Fundamental Form Research Articles

Characterising the Hierarchy of Multi-time Quantum Processes with Classical Memory

Memory is the fundamental form of temporal complexity: when present but uncontrollable, it manifests as non-Markovian noise; conversely, if controllable, memory can be a powerful resource for information processing. Memory effects arise from/are transmitted via interactions between a system and its environment; as such, they can be either classical or quantum. From a practical standpoint, quantum processes with classical memory promise near-term applicability: they are more powerful than their memoryless counterpart, yet at the same time can be controlled over significant timeframes without being spoiled by decoherence. However, despite practical and foundational value, apart from simple two-time scenarios, the distinction between quantum and classical memory remains unexplored. Here, we analyse multi-time quantum processes with memory mechanisms that transmit only classical information forward in time. Complementing this analysis, we also study two related – but simpler to characterise – sets of processes that could also be considered to have classical memory from a structural perspective, and demonstrate that these lead to remarkably distinct phenomena in the multi-time setting. Subsequently, we systematically stratify the full hierarchy of memory effects in quantum mechanics, many levels of which collapse in the two-time setting, making our results genuinely multi-time phenomena.

FIRST EIGENVALUE CHARACTERISATION OF CLIFFORD HYPERSURFACES AND VERONESE SURFACES

Abstract We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$ , where $\sigma $ is the square norm of the second fundamental form.

Harmonic maps on the tangent bundle according to the ciconia metric

The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection 
 π : TM → M, where M denotes a Riemannian manifold and TM its associated tangent bundle. Additionally, we delve into the harmonicity of vector fields ξ ∈ χ (M) , treated as mappings from M to TM. Moreover, our exploration extends to situations such as the case involving the map π : (TM, ˜g) → (M2n, J, g), where (M2n, J, g) represents an anti-paraKähler manifold and (TM, ˜g) its tangent bundle with the ciconia metric. In this context, we delve into the harmonicity relations between the ciconia metric ˜g and the Sasaki metric Sg, examining their mutual interactions. Furthermore, we delve into the Schoutan-Van Kampen connection and the Vranceanu connection, both associated with the Levi-Civita connection of the ciconia metric. We also undertake the computation of the mean connections for the Schoutan-Van Kampen and Vranceanu connections, thereby shedding light on their properties. Finally, our investigation extends to the second fundamental form of the identity mapping from 
 (TM, ˜g) to (TM,∇m) and (TM, ∇∗m). Here ∇m and ∇∗m represent the mean connections associated with the Schoutan-Van Kampen and Vranceanu connections, respectively.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

On a Class of Multivalent Functions with Negative Coefficients Involving (r, q)− Calculus

In this research, we focused on presenting a novel subclass of multivalent analytic functions situated in the open unit disk, characterized by the use of Jackson’s derivative operator. Our investigation systematically establishes the requisite inclusion conditions in this class, offering detailed coefficient characterizations. The exploration encompasses an array of significant properties intrinsic to this subclass, encompassing coefficient estimates, growth and distortion theorems, identification of extreme points, and the determination of the radius of starlikeness and convexity for functions falling within this specialized category. Expanding the preliminary findings, this research extended the inquiry to delve deeper into the intriguing features and implications associated with this new subclass of multivalent analytic functions. The research concentrated the light on the nuanced intricacies of coefficient estimates, providing a comprehensive understanding of how these functions evolve within the open unit disk through exploring the growth and distortion theorems, unraveling the underlying mathematical principles governing the behavior of functions in this subclass as they extend beyond the unit disk. The findings of this research contribute to the broader understanding of multivalent analytic functions, paving the way for further exploration and applications in diverse mathematical contexts.

A note on umbilic points at infinity

AbstractIn this note a definition of umbilic point at infinity is proposed, at least for surfaces that are homogeneous polynomial graphs over a plane in Euclidean 3-space. This is a stronger definition than that of Toponogov in his study of complete convex surfaces, and allows one to distinguish between different umbilic points at infinity. It is proven that all such umbilic points at infinity are isolated, that they occur in pairs and are the zeroes of the projective extension of the third fundamental form, as developed in Guilfoyle and Ortiz-Rodríguez (Math Proc R Ir Acad 123A(2), 63–94, 2023). A geometric interpretation for the definition is that an umbilic point at infinity occurs when the tangent to the level set at infinity is also an asymptotic direction at infinity.

Study of a generalized subclass of meromorphic functions

This paper is concerned with a new subclass of meromorphic close-to-convex functions defined by means of subordination. various properties of this class such as coefficient estimates, inclusion relationship, distortion property and radius of meromorphic convexity, are established. Some earlier known results follow as special cases.

