This paper examines how the degree of double-counting network effects (w) affects platform interior pricing mechanisms under bilateral partial multi-homing. A major result in the study of two-sided platforms is that platforms do not provide subsidies under the assumption of multi-homing without double-counting network effects on both sides. Addressing this benchmark, this study constructs a bilateral platform competition model based on the Hotelling framework that incorporates double-counting network effects (w ≠ 0) to analyze the interior equilibrium of two-sided competition and pricing strategies systematically. The findings reveal that the candidate stationary prices on one side become negative for sufficiently strong cross-side network effects. Comparative static analyses demonstrate that candidate user participation under bilateral partial multi-homing is positively correlated with network effects and stand-alone intrinsic value and negatively correlated with transportation costs. Numerical simulations further reveal the existence of non-monotonic interior equilibrium patterns in which the degree of double-counting network effects influences candidate user participation, interior stationary pricing, and optimal platform profit. It is contingent upon the relative magnitudes of the fundamental parameters of the two-sided market. This study not only delineates the boundaries of negative candidate stationary price under multi-homing, but also provides important practical insights for managers formulating digital platform pricing strategies.
We study two coupled two-sided quaternion matrix equations sharing an unknown. Because the map X↦AXB is real-linear but not generally quaternion-linear, an order-preserving real representation converts the problem into one real system. It yields an exact augmented-rank solvability criterion, a reduced shared-unknown criterion, and the complete Moore–Penrose solution. We also obtain fourteen necessary quaternion block-rank screening conditions and counterexamples proving that they are not jointly sufficient. For a side-information-assisted dual-watermarking application at embedding strength 0.05, the stored 8-bit image gives a peak signal-to-noise ratio (PSNR) of 35.99 dB and a global structural-similarity estimate of 0.9987; the no-further-processing normalized correlations are 0.3921 and 0.7179, respectively. The ideal path recovers to floating-point roundoff, whereas the stored-image results quantify sensitivity to clipping, quantization, and subsequent processing.
In this paper, we present an algorithmic modeling approach based on Disjunctive Programming using Boolean logic “OR” to solve the Optimal Phasor Measurement Unit Placement (OPP). We propose a framework for modeling the optimal PMU placement subject to disjunctive constraints. A convex objective function is minimized subject to a bilinear equality constraint with a piecewise linear structure. The polynomial constraint constitutes a union of linear segments conceptually analyzed in the two-dimensional continuous space, separating the infeasible from the feasible region. This work investigates the trajectory of iterates from infeasible initial points to stationary solutions and analyzes the convergence behavior using Interior-Point Method (IPM) and Sequential Quadratic Programming (SQP). Our algorithmic model addresses the progress of infeasible and feasible iterates, step computation using line-search and trust-region mechanisms. Combined with second-order correction (SOC) and filter methods, these mechanisms enable the algorithm to maintain a unit primal step, even when starting from an infeasible initial point. Network observability constraints are transformed from a Conjunctive Normal Form (CNF) into a Disjunctive Normal Form (DNF) via Balas’s theory. This geometry transformation reformulates the feasible set into a union of convex affine pieces, effectively eliminating constraint curvature issues. This affine reformulation ensures that gradient-based algorithms maintain a smooth optimization trajectory along a convex local manifold. This trajectory enables the algorithm to preserve the full Newton step, maintaining a superlinear convergence rate. Its underlying piecewise linear convexity inherently enables the gradient-based algorithm to avoid the Maratos effect. Numerical results on IEEE power systems validate the optimization problem. Our framework uses the IEEE-14 bus system to address the high-degree non-convexities. Monte Carlo simulations further enhance the argument that piecewise convexity enables IPM and SQP to converge to binary local minima. Depending on multiple-run initialization, these methods reach the same objective function value. These local minima can be characterized as non-strict optimum points that are structurally symmetric but exhibit unequal basins of attraction. Hence, this geometry-driven formulation enables gradient-based methods to reliably identify binary-valued optimal solutions for the OPP.
This paper presents criteria for Hurwitz stability and quasi-stability of regular matrix polynomials with possibly singular leading coefficients, for application to the stability analysis of differential-algebraic systems. The proposed framework involves two block Hankel matrices constructed from the coefficients of the Taylor expansions of two rational matrix-valued functions, which are well-defined under some natural assumptions. Building on the Markov parameters approach for the monic case, we establish correspondences between the stability of regular matrix polynomials and the positive semidefiniteness of these block Hankel matrices.
