Inertial iterative methods for solving split feasibility problems in real Hilbert spaces with applications to signal and image processing

In this paper, the concept of exponentially $m$-isometry \cite{Hedayatian} on a Hilbert space is generalized when an additional semi-inner product is considered. We present a comprehensive study of the algebraic properties and characterizations of operators within this extended class. Furthermore, we explore the dynamical behavior of these operators and conclude with an analysis of their spectral properties.

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order γ∈(0,1) and fully time-varying state and control delays. An explicit mild solution representation is derived using the fractional fundamental matrix, and a new controllability Gramian is introduced. Using analytic properties of the matrix-valued Mittag-Leffler function, we prove a fractional Kalman-type theorem showing that bounded time-varying delays do not change the algebraic controllability structure determined by (F,G,K). The minimum-energy control problem is solved in closed form through Hilbert space methods. Efficient numerical strategies and several examples—including delayed viscoelastic, neural, and robotic models—demonstrate practical applicability and computational feasibility.

Given Hilbert space operators A,B, and X, let ▵A,B and δA,B denote, respectively, the elementary operators ▵A,B(X)=I−AXB and the generalised derivation δA,B(X)=AX−XB. This paper considers the structure of operators Dd1,d2m(I)=0 and Dd1,d2m compact, where m is a positive integer, D=▵ or δ, d1=▵A*,B* or δA*,B* and d2=▵A,B or δA,B. This is a continuation of the work performed by C. Gu for the case where ▵δA*,B*,δA,Bm(I)=0, and the author with I.H. Kim for the cases where ▵δA*,B*,δA,Bm(I)=0 or ▵δA*,B*,δA,Bm is compact, and δ▵A*,B*,▵A,Bm(I)=0 or δ▵A*,B*,δA,Bm is compact. Operators Dd1,d2m(I)=0 are examples of operators with a finite spectrum; indeed, the operators A,B have at most a two-point spectrum, and if Dd1,d2m is compact, then (the non-nilpotent operators) A,B are algebraic. Dd1,d2m(I)=0 implies Dd1,d2n(I)=0 for integers n≥m: the reverse implication, however, fails. It is proved that Dd1,d2m(I)=0 implies Dd1,d2(I)=0 if and only if of A and B (are normal and hence) satisfy a Putnam–Fuglede commutativity property.

This study explores the application of symmetry analysis in solving fractional differential equations, a method often perceived as more labor-intensive compared to other techniques. Central to our research is the investigation of controllability issues within a specific class of nonlocal fractional stochastic evolution equations (FSEEs) set in a Hilbert space. Uniquely challenging in this context is the presence of a noncompact linear component that is instrumental in generating semigroups. Our approach integrates the Mönch fixed point theorem, coupled with advanced stochastic analytic techniques, and the measure of noncompactness, to derive significant insights. The synergy of these methodologies has led to the discovery of pivotal results that contribute to our understanding of FSEEs. To elucidate and substantiate our findings, we present a practical example that not only clarifies the theoretical aspects but also serves as a validation of our results. This example acts as a testament to the efficacy of our approach and provides a concrete application of the theoretical framework developed within this study. This research presents a novel approach to tackling complex differential equations and expands the scope of symmetrical analysis in mathematical studies.

This manuscript generalizes the concepts of hyperbolic distortion and boundary conformality from the unit disk to multidimensional complex domains such as polydiscs and bounded symmetric domains with intrinsic hyperbolic metrics. We extend strong and weak conformality to higher dimensions, characterize equality cases in multidimensional Schwarz–Pick-type inequalities, and develop invariant distortion metrics. By utilizing multidimensional reproducing kernel Hilbert spaces, we provide operator-theoretic characterizations and investigate applications in holomorphic dynamical systems, including the study of backward orbits and pre-model regularity. These results open pathways to further generalizations in quasiconformal mappings and their multidimensional rigidity properties.

