to a rational part and a finite rational linear combination of odd harmonic sums Hrodd=∑j=1r(2j−1)−1. For all integers n≥0 and p≥n+3, this proves F(n,p)∈Q. The threshold is sharp: at the adjacent boundary p=n+2 we obtain F(n,n+2)=Rn+Cnπ2 with Rn,Cn∈Q and Cn≠0, so rationality fails. We also prove, by analyticity and the independently derived formula, that the a→1 limit of the Shpot–Srivastava representation equals the present closed form. For computation, the exact assembly uses O(n+p) arithmetic operations under the unit-cost model, while a direct fixed-precision evaluation is ill-conditioned for large n. A stable series/closed-form hybrid for the 2F1 factor combined with adaptive positive-kernel Gauss–Jacobi quadrature gives relative errors at the machine-precision scale over the tested range up to n=64.
In an infinite-dimensional Hilbert space, we investigate a method for finding a solution of a variational inequality induced by an inverse strongly monotone set-valued mapping. Our results are established in the presence of summable and nonsummable computational errors. We show that our algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant.
In this work, we introduce new classes of lacunary sequence spaces based on Nörlund-type means, a sequence of modulus functions and the order parameter 0<α≤1. The main novelty is the incorporation of these components into a weighted Nörlund-lacunary framework. We establish inclusion and equivalence relations with classical Nörlund-type and lacunary sequence spaces under explicit conditions on the weight and lacunary sequences. We further characterize weighted lacunary statistical convergence of order α and investigate its relationships with the corresponding convergence spaces. Finally, we prove the completeness and topological properties of the resulting paranormed sequence spaces. These results extend existing frameworks for lacunary and Nörlund-type sequence spaces.
The Zero-inflated Poisson (ZIP) model has been widely adopted in quality control for near-zero-defect processes with sporadic nonconformities, while prior research predominantly focused on univariate ZIP processes for single defect types, modern high-precision manufacturing environments often involve two or more correlated defect categories. This paper proposes a Shiryaev–Roberts (SR) control chart for detecting parameter shifts in bivariate ZIP processes and compares its performance with the existing cumulative sum (CUSUM) chart under both zero-state and steady-state scenarios. Through Monte Carlo simulations, we establish the upper control limits (h) for both schemes while maintaining an identical in-control average run length (ARL) to ensure fair comparisons. Numerical simulations demonstrate that the SR chart exhibits superior out-of-control ARL performance compared to the CUSUM chart under steady-state conditions for detecting small to moderate shifts. The proposed methodology is further validated through a case study in LED packaging, highlighting its practicality in advanced manufacturing quality assurance.
In this paper, we establish a Berry–Esséen bound for the edge frequency polygon estimator based on linearly extended negative quadrant-dependent (LENQD) random variables under suitable conditions. The convergence rate is shown to be On−1/6, indicating that the edge frequency polygon estimator achieves a convergence performance that is comparable to that of the kernel estimator. In addition, a simulation study is conducted to validate the theoretical findings.
For 0≤α<1 and λ≥0, we introduce a filtered Janowski-rotated pseudo-Bazilević class of normalized analytic functions in the unit disk. The fractional power is defined through the unique analytic logarithm normalized at the origin, and a first-order differential filter is applied before the rotated Janowski subordination. Explicit formulae for a2 and a3 reduce the direct, logarithmic, inverse, and inverse-logarithmic Fekete–Szegő-type functionals to a common Schwarz coefficient template. We derive a fixed-a2 variability disk for a3, Janowski disk containment, a positive-real-part consequence, and branch-free reconstruction formulae. Sharpness and function-level realization are unconditional on the integrable slice α=0. For 0<α<1, the variability disk is exact at the Schwarz-data level and is realized by a function whenever the explicitly reconstructed auxiliary function is nonvanishing. Numerical results, including a two-parameter surface, show the damping effect of λ on the base coefficient scales and on representative complete bounds while also displaying their interaction with α, the rotation, and the Janowski parameters.
Quaternion-based moments and transforms have emerged as powerful tools for color image processing, attracting substantial research interest. Polar harmonic transforms (PHTs) have been shown to be a class of continuous orthogonal moments with good performance in image representation and reconstruction. In this paper, we propose a novel quaternion polar transform, termed the generalized quaternion polar linear canonical transform (GQPLCT). The proposed transform employs a six-parameter kernel, offering both flexibility and generality. By assigning special values to the parameters, it can degenerate into several existing quaternion polar harmonic transforms. Furthermore, we analyze its scaling invariance and establish its relationship to the complex GPLCT to enable efficient computation. Experimental results demonstrate the effectiveness of the proposed method and its favorable performance in terms of scale invariance, numerical stability, and image representation.