This paper investigates the stochastic behavior of Galton–Watson processes at the brink of extinction. Let {Zn:n≥0} denote a Galton–Watson process. We define the brink time of extinction as τ=infn−1:Zn=0, and set Sn=∑k=0nZk. We study the joint probability distributions and associated properties of the random pairs (τ,Zτ)) and (τ,Sτ). We further derive the joint distribution linking the brink time of extinction to the total number of particles that produce no offspring throughout their lifetimes. By constructing iterative bivariate probability generating functions, we obtain exact joint probability generating functions, probability mass functions and expectations for these pairs in the subcritical and critical regimes. These exact results complement the classical marginal laws of total progeny and leaf counts, as well as the asymptotic joint limit theorems available in the literature.
In response to the increasing quality demands of consumers amid technological advancements in manufacturing, manufacturers must manage the quality and lifespan of their products. In practical applications, various methods have been developed to assess product quality performance. This study employs process capability indices (PCIs) to evaluate product quality. This research explores scenarios involving products with multiple quality characteristics or multiple production lines under a Rayleigh lifetime distribution. Using the progressive Type-I interval censored sample, we assess whether the overall lifetime performance index meets a predetermined target. Given a specified significance level and statistical power, the minimum required sample size is derived. Under either fixed or unfixed total testing times, this study identifies the minimum number of observation intervals and the sample size that minimizes the total experimental cost, or the minimum number of observation intervals, equal-length interval time, and sample size. Finally, an illustrative practical example from two production lines is used to demonstrate how to apply the optimal experimental design proposed in this study to construct the progressive Type-I interval censored sample to test whether the overall lifetime performance index achieves the specified target.
We construct upward skip-free continuous-time Markov chains from finite-band recurrences associated with d-orthogonal polynomial systems. In the alternating-sign chamber, the recurrence yields an order-d generalized birth–death generator with one upward and at most d downward jump channels. In the non-negative chamber, an irreducible finite Hessenberg truncation admits a Perron–Doob normalization and a continuous-time uniformization. We give the stationary distribution explicitly in terms of the left and right Perron vectors and distinguish the asymptotic spectral rate from finite-time norm estimates for non-normal matrices. For Sheffer systems A(t)exB(t), we separate coefficientwise non-negativity, regular d-orthogonality, irreducibility at a fixed truncation size, and irreducibility for every truncation; this corrects the global admissibility conditions in the Sheffer and Appell chambers. For continuously observed paths, we specify the likelihood, identify the Perron-normalization nonidentifiabilities, and report a normalized Monte Carlo calibration. Finally, two recurrence-generated health generators are used directly in a product-state multiple-life valuation, and the dependence of retained-state rates on the number of living health categories is quantified. The numerical study is a simulation-based structural validation and does not claim empirical superiority on insurance data. In the actuarial interpretation, d is the largest permitted one-jump deterioration in the chosen severity ordering and m is the number of living health categories. Both are substantive modeling choices; finite-state truncation, asymmetric jump support, and the absence of an external-data calibration delimit the present practical conclusions.
This paper investigates an inverse problem of recovering the spatial component of a source term for a degenerate ultraslow diffusion equation with a Hadamard–Caputo fractional derivative. Nonlocal boundary conditions of the Samarskii–Ionkin type are imposed. A distinctive feature of the problem is that the system of eigenfunctions of the associated spectral problem, although complete and minimal, does not form an unconditional basis in L2(0,1). Hence the standard eigenfunction Fourier expansion cannot be applied directly. To overcome this difficulty, we construct a special auxiliary system from normalized linear combinations of asymptotically close eigenfunctions and prove that it forms a Riesz basis. Expansion with respect to this basis reduces the inverse problem to a cascade system of fractional differential equations for the modal coefficients. Using spectral asymptotics, estimates for the Le Roy function, and two-sided Riesz basis inequalities, we prove the existence, uniqueness, and conditional Lipschitz stability of a strong generalized solution and rigorously justify the convergence of the resulting series. We also quantify the high-frequency instability of the inverse mapping in the weaker L2 data topology. An analytical two-mode illustration shows explicitly how neglecting the coupling between asymptotically close spectral modes changes the reconstructed source coefficient.