Genomic prediction (GP) has catalyzed increased rates of genetic gain in animal and plant breeding. Recently, deep learning (DL) has been explored to increase GP accuracy by incorporating diverse data types and learning complex, non-linear patterns in datasets. However, DL consistently fails to significantly improvement prediction accuracy over gold standard genomic BLUP (gBLUP) models. In this study, we first review the theory behind neural networks and reproducing kernel Hilbert spaces (RKHS) regression to contextualize three claimed benefits of DL over linear models: incorporation of diverse data types, avoidance of feature engineering, and universal approximation behavior. We also propose a taxonomy of prediction problems so that model comparisons do not confound differences in the predictive skill of different model classes with differences in the input data. Second, we leverage a maize multi-environment trial dataset to train DL and RKHS models that implicitly capture non-linear patterns between genomic, soil, weather, and management inputs and grain yield. The results demonstrate that feature engineering using principal components of SNPs generally degrades prediction accuracy across model classes. Furthermore, DL models persistently fail to outperform RKHS models across prediction problems. Finally, we evaluate the theoretical critiques with the empirical results, confirming the theoretical arguments. Nevertheless, a small portion of the possible DL model space has been explored, leaving open the possibility of DL making significant contributions to GP problems through additional aspects not considered here. We conclude by suggesting several avenues for further theoretical and practical research, including the resolution of several disciplinary differences.

While quantum statistical mechanics triumphs in explaining many equilibrium phenomena, there is an increasing focus on going beyond conventional scenarios of thermalization. Traditionally examples of nonthermalizing systems are either integrable or disordered. Recently, examples of translationally invariant physical systems have been discovered whose excited energies avoid thermalization either due to local constraints (whether exact or emergent) or due to higher-form symmetries. In this article, we extend these investigations for the case of 3D U ( 1 ) quantum dimer models, which are lattice gauge theories with finite-dimensional local Hilbert spaces (also generically called quantum link models) with staggered charged static matter. Using a combination of analytical and numerical methods, we uncover a class of athermal states that arise in large winding sectors, when the system is subjected to external electric fields. The polarization of the dynamical fluxes in the direction of applied field traps excitations in 2D planes, while an interplay with the Gauss law constraint in the perpendicular direction causes exotic athermal behavior due to the emergence of new conserved quantities. This causes a geometric fragmentation of the system. We provide analytical arguments showing that the scaling of the number of fragments is exponential in the linear system size, leading to weak fragmentation. Further, we identify sectors which host fractonic excitations with severe mobility restrictions. The unitary evolution of fragments dominated by fractons is qualitatively different from the one dominated by nonfractonic excitations.

Error-correcting codes and absolutely maximally entangled states for mixed-dimensional Hilbert spaces

Abstract Scalable quantum information processing with limited physical resources is a key challenge in the pursuit of practical quantum advantage. Here, we introduce an approach that encodes two qubits per ion, enabling an $n$-ion--$2n$-qubit quantum processor by harnessing four internal levels of trapped ions. As a proof of principle, we demonstrate a universal 1-ion--2-qubit processor using the valence electron spin and nuclear spin of a single $^{171}$Yb$^+$ ion, achieving gate fidelities exceeding 98\% for both single- and two-qubit operations via quantum process tomography. Furthermore, we implement Grover's algorithm with a success rate surpassing 99\%, showing the system's computational capability. Through robust optimal quantum control, we enhance gate robustness against amplitude and frequency fluctuations, critical for large-scale operation. We present scalable architectures leveraging both laser-free and laser-based entangling gates, revealing that intra-atomic electron-nuclear spin interactions can reduce the complexity of inter-atomic operations. By substituting inter-atomic gates with high-fidelity intra-atomic ones, our scheme significantly improves circuit performance. This work establishes a pathway to exponentially expand the Hilbert space of quantum processors, and represents an important advance toward scalable, high-capacity quantum computing.

A bstract Big bang/big crunch closed universes can be realized in AdS/CFT, even though they lack asymptotically AdS boundaries. With enough bulk entanglement, the bulk Hilbert space of a closed universe can be holographically encoded in the CFT. We clarify the relation of this encoding to observer-clone proposals and refute recent arguments about the breakdown of semiclassical physics in such spaces. In the limit of no bulk entanglement, the holographic encoding breaks down. The oft-cited one-dimensional nature of the closed universe Hilbert space represents the limitation of the external (CFT) Hilbert space to access the quantum information in the closed universe, similar to the limitations imposed on observers outside a perfectly isolated quantum lab. We advocate that the CFT nevertheless continues to determine the physical properties of the closed universe in this regime, showing how to interpret this relationship in terms of a final state projection in the closed universe. We provide a dictionary between the final state wavefunction and CFT data. We propose a model of the emergence of an arrow of time in the universe with a given initial or final state projection. Finally, we show that the conventional EFT in the closed universe, without any projection, can be recovered as a maximally ignorant description of the final state. This conventional EFT is encoded in CFT data, and it can be probed by computing coarse-grained observables. We provide an example of one such observable. Taken together, these results amount to a clean bill of health for baby universes born of AdS/CFT. A video abstract is available at https://youtu.be/s_9VqF-N8uQ .

