Fundamental Form Research Articles

Contact CR-Warped Product Submanifold of a Sasakian Space Form with a Semi-Symmetric Metric Connection

The main goal of this research paper is to investigate contact CR-warped product submanifolds within Sasakian space forms, utilizing a semi-symmetric metric connection. We conduct a comprehensive analysis of these submanifolds and establish several significant results. Additionally, we formulate an inequality that establishes a relationship between the squared norm of the second fundamental form and the warping function. Lastly, we present a number of geometric applications derived from our findings.

Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

AbstractWe study hypersurfaces of the four‐dimensional Thurston geometry , which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.

Rigidity of self-shrinkers with constant squared norm of the second fundamental form

In this paper, we investigate the rigidity of self-shrinkers in a Euclidean space [Formula: see text]. We first prove that any self-shrinker [Formula: see text] with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is an open part of a hyperplane [Formula: see text], a cylinder [Formula: see text] or the round sphere [Formula: see text]. Then, it can be applied to show that any complete self-shrinker [Formula: see text] with constant squared norm of the second fundamental form and with at most two distinct principal curvatures is isometric to a hyperplane [Formula: see text], a cylinder [Formula: see text] [Formula: see text] or the round sphere [Formula: see text]. Finally, some characterizations of self-shrinkers with constant mean curvature in [Formula: see text] are also obtained.

Numerical investigation of pull in control of rectangular AA1050 rolling ingot through the design of convex mould during direct chill casting process

The pull in effect/concavity on the lateral faces of the ingot is formed during the solidification of the ingot due to the thermal stresses induced in the ingot. A part of rolling ingots with pull in effect are cut by a conventional cutting process which affects the recovery and increases wastage and cost of production. In the present work, an elemental death and birth technique was used to develop a numerical model to study mechanical and thermal behavior during the solidification of AA1050 rolling ingot. The results of the model are validated with the results of the Williams model available in the literature. Furthermore, different mould designs provided with convexity on lateral surfaces with a convexity radius of 16 mm to 26 mm were numerically analyzed. A series of numerical simulations were conducted with varying mould convexity with an increment value of one. The results revealed that, providing convexity to the mould is an effective method to control the pull in effect during the direct chill casting process. The convexity radius of 26 mm is enough to completely eliminate the pull in radius of 23.34 mm for the rectangular AA1050 ingot.

Clairaut slant Riemannian maps to Kähler manifolds

The aim of this paper is to study the idea of Clairaut slant Riemannian maps from Riemannian manifolds to Kähler manifolds. First, for the slant Riemannian map, we obtain the necessary and sufficient conditions for a curve to be a geodesic on the base manifold. Further, we find the necessary and sufficient conditions for the slant Riemannian map to be a Clairaut slant Riemannian map; for Clairaut slant Riemannian map to be totally geodesic; for the base manifold to be a locally product manifold. Further, we obtain the necessary and sufficient condition for the integrability of range of derivative map. Also, we investigate the harmonicity of Clairaut slant Riemannian map. Finally, we get two inequalities in terms of second fundamental form of a Clairaut slant Riemannian map and check the equality case.

Eyes meet, hands greet: The art of timing in social interactions.

Shaking hands is a fundamental form of social interaction. The current study used high-definition cameras during a university graduation ceremony to examine the temporal sequencing of eye contact and shaking hands. Analyses revealed that mutual gaze always preceded shaking hands. A follow up investigation manipulated gaze when shaking hands, and found that participants take significantly longer to accept a handshake when an outstretched hand precedes eye contact. These findings demonstrate that the timing between a person's gaze and their offer to shake hands is critical to how their action is interpreted.

Geometry and evolution of Hasimoto surface in Minkowski 3-space.

The main goal of this paper is to introduce the evolution equations for a timelike Hasimoto surface from its fundamental form coefficients in Minkowski 3-space [Formula: see text]. By utilizing the evolved quasi-curve (q-curve), we present and analyze three types of Hasimoto surfaces, attributed to the quasi-tangent, quasi-normal, and quasi-binormal vectors of the curve. Finally, we provide an illustrated example to strengthen our main results.

