Search Results (105)

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map $\overline{R}:{\mathcal{C}}_R\to \mathcal{E}$ is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C^*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on ${\mathbb{R}}^n$, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency. Full article

Discovering the compactness properties in generalized-type metric spaces opens up a fascinating area of research. The present study tries to develop a theoretical framework for compactness with key properties in the recently developed interval metric space. This work begins with explaining the covers and open covers to define compact interval metric spaces and their main features. Next, a similar definition of compactness using the finite intersection property is introduced. Then, the famous Heine–Borel theorem for compactness is extended in the case of interval metric spaces. Also, the concepts of sequential-type compactness and Bolzano–Weierstrass (BW)-type compactness for interval metric spaces are introduced with their equivalency relationship. Finally, the notion of total boundedness in interval metric spaces and its connection with compactness is introduced, providing new insights into these mathematical concepts. Full article

Biophysics-based, patient-specific modeling remains challenging for clinical translation, particularly for cerebrospinal fluid (CSF) flow where anatomical detail and computational cost are tightly coupled. We present a computational framework for steady net CSF redistribution in an MRI-derived cranial CSF domain reconstructed from T₂-weighted imaging, including the ventricular system, cranial subarachnoid space, and periarterial pathways, to the extent resolvable by clinical MRI. Cranial CSF spaces were segmented in 3D Slicer and a steady Darcy formulation with prescribed CSF production/absorption was solved in COMSOL Multiphysics^®. Geometrical and flow descriptors were quantified using region-based projection operations. We assessed discretization cost–accuracy trade-offs by comparing first- and second-order finite elements. First-order elements produced a 1.4% difference in transmantle pressure and a <10% difference in element-wise mass-weighted velocity metric for 90% of elements, while reducing computation time by 75% (20 to 5 min) and peak memory usage five-fold (150 to 30 GB). This proof-of-concept framework provides a computationally tractable baseline for studying steady net CSF pathway redistribution and sensitivity to boundary assumptions, and may support future patient-specific investigations in pathological conditions such as subarachnoid hemorrhage, hydrocephalus and brain tumors. Full article

This paper relies on orthogonal metric spaces related to cyclic self-mappings and some of their relevant properties. The involved binary relation is not symmetric, and then the term pseudo-orthogonality will be used for the relation used in the article to address the established results on cyclic self-mappings. Firstly, some orthogonal binary relations are given through examples to fix some ideas to be followed in the main body of the article. It is seen that the orthogonal elements of the orthogonal sets are not necessarily singletons. Secondly, “ad hoc” specific orthogonality binary relations are also described through examples related to the investigation of stability and controllability problems in dynamic systems. The main objective of this paper is to investigate the properties of cyclic single-valued self-mappings on the union of any finite number $p\ge 2$ of nonempty closed subsets of a metric space in a cyclic disposal under an “ad hoc” defined pseudo-orthogonality condition. Such a condition is defined on certain subsequences, referred to as pseudo-orthogonal sequences, rather than on the whole generated sequences under the self-mapping. It basically consists of a cyclic, in general iteration-dependent, contractive condition just for such subsequences which, on the other hand, are not forced as a constraint to be fulfilled by the whole sequences. Furthermore, the whole sequences in which those sequences are contained are allowed to be locally non-contractive or even locally expansive. The boundedness and the convergence properties of distances between pseudo-orthogonal subsequences and sequences are investigated under the condition that one of the subsets has a unique best proximity point to its adjacent subset in the cyclic disposal to which the pseudo-orthogonal subsequences converge. The pseudo-orthogonal metric subspace of the given metric space is proved to be complete although the whole metric space is not assumed to be complete. The pseudo-orthogonal element is seen to be a set of best proximity points, one per subset of the cyclic disposal, although it is not required for all the best proximity sets to be singletons. It is proved that the pseudo-orthogonal subsequences converge to a limit cycle, consisting of a best proximity point per subset of the cyclic disposal, which is also the pseudo-orthogonal element. The whole sequences are also proved to be bounded and the distances between their elements in adjacent subsets are also proved to converge to the distance between adjacent subsets. In the event that the metric space is a uniformly convex Banach space, it suffices that one of the subsets of the cyclic disposal be boundedly compact with its best proximity set being a singleton. In this case, the pseudo-orthogonal sequences converge to their best proximity set to their adjacent subset provided that such a best proximity set is a singleton. Full article

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning $\sigma$-algebra, become inadequate when the available information is restricted to a $\sigma$-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning $\sigma$-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces. Full article

The field of probabilistic metric spaces has an intrinsic interest based on a blend of ideas drawn from metric space theory and probability theory. The goal of the present paper is to introduce and study new ideas in this field. In general terms, we investigate the following concepts: linearly ordered families of distances and associated continuity properties, geometric properties of distances, finite range weak probabilistic metric spaces, generalized Menger spaces, and a categorical framework for weak probabilistic metric spaces. Hopefully, the results will contribute to the foundations of the subject. Full article