We introduce Causal Incentive Design (CID), a framework that applies causal inference to canonical single-stage principal–agent problems (PAPs) characterized by bilateral private information. Within CID, the operating rules of PAPs are formalized using an additive-noise causal graphical model (CGM). Incentives are modeled as interventions on a function space variable, Γ, which correspond to policy interventions in the principal–follower causal relation. The causal inference target estimand V(Γ) is defined as the expected value of the principal’s utility variable under a specified policy intervention in the post-intervention distribution. In the context of additive-Gaussian independent noise, the estimand V(Γ) decomposes into a two-layer expectation: (i) an inner Gaussian smoothing of the principal’s utility regression; and (ii) an outer averaging over the conditional probability of the follower’s action given the incentive policy. A Gauss–Hermite quadrature method is employed to efficiently estimate the first layer, while a policy-local kernel reweighting approach is used for the second. For offline selection of a single incentive policy, a Functional Causal Bayesian Optimization (FCBO) algorithm is introduced. This algorithm models the objective functional γ↦V(γ) using a functional Gaussian process surrogate defined on a Reproducing Kernel Hilbert Space (RKHS) domain and utilizes an Upper Confidence Bound (UCB) acquisition functional. Consequently, the policy value V(γ) becomes an interventional query that can be answered using offline observational data under standard identifiability assumptions. High-probability cumulative-regret bounds are established in terms of differential information gain for the proposed FBO algorithm. Collectively, these elements constitute the central contributions of the CID framework, which integrates causal inference through identification and estimation with policy search in principal–agent problems under private information. This approach establishes a causal decision-making pipeline that enables commitment to a high-performing incentive in a single-shot game, supported by regret guarantees. Provided that the data used for estimation is sufficient, the resulting offline pipeline is appropriate for scenarios where adaptive deployment is impractical or costly. Beyond the methodological contribution, this work introduces a novel application of causal graphical models and causal reasoning to incentive design and principal–agent problems, which are central to economics and multi-agent systems.

We develop a unified and exact quantum geometric framework to understand and model molecular reactive quantum dynamics. The critical roles of quantum geometry of adiabatic electronic states in both adiabatic and nonadiabatic quantum dynamics are unveiled. A numerically exact geometric quantum molecular dynamics method is developed via discrete local trivialization of the projected electronic Hilbert space bundle over nuclear configuration space, eliminating all singularities from nonanalytic adiabatic electronic states. In it, the singular electronic quantum geometric tensor-Abelian for adiabatic dynamics and non-Abelian for nonadiabatic dynamics-is fully encoded in the global electronic overlap matrix. Numerical illustrations demonstrate that atomic motion, whether adiabatic or nonadiabatic, is governed not only by variations in electronic energies (potential energy surfaces) but also by variations in electronic states (electronic quantum geometry). Beyond quantum molecular dynamics, the strategy of discrete local trivialization can be extended to describe quantum dynamics, possibly non-Hermitian, on arbitrary, especially nondifferential fiber bundles.

For the chain $H_+\subset H_0\subset H_-$ of rigged Hilbert spaces we solve the problem of finding sharp Taikov type inequalities that estimate the value of generalized element $\alpha\in H_-$ on smooth elements $u\in H_+$ in terms of the norm of the element $u$ in $H_+$. Also, we solve the problem of approximating generalized elements $\alpha\in H_-$ by regular elements $x\in H_0$ on the class $W_+ = \{u\in H_+\,:\,\|u\|_{H_+}\le 1\}$. We consider some applications of these results, in particular, to finding sharp Taikov-type inequalities for partial derivatives of multi-variate periodic functions.