In this paper, the scale mixture of Akash (SMAK) distribution is introduced. This new distribution results from a scale mixture of the Akash distribution and the exponential distribution. The SMAK distribution is an alternative to two-parameter distributions with a heavy right tail. We study its representation, some basic properties, and maximum likelihood inference. We carry out three applications with real data, in which the SMAK distribution exhibits better performance than other distributions.
Motivated by the interplay between discrete orthogonal polynomials and geometric function theory, we introduce and investigate some new Ma-Minda-type subclasses of bi-univalent functions generated by the Poisson-Charlier polynomials through their following analytic generating function: Ea(x,z)=(1+z)xe−az(|z|<1). Within the classical class Σ of analytic and bi-univalent functions, we first define the Poisson-Charlier-generated bi-starlike and bi-convex families via the subordination relations involving Ea(x,·) for both a function f and its inverse f−1. By combining the Carathéodory representation with the series expansion of Ea(x,z) and the Lagrange inversion formula for f−1, we derive coefficient estimates for the initial Taylor-Maclaurin coefficients a2 and a3 of functions in the starlike class Σ𝒮*E(a,x) and in the convex class Σ𝒦E(a,x). In addition, we obtain corresponding Fekete-Szegö type inequalities of the form a3−μa22 for a real parameter μ in both settings, which are expressed explicitly in terms of the parameters a and x of the Poisson-Charlier framework. To the best of our knowledge, these Poisson-Charlier-generated bi-starlike and bi-convex classes have not previously been investigated, so the resulting coefficient and Fekete-Szegö estimates constitute a distinct contribution rather than direct special cases of previously studied Ma-Minda families.
Planar knotoids contain endpoint position information because the forbidden endpoint moves prevent the leg and the head from passing across arcs. Existing polynomial invariants based on Gauss diagrams and winding data record important parts of this information, but they may separate endpoint winding data from affine index data or combine winding contributions only after summation. We introduce an index refined winding pair polynomial in three variables for oriented planar knotoids. At each crossing, the invariant records the ordered winding pair of the crossing lobe together with the affine index weight of the same crossing. We prove invariance under planar knotoid equivalence, derive formulas for orientation reversal, mirror image and planar product, and show that the invariant is a Vassiliev invariant of degree one. We also obtain lower bounds for crossing number, Gordian distance and unknotting number from a nonconstant coefficient norm. Finally, explicit computations show that the winding signed sum polynomial and the affine index polynomial, even when considered together, do not determine the new invariant.
In longitudinal studies, a terminal event such as death typically halts data collection, and patients may exhibit marked changes as they approach the event. Existing methods face three related challenges: interpreting the effect of residual life on the longitudinal response, accommodating time-varying covariates and coefficients, and retaining information from censored individuals. We propose a joint model that addresses these challenges in a unified likelihood framework. The longitudinal submodel includes an explicit residual-life effect g(Ti−t,ξ), time-varying covariates Xi(t) with time-varying coefficients β(t), and shared random effects. Exponential-decay and Gaussian-kernel specifications are considered for g(·). The survival submodel depends on the latent state mi(t)=Xi(t)⊤β(t)+Zi(t)⊤bi. The coefficient functions are approximated by B-splines, and the full observed-data likelihood is maximized numerically with random-effects integrals evaluated by Gauss–Hermite quadrature. Under the stated regularity conditions, we establish consistency, root-n asymptotic normality of the finite-dimensional parameters, and the sieve convergence rate of the time-varying coefficient functions. Simulation studies demonstrate accurate recovery of the coefficient and residual-life functions. In the MADIT application, the fitted hazard ratio for ICD implantation is 0.433, and the residual-life effect indicates an increase in medical costs approximately three weeks before death.
In this paper, we investigate the real reverse-space–time nonlocal modified Korteweg–de Vries equation using the ∂¯-dressing representation established by Luo and Fan. By specializing the purely imaginary discrete spectral data, we obtain explicit one-, two- and three-soliton expressions from the general determinant formula. We then examine their profiles and parameter-dependent evolution. For regular one-soliton branches, the characteristic velocity and exponential amplitude rate are derived analytically. Depending on the spectral parameters, the amplitude may grow in one temporal direction and attenuate in the reverse direction, while the solution remains spatially localized at each finite time. The nonlocal product q(x,t)q(−x,−t) is shown to be invariant under joint space–time reflection, highlighting the dynamical role of the reverse-space–time coupling and its difference from the real local mKdV equation.
