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In this study, the theory of curves is reconstructed with fractional calculus. The condition of a naturally parametrized curve is described, and the orthonormal conformable frame of the naturally parametrized curve at any point is defined. Conformable helix and conformable slant helix curves are defined with the help of conformable frame elements at any point of the conformable curve. The characterizations of these curves are obtained in parallel with the conformable analysis Finally, examples are given for a better understanding of the theories and their drawings are given with the help of Mathematics. [ABSTRACT FROM AUTHOR]

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included. [ABSTRACT FROM AUTHOR]

Sophus Lie was arrested in 1870 because of his mathematics notebook. In his notes, Lie used geometry to solve differential equations by identifying transformations that leave solutions unchanged. These transformations suggested a change of variable that made the differential equations easier to solve. Lie groups were born in those notes. This article reveals Lie's unlawful methods that could lead to a life of crime. [ABSTRACT FROM AUTHOR]

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relics on and answers a question in a recent work by Fujiwara and Papasoglu. Wc conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts. [ABSTRACT FROM AUTHOR]

We construct a concrete example of constant Gauss curvature K = 1 on the 2-sphere having all geodesics closed and of same length. [ABSTRACT FROM AUTHOR]

The notions of k-regularity and intra k-regularity in bi-semimodules of a Morita context of semirings have been introduced. Some characterizations of k-regular and intra kregular idempotent bi-semimodules of a Morita context of semirings have been obtained in terms of k-b-subsemimodules. Then the study of the interplay among various components of a Morita context of semirings in terms of k-regularity and intra k-regularity has been accomplished. Finally it has been shown that k-regularity and intra k-regularity are Morita invariant properties. [ABSTRACT FROM AUTHOR]

In my 2018 article in this journal, I described 15th-century Italian astronomer Giovanni Bianchini's treatment of the problem of stellar coordinate conversion in his Tabulae primi mobilis, the first correct European solution. In this treatise Bianchini refers to a book he had written previously, containing the same error that had plagued his predecessors' work on the problem. In this article, we announce the discovery of this earlier treatise. We compare its canons and tables to Bianchini's later work, noting the places where the contents overlap (roughly one quarter of the text). We analyze his mathematical methods and the unique tables he constructed for converting stellar coordinates, including the earliest known European arc sine table, that he would abandon only a few years later. [ABSTRACT FROM AUTHOR]

We give a q -analogue of the supercongruence (I.2) of Van Hamme. As a conclusion, we confirm a recent conjecture of Swisher. We also give a q -analogue of the corresponding π series (I.1) along with some similar results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some general Kastler–Kalau–Walze type theorems for any dimensional manifolds with boundary which generalize the results in [J. Wang and Y. Wang, The Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary, J. Math. Phys.56 (2015), Article ID: 052501, 14 pp.]. [ABSTRACT FROM AUTHOR]

Abstract In this work, we provide a new post-processing procedure for automatically adjusting node locations of an all-hex mesh to better match the volume of a reference geometry. This process is particularly well-suited for mesh-first approaches, as overlay grid ones. In practice, hexahedral meshes generated via an overlay grid procedure, where a precise reference geometry representation is unknown or is impractical to use, do not provide for precise volumetric preservation. A discrete volume fraction representation of the reference geometry M I on an overlay grid is compared with a volume fraction representation of a 3D finite element mesh M O. This work introduces the notion of localized discrepancy between M I and M O and uses it to design a procedure that relocates mesh nodes to more accurately match a reference geometry. We demonstrate this procedure on a wide range of hexahedral meshes generated with the Sculpt code and show improved volumetric preservation while still maintaining acceptable mesh quality. Highlights • Definition of a localized discrepancy. • Volume preservation improvement of a mesh. • Different kinds of inputs were tested. [ABSTRACT FROM AUTHOR]

The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex, involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form, and lacks smoothness near the collision point. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to investigate three applications: 1) solutions of the Kepler function, 2) calculation of orbits including time-of-flight data based on the Delaunay Hamiltonian, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer–Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid. [ABSTRACT FROM AUTHOR]

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions. [ABSTRACT FROM AUTHOR]

In this paper we prove the containment I(nm) ⊂ M(n-1)mIm, for a radical ideal I of s general points in Pn, where s ≥ 2n. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least 2n points in Pn. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to survey some recent results on the Chern-Moser question concerning existence of umbilical points on three-dimensional CR submanifolds in . [ABSTRACT FROM AUTHOR]

The object of the present paper is to study mixed quasi-Einstein manifolds. Some geometric properties of mixed quasi-Einstein manifolds have been studied. We also discuss spacetime with space-matter tensor and some properties related to it. Finally, we construct an example of a mixed quasi-Einstein spacetime. [ABSTRACT FROM AUTHOR]

The paper examines so-called wave boundary layers arising in a bounded stratified fluid for large times. Each the layer is a narrow domain in the vicinity of the fluid surface and/or bottom characterized by sharp, growing with increasing time, vertical gradients of the buoyancy and horizontal velocity. The layers arise as a result of free linear wave evolution of the initial fields if the initial buoyancy at the boundaries depends on the horizontal coordinates. An asymptotic solution for the boundary layer for large times is presented, and it is shown that this solution describes exact fields fairly well even for moderate times. [ABSTRACT FROM AUTHOR]

