Search Results (18)

In this paper, we consider the existence of normalized solutions for the fractional Kirchhoff system with Hardy critical nonlinearities. The problem combines the challenges of nonlocal operators, singular critical effects, and a variational framework with fixed mass conditions. By employing the minimax theorem, analyzing the Palais–Smale sequence and restricting to radial functions, we overcome the loss of strong convergence at critical energy levels. Under $L^2$-supercritical case, we prove the existence of positive radial normalized solutions, along with the positivity of the corresponding Lagrange multipliers. Full article

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically. Full article

Von Neumann’s minimax theorem defines optimal strategic unpredictability in zero-sum games. Empirical evidence from professional sports has been interpreted as positive behavioral evidence for minimax. In this article, we analyze the strategic optimality of offensive plays in the basketball endgame when a team has a final possession and trails by no more than a single basket. This final moment of the game most closely approximates the simultaneous-move conditions of a game where minimax theory applies. Using comprehensive NBA data from 2010 to 2025, we test for equality of success rates across shooter types (star vs. non-stars) and shot selection (two-point vs. three-point). Our analysis reveals systematic violations of minimax play that have intensified with basketball’s shift to three-pointers and higher expected points. In the final decisive moment of the game, we find that teams systematically overuse three-point shots even though the two-point attempt yields higher field goal percentages. In addition, teams over-rely on star players for the final shot; non-star two-point shots have been the top-performing endgame option in 2022–2025. Full article

We consider the lossless compression bound of any individual data sequence. Conceptually, its Kolmogorov complexity is such a bound yet uncomputable. According to Shannon’s source coding theorem, the average compression bound is $nH$, where n is the number of words and H is the entropy of an oracle probability distribution characterizing the data source. The quantity $nH\left({\widehat{\theta }}_n\right)$ obtained by plugging in the maximum likelihood estimate is an underestimate of the bound. Shtarkov showed that the normalized maximum likelihood (NML) distribution is optimal in a minimax sense for any parametric family. Fitting a data sequence—without any a priori distributional assumption—by a relevant exponential family, we apply the local asymptotic normality to show that the NML code length is $nH\left({\widehat{\theta }}_n\right)+{\displaystyle \frac{d}{2}}log{\displaystyle \frac{n}{2\pi }}+log{\int }_{\Theta }{\left|I\left(\theta \right)\right|}^{1/2}d\theta +o\left(1\right)$, where d is dictionary size, $\left|I\right(\theta \left)\right|$ is the determinant of the Fisher information matrix, and $\Theta$ is the parameter space. We demonstrate that sequentially predicting the optimal code length for the next word via a Bayesian mechanism leads to the mixture code whose length is given by $nH\left({\widehat{\theta }}_n\right)+{\displaystyle \frac{d}{2}}log{\displaystyle \frac{n}{2\pi }}+log{\displaystyle \frac{|I\left({\widehat{\theta }}_n\right){|}^{1/2}}{w\left({\widehat{\theta }}_n\right)}}+o\left(1\right)$, where $w\left(\theta \right)$ is a prior. The asymptotics apply to not only discrete symbols but also continuous data if the code length for the former is replaced by the description length for the latter. The analytical result is exemplified by calculating compression bounds of protein-encoding DNA sequences under different parsing models. Typically, compression is maximized when parsing aligns with amino acid codons, while pseudo-random sequences remain incompressible, as predicted by Kolmogorov complexity. Notably, the empirical bound becomes more accurate as the dictionary size increases. Full article

Stochastic vibration control of uncertain structures under random loading is an important problem and its minimax optimal control strategy remains to be developed. In this paper, a stochastic optimal control strategy for uncertain structural systems under random excitations is proposed, based on the minimax stochastic dynamical programming principle and the Bayes optimal estimation method with the combination of stochastic dynamics and Bayes inference. The general description of the stochastic optimal control problem is presented including optimal parameter estimation and optimal state control. For the estimation, the posterior probability density conditional on observation states is expressed using the likelihood function conditional on system parameters according to Bayes’ theorem. The likelihood is replaced by the geometrically averaged likelihood, and the posterior is converted into its logarithmic expression to avoid numerical singularity. The expressions of state statistics are derived based on stochastic dynamics. The statistics are further transformed into those conditional on observation states based on optimal state estimation. Then, the obtained posterior will be more reliable and accurate, and the optimal estimation will greatly reduce uncertain parameter domains. For the control, the minimax strategy is designed by minimizing the performance index for the worst-parameter system, which is obtained by maximizing the performance index based on game theory. The dynamical programming equation for the uncertain system is derived according to the minimax stochastic dynamical programming principle. The worst parameters are determined by the maximization of the equation, and the optimal control is determined by the minimization of the resulting equation. The minimax optimal control by combining the Bayes optimal estimation and minimax stochastic dynamical programming will be more effective and robust. Finally, numerical results for a five-story frame structure under random excitations show the control effectiveness of the proposed strategy. Full article