Text-to-image generation has rapidly advanced the creation of digital artwork, yet most existing models rely on centralized training pipelines that require collecting large-scale image–text pairs from artists, design studios, or online communities. Such centralized practice raises serious privacy, ownership, and style leakage concerns, especially when local datasets contain identifiable artistic signatures or proprietary visual assets. To address this problem, this paper proposes FedMuse, a privacy-aware federated framework for multi-style text-to-image art generation, in which distributed clients collaboratively train a shared generative model while keeping their private art data local. The proposed framework decomposes the learning process into three coordinated components: a global semantic alignment module that captures cross-client text–image correspondence, a local style adapter that preserves client-specific artistic characteristics, and a privacy-calibrated aggregation mechanism that suppresses sensitive style leakage during model update exchange. To further improve multi-style generation, we design a style-disentangled federated optimization algorithm that separates content-relevant knowledge from client-private stylistic representations, allowing the global model to generalize across diverse artistic domains without directly absorbing private local styles. In addition, an adaptive privacy regularizer is introduced to reduce memorization risk while maintaining visual quality and prompt consistency. Experiments on real-world text-to-image art datasets demonstrate competitive generation quality, stronger personalization, and improved empirical resistance to membership and style-leakage attacks. Because the calibration statistics are data-dependent, FedMuse does not claim a certified end-to-end differential-privacy budget. The results suggest that privacy-aware collaboration can be a practical direction for distributed AI art generation.
Distributed network security requires multiple defenders to make coupled resource allocation and coordination decisions under limited budgets, connectivity-dependent risks, and non-negligible coordination costs. We develop a mathematical framework for jointly optimizing defensive resource allocation and coalition formation in a multi-defender network security game. Nodes have heterogeneous values and protection thresholds, attacks propagate through vulnerable connected components, and defenders retain individual budgets while forming coalitions for joint decision-making. For each defender partition, the induced allocation game is evaluated by its worst pure-Nash-equilibrium loss together with a supermodular coordination cost. We prove that optimal protection is NP-hard even in a restricted setting, that every fixed partition induces an exact potential game with finite best-response convergence, and that decentralized equilibria can have an unbounded price of anarchy. We further characterize merge and split thresholds, the piecewise-constant dependence of optimal partitions on the coordination weight, the monotonicity of the selected coordination cost, and conditions for diminishing net merge benefits. Based on these properties, we propose Cost-Aware Coalition Search (CACS), which combines a tractable allocation oracle with multi-start local search over merge, split, and move operations. Experiments on synthetic and real networks show that selective coalitions provide a scalable tradeoff between residual attack loss and coordination cost across different resource levels, coordination weights, and defender populations. The framework provides a mathematical basis for organizing distributed defenders in networked security environments.
Rising renewable generation is increasing computational demands on power system simulation, motivating physics-informed surrogates for classical solvers. This study identifies a publicly released synchronous generator dataset in which explicit Euler integration (step 1/480 s) diverges near the system’s pre-disturbance equilibrium while implicit, symplectic, and adaptive Runge–Kutta solvers remain bounded; a data-driven Jacobian estimate confirms that the dominant eigenvalues lie outside the explicit Euler stability region. A physics-informed neural corrector with a spectral-normalization-certified Lipschitz constant is proposed, providing a provable, data-independent bound. Evaluated on the original case and on 100 synthetic machines spanning the stability boundary, a machine-level, seed-averaged analysis (14 of 25 held-out test machines diverge under explicit Euler) shows that it underperforms classical integration where explicit Euler is stable (p = 0.001) but significantly improves accuracy where it diverges (p < 0.001). Long-horizon and out-of-distribution tests reveal no stability difference between the certified corrector and an unconstrained ablation, but direct measurement shows that the certified model remains within its theoretical Lipschitz bound (0.128) throughout, whereas the unconstrained variant reaches values up to seven times larger (p < 0.001). These results suggest that the guarantee’s value lies not in improved accuracy but in provably eliminating an operating regime where numerical divergence would otherwise remain possible.
Climate change poses several challenges for the economic system. The financial and insurance industries are called on to play a pivotal role, in particular in coming up with innovative solutions such as sustainable products. According to the EU’s Sustainable Finance Disclosure Regulation, sustainable products have to meet the requirement of pursuing a sustainable investment objective. Furthermore, they need to have measurable goals in place aligned with the investment objective, which can be reported to clients, in order to achieve an advanced understanding of the potential performance to obtain more effective stress scenario analysis. Accordingly, the design of insurance products requires suitable risk and profitability analytics. In our paper, we provide an innovative risk analytical tool for the design of sustainable insurance products environmentally linked, that is, a Climate Conditional Value at Risk (CCVaR), based on Random Forests with a Canonical Correlation Analysis algorithm, building a VaR of climate-related portfolio volatility that is obtained by assessing how the relationships between volatilities of portfolio assets modify according to the bioclimatic indicators. In particular, we propose a splitting rule for the Random Forest algorithm based on OVERALS, a generalisation of Canonical Correlation Analysis based on similarities and not on correlations, considering a more general perspective of non-linear relationships.