The Gauss map of a projective variety X ⊂ PN is a rational map from X to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties F and Y, we construct a projective variety X whose Gauss map has F as its general fiber and has Y as its image. More generally, we give such construction for families of varieties over Y instead of fixed F. (2) At least in the case when the characteristic is not equal to 2, any inseparable field extension appears as the extension induced from the Gauss map of some X. [ABSTRACT FROM AUTHOR]

We derive a formula of the greatest lower bounds on Ricci curvature of Fano homogeneous toric bundles. A criteria for the ampleness of line bundles over general homogeneous toric bundles is also obtained. [ABSTRACT FROM AUTHOR]

Special coordinate systems are constructed in a neighborhood of a point or of a curve. Taylor expansions can then be easily inferred for the metric, the connection, or the Finsler Lagrangian in terms of curvature invariants. These coordinates circumvent the difficulties of the normal and Fermi coordinates in Finsler geometry, which in general are not sufficiently differentiable. They are obtained applying the usual constructions to the pullback of a horizontally torsionless connection. The results so obtained are easily specialized to the Berwald or Chern–Rund connections and have application in the study of the equivalence principle in Finslerian extensions of general relativity. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study warped product pseudo-slant submanifolds of (LCS)n-manifolds. We study the existence or non-existence of such submanifolds. The existence is also ensured by an example. [ABSTRACT FROM AUTHOR]

In this paper, we study the hemi-slant submanifolds of cosymplectic manifolds. Necessary and sufficient conditions for distributions to be integrable are worked out. Some important results are obtained in this direction. We study the geometry of leaves of hemi-slant submanifolds. [ABSTRACT FROM AUTHOR]

We characterise sets of uniformly absolutely continuous norm in strongly symmetric spaces of τ-measurable operators. Applications are given to the study of relatively weakly compact and relatively compact sets and to compactness properties of operators dominated in the sense of complete positivity by compact or by Dunford-Pettis operators. [ABSTRACT FROM AUTHOR]

The communication processes of knowledge creation represent a particular class of human dynamics where the expertise of individuals plays a substantial role, thus offering a unique possibility to study the structure of knowledge networks from online data. Here, we use the empirical evidence from questions-and-answers in mathematics to analyse the emergence of the network of knowledge contents (or tags) as the individual experts use them in the process. After removing extra edges from the network-associated graph, we apply the methods of algebraic topology of graphs to examine the structure of higher-order combinatorial spaces in networks for four consecutive time intervals. We find that the ranking distributions of the suitably scaled topological dimensions of nodes fall into a unique curve for all time intervals and filtering levels, suggesting a robust architecture of knowledge networks. Moreover, these networks preserve the logical structure of knowledge within emergent communities of nodes, labeled according to a standard mathematical classification scheme. Further, we investigate the appearance of new contents over time and their innovative combinations, which expand the knowledge network. In each network, we identify an innovation channel as a subgraph of triangles and larger simplices to which new tags attach. Our results show that the increasing topological complexity of the innovation channels contributes to network’s architecture over different time periods, and is consistent with temporal correlations of the occurrence of new tags. The methodology applies to a wide class of data with the suitable temporal resolution and clearly identified knowledge-content units. [ABSTRACT FROM AUTHOR]

Given the amount of knowledge and data accruing in the neurosciences, is it time to formulate a general principle for neuronal dynamics that holds at evolutionary, developmental, and perceptual timescales? In this paper, we propose that the brain (and other self-organised biological systems) can be characterised via the mathematical apparatus of a gauge theory. The picture that emerges from this approach suggests that any biological system (from a neuron to an organism) can be cast as resolving uncertainty about its external milieu, either by changing its internal states or its relationship to the environment. Using formal arguments, we show that a gauge theory for neuronal dynamics—based on approximate Bayesian inference—has the potential to shed new light on phenomena that have thus far eluded a formal description, such as attention and the link between action and perception. [ABSTRACT FROM AUTHOR]

The aim of this article is to try to determine whether there’s a statistically significant difference between coordinates obtained under different elevation angles and measuring epochs through different correction methods in Network Real Time Kinematic Applications. For this purpose, analysis of variance was used. When the statistical data obtained according to different variables are compared, the accuracy seems to be less in the direction of h coordinate axis. There is no a statistically significant difference among the values obtained according to different epoch numbers in all directions of the coordinate axes. The subsets formed according to satellite elevation angles show similar characteristics along Y and h coordinate axes while they differ 30° according to other angles in the direction of X coordinate axis. Based on the correction methods, while FKP and VRS method show different characteristics with respect to each other, MAC method shows similar characteristics compared to other methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of fast and efficient indexing techniques for sequences evolving in non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search, and recognition in very high dimensional spaces. The problem is made more challenging when representations such as landmarks, contours, and human skeletons etc. are naturally studied in a non-Euclidean setting where even simple operations are much more computationally intensive than their Euclidean counterparts. We propose a geometry and data adaptive symbolic framework that is shown to enable the deployment of fast and accurate algorithms for activity recognition, dynamic texture recognition, motif discovery. Toward this end, we present generalizations of key concepts of piece-wise aggregation and symbolic approximation for the case of non-Euclidean manifolds. We show that one can replace expensive geodesic computations with much faster symbolic computations with little loss of accuracy in activity recognition and discovery applications. The framework is general enough to work across both Euclidean and non-Euclidean spaces, depending on appropriate feature representations without compromising on the ultra-low bandwidth, high speed and high accuracy. The proposed methods are ideally suited for real-time systems and low complexity scenarios. [ABSTRACT FROM AUTHOR]