A girth cycle refers to a cycle that has the minimum length within a graph. The graphs D(k,q) form an important family of algebraically defined bipartite graphs over finite fields, and their short-cycle structure is closely related to questions in extremal graph theory and finite geometry. Motivated by the problem of determining the edge-girth-regular parameter of D(k,q), we determine the exact number of girth cycles in D(4,q) for every prime power q. Our proof uses a convenient isomorphic model Γ(4,q) and edge-transitivity to reduce the global enumeration to counting the girth cycles containing one fixed edge. Specifically, we prove that when q is odd with q>3, the number of girth cycles in D(4,q) is q5(q−1)2(q−3)/8. Moreover, when q is even, the number of girth cycles in D(4,q) is q5(q−1)2(2q−3)/8. When q=3, the girth is 12 and the number of girth cycles is 729. Together, these results resolve the case k=4 of the problem posed in our earlier work.
This paper focuses on three finite difference schemes for the interval advection equation based on a novel interval calculus framework. Different from classical interval arithmetic, this innovative framework equips the interval number space with a rigorous Hilbert space structure and enables a critical decoupling of the derivative for interval-valued functions, where the center component obeys classical differentiation rules and the radius component complies with multiplicative differentiation principles. Using this unique decoupling property, we construct interval counterparts of three representative classical finite difference schemes, namely the Upwind, Lax–Friedrichs, and Lax–Wendroff schemes, and conduct a comprehensive and rigorous theoretical assessment of their numerical properties. Utilizing the inherent isometric isomorphism between the interval space and R2, we rigorously establish the consistency of the proposed schemes and adopt the von Neumann method for systematic stability analysis. A sharp, explicit Courant–Friedrichs–Lewy (CFL) condition is derived, which jointly accounts for the center velocity and logarithmic radius velocity of the interval advection field, and the scheme convergence is strictly guaranteed via the Lax equivalence theorem. Extensive numerical experiments are carried out to validate the theoretical conclusions and verify the practical merits of the developed interval schemes. The numerical results demonstrate that the proposed methods retain nearly constant interval width in long-duration simulations, completely bypass the switching-point complexity that intrinsically exists in traditional generalized Hukuhara (gH)-based interval approaches, and deliver competitive computational efficiency. This work corroborates that the newly proposed interval calculus framework serves as an elegant, solid, and versatile foundation for the numerical computation and analysis of interval partial differential equations.
In social and economic systems, collective returns often depend nonlinearly on group size, exhibiting a curvature that can be convex or concave, yet how this curvature reshapes cooperation transitions in higher-order social dilemmas is not fully understood. In this work, we study a controlled public goods game on hypergraphs with minimal ad hoc parameters, where linear pairwise interactions are supplemented by a three-body power-law payoff whose exponent γ controls the curvature of the collective return. Through systematic numerical simulations and microscopic diagnostics, we show that this curvature qualitatively reshapes cooperation transitions: for γ<1, the transition is continuous-like, similar to the linear benchmark γ=1, whereas for γ>1, the transition becomes abrupt and first-order-like. This distinction remains robust across different system sizes and multiple hypergraph topologies. We further show that cooperation in the γ>1 regime is primarily driven by a cascade-like proliferation of fully cooperative triads, accompanied by the depletion of partially cooperative configurations. Based on this mechanism, we predict and verify that a strict three-body product-form payoff—although motivated by a completely different social mechanism—produces the same discontinuous cooperation transition.
Based on the new concept of 2-tuples of noncommutative Orlicz sequence spaces, we establish the Riesz–Thorin interpolation theorem. As applications, bounds for the nonsquare and von Neumann–Jordan constants of the noncommutative Orlicz space are given.
We establish a cohomology theory for Reynolds Lie triple systems by first developing their representation theory and then assembling the corresponding cochains into a suitable complex. This construction is subsequently applied to deformation and extension problems. In particular, infinitesimal formal deformations are described by degree-three cocycles, while equivalence classes of abelian extensions are determined by the associated cohomology classes. We further introduce Reynolds Lie triple 2-systems and show that the skeletal case is governed by degree-five cocycles of Reynolds Lie triple systems.
