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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]

Differential geometry is a strong and highly effective mathematical subject for robot gripper design when grasping within the predetermined trajectories of path planning. This study in grasping focuses on differential geometry analysis utilizing the Lie algebra, geodesic, and Riemann Curvature Tensors (RCT). The novelty of this article for 2RR robot mechanisms lies in the approach of the body coordinate with the geodesic and RCT. The importance of this research is significant especially in grasping and regrasping objects with varied shapes. In this article, the types of workspaces are clarified and classified for grasping and regrasping kinematics. The regrasp has not been sufficiently investigated of body coordinate systems in Lie algebra. The reason for this is the difficulty in understanding relative coordinates in Lie algebra via the body coordinate system. The complexity of the equations has not allowed many researchers to overcome this challenge. The symbolic mathematics toolbox in the Maxima, on the other hand, aided in the systematic formulation of the workspaces in Lie algebra with geodesic and RCT. The Lie algebra se(3) equations presented here have already been developed for robot kinematics from many references. These equations will be used to derive the followingworkspace types for grasping and regrasping. Body coordinate workspace, spatial coordinate workspace with constraints, body coordinate workspace with constraints, spatial coordinate workspace with constraints are the workspace types. The RCT and geodesic solutions exploit these four fundamental workspace equations derived using Lie algebra.

The article provides a brief overview of the main directions of the history of scientific and pedagogical activity and the achievements of the Department of Geometry of Irkutsk State University for 1938-2012 years.

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]

Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of a lot of discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple method of generating ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, as these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.

This manuscript endeavors to establish a framework for the mapping of music onto a three-dimensional structure. Our objective is to transform the guitar choruses of Beatles songs into curves, with each chorus corresponding to its respective curve. We aim to investigate and characterize the intricacy of each song by employing mathematical techniques derived from differential geometry, specifically focusing on the total curvature of the chorus curve. Given that a single song may possess varying chord progressions in different verses, the performer can determine the geometric representation they aim to convey through the number of loops and the direction of the curve. The overarching objective of our study is to enable viewers to identify specific songs or motives by visually examining an object and exploring its geometric properties. Furthermore, we posit that these ideas can provide composers with a fresh perspective on their own musical compositions while also granting non-professional audiences a glimpse into the intricacies involved in the process of composing.

Modeling has become firmly established as a methodology to close the Reynolds-averaged Navier–Stokes (RANS) equations, owing to theoretical and empirical efforts towards a complete formulation of the Reynolds stress tensor and, recently, breakthroughs in data-processing technology. However, mathematical exactness is not generally ensured by modeling, which is an intrinsic reason why the reliability of RANS closure models is not supposed to be consistent for all kinds of turbulent flow. Rather than straightforwardly overcoming this inherent limitation, most of the studies to date were reasonably directed towards broadening the range of turbulent flows, where reliable prediction accuracy can be obtained via modeling. In this paper, we present three identities between components of the Reynolds stress tensor, constructed via spatial mapping on the basis of the differential version of the Gauss–Bonnet formula. Further, we present a constraint condition that gives a set of equations as numerous as the parameters within a RANS model.

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]

This paper proposes an efficient method for active unsupervised texture segmentation. A new descriptor for texture features extractions based on Gaussian and mean curvature is constructed. Then the optimization of a functional who uses the R´enyi divergence measure and our descriptor is proposed in order to design an active contour model for texture segmentation. To get a global solution and efficient, fast algorithm, the optimization problem is redefined. The algorithm associated with this last optimization problem avoids local minimums and the run-time consuming compared to the level-set representation of our active contour model. In order to illustrate the performance of the technique, some results are presented showing the effectiveness and robustness of our approach.

In order to extend the sphere theorem for Finsler metrics, the concept of reversibility introduced by H-B. Rademacher for a compact Finsler manifold. In this paper, we extend this notion to the general Finsler manifolds. Then we find an upper bound for the reversibility of some important spherically symmetric Finsler metrics. Furthermore, we introduce the concept of sub-reversibility for a general Finsler manifold and obtain a non-zero lower bound for this new quantity. [ABSTRACT FROM AUTHOR]

In this paper, we study conformallyat 5-th root (α,β)-metrics. We prove that every conformallyat 5-th root (α,β)-metric with relatively isotropic mean Landsberg curvature must be either Riemannian metrics or locally Minkowski metrics. [ABSTRACT FROM AUTHOR]

