Trigonometrically convex functions have recently emerged as an important class of functions in the field of convex analysis. Despite the growing interest, several aspects of their theory remain unexplored. In this article, we first establish lower and upper bounds for the mean integral of the reciprocal of a function that is either trigonometrically convex or trigonometrically concave. Building on these results, we further investigate the mean integral of the ratio of two functions, each of which is assumed to be either trigonometrically convex or concave. These investigations provide new knowledge about the integral behavior and comparative structure of such functions.

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies n i of orbitals φ i according to 0 ≤ n i ≤ 2 . In this work, we first refine the underlying one-body N -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness w of the N -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope Σ N , S ( w ) ⊂ [ 0 , 2 ] d . These constraints are independent of M and the number d of orbitals, while their dependence on N , S is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

In the paper, we consider a class dynamic control system with a discrete parameter and under conditions of uncertainty in the initial data. The optimal control problem of the minimax type is formulated for a non-smooth terminal functional using the principle of the best-guaranteed result. This problem is studied by methods of multivalued and convex analysis. For this non-smooth control problem the necessary and sufficient conditions for optimality are obtained.

This study focuses on the formulation and analysis of problems that are dual to those defined by convex set-valued mappings. Various important classes of optimization problems—such as the classical problems of mathematical and linear programming, as well as extremal problems arising in economic dynamics models—can be reduced to problems of this type. The dual problem proposed in this work is constructed on the basis of the duality theorem connecting the operations of addition and infimal convolution of convex functions, a result that has been previously applied to compact-valued mappings. It appears that, under the so-called nondegeneracy condition, this construction serves as a fundamental approach for deriving duality theorems and establishing both necessary and sufficient optimality conditions. Furthermore, alternative conditions that partially replace the nondegeneracy assumption may also prove valuable for addressing other issues within convex analysis.

The upper bound estimate for the convergence rate of two-stage sampling distribution regression is investigated. The excess risk is decomposed as the two-stage sample error and the approximation error, the approximation error is bounded by a [Formula: see text]-functional. The two-stage sample error is bounded with a convex analysis method whose advantage lies separating the two-stage optimal solution and the one-stage optimal solution from the mean optimal solution. On this basis, the two-stage sample error is attributed to the decay of a modulus of smoothness whose convergence rate has been investigated in approximation theory. Finally, a kind of explicit capacity independent convergence rate for the excess risk is provided. The feature of the investigation lies that the usual regularity condition is replaced by a decay assumption related to the modulus of smoothness.

This research offers a fresh lens on Iranian cultural heritage houses by interrogating the overlooked role of Orosi windows in shaping socio-spatial accessibility and visual permeability. While these decorative stained-glass features are traditionally appreciated for their artistry and environmental performance, their functional impact on visibility and spatial interaction remains underexplored. The study aims to assess how window visual permeability influences socio-spatial accessibility within the hierarchical layouts of historic houses in Iran. To this end, a quantitative approach was adopted, applying convex space analysis to examine socio-spatial dynamics and visibility graph analysis (VGA) to study visual permeability within the space syntax framework. Fifteen heritage houses were analysed under two conditions using VGA: their current status quo, and a hypothetical model in which windows were treated as fully transparent, allowing unobstructed sightlines. The analyses demonstrated that removing window barriers enhanced visual integration and connectivity across all cases. Statistical t-tests further confirmed that these differences were significant, establishing that Orosi windows exert a profound influence on visual permeability. Beyond their ornamental and climatic roles, this study redefines Orosi windows as dynamic cultural devices that actively script human visibility, privacy, and interaction, revealing how historical design intelligence can inform sustainable, culturally responsive architectural practices.

Abstract In this paper, we first establish a novel integral identity involving functions of two variables by using the Riemann–Liouville fractional integrals. By taking the modulus of this newly derived identity, we obtain a new form of Newton-type inequality specifically for differentiable co-ordinated convex functions. Moreover, we give example using graph in order to show that our main results are correct. In addition, we derive several new inequalities by employing Hölder’s inequality. Furthermore, we present previously achieved results and new results by using special cases of the obtained theorems. These results not only extend existing inequalities in the literature but also offer new insights into the interplay between fractional calculus and convex analysis.

