Study optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
problem Optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
method Using Gauss and Codazzi equations, prove an optimal inequality.
result Prove an optimal inequality for contact CR-warped product submanifolds.
The article surveys recent results on warped and CR-warped product submanifolds in Kaehler manifolds.
problem Characterizing submanifolds in Kaehler manifolds using warped and CR-warped products.
method Analyzing properties of warped and CR-warped products in the context of Kaehler manifolds.
result Presented several results on the geometry of warped and CR-warped product submanifolds.
The paper proves a new inequality for CR-warped product submanifolds in complex space forms.
problem Proving a new inequality for CR-warped product submanifolds in complex space forms.
method Developed a first Chen inequality for CR-warped product submanifolds in complex space forms.
result The bound is sharp and uniform in the sign of the holomorphic sectional curvature.
Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?φ…
The paper studies geometric properties of a specific type of submanifolds in Kaehler manifolds.
problem Analyzing the geometry of pointwise semi-slant warped products in locally conformal Kaehler manifolds.
method The study extends Chen's inequality for CR-warped product submanifolds and investigates equality cases.
result Several results extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds.
n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for…
Study on warped products in contact skew-CR submanifolds with inequality and examples.
problem Analyzing warped products in contact skew-CR submanifolds.
method Established an inequality for the squared norm of the second fundamental form.
result Derived inequality and provided non-trivial examples.
Study on warped product pointwise semi-slant submanifolds in Sasakian and cosymplectic manifolds.
problem Investigate the geometry of pointwise semi-slant submanifolds and their warped products.
method Using the notion of pointwise slant submanifolds, investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds.
result Prove the existence of no proper pointwise semi-slant warped product submanifolds other than contact CR-warped products in Sasakian manifolds.
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called D1-minimality, leading to an optimal general inequality. The paper characterizes warped product submanifolds in LCS-manifolds.
problem Characterizing warped product submanifolds in LCS-manifolds.
method Analyzing contact CR-warped product and pseudo-slant submanifolds, and constructing examples.
result There do not exist any proper warped product bi-slant submanifolds of LCS-manifolds.
B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product subman…
In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if M=N⊥×fNT is a warped CR-submanifold such that N⊥ is φ?-anti-invariant and NT is φ?-invariant, then M is a CR-product. We…
Paper explores existence and nonexistence of submanifolds in contact geometry.
problem Existence and nonexistence of warped product submanifolds in almost contact manifolds.
method Generalizes results for existence and nonexistence, constructs examples, addresses open problems.
result Offers decisive results for existence and nonexistence of submanifolds.
The CR δ-invariant for CR-submanifolds was introduced in a recent article [B. Y. Chen, An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23} (2012), no. 3, 1250045 (17 pages)]. In this paper, we prove two new optimal inequalities for anti-holomorphic su…
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Characterizes metallic structures on product manifolds and provides conditions for warped products to be metallic.
problem Understanding metallic structures on product manifolds and conditions for warped products to be metallic.
method Characterization through metallic maps and necessary/sufficient conditions for warped products.
result Characterizes metallic structures and provides conditions for warped products to be metallic.
Productivity and credit limits affect aggregate production in non-monotonic ways.
problem Understanding how aggregate production is influenced by individual characteristics and financial constraints.
method Analytical proof of non-monotonic effects of productivity and credit limits on aggregate production in a general equilibrium model.
result Equilibrium aggregate production can be non-monotonic in both individual productivity and credit limit.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
ProductNet curates high-quality product datasets for better product understanding.
problem Lack of high-quality product datasets for product representation learning.
method Curated high-quality product datasets with a multi-modal deep neural network and active learning.
result Master model yields high categorization accuracy (94.7% top-1 accuracy for 1240 classes).
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Bo…
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.
Generalizes warped product submersion to conformal case.
problem Understanding angles preservation in submersions.
method Introduces conformal warped product submersion.
result Fundamental tensors derived for conformal submersion.
Model learns product vectors from baskets and browsing sessions for better complementary product recommendations.
problem Inferring complementary products from basket and browsing data.
method Proposes BB2vec model that learns product vectors from both baskets and browsing sessions.
result The BB2vec model improves complementary product recommendations and alleviates the cold start problem.
The paper explores geometric properties of Riemannian warped product maps and their curvature.
problem Investigating the geometric properties of Riemannian warped product maps.
method The approach involves establishing conditions for geodesics, deriving curvature tensors, and examining various types of maps.
result Derivation of integral formula for scalar curvature of conformal Riemannian warped product maps.
In this article we obtain classification results on the quasi-product production functions in terms of the geometry of their associated graph hypersurfaces, generalizing in a new setting some recent results concerning basic production models. In particular, we obtain several results on the geometry of Spillman-Mitscher…
Constructing a conformal product structure on S3 using the Reeb foliation.
problem First example of conformal product structure on compact simply connected manifold.
method Reeb foliation
result Conformal product structure on S3 The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
Defines a new free product for coarse spaces.
problem No specific problem stated; focuses on a new mathematical concept.
method Defines and analyzes free products for coarse spaces.
result Free products preserve coarse properties and have a dimension bound.
The study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
The study determines dominations between manifold products and semi-norm finiteness.
problem Understanding dominations between different products of manifolds.
method Analyzing the finiteness of product-associated semi-norms on fundamental classes.
result Partial answers to M. Gromov's questions on manifold product dominations and semi-norms.
Production networks amplify economic growth through technology diffusion.
problem Understanding how technology improvements propagate through production networks.
method Analyzing a production network model to study the effects of technological improvements.
result Longer production chains lead to faster price reduction and GDP growth.
In this paper, we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat. As an immediate consequence, we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable.
A new method for analyzing product competition using low-dimensional embeddings.
problem Computational challenges in studying product-level competition for millions of products.
method Product2Vec, a method based on representation learning algorithm Word2Vec.
result The method produces more accurate demand forecasts and price elasticities compared to state-of-the-art models.
Economies grow by upgrading the type of products they produce and export. The technology, capital, institutions and skills needed to make such new products are more easily adapted from some products than others. We study the network of relatedness between products, or product space, finding that most upscale products a…
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Product formula for K-stability proved.
problem K-stability of products of Fano varieties.
method Proved a product formula for δ-invariant.
result Product of K-stable Fano varieties is K-stable.
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Method finds reference products for a given item.
problem Finding relevant products for a given item.
method Product representation learning and fingerprint-type vector searching.
result The method outperforms peer services in search return rate and precision.
Paper reinterprets marginal productivity theory using vectorial products, challenging traditional ethical interpretations.
problem Challenges traditional ethical interpretations of marginal productivity theory.
method Formulates marginal productivity theory using vectorial marginal products, contrasting with traditional scalar approach.
result Vectorial marginal products conflict with traditional distributive shares picture of property.
Paper calculates volatility distribution for cumulative production.
problem Volatility distribution for cumulative production.
method Generalizes study of volatility with arbitrary distribution function.
result Exact probability distribution function for volatility.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our …
Dynamic pricing improves sales of low-sale products using online clustering.
problem Low-sale products lack sufficient data for traditional dynamic pricing algorithms.
method Online clustering of product demand and dynamic pricing decisions based on cluster analysis.
result The proposed algorithms significantly outperform traditional single-product pricing policies in increasing revenue.