The Conformally Symmetric K-Contact Manifolds

In this paper, we establish a differential equation about scalar curvature of conformally flat K-contact manifolds, and prove that a conformally symmetric K-contact manifold is a Riemann manifold with constant curvature 1. At the same time, the results on Sasaki manifolds which are given by Miyazaawa and Yamagushi are generalized to K-contact manifolds.

DISTRIBUTIONS ON ALMOST CONTACT MANIFOLDS

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to deter- mine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G2-structure.

Conformal Human–Machine Integration Using Highly Bending‑Insensitive,Unpixelated,and Waterproof Epidermal Electronics Toward Metaverse

Efficient and flexible interactions require precisely converting human intentions into computer-recognizable signals,which is critical to the breakthrough development of metaverse.Interactive electronics face common dilemmas,which realize highprecision and stable touch detection but are rigid,bulky,and thick or achieve high flexibility to wear but lose precision.Here,we construct highly bending-insensitive,unpixelated,and waterproof epidermal interfaces(BUW epidermal interfaces)and demonstrate their interactive applications of conformal human–machine integration.The BUW epidermal interface based on the addressable electrical contact structure exhibits high-precision and stable touch detection,high flexibility,rapid response time,excellent stability,and versatile“cut-and-paste”character.Regardless of whether being flat or bent,the BUW epidermal interface can be conformally attached to the human skin for real-time,comfortable,and unrestrained interactions.This research provides promising insight into the functional composite and structural design strategies for developing epidermal electronics,which offers a new technology route and may further broaden human–machine interactions toward metaverse.

Killing Operator for the Kerr Metric

When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) →F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra

Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O  to be a minimal submanifold of M 6.

The U-Kenmotsu Hypersurfaces Axiom and Six-Dimensional Hermitian Submanifolds of Cayley Algebra

Six-dimensional Hermitian submanifolds of Cayley algebra are considered.It is proved that if such a submanifold of the octave algebta complies with the U-Kenmotsu hypersurfaces axiom,then it is Khlerian.

Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties

In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by threedimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries(mKdV) equation, generalized coupled mKdV integrable system and nonisospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit N-fold Darboux transformation for integrable system(2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

Impact of source and drain contact thickness on the performance of organic thin film transistors

This paper analyzes the impact of source（ts） and drain（td） contact thicknesses on top contact（TC） and bottom contact（BC） organic thin film transistors（OTFTs） with a gate in the bottom, using a benchmarked industry standard Atlas 2-D numerical device simulator. The parameters including drive current（Ids）, mobility（μ）, threshold voltage（Vt）and current on-off ratio（ION/IOFF） are analyzed from the device physics point of view on different electrode thicknesses, ranging from infinitesimal to 50 nm, for both top and bottom contact structures. Observations demonstrate that the performance of the BC structure is more affected by scaling of ts=din comparison to its counterpart. In the linear region, the mobility is almost constant at all the values of ts=dfor both structures. However,an increment of 18% and 83% in saturation region mobility is found for TC andBC structures, respectively with scaling down ts=dfrom 50–0 nm. Besides this, the current on-off ratio increases more sharply in the BC structure.This analysis simplifies a number of issues related to the design and fabrication of organic material based devices and circuits.