Multidisciplinary Design Optimization, also known as MDO, deals with the optimization of complex but typically single-objective design problems. As such, it would be valuable to combine it with Multi-Objective Optimization (MOO). Within MOO, Weighted Sums (WS) is a standard solution method, but it may not produce a good spacing of solution points along the Pareto Front (PF). Using our differential geometry MDO framework, we present a technique for intelligently spacing out Pareto solutions produced with WS in a multidisciplinary MOO formulation. Following an initial demonstration, we present modifications to our method to handle ill-conditioning better, to produce fewer dominated points in PF refinement, and to use MOO solution methods other than WS. [ABSTRACT FROM AUTHOR]

The delayed feedback stabilization of rigid spacecraft attitude dynamics in the presence of an unknown time-varying delay in the measurement is addressed. The attitude representation is parameterized using minimal attitude coordinates. The time-varying delay and its derivative are assumed to be bounded. By employing a linear state feedback controller via a Lyapunov–Krasovskii functional, a general delay-dependent stability condition is characterized for the closed-loop parameterized system in terms of a linear matrix inequality (LMI) whose solution gives the suitable controller gains. An estimate of the region of attraction of the controlled system is also obtained, inside which the asymptotic stability of parameterized system is guaranteed. [ABSTRACT FROM AUTHOR]

The purpose of this note is to provide some volume estimates for Einstein warped products similar to a classical result due to Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. To do so, we make use of the approach of quasi-Einstein manifolds which is directly related to Einstein warped products. In particular, we present an obstruction for the existence of such a class of manifolds. [ABSTRACT FROM AUTHOR]

We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. [ABSTRACT FROM AUTHOR]

Let m ≥ 2 be a positive integer. An m-regular partition of a positive integer n is a partition none of whose parts is divisible by m. In this paper, we prove infinite families of congruences modulo 2 for 25-regular partitions. [ABSTRACT FROM AUTHOR]

In order to accelerate the algorithm speed of dynamic model of helicopter, an IDRA approach for modeling helicopter was proposed and a dynamic model of six degrees of freedom (6 DOF) combining flying and landing was established in this paper. As the dynamic model is derived from the Lagrange equation of a non-conservative system, treating micro displacements and rotating angles of airframe in a simulation step as generalized coordinates, the relationship between energy dissipation, kinetic energy, potential energy and the generalized coordinates was analyzed. Finally, the reliability of this method and real-time of dynamic model were verified on the experimental platform of helicopter flight simulator, which could provide reliable theoretical foundation for solving the retardance of data transfer of helicopter simulator. [ABSTRACT FROM AUTHOR]

For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasi-geodesics. Brock-Masur- Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/ unstable laminations of a pseudo-Anosov, all have bounded combinatorics, outside of annuli. In this paper it is shown that there exist geodesics that also have bounded combinatorics within annuli. These geodesics are shown to have finiteness properties analogous to those of tight geodesics in the complex of curves, from which rationality of stable lengths of pseudo-Anosovs acting on the pants graph then follows from the arguments of Bowditch for the curve complex. [ABSTRACT FROM AUTHOR]

An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If X is a weakly conformal complete vector field on a connected Finsler space ( M, F) of dimension [ABSTRACT FROM AUTHOR]

Keywords: Differential geometry; Symplectic geometry; Symplectic aspects of mirror symmetry; Homological mirror symmetry; Derived categories; Fukaya category EN Differential geometry Symplectic geometry Symplectic aspects of mirror symmetry Homological mirror symmetry Derived categories Fukaya category N.PAG N.PAG 1 03/04/21 20210416 NES 210416 The publisher regrets there were errors in the printed version of this article, affecting a number of the commutative diagrams, specifically those appearing on pp. 4, 85, 95 and 100. Differential geometry, Symplectic geometry, Derived categories, Symplectic aspects of mirror symmetry, Homological mirror symmetry, Fukaya category. [Extracted from the article]

The article examines an infinitesimal group of the identity element of synthetic Lie theory through the use of an algebraic structure, such as a Lie algebra, as a representation. Topics discussed include use of the theory of synthetic differential geometry to reformulate Lie theory in terms of infinitesimal, significance of synthetic differential geometry when thinking heuristically about modern differential geometry, and the types of groupoid which generalize the classical Lie correspondence.

We consider equivariant wave maps from to in supercritical dimensions . Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability. To probe self-similar blowup, we develop a novel numerical method, based on an adaptive rescaling of coordinates, which may be of independent interest in numerical studies of singularity formation. [ABSTRACT FROM AUTHOR]