This paper is concerned with a three-level multi-leader-follower incentive Stackelberg game with $H_\infty$ constraint. Based on $H_2/H_\infty$ control theory, we firstly obtain the worst-case disturbance and the team-optimal strategy by dealing with a nonzero-sum stochastic differential game. The main objective is to establish an incentive Stackelberg strategy set of the three-level hierarchy in which the whole system achieves the top leader's team-optimal solution and attenuates the external disturbance under $H_\infty$ constraint. On the other hand, followers on the bottom two levels in turn attain their state feedback Nash equilibrium, ensuring incentive Stackelberg strategies while considering the worst-case disturbance. By convex analysis theory, maximum principle and decoupling technique, the three-level incentive Stackelberg strategy set is obtained. Finally, a numerical example is given to illustrate the existence of the proposed strategy set.

Fractional Hermite-Hadamard-type inequalities represent a significant area of study in convex analysis due to their extensive applications in mathematical and applied sciences. These inequalities provide powerful tools for estimating the integral mean of a convex function in terms of its values at the endpoints of a given interval. In this paper, we focus on the development and refinement of fractional Hermite-Hadamardtype inequalities for the class of twice differentiable m-convex functions. Utilizing advanced analytical techniques, such as Ho¨lder’s inequality and the power mean integral inequality, we derive new bounds that generalize existing results in the literature. These findings not only extend classical inequalities to a broader class of convex functions but also provide sharper and more versatile estimations. The presented results are expected to have significant implications in various mathematical domains, including fractional calculus, optimization, and mathematical modeling. This work contributes to the ongoing efforts to generalize and refine integral inequalities by incorporating fractional operators and broader convexity assumptions, offering a deeper understanding of the behavior of m-convex functions under fractional integration.

Purpose The aim of this work is to prove a strong law of large numbers for a sequence of independent compactly uniformly integrable random sets with values in the family of convex closed subsets of a separable Banach space E, again without requiring any geometric conditions on E. Design/methodology/approach Our approach in this work is based on several theories; probability and its application to strong law of large numbers, properties of random sets, convex analysis and functional analysis. Findings This article establishes two strong laws of large numbers for independent, compactly uniformly integrable random sets. Originality/value This paper presents original results concerning the Strong Law of Large Numbers (SLLN) for random sets, specifically focusing on compactly uniformly integrable random sets in separable Banach spaces.

Purpose of the Study: The study aims to explore the structure of the dual space corresponding to a subspace of a reflexive Banach space equipped with a strictly convex norm. It further seeks to analyze properties of linear continuous operators, the separation of convex subsets, and the existence of invariant subspaces. Methodology: The research is based on theoretical and functional analysis techniques. It constructs the dual space under the given conditions and uses analytical methods to examine operator properties and convex set separability. Fixed-point and existence theorems are formulated using general mapping principles. Main Findings: A dual space corresponding to a subspace of a reflexive Banach space with a strictly convex norm is developed. Several properties of linear continuous operators and convex subset separation are identified. Fixed-point and existence theorems for general maps are established. Applications of this Study: The results can be applied in advanced functional analysis, particularly in operator theory, optimization problems, and mathematical modeling where the behavior of linear operators in Banach spaces is critical. These findings are also relevant to areas involving fixed-point theory and convex analysis. Novelty/Originality of this Study: This study offers a novel construction of the dual space in a specific setting of Banach spaces—reflexive with strictly convex norms—where such structures are not commonly analyzed in depth.

Reshaping anisotropic behavior in metallic sheets under complex stress states: Symmetric and asymmetric polynomial models with advanced convexity analysis approach

Abstract Averaged operators are important in Convex Analysis and Optimization Algorithms. In this article, we propose classifications of averaged operators, firmly nonexpansive operators, and proximal operators using the Bauschke–Bendit–Moursi modulus of averagedness. We show that if an operator is averaged with a constant less than $1/2$ , then it is a bi-Lipschitz homeomorphism. Amazingly the proximal operator of a convex function has its modulus of averagedness less than $1/2$ if and only if the function is Lipschitz smooth. Some results on the averagedness of operator compositions are obtained. Explicit formulae for calculating the modulus of averagedness of resolvents and proximal operators in terms of various values associated with the maximally monotone operator or subdifferential are also given. Examples are provided to illustrate our results.

In this paper, our main focus is on the Landweber–Kaczmarz iteration with the general uniform convex penalty function. Based on the convex analysis, we generalized that the penalty term can be non-smooth to include L 1 function space and total variation-like penalty function. By the use of subdifferential calculus, Bregman distance and Fréchet differentiable, we get monotony results. Furthermore, we obtain some results of the stability and convergence about the Landweber–Kaczmarz iterative algorithm under exact data and noisy data.

