Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

17.9K
It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
17.9K
Cartesian Form for Vector Formulation01:26

Cartesian Form for Vector Formulation

910
The Cartesian form for vector formulation is a process to calculate  the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.
910
Vector Algebra: Graphical Method01:10

Vector Algebra: Graphical Method

15.9K
Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
We use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at arbitrary positions. Translate either one of...
15.9K
Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

45
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the Complete...
45
Graphs of Polar Equations01:17

Graphs of Polar Equations

75
The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
75
Cartesian Vector Notation01:28

Cartesian Vector Notation

1.2K
Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
1.2K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Sex-specific multimorbidity clusters and all-cause mortality in relatively healthy older adults: findings from the ASPREE cohort.

medRxiv : the preprint server for health sciences·2026
Same author

Therapeutic effect of mitochondrial transfer on bone tissue diseases: treatment strategy of mitochondrial transplantation and delivery technology.

Journal of orthopaedic translation·2026
Same author

Relationship between oral microbiota and chronic kidney disease: facts and perspectives.

Journal of oral microbiology·2026
Same author

Metabolic dysfunction-associated fatty liver disease and cardiac remodeling in non-ischemic dilated cardiomyopathy.

Nutrition, metabolism, and cardiovascular diseases : NMCD·2026
Same author

DEB-TACE-HAIC combined with donafenib and camrelizumab in the treatment of unresectable hepatocellular carcinoma: a multicenter retrospective study.

Frontiers in immunology·2026
Same author

Nerve Growth Factor: A Multifaceted Regulator of Pain, Inflammation, and Tissue Remodeling in Bone and Joint Disorders.

Cell biochemistry and function·2026
Same journal

The nonlinear porous medium equation for the <i>f</i>-Laplacian: Hamilton-Souplet-Zhang type gradient estimates and implications.

Annali di matematica pura ed applicata·2026
Same journal

On the instability of syzygy bundles on toric surfaces.

Annali di matematica pura ed applicata·2026
Same journal

Counting function estimates for coherent frames and Riesz sequences.

Annali di matematica pura ed applicata·2025
Same journal

Partial regularity for minima of higher-order quasiconvex integrands with natural Orlicz growth.

Annali di matematica pura ed applicata·2025
Same journal

Optimal and typical <math></math> discrepancy of 2-dimensional lattices.

Annali di matematica pura ed applicata·2024
Same journal

Higher integrability for singular doubly nonlinear systems.

Annali di matematica pura ed applicata·2024
See all related articles

Related Experiment Video

Updated: Nov 8, 2025

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.3K

Vector bundles on rational homogeneous spaces.

Rong Du1, Xinyi Fang2, Yun Gao3

  • 1School of Mathematical Sciences Shanghai Key Laboratory of PMMP, East China Normal University, Rm. 312, Math. Bldg, No. 500, Dongchuan Road, Shanghai, 200241 People's Republic of China.

Annali Di Matematica Pura Ed Applicata
|April 19, 2021
PubMed
Summary
This summary is machine-generated.

We show that uniform r-bundles on rational homogeneous spaces are direct sums of line bundles or unstable under certain conditions. This work partially answers a problem by Muñoz et al. and improves prior results for generalized Grassmannians.

Keywords:
Generalized GrassmannianRational homogeneous spaceVector bundle

More Related Videos

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
09:33

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

28.8K
Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

11.8K

Related Experiment Videos

Last Updated: Nov 8, 2025

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.3K
Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
09:33

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

28.8K
Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
11:00

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section

Published on: July 19, 2016

11.8K

Area of Science:

  • Algebraic Geometry
  • Differential Geometry

Background:

  • Complex rational homogeneous spaces are fundamental objects in algebraic geometry.
  • Understanding the properties of vector bundles on these spaces is crucial for various mathematical applications.

Purpose of the Study:

  • To investigate the conditions under which a uniform r-bundle on a complex rational homogeneous space decomposes into line bundles.
  • To partially answer a problem posed by Muñoz et al. regarding the properties of such bundles.
  • To improve existing theorems for generalized Grassmannians and provide explicit bounds for the generalized Grauert-Mülich-Barth theorem.

Main Methods:

  • Analysis of poly-uniform vector bundles with respect to special families of lines.
  • Rank analysis based on the specific rational homogeneous space.
  • Calculation of relative tangent bundles between rational homogeneous spaces.

Main Results:

  • A uniform r-bundle E on a complex rational homogeneous space X, under specific conditions on its rank and poly-uniformity, is either a direct sum of line bundles or unstable.
  • For generalized Grassmannians, the r-bundle E splits as a direct sum of line bundles if its rank is sufficiently small.
  • Explicit bounds for the generalized Grauert-Mülich-Barth theorem are derived.

Conclusions:

  • The study provides a deeper understanding of vector bundle decomposition on rational homogeneous spaces.
  • The results refine and extend previous theorems in algebraic geometry.
  • New explicit bounds for a significant theorem are established, opening avenues for further research.