We investigate the nonrelativistic quantum dynamics of a spinless particle in a screw-type spacetime endowed with two independent twist controls that interpolate between a pure screw dislocation and a homogeneous twist. From the induced spatial metric, we build the covariant Schrödinger operator, separate variables to obtain a single radial eigenproblem, and include a uniform axial magnetic field and an Aharonov–Bohm (AB) flux by minimal coupling. Analytically, we identify a clean separation between a global, AB-like reindexing set by the screw parameter and a local, curvature-driven mixing generated by the distributed twist. We derive the continuity equation and closed expressions for the azimuthal and axial probability currents, establish practical parameter scalings, and recover limiting benchmarks (AB, Landau, and flat space). Numerically, a finite-difference Sturm–Liouville solver (with core excision near the axis and Langer transform) resolves spectra, wave functions, and currents. The results reveal AB periodicity and reindexing with the screw parameter, Landau fan trends, twist-induced level tilts and avoided crossings, and a geometry-induced near-axis backflow of the axial current with negligible weight in cross-section integrals. The framework maps the geometry and fields directly onto measurable spectral shifts, interferometric phases, and persistent-current signals. Full article

This paper analyzes a finite-capacity $GI/M/2/N$ queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the stationary distributions of the system size as observed at pre-arrival instants and at arbitrary epochs. From these, we obtain explicit expressions for key performance metrics, including blocking probability, average reneging rate, mean queue length, mean sojourn time, throughput, and server utilizations. We then embed these metrics in an economic cost function and determine service-rate settings that minimize the total expected cost via the Bat Algorithm. Numerical experiments implemented in R validate the analysis and quantify the managerial impact of the vacation, feedback, and impatience parameters through sensitivity studies. The framework accommodates general renewal arrivals ($GI$), thereby extending classical ($M/M/2/N$) results to more realistic input processes while preserving computational tractability. Beyond methodological interest, the results yield actionable design guidance: (i) they separate Palm and time-stationary viewpoints cleanly under non-Poisson input, (ii) they retain heterogeneity throughout all formulas, and (iii) they provide a cost–optimization pipeline that can be deployed with routine numerical effort. Methodologically, we (i) characterize the generator of the augmented piecewise–deterministic Markov process and prove the existence/uniqueness of the stationary law on the finite state space, (ii) derive an explicit Palm–time conversion formula valid for non-Poisson input, (iii) show that the boundary-value recursion for the Laplace–Stieltjes transforms runs in linear time $\mathcal{O}\left(N\right)$ and is numerically stable, and (iv) provide influence-function (IPA) sensitivities of performance metrics with respect to $({\mu }_1,{\mu }_2,\nu ,\alpha ,\varphi ,\beta )$. Full article

In this paper, heat dissipation through a passive vertical plate fin heat sink via natural convection was numerically investigated. The influence of two nondimensional geometric parameters, fin spacing-to-thickness ratio and fin height-to-spacing ratio, on the heat sink’s thermal performance was evaluated. A mathematical model describing the three-dimensional steady-state problem of buoyancy-driven flow and heat transfer was formulated. The solution was obtained numerically using the finite volume method in Ansys Fluent. The model and numerical procedure were validated by comparing the numerical predictions with measurements acquired on a constructed experimental apparatus. The heat sink thermal performance was assessed based on a series of performance metrics: heat dissipation rate, heat transfer coefficient, overall thermal resistance, and fin efficiency. Fin spacing-to-thickness ratio was varied between 1.86 and 4.8. Heat dissipation rate displayed a clear peak at a value of approximately 2.6, which coincided with a minimum in the overall thermal resistance. The heat transfer coefficient initially grew steadily, but at higher values of fin spacing-to-thickness ratio, it began to stagnate. Fin efficiency consistently decreased across the investigated range. Fin height-to-spacing ratio was varied between 1.11 and 7.78. The heat dissipation rate increased almost linearly across this range, but when the mass-specific heat dissipation rate was considered, it yielded diminishing returns. The heat transfer coefficient likewise exhibited a plateauing trend, while fin efficiency decreased steadily across the investigated range of fin height-to-spacing ratio. The obtained numerical results provide guidelines for geometry selection and can serve as a foundation for further analyses and optimizations of passive heat sinks’ thermal performance. Full article

In this research, a few metrical fixed-point outcomes consisting of an almost nonlinear Pant-type contraction employing a locally finitely transitive relation have been established. The findings of our research extrapolate, unify, develop, and improve a number of previously mentioned results. In the present investigation, we formulate a fixed-point finding for almost nonlinear Pant-type contractions in abstract metric space. To assist our study, we formulate numerous examples to illustrate our outcomes. Using our findings, we describe the existence and uniqueness of solutions to a nonlinear Fredholm integral equation. Full article