Characterization of Tubular Surfaces in Terms of Finite III-type

In this paper, we first define relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E 3 . Then, we will characterize the tubular surfaces in terms of their coordinate finite type.

Understanding in mathematics: The case of mathematical proofs

AbstractAlthough understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on Bratman's theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency.

Abelian covers and the second fundamental form

We give some conditions on a family of abelian covers of P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb P}^1$$\\end{document} of genus g curves, that ensure that the family yields a subvariety of Ag\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathsf A}_g$$\\end{document} which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if g>M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g >M$$\\end{document}, then the family yields a subvariety of Ag\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathsf A}_g$$\\end{document} which is not totally geodesic. We prove then analogous results for families of abelian covers of C~t→P1=C~t/G~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{C}}_t \\rightarrow {\\mathbb P}^1 = {\ ilde{C}}_t/{\ ilde{G}}$$\\end{document} with an abelian Galois group G~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{G}}$$\\end{document} of even order, proving that under some conditions, if σ∈G~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma \\in {\ ilde{G}}$$\\end{document} is an involution, the family of Pryms associated with the covers C~t→Ct=C~t/⟨σ⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{C}}_t \\rightarrow C_t= {\ ilde{C}}_t/\\langle \\sigma \\rangle $$\\end{document} yields a subvariety of Apδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathsf A}_{p}^{\\delta }$$\\end{document} which is not totally geodesic. As a consequence, we show that if G~=(Z/NZ)m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{G}}=(\\mathbb Z/N\\mathbb Z)^m$$\\end{document} with N even, and σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document} is an involution in G~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{G}}$$\\end{document}, there exists an integer M(N) which only depends on N such that, if g~=g(C~t)>M(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{g}}= g({\ ilde{C}}_t) > M(N)$$\\end{document}, then the subvariety of the Prym locus in Apδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathsf A}}^{\\delta }_{p}$$\\end{document} induced by any such family is not totally geodesic (hence it is not Shimura).

On Certain Subclass of Meromorphic Multivalent Functions Associated with Fractional Calculus Operator

In this paper, the class of meromorphic multivalent functions of the form by using fractional differ-integral operators is introduced. We get Coefficients estimates, radii of convexity and star likeness. Also closure theorems and distortion theorem for the class , is calculaed.

On Generalized Kenmotsu Manifolds as Hypersurfaces of Vaisman-Gray Manifolds

In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's tensors in the Vaisman--Gray class are equal to zero. On the other hand, the minimality of totally umbilical, totally geodesic hypersurfaces of Vaisman--Gray manifolds with generalized Kenmotsu structures are investigated. In addition, we deduced that the hypersurface of Vaisman--Gray manifold that have generalized Kenmotsu structure is totally geodesic if and only if it is totally umbilical and some components of Lee form are constants.

Post-Buckling Analysis of Arch and Serpentine Structures Under End-to-End Compression

Abstract Arch and serpentine structures are two fundamental structural forms with significant applications in various fields. When subjected to compressive loading at both ends, these structures undergo flexural-torsional post-buckling, resulting in complex deformation modes that are challenging to describe using basic functions (e.g., trigonometric functions and polynomial functions), posing significant challenges in finding analytical solutions. In this study, we propose a novel approach to address this issue. By representing the lateral displacement with a trigonometric series expansion and utilizing the equilibrium equation, the angular displacement is expressed in terms of special functions known as Mathieu functions. Furthermore, the energy method is employed to obtain analytical solutions for the flexural-torsional post-buckling deformation components. The theoretical findings are validated through experiments and finite element analysis. Based on the theoretical results, explicit analytical expressions for the maximum principal strain and the bending-torsion ratio of the structures are derived, offering valuable insights for the design of arch and serpentine structures in practical applications.

Формы PR-биографии в контексте жанровой системы

The aim of the research is to identify the main genre forms of the PR biography, their specificity in application, the features of interaction in communication processes and their effective utilization in PR activities. The paper examines the features of using fundamental forms of the PR biography such as summary biographies, story biographies, and narrative biographies, studying their properties in both theoretical and practical aspects. The scientific novelty lies in contributing to science through a comprehensive analysis of PR biographies as a tool for shaping public opinion and reputation management. This enriches the theoretical database of genres with new aspects of their structural-stylistic diversity and functionality. For the first time in the realm of Russian communicative culture, there has been an in-depth study of the relationships between various forms of PR biographies and their effectiveness in goal setting and achieving communicative tasks. Furthermore, the scientific novelty lies in the application of a comprehensive approach combining analytical, comparative, and empirical methods to assess the practical use of PR biographies, revealing new trends and techniques in creating this genre of texts not previously explored in scientific literature. This research emphasizes the importance of the PR biography genre in the context of communication strategies and public relations. The research findings show that PR biographies can take various forms and combine genre models, enhancing text perception efficiency through the use of established forms from the practice of public relations. In addition, mechanisms from other discourses, such as interviews, confessions, newspaper articles, and chronicles, are actively involved in PR biographies, highlighting the need for systematic text updating and maintenance to keep them current.