Stability of isometric immersions of hypersurfaces

Abstract We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$ -perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$ , whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$ , removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

Sufficiency criteria for a class of convex functions connected with tangent function

<abstract><p>The research here was motivated by a number of recent studies on Hankel inequalities and sharp bounds. In this article, we define a new subclass of holomorphic convex functions that are related to tangent functions. We then derive geometric properties like the necessary and sufficient conditions, radius of convexity, growth, and distortion estimates for our defined function class. Furthermore, the sharp coefficient bounds, sharp Fekete-Szegö inequality, sharp 2nd order Hankel determinant, and Krushkal inequalities are given. Moreover, we calculate the sharp coefficient bounds, sharp Fekete-Szegö inequality, and sharp second-order Hankel determinant for the functions whose coefficients are logarithmic.</p></abstract>

How is a black hole created from nothing?

Using the synchronous coordinates, the creation of a Schwarzschild black hole immersed in a de Sitter spacetime can be viewed as a coherent creation of a collection of timelike geodesics. The previously supposed conical singularities do not exist at the horizons of the constrained instanton. Instead, the unavoidable irregularity is presented as a non-vanishing second fundamental form elsewhere at the quantum transition 3-surface. The same arguments can be applied to charged, topological or higher dimensional black hole cases.

Concepts of statistical learning and classification in machine learning: An overview

Statistical learning theory serves as the foundational bedrock of Machine learning (ML), which in turn represents the backbone of artificial intelligence, ushering in innovative solutions for real-world challenges. Its origins can be linked to the point where statistics and the field of computing meet, evolving into a distinct scientific discipline. Machine learning can be distinguished by its fundamental branches, encompassing supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning. Within this tapestry, supervised learning takes center stage, divided in two fundamental forms: classification and regression. Regression is tailored for continuous outcomes, while classification specializes in categorical outcomes, with the overarching goal of supervised learning being to enhance models capable of predicting class labels based on input features. This review endeavors to furnish a concise, yet insightful reference manual on machine learning, intertwined with the tapestry of statistical learning theory (SLT), elucidating their symbiotic relationship. It demystifies the foundational concepts of classification, shedding light on the overarching principles that govern it. This panoramic view aims to offer a holistic perspective on classification, serving as a valuable resource for researchers, practitioners, and enthusiasts entering the domains of machine learning, artificial intelligence and statistics, by introducing concepts, methods and differences that lead to enhancing their understanding of classification methods.

Family of right conoid hypersurfaces with light-like axis in Minkowski four-space

<abstract><p>In the realm of the four-dimensional Minkowski space $ \mathbb{L}^{4} $, the focus is on hypersurfaces classified as right conoids and defined by light-like axes. Matrices associated with the fundamental form, Gauss map, and shape operator, all specifically tailored for these hypersurfaces, are currently undergoing computation. The intrinsic curvatures of these hypersurfaces are determined using the Cayley-Hamilton theorem. The conditions of minimality are addressed by the analysis. The Laplace-Beltrami operator for such hypersurfaces is computed, accompanied by illustrative examples aimed at fostering a more profound understanding of the involved mathematical principles. Additionally, scrutiny is applied to the umbilical condition, and the introduction of the Willmore functional for these hypersurfaces is presented.</p></abstract>

Complete $\lambda $-hypersurfaces with constant squared norm of the second fundamental form in the Euclidean space $\mathbb R^{4}$

Complete $\lambda $-hypersurfaces with constant squared norm of the second fundamental form in the Euclidean space $\mathbb R^{4}$

Generalized warped product submanifolds of Lorentzian concircular structure manifolds

<abstract><p>We began by considering invariant, anti-invariant, proper slant, and pointwise slant submanifolds of a Lorentzian concircular structure manifold. Subsequently, we explored two distinct categories of warped product submanifolds. The first category encompassed the fiber submanifold as an anti-invariant submanifold, while the second category included the fiber submanifold as a pointwise slant submanifold. We established several fundamental results concerning these submanifold classes. Additionally, we demonstrated the existence of such submanifold classes through specific examples. Moreover, we derived inequalities for the squared norm of the second fundamental form.</p></abstract>

Chacinas Estatais uma análise crítica sobre a vingança policial

The Brazilian State, through the police, is one of the most repressive in the world. This is concretely expressed in the number of people executed by state agents throughout its history, reaching levels never seen before. In the present study, we will seek to understand the specificities of state repression in contemporary society, which has the neoliberal State as its fundamental form of social regularization. We will analyze state massacres, phenomena that have become more frequent in recent years, and one of their specificities: revenge for the murder of a police officer as a motivation. Through selected cases, we intend to contribute to a critical analysisof this phenomenon in Brazil.