Extreme financial events are usually defined by loss magnitude alone, which quantifies severity but leaves open whether tail losses arrive in isolation or in temporally concentrated patterns. This paper develops a fractional tail-event timing diagnostic combining peaks-over-threshold (POT) generalized Pareto distribution (GPD) severity with Mittag-Leffler recurrence of interarrival times. We show that when exceedance times are driven by a stationary and ergodic latent intensity, the log-moment recurrence estimator has an explicit probability limit below the Poisson boundary that depends on the intensity only through its marginal log-variance, so volatility-driven dispersion of the arrival rate suffices to produce sub-Poisson estimates. In six daily series through 2026, the estimator is calibrated against a permutation null preserving the marginal loss distribution and the integer-gap discretization. That null sits near 1.05 rather than at unity, so the nominal Poisson boundary is not the correct reference for trading-day gaps. Against the calibrated null, the Deutscher Aktienindex (DAX), Nasdaq Composite, Nikkei 225, platinum, and the Standard & Poor’s 500 (S&P 500) display significant temporal concentration and coffee does not, whereas the monotone decline of the estimator across thresholds proves to be mechanical. Once goodness-of-fit tests are bootstrap-calibrated, no waiting-time law survives for four assets. The Mittag-Leffler parameter is therefore used only as an interpretable diagnostic index of temporal concentration; it is not a fitted-law claim and should not be interpreted as evidence of structural long memory or of a fractional data-generating mechanism.
Weakly nonlinear free-surface flows past disturbances are traditionally modeled using the forced Korteweg–de Vries (fKdV) equation with a prescribed instantaneous pressure field. However, physical wake responses possess finite relaxation times and advection scales that diagnostic algebraic closures fail to capture. This paper introduces a novel coupled system in which the surface pressure is a dynamical field governed by an advection–reaction–diffusion equation driven by band-limited curvature. Using linear spectral theory and numerical validation, we derive a phase-speed criterion demonstrating that energy transfer is determined by the comparison between the pressure drift speed and the surface phase speed. A sharp stability theorem proves that, to leading order in the coupling strength and for a non-negative even response transfer function whose drift speed exceeds the Froude detuning, the system is spectrally stable if and only if the response is band-limited below a critical wavenumber kc. Furthermore, an exact energy identity establishes that passivity and linear stability are equivalent. Finally, we demonstrate resonance steering: while coupling typically increases the wave resistance for monotone spectra, tuning the response to a spectral zero of a multi-lobe footprint reduces the drag significantly relative to its classical value. This result identifies an explicit performance–strongness trade-off, providing a mathematically strong structure for wave drag minimization through dynamic pressure control.
This paper examines directional quantile dependence from green bonds, clean energy markets, and carbon markets to the return states of five literature-classified green cryptocurrencies. Using daily returns from 25 September 2019 to 23 May 2025, we estimate static cross-quantilograms on a 19×19 quantile grid at lags 1, 5, and 22, 500-observation rolling cross-quantilograms, bootstrap surface tests, and green-specificity comparisons with five cryptocurrencies that used proof-of-work (PoW) consensus throughout the comparison sample. Point estimates display heterogeneous short-run patterns in selected green-bond and carbon-market pairs, but none of the 45 forward surfaces rejects the omnibus null at the 5% level. Rolling estimates vary across windows and tail cutoffs. The largest raw green-group contrast occurs for carbon quota prices at lag five, although its time-series bootstrap interval includes zero and factor-adjusted tests do not detect systematic green-minus-PoW separation. Descriptive quantile-on-quantile connectedness estimates are higher at extreme quantiles than at the median–median state. Overall, the evidence is more consistent with broad cryptocurrency-market conditions than with a uniform dependence pattern associated with the environmental label.
