We extend the previous work [Ann. of Math. (2) 201 , 225–289 (2025)] by building a smooth complete manifold (M^{6},g,p) with \operatorname{Ric}\geq 0 and whose fundamental group \pi_1(M^6)=\mathbb{Q}/\mathbb{Z} is infinitely generated. The example is built with a variety of interesting geometric properties. To begin, the universal cover \widetilde M^{6} is diffeomorphic to S^{3}\times \mathbb{R}^{3} , which turns out to be rather subtle as this diffeomorphism is increasingly twisting at infinity. The curvature of M^{6} is uniformly bounded and in fact decaying polynomially. The example is locally noncollapsed, in that \operatorname{Vol}(B_{1}(x))>v>0 for all x\in M . Finally, the space is built so that it is almost globally noncollapsed. Precisely, for every \eta>0 there exist radii r_{j}\to \infty such that \operatorname{Vol}(B_{r_j}(p))\geq r_j^{6-\eta} . The broad outline for the construction of the example will closely follow the scheme introduced in [Ann. of Math. (2) 201 , 225–289 (2025)]. The six-dimensional case requires a couple of new points, in particular the corresponding Ricci curvature control on the equivariant mapping class group is harder and cannot be done in the same manner.

Introduction: Fixed point theory is a fundamental area in mathematical analysis with wide-ranging applications. This research aims to extend fixed point results in the framework of neutrosophic metric spaces, which incorporate degrees of membership, non-membership, and indeterminacy. The study focuses on the development of compatible maps to generalize fixed point theorems under more realistic conditions. Methods: We introduce two novel classes of compatible maps—types (α) and (β)—defined via implicit relations in neutrosophic metric spaces. New fixed point theorems are established by using these map types and extending existing theories from fuzzy and intuitionistic fuzzy metric spaces. The work includes formal definitions, interrelations among mappings, and proofs of key results. Results: The newly established theorems offer broader conditions for ensuring the existence of fixed points in neutrosophic metric spaces. These generalizations effectively account for uncertainty, expanding the applicability of fixed point theory to more complex and indeterminate settings. Discussion: The inclusion of implicit relations and compatible maps enhances the theoretical flexibility of fixed point results. To demonstrate the applicability, the proposed theorems are used to solve the Fredholm integral equation under a neutrosophic framework, validating their relevance in handling real-world problems characterized by imprecise information. Conclusion: This study significantly advances fixed point theory by developing and analyzing compatible maps of types (α) and (β) in neutrosophic metric spaces. The findings provide a robust foundation for future mathematical research and practical applications involving uncertainty and incomplete data.

In this research article, we prove the existence and uniqueness of common best proximity points for a given class of discontinuous mappings. For that, we have introduced the notions of neutrosophic proximally compatible mappings, neutrosophic proximally reciprocal and weak reciprocal mappings, and R-proximally weak reciprocal commuting mappings of type I and type II. We have given examples to validate our findings.

The present paper is devoted to the introduction and development of the notions of multivalued graphic contractions and multivalued GF-contractions in the setting of F-metric spaces. By extending the idea of contractions to multivalued mappings associated with an underlying graph structure, we aim to enrich the existing theory of fixed point results in generalized metric frameworks. The main contribution of this study is the establishment of new fixed point theorems for these classes of mappings in F-metric spaces, which provide a natural extension of classical fixed point principles. Furthermore, in order to demonstrate the validity and applicability of our theoretical findings, we construct a non-trivial illustrative example that highlights how the proposed conditions can be effectively utilized. These results not only advance the fixed point theory in abstract metric settings but also open potential avenues for applications in mathematical analysis and applied sciences.

The aim of this paper is to obtain several properties of α -harmonic mapping with f ∗ ∈ L p ( T ) and p ∈ [ 1 , ∞ ] . After estimating the coefficients of α -harmonic mapping, we give a new Landau-type theorem for this class of mappings, which is a kind of generalization from the case of α = 0 to the case -1 $ ]]> α > − 1 given by Shi et al. After establishing a Schwarz-Pick type inequality, we also build another Landau-type theorem for the solutions to an inhomogeneous equation Δ α f = g , here g : D ¯ → C is a continuous function.