Заметка об аксиомах почти контактных метрических гиперповерхностей для почти эрмитовых многообразий

From 1950s, it is known that an almost contact metric structure is in­duced on an arbitrary oriented hypersurface in an almost Hermitian mani­fold. In accordance with the definition, an almost Hermitian manifold satisfies the axiom of almost contact hypersurfaces endowed with a some property, if an almost contact hypersurface with this property passes through every point of considered almost Hermitian manifold. In the present note, we discuss some problems related to almost con­tact metric hypersurfaces axioms for almost Hermitian manifolds. In par­ticular, we select some special types of almost contact metric hypersur­faces axioms for almost Hermitian manifolds. We mark out the axioms consisting of the conditions for the almost contact metric structure on the hypersurface of an almost Hermitian manifold to belong to a special class (for example, to the class of Sasakian or quasi-Sasakian structures). We also mark out the axioms that are related to the second fundamental form of the immersion of the almost contact metric hypersurface into an almost Hermitian manifold.

The Introduction of Modal Verbs in English Textbooks Across Levels in China

The introduction of modal verbs in English textbooks across levels in China was surveyed and compared with Chinese learners’ use in writing to explore the potential impact of textbook introduction on learners’ real use. The results have revealed (1) divergences between the sequence of modal verb presentation in textbooks and their frequency order in native speakers’ use, the unusual lower coverage of WOULD and early introduction of SHALL in particular; (2) the imbalanced coverage of different forms and senses of modal verbs in textbooks, notwithstanding that learners deploy some senses or forms in writing even if these usages have not been introduced; and (3) the constrained way of introduction—English to Chinese translation in vocabulary, simple explanation and exemplification in sentence context. Core senses and fundamental forms introduced early in textbooks are used most frequently in learners’ writing. Modal verbs with the perfect aspect structure and their associated senses introduced in textbooks at university levels rarely occur in learners’ writing even at advanced levels. Limited agreement between textbook introduction and learners’ use has been detected.

Elucidation of Kinetic Studies in Biosurfactant Fermentative Production and Their Behaviour: A Mini Review

Bioprocess engineering, which include kinetic behaviour, is a fundamental form of developing effective product performance and functionality. Kinetic studies are one of the most important steps in any bioprocess and bioproduct development to elucidate the production phase and product behaviour. The lack of comprehensive understanding and limited knowledge of the kinetic behaviour of biosurfactant production, especially in the complex fermentation process involving microorganisms, substrate, and product formation, is major challenge and hinder the development of optimized biosurfactant in the industrial scale. Kinetic studies in the field of biosurfactants need to be emphasized to better understand the mechanisms of biosurfactant formation through biomass growth and substrate consumption. In this mini-review, various kinetic models used in biosurfactant work were critically discussed. The objective of this study is to analyze the kinetic studies of biomass growth, biosurfactant formation and substrate consumption focusing on the biosurfactant itself. This review revealed the various kinetic models used in fermentative biosurfactant production and how the different models were used and interpreted based on different substrates and biosurfactant-producing microorganisms. A key feature of this review is its focus on the biosurfactant, which provides valuable insights into the factors that maximize productivity and improve scale-up. Based on previous works, the logistic model is preferred to represent biomass growth, biosurfactant formation and substrate consumption due to its simplicity and rationale. Throughout this study, the kinetic profile of the biosurfactant able to be established and serve as a foundation for the prediction of the biosurfactant behaviour.

Geometry of Warped Product Hemi-Slant Submanifolds of an S-Manifold

The purpose of this paper is to investigate a warped product of hemi-slant submanifolds on an S-manifold. We prove many interesting results for the existence of warped product hemi-slant submanifold of the type Mθ×fM⊥ with ξα∈Mθ of an S-manifold. For such submanifolds, a characterization theorem is proven. In addition, we form an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. We also provide some examples, and the equality case is also considered.

RIGHT CONOID HYPERSURFACES IN FOUR-SPACE

The right conoid hypersurfaces in the four-dimensional Euclidean space $\mathbb{E}^{4}$ are introduced. The matrices corresponding to the fundamental form, Gauss map, and shape operator of these hypersurfaces are calculated. By utilizing the Cayley--Hamilton theorem, the curvatures of these specific hypersurfaces are determined. Furthermore, the conditions for minimality are presented. Additionally, the Laplace--Beltrami operator of this family is computed, and some examples are provided.