People who benefit from valuable information do not always pass it on. We develop a delayed IHSCR information propagation model in which a beneficiary can either continue spreading the information or hold it after a behavioral decision lag. The key modeling distinction is that information acquisition and the subsequent sharing-or-holding decision are treated as separate behavioral stages, while information holders can also suppress active spreaders. The delayed transitions are written as outflow rates, so arbitrary nonnegative histories do not automatically preserve positivity. We therefore work with nonnegative-feasible histories and show that such histories exist near each positive equilibrium on any fixed finite interval. For zero delay, we derive the basic reproduction number, prove global asymptotic stability of the information-free equilibrium when R0<1, and give Routh–Hurwitz conditions for local stability of the positive equilibrium. With the delay as a bifurcation parameter, the linearized system gives a transcendental characteristic equation and a quartic frequency equation. The critical delay is recovered from an atan2-based phase condition, and the transversality condition identifies the first spectral stability switch. For the stability-switching parameter set, an independent characteristic-root computation verifies a unique simple crossing, and a characteristic-matrix normal-form calculation gives a negative first Lyapunov coefficient, classifying the local Hopf bifurcation as supercritical with a locally orbitally stable periodic branch. We also prove the existence of a delayed optimal control on the nonnegative-feasible set and derive the optimality system, including advanced adjoint terms. At the baseline cost weights, the computed dynamic control gives a modestly higher net objective than a numerically optimized constant-control benchmark, and this ordering persists when the relative cost ratio c1/c2 is varied from 1 to 100. A two-parameter sensitivity analysis shows how the beneficiary-to-spreader and beneficiary-to-holder transition rates shift the first critical delay.
This editorial introduces the Special Issue of Mathematics “Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Professor Emanuele Munarini,” which collects ten peer-reviewed papers on theoretical and applied aspects of enumerative combinatorics, combinatorial matrices, and polynomial sequences. The contributions cover a variety of topics and methods, including generating functions, Riordan arrays, recurrence relations, and related combinatorial structures. The Special Issue’s scope and individual contributions are briefly discussed.
This paper studies candidate intermediary node deployment for influence diffusion under the linear threshold model (LTM). Given fixed diffusion sources, target nodes, and a budget, the decision maker selects candidate intermediary nodes to maximize the expected total weight of activated targets. Once deployed, a candidate node enables its associated potential arcs whose other endpoints belong to the effective network. Using the LTM live-arc representation, we establish distributional equivalence between sampling on the potential graph and then restricting each scenario to the deployed induced network, and sampling directly on the deployed network. This leads to a finite-scenario sample-average approximation (SAA) mixed-integer formulation based on canonical live paths; the resulting deployment objective is monotone and supermodular but is generally not submodular, so the classical greedy-approximation guarantee for monotone submodular maximization does not apply in general. Since the compact SAA formulation contains many scenario–target variables and covering constraints, solving the formulation directly can be computationally demanding. We therefore propose a scenario-decomposed branch-and-Benders-cut algorithm that solves the finite-scenario SAA model to optimality. Each scenario subproblem is separable by target and has a closed-form dual optimum, so Benders cuts are separated by scanning required-node sets rather than solving linear programs inside callbacks. On five real networks and 225 SAA instances, the algorithm solves all instances within one hour, averaging 27.46 s; the compact SAA formulation solves 172 instances, with an average capped time of 1444.11 s.
This paper investigates a distributed-order fractional SEIRS epidemic model with temporary immunity, vaccination, disease-induced mortality, and density-dependent natural mortality. The distributed-order formulation provides a flexible framework for incorporating heterogeneous memory effects into epidemic dynamics. The basic reproduction number is derived, the disease-free and endemic equilibria are characterized, and their local stability properties are investigated through the distributed-order characteristic equation. In addition, a sufficient global disease extinction criterion is established using a distributed-order Lyapunov argument, yielding global asymptotic stability of the disease-free equilibrium under an explicit transmission bound. Numerical spectral analysis shows that no oscillatory stability transition occurs within the considered parameter ranges and that the endemic equilibrium remains asymptotically stable within these ranges. Numerical simulations examine the effects of the memory interval and memory density shape on epidemic transients. Broadening the memory interval toward lower fractional orders slows convergence and increases transient persistence. For a fixed memory interval, densities concentrated near lower fractional orders produce the slowest relaxation, whereas those concentrated near higher orders, closer to the classical first-order limit, yield the fastest decay. These results show that distributed-order memory primarily regulates transient epidemic dynamics while preserving endemic equilibrium stability within the investigated parameter regimes, providing new insight into heterogeneous memory effects in fractional epidemic models.
In this paper, an efficient numerical method is presented for solving the generalized time-fractional Boussinesq equation. The Caputo fractional derivative is approximated by the L1 formula and the space is discretized by using the Fourier spectral method. The a priori estimates, solvability and convergence of the numerical scheme are rigorously established. The presented numerical results demonstrate the effectiveness of the Fourier spectral method.