Impact of energy-momentum squared gravity on the geometry of stellar objects

This paper investigates the impact of energy–momentum squared gravity on the geometry of anisotropic compact stellar objects. In this perspective, we use the Tolman IV solution to examine the configuration of static spherically symmetric structures. The values of unknown constants are found by the first fundamental form of Darmois junction conditions. We consider four different models of this theory to analyze the behavior of physical quantities such as effective fluid parameters, anisotropy, energy constraints and equation of state parameters in the interior of considered compact stars. Sound speed and adiabatic index methods are used to determine the stability of proposed compact stars. It is found that the proposed compact stars are viable and stable in energy–momentum squared gravity.

Repetition Suppression Reveals Cue-Specific Spatial Representations for Landmarks and Self-Motion Cues in the Human Retrosplenial Cortex.

The efficient use of various spatial cues within a setting is crucial for successful navigation. Two fundamental forms of spatial navigation, landmark-based and self-motion-based, engage distinct cognitive mechanisms. The question of whether these modes invoke shared or separate spatial representations in the brain remains unresolved. While nonhuman animal studies have yielded inconsistent results, human investigation is limited. In our previous work (Chen et al., 2019), we introduced a novel spatial navigation paradigm utilizing ultra-high field fMRI to explore neural coding of positional information. We found that different entorhinal subregions in the right hemisphere encode positional information for landmarks and self-motion cues. The present study tested the generalizability of our previous finding with a modified navigation paradigm. Although we did not replicate our previous finding in the entorhinal cortex, we identified adaptation-based allocentric positional codes for both cue types in the retrosplenial cortex (RSC), which were not confounded by the path to the spatial location. Crucially, the multi-voxel patterns of these spatial codes differed between the cue types, suggesting cue-specific positional coding. The parahippocampal cortex exhibited positional coding for self-motion cues, which was not dissociable from path length. Finally, the brain regions involved in successful navigation differed from our previous study, indicating overall distinct neural mechanisms recruited in our two studies. Taken together, the current findings demonstrate cue-specific allocentric positional coding in the human RSC in the same navigation task for the first time and that spatial representations in the brain are contingent on specific experimental conditions.

Umbilicity of constant mean curvature hypersurfaces into space forms

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

AbstractIn this paper, we investigate the rigidity and vanishing properties of compact submanifolds with free boundary of arbitrary codimension in the unit ball. We first show that a minimal submanifold with free boundary in the unit ball satisfying a pointwise or integral curvature pinching condition on the second fundamental form is a flat equatorial disk. Then we prove a vanishing theorem for cohomology groups for submanifolds with free boundary in the unit ball under an integral curvature pinching condition on the trace‐free second fundamental form.

Complete real Kähler submanifolds

AbstractLet denote an isometric immersion of a Kähler manifold with complex dimension into Euclidean space with codimension . We show that generic rank conditions on the second fundamental form of a non‐minimal complete real Kähler submanifold imply that is a cylinder over a real Kähler submanifold .

A Review on Interactive Air Canvas Application

In today's dynamic landscape of human- computer interaction, there's a growing demand for interfaces that seamlessly merge the digital and physical realms. Traditional input methods like keyboards and mice, though reliable, fall short in meeting the expectations of users seeking more intuitive engagement with technology. The Air Canvas project, leveraging OpenCV and MediaPipe, addresses this need by creating an interactive drawing platform that monitors real- time hand movements. Users can paint in mid-air, translating hand gestures into vibrant strokes on a digital canvas. This work highlights the potential of computer vision and gesture recognition to bridge the gap between technology and art, offering immersive and creative applications. Writing, a fundamental form of communication, is explored through a motion-to-text converter for smart wearables. By tracking finger movements using computer vision, this system generates text for purposes like texting and emails, presenting a valuable communication tool, especially for the deaf community. Keywords— OpenCV, MediaPipe, Interfaces, Real-time hand movements, Gesture Recognition.