We introduce a novel portfolio optimization framework—Distributionally Robust Multivariate Stochastic Cone Order (DR-MSCO)—which integrates partial orders on random vectors with Wasserstein-metric ambiguity sets and adaptive cone structures to model multivariate investor preferences under distributional uncertainty. Grounded in measure theory and convex analysis, DR-MSCO employs data-driven cone selection calibrated to market regimes, along with coherent tail-risk operators that generalize Conditional Value-at-Risk to the multivariate setting. We derive a tractable second-order cone programming reformulation and demonstrate statistical consistency under empirical ambiguity sets. Empirically, we apply DR-MSCO to 23 Borsa Istanbul equities from 2021–2024, using a rolling estimation window and realistic transaction costs. Compared to classical mean–variance and standard distributionally robust benchmarks, DR-MSCO achieves higher overall and crisis-period Sharpe ratios (2.18 vs. 2.09 full sample; 0.95 vs. 0.69 during crises), reduces maximum drawdown by 10%, and yields endogenous diversification without exogenous constraints. Our results underscore the practical benefits of combining multivariate preference modeling with distributional robustness, offering institutional investors a tractable tool for resilient portfolio construction in volatile emerging markets.

This Special Issue of the journal Mathematics was dedicated to the study of properties of convex functions and convex sets [...]

This paper addresses the median line location problem from a facility location perspective, where the objective is to find the optimal location of a straight line in the plane with respect to weighted demand points, such that the sum of the weighted distances from the line to these points is minimised. Various algorithms have been proposed to determine the optimal median line to date. However, despite the importance of convexity of the problem and sensitivity analysis of the optimal line in real-world scenarios, these issues have not been thoroughly explored in the literature. Accordingly, this paper investigates the convexity of the median line location problem and examines how the optimal line maintains its optimality in light of changes in the weights of demand points. Additionally, it investigates the extent to which increases or decreases in the weights of the demand points can occur without affecting the optimisation of the service line. This sensitivity analysis aids decision-makers in taking necessary precautions and making predictions before the linear facility used is actually disrupted. For this purpose, mathematical modelling of the sensitivity analysis of the median line is performed. Then, an efficient algorithm along with some examples are proposed to handle the presented models.

Three-dimensional reliability analysis of convex turning corner slopes considering spatial variability of soil parameters

Breast cancer, the most prevalent malignant tumor among women worldwide, exhibits substantial heterogeneity, which manifests as different molecular subtypes with different therapeutic and prognostic implications. Owing to existing studies focusing on either kinetic or spatial heterogeneity in isolation, this study proposed a Dual Heterogeneity Fusion Network (DHF-Net) that integrated both the kinetic and spatial heterogeneities from DCE-MRIs for diagnosing breast cancer molecular subtypes. Initially, a convex analysis of mixtures algorithm was employed to identify dynamic heterogeneity subregions by analyzing contrast enhancement patterns over time. Meanwhile, K-Means clustering was utilized for spatial analysis to delineate spatial heterogeneity subregions that reflected structural diversity within tumors. Then, the dynamic and spatial heterogeneity features obtained from a ResNet-based feature extractor were integrated using a dual-attention module that incorporated both cross- and self-attentions. Final molecular subtype diagnosis was performed by a Mixture of Experts (MoE) framework. Experimental results demonstrated the effectiveness of the DHF-Net on a publicly available TCIA dataset in two molecular subtype classification tasks.Clinical Relevance-This study preliminary exploits both the kinetic and spatial heterogeneity to predict breast cancer molecular subtypes, contributing to personalized treatment for breast cancer patients with different molecular subtypes.

Convex analysis theory has found extensive applications in optimization, information science, and economics, leading to numerous generalizations of convex functions. However, a drawback in the vast literature on convex functions is that only a limited number of these notions significantly impact practical applications. With this context, we explore a novel convexity notion known as k-harmonically convex function (k-HCF) using two approaches and present applications in information science. First, we propose an r-parameterized extension of k-HCF, broadening its applicability. Secondly, we extend this concept to interval-valued functions (IVFs), based on a complete order relation on closed bounded intervals. We then investigate properties and inequalities for both extensions to derive lower bounds for information-theoretic measures such as Tsallis entropy, Shannon entropy, and Tsallis relative entropy, using the new parametric extensions of these functions. Additionally, we prove inequalities of the Jensen, Mercer, and Hermite-Hadamard types for the Cr-order-based extension of k-HCFs. Our findings reproduce known results while introducing significant new insights into the field, showing the broader usefulness of k-HCFs in information science.