In this study, the main aim is to establish a set of novel inequalities that enhance the mathematical inequalities discussed. This paper introduces a new fuzzy fractional integral framework along with associated inequalities by defining a novel class of fuzzy-valued convex mappings, termed up and down (U ·D) fuzzy-valued generalized strong ƛ-convex mappings. Some new and classical exceptional cases are also obtained for generalized fuzzy fractional integral operators and U ·D-fuzzy-valued generalized strong A convex mapping. Some new forms of Hermite-Hadamard inequalities are also derived through fuzzy generalized fractional integrals and extends Pachpatte-type inequalities by applying products of U ·D-fuzzy-valued generalized strong ƛ-convex mappings. Additionally, several midpoint inequalities are introduced. Some open problems are also presented for future discussion.

In this paper, we introduce and investigate a new class of mappings called generalised graph \phi-contractions within the setting of Generalised Hausdorff Controlled Partial Metric (GHCPM) spaces. This framework integrates the structure of a graph with a controlled partial metric, providing a natural generalization of classical fixed point theories. Our study extends previous results by incorporating mappings defined on collections of non-empty closed and bounded subsets of a GHCPM space, and introducing contractive conditions governed by an upper semi-continuous and non-monotonic function. By leveraging the graph structure on GHCPM, we define a generalised graph contraction as a mapping that respects the connectivity induced by the graph while satisfying a contractive inequality involving the Hausdorff controlled partial metric. We establish novel fixed point theorems for such contractions, which unify and extend several existing results in the literature. To illustrate the applicability and generality of our results, we demonstrate the existence of solutions for nonlinear integral equations of Fredholm type. Concrete examples demonstrating the existence of fixed points under the proposed framework are also provided. These results open new directions in the study of fixed point theory in generalized metric spaces with additional structure.

We introduce and study the concept of factorable strongly p-nuclear Bloch maps, a novel class of mappings in the category of Bloch functions. We provide several characterizations of these maps, including a Pietsch-type domination theorem and connections to p-nuclear linear operators via their linearization and transposition. Key properties such as Möbius invariance and duality by applying the theory on tensor products are established. We also investigate the injective Banach ideal structure of these maps and their Bloch weak compactness properties. The results extend known theory on Bloch mappings, offering new insights into their interplay.

This paper introduces a novel accelerated shrinking projection algorithm for approximating Cesàro mean sequences and solving split equilibrium problems in real Hilbert spaces. The iterative scheme is constructed using finite families of commutative, normally m-generalized hybrid mappings, with a step size chosen independently of the spectral radius to facilitate computation. We prove that the generated sequence converges strongly to a common element in the intersection of the fixed point sets of the mappings, which also solves the associated split equilibrium problem. The proposed method yields new and extended strong convergence theorems for various classes of hybrid mappings, including normally generalized hybrid, m-generalized hybrid, and normally 2-generalized hybrid mappings. A numerical example is provided to demonstrate the superior convergence rate of our algorithm compared to existing methods. These results generalize and unify several known findings in this direction.

We extend the notion of asymptotically nonexpansive mapping to the more general class, namely, e-enriched asymptotically nonexpansive mappings. It is shown, with an example, that the class of e-enriched asymptoticallynonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. Certain weak and strong convergence theorems are then proved for the iterative approximation of split common fixed point problem involving the class of(e,ϑ)-enriched strictly quasi-pseudocontractive mappings and the class of e-enriched asymptotically nonexpansive mappings in the domain of two Banach spaces. Furthermore, a significant result for the hierarchical variational inequality problem is obtained as a consequence of our main result.

This paper explores the landscape of fixed point theory by introducing two novel classes of contraction mappings: the (αs, νs,(Q, h)-F)-contraction and the (αs, ηs, νs,(Q, h)-F)-contraction, defined within the rich structure of triple-controlled S-metric type spaces. These mappings are constructed using a blend of αs- and ηs-admissibility, νs-subadmissibility, and a pair of upper-class functions (Q, h), integrated with Wardowski’s powerful F-contraction approach. Our results significantly extend the classical (αs F)-contraction framework by proving the existence and uniqueness of fixed points under these generalized settings. Furthermore, we derive meaningful corollaries by specifying various (Q, h) pairs, illustrating the versatility and depth of the proposed theory and its contribution to the advancement of fixed point results in generalized metric environments.