Weak and strong convergence theorems for the mixed split feasibility problem in Hilbert spaces

BackgroundBread wheat (Triticum aestivum L.) is a vital global crop, supplying 20% of the protein in the human diet. Improving its productivity and resilience, particularly under water-limited conditions, is a major breeding priority. Genomic selection offers a promising approach to accelerate genetic gains by predicting complex traits using genome-wide marker data. This study evaluated the performance of various genomic selection (GS) models in predicting key agronomic traits under contrasting well-watered (WW) and water-stressed (WS) conditions, with the goal of enhancing drought adaptation in wheat breeding programs.ResultsA panel of 179 wheat lines was evaluated for grain yield, yield components, and grain protein content. Models were trained on data from well-watered and water-stressed regimes independently, as well as on combined data from both conditions. Predictive approaches included linear models (Ridge Regression Best Linear Unbiased Prediction and Bayesian methods), semi-parametric models (Reproducing Kernel Hilbert Space Regression), and machine learning algorithms (Random Forest, Support Vector Machine, and Extreme Gradient Boosting). Ridge regression consistently delivered strong performance across all traits and conditions, with mean rMG of 0.70 (water-stressed), 0.64 (well-watered), and 0.65 (combined). Machine learning models, especially Random Forest and Extreme Gradient Boosting, performed competitively for complex traits such as grain yield and thousand kernel weight. Random Forest achieved a rMG of 0.81 for grain yield and 0.73 for thousand kernel weight under well-watered conditions. Trait stability was observed under well-watered conditions for thousand kernel weight and plant height, supported by moderate heritability estimates (0.69–0.74). Cross-validation comparisons showed consistent model performance across validation schemes, with leave-one-out cross-validation offering slightly improved accuracy for select traits and models. Notably, models trained under water-stressed conditions generalized better when tested on well-watered data than the reverse, highlighting the value of diverse training environments.ConclusionsGenomic selection models, particularly ridge regression and machine learning approaches, demonstrated reliable predictive performance across environments and traits. Incorporating data from multiple environmental conditions improves prediction accuracy and supports the development of drought-resilient wheat lines. These results reinforce the utility of genomic selection in modern wheat breeding programs for enhancing both productivity and stress tolerance.Supplementary InformationThe online version contains supplementary material available at 10.1186/s13007-025-01467-5.

The weighted MAX k -CUT problem involves partitioning a weighted undirected graph into k subsets, or colors, to maximize the sum of the weights of edges between vertices in different subsets. This problem has significant applications across multiple domains. This study explores encoding methods for MAX k -CUT on qubit systems by utilizing quantum approximate optimization algorithms (QAOA) and addressing the challenge of encoding integer values on quantum devices with binary variables. We examine various encoding schemes and evaluate the efficiency of these approaches. The study presents a systematic and resource-efficient method to implement the phase separation operator for the cost function of the MAX k -CUT problem. When encoding the problem into the full Hilbert space, we show the importance of encoding the colors in a balanced way. We also explore the option of encoding the problem into a suitable subspace by designing suitable state preparations and constrained mixers (LX- and Grover-mixer). Numerical simulations on weighted and unweighted graph instances demonstrate the effectiveness of these encoding schemes, particularly in optimizing circuit depth, approximation ratios, and computational efficiency.

In this paper, we propose an efficient inertial iterative algorithm for solving a system of generalized quasi-variational inequalities (SGQVI) in Hilbert spaces. Using the projection operator technique, we establish an equivalence between SGQVI and fixed-point problems, thus developing a novel inertial method. The algorithm introduces an inertial term to accelerate convergence, and its performance is rigorously analyzed under some mild conditions, including relaxed co-coercivity and Lipschitz continuity of the involved mappings. Our framework unifies and extends several existing models, such as classical variational inequalities, quasi-variational inequalities, and related optimization problems. Some experiments demonstrate the effectiveness of the inertial method, which shows an improvement in convergence speed compared to noninertial methods. Our results generalize and enhance previous research results in the literature, making it more widely applicable in computational mathematics, engineering, and economics.

In this paper, we design some generalized split problems which can be seen as an extended form of the split variational inequality problems. We present several iterative algorithms for solving generalized split problems and demonstrate the weak convergence results under some appropriate assumptions within the context of real Hilbert spaces. Finally, we support these results with the help of numerical examples in both the finite and infinite dimensional spaces. As a result of this work, a new direction will be opened in studying split problems.

A Two-Step Inertial Method with a New Step-Size Rule for Quasimonotone Variational Inequalities in Hilbert Spaces