Every Landsberg metric and every Landsbeg metric is a weakly Landsberg metric, but the converse is not true generally. Let (M; F) be a 3-dimensional Finsler manifold. In this paper, we find a condition under which the notions of weakly Landsberg metric and Landsberg metric are equivalent. [ABSTRACT FROM AUTHOR]

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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]

The LeBrun ansatz was designed for scalar-flat K\"ahler metrics with a continuous symmetry; here we show it is generalizable to much broader classes of metrics with a symmetry. We state the conditions for a metric to be (locally) expressible in LeBrun ansatz form, the conditions under which its natural complex structure is integrable, and the conditions that produce a metric that is K\"ahler, scalar-flat, or extremal K\"ahler. Second, toric K\"ahler metrics (such as the generalized Taub-NUTs) and $U(2)$-invariant metrics (such as the Fubini-Study or Page metrics) are certainly expressible in the LeBrun ansatz. We give general formulas for such transitions. We close the paper with examples, and find expressions for two examples---the exceptional half-plane metric and the Page metric---in terms of the LeBrun ansatz.

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]

Let M be a smooth manifold of dimension m ≤ 1 and P be a G-structure on M, where G is a Lie subgroup of linear group GL(m). In [8], it has been defined the prolongations of G-structures related to tangent functor of higher order and some properties have been established. The aim of this paper is to generalize these prolongations to a Weil bundles. More precisely, we study the prolongations of G-structures on a manifold M, to its Weil bundle TAM (A is a Weil algebra) and we establish some properties. In particular, we characterize the canonical tensor fields induced by the A-prolongation of some classical G-structures. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study a semiconformally flat on the Riemannian manifolds with a perfect fluid spacetime that satisfies Einstein's field equation without cosmological constant. Many geometric features related to weakly semiconformally symmetric manifolds have been studied with the weakly symmetric and the weakly Ricci symmetric manifolds. [ABSTRACT FROM AUTHOR]

The congruence of certain restricted partition functions known as regular bipartition is discussed in this paper. We particularly investigate the (2,β)-regular bipartitions of n, denoted by B2,β(n), and establish certain congruences for B2,β(n) when β ≥ 3. We derive infinite families of congruences modulo 4 for the (2; 3)-regular bipartition. We also obtain a generalisation of the regular bipartition for modulo p and p². [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]

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.

The term ‘mathematical chemistry’ is mostly associated with applications of graph theory in topological issues of 3D chemical structures, thought of as a collection of atoms as dots and bonds as lines. We propose here new directions in this field, coming from the side of theoretical chemistry approached with modern computational tools. Possible challenges are proposed in using ancillary tools of differential geometry for examining the potential energy surfaces of certain specific structural prototypes. Concretely, we describe here the geodesics on the surfaces related to the potential energy functions of the so-called E⊗e Jahn–Teller effect, a spontaneous symmetry-breaking phenomenon also known as a case of conical intersection. To illustrate the case, first-principles (ab initio) quantum chemical calculations are performed on the cyclo-propenyl molecular radical C3H3.

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]

A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invariant Riemannian metrics have a perfect theoretical framework and avoid the drawbacks of the Euclidean inner product, their complex formulas also lead to sophisticated and time-consuming algorithms. To make up for this limitation, log-Euclidean metrics with simpler formulas and faster calculations are employed in this manuscript. Our new approach is to transform a symmetric positive-definite matrix into a symmetric matrix via logarithmic maps, and then to transform the results back to the Lie group through exponential maps. Moreover, the present method does not need to compute the mean matrix and retains the usual Euclidean operations in the domain of matrix logarithms. In addition, for some randomly generated positive-definite matrices, the method is compared using experiments with that induced by the classical affine-invariant Riemannian metric. Finally, our proposed method is applied to denoise the point clouds with high density noise via the K-means clustering algorithm.

In this paper, a simple flatness condition for the two degree of freedom underactuated mechanical system has been derived. Differential geometry was used as a mathematical tool in the derivation of the flatness condition. The flatness condition has been found as a direct inner product between the covariant derivative of a vector field which annihilate the codistribution, that spanned by the force matrix and the force matrix itself. Several examples of underactuated mechanical systems are 2DOF systems, or mechanical systems underactuated by one control which can be reduced to 2DOF system. Systems that are classified as differentially flat have many useful features, which can be used in designing an effective controller for the nonlinear systems. The Translational Oscillator with Rotational Actuator (TORA) system is considered here as an example of a typical flat 2DOF underactuated mechanical system. The flat output is derived based on the obtained result here, and then a nonlinear controller is designed for a TORA system based on the flatness property, and using a Backstepping method to overcome the under actuation problem. The simulation results demonstrate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics.

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]