This paper investigates optimal solutions for best proximity points through the framework of generalized interpolative proximal contractions. We introduce a new method that uses interpolation techniques to handle a wider class of mappings by expanding the concepts of classical proximal contraction. In the absence of a precise solution, best proximity point theorems investigate the existence of such best proximity points for approximate solutions to the fixed point problem. This article aims to develop the best proximity point theorems for contractive non-self mappings via interpolation to generate global optimal approximate solutions to particular fixed point equations. In addition to demonstrating the existence of the optimal proximity points, iterative techniques are also offered to locate such optimal approximative solutions. We illustrate the utility of our findings with a few instances. The value of our research is illustrated with a few examples and applications.

The complexity of terms of type [Formula: see text] of an algebra of type [Formula: see text] can be measured by several methods, including the depth of a term, the total number of variables or operation symbols that appear in a term and the length of a term. Generally, algebraic systems can be viewed as a generalization of algebras. To describe properties of algebraic systems of arbitrary types, quantifier free formulas of type [Formula: see text] induced by terms, relation symbols, and logical connectors are essential. In this paper, we aim to define various measurements of the complexity of quantifier free formulas and then formalize these measurements under different operations. Applications of these explicit formulas for some classes of mappings are provided.

Abstract Building on work of Harer, we construct a spine for the decorated Teichmüller space of a non‐orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non‐orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non‐orientable surface, which is of minimal dimension in the case there is a single puncture.

Homomorphisms between pure mapping class groups

Symplectic groups, mapping class groups and the stability of bounded cohomology

We extend the notion of (a, k)-enriched strictly pseudocontractive mappings to the notion of the more general aenriched pseudocontractive mappings. It is shown with examples that the class of a-enriched pseudocontractive mappings is more general than the classes of (a, k)-enriched strictly pseudocontractive and pseudocontractive mappings. Some fundamental properties of the class a-enriched pseudocontractive mappings are proved. In particular, it is shown that the fixed point set of certain class of a-enriched pseudocontractive self-mappings of a nonempty closed convex subset of a real Hilbert space is closed and convex. Demiclosedness property of such class of a-enriched pseudocontractive mappings is proved. Certain strong convergence theorems are then proved for the iterative approximation of fixed points of the class a-enriched pseudocontractive mappings.

Multitwists in big mapping class groups

We compute the wrapped Fukaya category \mathcal{W}(T^{*}S^{1}, D) of a cylinder relative to a divisor D= \{p_{0},\dots,p_{n}\} of n+1 points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over k\llbracket t_{0},\dots,t_{n}\rrbracket ) of the singularity uv=t_{0}t_{1}\cdots t_{n} . Upon making the base-change t_{i}= f_{i}(x,y) , we obtain the derived category of any crepant resolution of the cA_{n} singularity given by the equation uv= f_{0}\cdots f_{n} . These categories inherit braid group actions via the action on \mathcal{W}(T^{*}S^{1},D) of the mapping class group of T^{*}S^{1} fixing D . We also give geometric models for the derived contraction algebras associated to a cA_{n} singularity in terms of the relative Fukaya category of the disc.

Simulation of interacting Fermionic Hamiltonians is one of the most promising applications of quantum computers. However, the feasibility of analyzing Fermionic systems with a quantum computer hinges on the efficiency of Fermion-to-qubit mappings that encode nonlocal Fermionic degrees of freedom in local qubit degrees of freedom. While recent studies have highlighted the importance of designing Fermion-to-qubit mappings that are tailored to specific problem Hamiltonians, the methods proposed so far either are restricted to a narrow class of mappings or they use computationally expensive and unscalable brute-force search algorithms. Here, we address this challenge by designing a heuristic numerical optimization framework for Fermion-to-qubit mappings. To this end, we first translate the Fermion-to-qubit mapping problem to a Clifford circuit optimization problem and then use simulated annealing to optimize the average Pauli weight of the problem Hamiltonian. For all Fermionic Hamiltonians we have considered, the numerically optimized mappings outperform their conventional counterparts, including ternary-tree-based mappings that are known to be optimal for single creation and annihilation operators. We find that our optimized mappings yield between 15% and 40% improvements on the average Pauli weight when the simulation Hamiltonian has an intermediate level of complexity. Most remarkably, the optimized mappings improve the average Pauli weight for 6 × 6 nearest-neighbor hopping and Hubbard models by more than 40% and 20%, respectively. Surprisingly, we also find specific interaction Hamiltonians for which the optimized mapping outperforms any ternary-tree-based mapping. Our results establish heuristic numerical optimization as an effective method for obtaining mappings tailored for specific Fermionic Hamiltonian.