This paper focuses on dealing with several types of enriched cyclic contractions defined in the union of a set of non-empty closed subsets of normed or metric spaces. In general, any finite number $p\ge 2$ of subsets is permitted in the cyclic arrangement. The types of examined single-valued enriched cyclic contractions are, in general, less stringent from the point of view of constraints on the self-mappings compared to $p$-cyclic contractions while the essential properties of these last ones are kept. The convergence of distances is investigated as well as that of sequences generated by the considered enriched cyclic mappings. It is proved that, both in normed spaces and in simple metric spaces, the distances of sequences of points in adjacent subsets converge to the distance between such subsets under weak extra conditions compared to the cyclic contractive case, which is simply that the contractive constant be less than one. It is also proved that if the metric space is a uniformly convex Banach space and one of the involved subsets is convex then all the sequences between adjacent subsets converge to a unique set of best proximity points, one of them per subset which conform a limit cycle, although the sets of best proximity points are not all necessarily singletons in all the subsets. Full article

In this paper, we investigate the geometric realization of $\mathrm{Spin}\left(8\right)$ triality through vector fields on the octonionic algebra $\mathbb{O}$. The triality automorphism group of $\mathrm{Spin}\left(8\right)$, isomorphic to ${\mathfrak{S}}_3$, cyclically permutes the three inequivalent 8-dimensional representations: the vector representation V and the spinor representations $S_{+}$ and $S_{-}$. While triality automorphisms are well known through representation theory, their concrete geometric realization as flows on octonionic space has remained unexplored. We construct explicit smooth vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ on $\mathbb{O}\cong {\mathbb{R}}^8$ whose flows generate infinitesimal triality transformations. These vector fields have a linear structure arising from skew-symmetric matrices that implement simultaneous rotations in three orthogonal coordinate planes, providing the first elementary geometric description of triality symmetry. The main results establish that these vector fields preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$, demonstrating fundamental compatibility between geometric flows and octonionic algebra. We prove that the standard Euclidean metric on $\mathbb{O}$ is triality-invariant and classify all triality-invariant Riemannian metrics as conformal to the Euclidean metric with a conformal factor depending only on the isotonic norm. This classification employs Schur’s lemma applied to the irreducible $\mathrm{Spin}\left(8\right)$ action, revealing the rigidity imposed by triality symmetry. We provide a complete classification of triality-symmetric minimal surfaces, showing they are locally isometric to totally geodesic planes, surfaces of revolution about triality-fixed axes, or surfaces generated by triality orbits of geodesic curves. This trichotomy reflects the threefold triality symmetry and establishes correspondence between representation-theoretic decomposition and geometric surface types. For complete surfaces with finite total curvature, we establish global classification and develop explicit Weierstrass-type representations adapted to triality symmetry. Full article

Maximum Rank-Distance (MRD) codes are a class of optimal error-correcting codes that achieve the Singleton-like bound for rank metric, making them invaluable in applications such as network coding, cryptography, and distributed storage. While algebraic constructions of MRD codes (e.g., Gabidulin codes) are well-studied for specific parameters, a comprehensive theory for their existence and structure over arbitrary finite fields remains an open challenge. Recent advances have expanded MRD research to include twisted, scattered, convolutional, and machine-learning-aided approaches, yet many parameter regimes remain unexplored. This paper introduces a computational optimization framework for constructing MRD codes using Particle Swarm Optimization (PSO), a bio-inspired metaheuristic algorithm adept at navigating high-dimensional, non-linear, and discrete search spaces. Unlike traditional algebraic methods, our approach does not rely on prescribed algebraic structures; instead, it systematically explores the space of possible generator matrices to identify MRD configurations, particularly in cases where theoretical constructions are unknown. Key contributions include: (1) a tailored finite-field PSO formulation that encodes rank-metric constraints into the optimization process, with explicit parameter control to address convergence speed and global optimality; (2) a theoretical analysis of the adaptability of PSO to MRD construction in complex search landscapes, supported by experiments demonstrating its ability to find codes beyond classical families; and (3) an open-source Python toolkit for MRD code discovery, enabling full reproducibility and extension to other rank-metric scenarios. The proposed method complements established theory while opening new avenues for hybrid metaheuristic–algebraic and machine learning–aided MRD code construction. Full article

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. For this class, we provide a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. Intermediate results include the generalization of the decomposition theory introduced by Bandelt and Dress in 1992 as well as results on the tight span of totally split-decomposable metric spaces proved by Huber, Koolen, and Moulton in 2006. Next, using results of Lang from 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric and for which the associated splits satisfy a simple combinatorial property. In the case of Gromov hyperbolic groups, the obtained action is both proper aand co-compact. Finally, we obtain as an application that injective hulls of odd cycles are cell complexes isomorphic to CAT(0) cube complexes. Full article

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article