In this paper, we apply the classical FKKM lemma to obtain the Ky Fan minimax inequality defined on nonempty non-compact convex subsets in reflexive Banach spaces, and then we apply it to game theory and obtain Nash’s existence theorem for non-compact strategy sets, which can be regarded as a new, simple but interesting application of the FKKM lemma and the Ky Fan minimax inequality, and we can also present another proof about the famous John von Neumann’s existence theorem in two-player zero-sum games. Due to the results of Li, Shi and Chang, the coerciveness in the conclusion can be replaced with the P.S. or G.P.S. conditions. Full article

The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and $\sigma$-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems. Full article

Particle swarm optimization (PSO) is an attractive, easily implemented method which is successfully used across a wide range of applications. In this paper, utilizing the core ideology of genetic algorithm and dynamic parameters, an improved particle swarm optimization algorithm is proposed. Then, based on the improved algorithm, combining the PSO algorithm with decision making, nested PSO algorithms with two useful decision making criteria (optimistic coefficient criterion and minimax regret criterion) are proposed . The improved PSO algorithm is implemented on two unimodal functions and two multimodal functions, and the results are much better than that of the traditional PSO algorithm. The nested algorithms are applied on the Michaelis–Menten model and two parameter logistic regression model as examples. For the Michaelis–Menten model, the particles converge to the best solution after 50 iterations. For the two parameter logistic regression model, the optimality of algorithms are verified by the equivalence theorem. More results for other models applying our algorithms are available upon request. Full article

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. These techniques are based on two computing steps of a proximal-like mapping with inertial terms. We investigated two simplified stepsize rules that do not require a line search, allowing the technique to be carried out more successfully without knowledge of the Lipschitz-type constant of the cost bifunction. Once control parameter constraints are put in place, the iterative sequences converge on a particular solution to the problem. We prove strong convergence theorems without knowing the Lipschitz-type bifunction constants. A sequence of numerical tests was performed, and the results confirmed the correctness and speedy convergence of the new techniques over the traditional ones. Full article

This paper is devoted to provide sufficient Karush Kuhn Tucker (in short, KKT) conditions of optimality of second-order for a set-valued fractional minimax problem. In addition, we define duals of the types Mond-Weir and Wolfe of second-order for the problem. Further we obtain the theorems of duality under contingent epi-derivative together with generalized cone convexity suppositions of second-order. Full article

Learning and making inference from a finite set of samples are among the fundamental problems in science. In most popular applications, the paradigmatic approach is to seek a model that best explains the data. This approach has many desirable properties when the number of samples is large. However, in many practical setups, data acquisition is costly and only a limited number of samples is available. In this work, we study an alternative approach for this challenging setup. Our framework suggests that the role of the train-set is not to provide a single estimated model, which may be inaccurate due to the limited number of samples. Instead, we define a class of “reasonable” models. Then, the worst-case performance in the class is controlled by a minimax estimator with respect to it. Further, we introduce a robust estimation scheme that provides minimax guarantees, also for the case where the true model is not a member of the model class. Our results draw important connections to universal prediction, the redundancy-capacity theorem, and channel capacity theory. We demonstrate our suggested scheme in different setups, showing a significant improvement in worst-case performance over currently known alternatives. Full article

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results. Full article

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones. Full article

This paper considers the binary Gaussian distribution robust hypothesis testing under a Bayesian optimal criterion in the wireless sensor network (WSN). The distribution covariance matrix under each hypothesis is known, while the distribution mean vector under each hypothesis drifts in an ellipsoidal uncertainty set. Because of the limited bandwidth and energy, we aim at seeking a subset of p out of m sensors such that the best detection performance is achieved. In this setup, the minimax robust sensor selection problem is proposed to deal with the uncertainties of distribution means. Following a popular method, minimizing the maximum overall error probability with respect to the selection matrix can be approximated by maximizing the minimum Chernoff distance between the distributions of the selected measurements under null hypothesis and alternative hypothesis to be detected. Then, we utilize Danskin’s theorem to compute the gradient of the objective function of the converted maximization problem, and apply the orthogonal constraint-preserving gradient algorithm (OCPGA) to solve the relaxed maximization problem without 0/1 constraints. It is shown that the OCPGA can obtain a stationary point of the relaxed problem. Meanwhile, we provide the computational complexity of the OCPGA, which is much lower than that of the existing greedy algorithm. Finally, numerical simulations illustrate that, after the same projection and refinement phases, the OCPGA-based method can obtain better solutions than the greedy algorithm-based method but with up to $48.72\%$ shorter runtimes. Particularly, for small-scale problems, the OCPGA -based method is able to attain the globally optimal solution. Full article

In this paper, we will consider a minimax fractional programming in complex spaces. Since a duality model in a programming problem plays an important role, we will establish the second-order Mond–Weir type and Wolfe type dual models, and derive the weak, strong, and strictly converse duality theorems. Full article