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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]

The classical Bonnet problem is to classify all immersions f : M 2 → R 3 into Euclidean three-space that are not determined, up to a rigid motion, by their induced metric and mean curvature function. The natural extension of Bonnet problem for Euclidean hypersurfaces of dimension n ≥ 3 was studied by Kokubu (Tôhoku Math J 44:433–442, 1992). In this article, we investigate an infinitesimal version of Bonnet problem for hypersurfaces with dimension n ≥ 3 of any space form, namely, we classify the hypersurfaces f : M n → Q c n + 1 , n ≥ 3 , of any space form Q c n + 1 of constant curvature c, for which there exists a (non-trivial) one-parameter family of immersions f t : M n → Q c n + 1 , with f 0 = f , whose induced metrics g t and mean curvature functions H t coincide "up to the first order," that is, ∂ / ∂ t | t = 0 g t = 0 = ∂ / ∂ t | t = 0 H t. [ABSTRACT FROM AUTHOR]

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples. [ABSTRACT FROM AUTHOR]

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds M = G / H such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition g = h ⊕ n (vector space direct sum) with n nilpotent. When the metric is nondegenerate on [ n , n ] , we show that n is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on [ n , n ] , we show that n is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric. We obtain two results on such hypersurfaces which admit at most one affine principal curvature of multiplicity one: (1) classify these being not affine hyperspheres; (2) classify these affine hyperspheres with constant scalar curvature. For the latter, by proving the parallelism of their cubic forms we translate the classification into that of affine hypersurfaces with parallel cubic form, which has been completed by Hu-Li-Vrancken (J Differ Geom 87:239–307, 2011). [ABSTRACT FROM AUTHOR]

We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step 2, we prove a Blaschke–Privaloff–Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces. [ABSTRACT FROM AUTHOR]

The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient γ > 3 . Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper. [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]

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]

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]