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Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form).

In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.

The purpose of this paper is to study the geometry of screen real lightlike submanifolds of metallic semi-Riemannian manifolds. Also, the authors investigate whether these submanifolds are warped product lightlike submanifolds or not.

The paper is design as follows: In Section 3, the authors introduce screen-real lightlike submanifold of metallic semi Riemannian manifold. In Section 4, the sufficient conditions for the radical and screen distribution of screen-real lightlike submanifolds, to be integrable and to be have totally geodesic foliation, have been established. Furthermore, the authors investigate whether these submanifolds can be written in the form of warped product lightlike submanifolds or not.

The geometry of the screen-real lightlike submanifolds has been studied. Also various results have been established. It has been proved that there does not exist any class of irrotational screen-real r-lightlike submanifold such that it can be written in the form of warped product lightlike submanifolds.

All results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

Bi-slant submanifolds of S-manifolds are introduced, and some examples of these submanifolds are presented.

Some properties of D_i-geodesic and D_i-umbilical bi-slant submanifolds are examined.

The Riemannian curvature invariants of these submanifolds are computed, and some results are discussed with the help of these invariants.

The topic is original, and the manuscript has not been submitted to any other journal.

In this paper some characterizations for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms are obtained. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, the equality case is also discussed. By the application of derived inequality, we compute an expression for Dirichlet energy of the involved warping function. Finally, we also proved some classifications for these warped product submanifolds in terms of Ricci solitons and Ricci curvature. A non-trivial example of these warped product submanifolds is provided.

The author considers an invariant lightlike submanifold M, whose transversal bundle tr(TM) is flat, in an indefinite Sasakian manifold M¯(c) of constant φ¯-sectional curvature c. Under some geometric conditions, the author demonstrates that c=1, that is, M¯ is a space of constant curvature 1. Moreover, M and any leaf M′ of its screen distribution S(TM) are, also, spaces of constant curvature 1.

The author has employed the techniques developed by K. L. Duggal and A. Bejancu of reference number 7.

The author has discovered that any totally umbilic invariant ligtlike submanifold, whose transversal bundle is flat, in an indefinite Sasakian space form is, in fact, a space of constant curvature 1 (see Theorem 4.4).

To the best of the author’s findings, at the time of submission of this paper, the results reported are new and interesting as far as lightlike geometry is concerned.

This paper aims to use the redundancy of a 7-DOF (degree of freedom) serial manipulator to solve motion planning problems along a given 6D Cartesian tool path, in the presence of geometric constraints, namely, obstacles and joint limits.

This paper describes an explicit expression of the task submanifolds for a 7-DOF redundant robot, and the submanifolds can be parameterized by two parameters with this explicit expression. Therefore, the global search method can find the feasible path on this parameterized graph.

The proposed planning algorithm is resolution complete and resolution optimal for 7-DOF manipulators, and the planned path can satisfy task constraint as well as avoiding singularity and collision. The experiments on Motoman SDA robot are reported to show the effectiveness.

This algorithm is still time-consuming, and it can be improved by applying parallel collision detection method or lazy collision detection, adopting new constraints and implementing more effective graph search algorithms.

Compared with other task constrained planning methods, the proposed algorithm archives better performance. This method finds the explicit expression of the two-dimensional task sub-manifolds, so it’s resolution complete and resolution optimal.

To describe and extend existing concepts of discrete electromagnetism in a unified formalism; to give examples for the usefulness of the presented ideas for our theoretical work, especially with regard to energy.

After a concise introduction to the mathematical concepts of discrete electromagnetism, we introduce continuous de Rham currents and give their discrete counterpart. We define operators acting upon discrete currents, and apply the theory to electromagnetism.

de Rham current theory yields a mathematical framework for the discussion of discrete electromagnetic problems: The focus is on energy‐balance equations; a discrete Lagrangian can be defined for various modeling problems; the Galerkin approach fits nicely into the proposed formalism; boundary terms in discrete formulations are an implicit feature to the theory.

In this paper, we use the interpolation of discrete fields by Whitney forms on a simplicial cell complex. The resulting discrete formulation is identical to a Galerkin finite‐element method. Other numerical techniques that do not resort to Whitney‐form interpolation can equally be discussed in de Rham‐current terminology.

Rather than a novel numerical technique, the paper presents a unified mathematical framework for the discussion of different practical approaches. We advocate a canonical treatment of energy‐related quantities and of boundary terms in discrete formulations.

By means of topological conjugate transformation, the previous theory of abstract neural automata (ANA) on d‐dimensional (d≥1) integer lattice is extended to compact Riemannian manifold. This paper points out emphatically that intelligence of ANA is related to the geometrical features. The greater the volume of relative plane, the stronger the intelligence; curved Riemannian manifold X˜ configuration space of ANA are locally flat such that the cognitive process of NAN limits the Gibbs' probability measure for a sufficiently small time i.e. the cognitive process of ANA can determine the solution in a sufficiently small time the problem. This hypothesis was supported by studying the human brain, in particular by studying Einstein's brain.

The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3 + 1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries.

The framework of differential‐form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike.

The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric‐incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates.

The present formalism can be used to generate concise formulations of complex field problems. The differential‐form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation.

The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.

Believes that in the view of philosophy, a concept is the highest form of activity of human brain. This paper demonstrates Abstract Neural Automata and a more perfect brain's models that have the ability of transition of concept‐ability of thought. The transition of the concept of Abstract Neural automata results from the non‐uniqueness of its limit Gibbs measure‐variability of the structure of Abstract Neural Automata.By means of topological conjugate transformation, the previous theory of Abstract Neural Automata on a d‐dimensional (d≥1) integer lattice is extended to the compact Riemannian manifold. We have pointed out emphatically that functions of cognition and thought of Abstract Neural Automata depend crucially on its topological and the Riemannian structure, particularly, on its Riemannian volume of some relative places which are relative learning, memory, cognition and thought. Furthermore, the larger the Riemannian volume, the stronger the intelligent function. In the study of the human brain, and in particular, Einstein's brain, one has discovered such information.