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

Uniform Distribution01:19

Uniform Distribution

6.0K
The uniform distribution is a continuous probability distribution of events with an equal probability of occurrence. This distribution is rectangular.
Two essential properties of this distribution are
6.0K
Uniform Circular Motion01:14

Uniform Circular Motion

21.6K
Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is undergoing uniform circular motion. The second, minute, and hour hands of a watch also undergo uniform circular motion. It is hard to believe that points on these rotating objects are actually accelerating, even though the rotation rate is constant. To understand this, we must analyze the motion in terms of...
21.6K
Non-uniform Circular Motion01:22

Non-uniform Circular Motion

9.5K
In uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all circular motion occurs at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of motion. In that case, the motion is called non-uniform circular motion, and an additional acceleration is introduced, which is in the direction tangential to the circle. 
For example, such...
9.5K
Uniform Depth Channel Flow01:27

Uniform Depth Channel Flow

536
Uniform depth channel flow keeps fluid depth consistent along channels such as irrigation canals. In natural channels, such as rivers, approximate uniform flow is often assumed. This condition occurs when the channel’s bottom slope matches the energy slope, balancing potential energy lost from gravity with head loss due to shear stress. This balance prevents depth changes along the channel length, resulting in a steady, uniform flow.Uniform flow in open channels with a constant cross-section...
536
Drug Distribution: Volume of Distribution01:25

Drug Distribution: Volume of Distribution

7.3K
The volume of distribution refers to the theoretical volume necessary to contain the entire amount of an administered drug at the same concentration observed in the blood plasma. The body's intracellular fluid compartment, which makes up two-thirds of the total body water, is contrasted with the extracellular fluid compartment—comprising plasma and interstitial fluid—that accounts for one-third. The volume of distribution can vary depending on the characteristics of the drug.
7.3K
Uniform Depth Channel Flow: Problem Solving01:18

Uniform Depth Channel Flow: Problem Solving

451
To calculate the flow rate for a trapezoidal channel, first, identify the bottom width, side slope, and flow depth of the channel. The cross-sectional area (A) corresponding to the depth of flow (y), channel bottom width (B), and side slope (θ) is determined by:Next, calculate the wetted perimeter, which includes the bottom width and the sloped side lengths in contact with the water. Using the values of the cross-sectional area and the wetted perimeter, determine the hydraulic radius by...
451

You might also read

Related Articles

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

Sort by
Same author

A central limit theorem for integer partitions into small powers.

Monatshefte fur Mathematik·2024
Same author

Quantum invariants of hyperbolic knots and extreme values of trigonometric products.

Mathematische Zeitschrift·2022
Same author

Exact order of extreme <math></math> discrepancy of infinite sequences in arbitrary dimension.

Archiv der Mathematik. Archives of mathematics. Archives mathematiques·2022
Same author

Stieltjes constants of <i>L</i>-functions in the extended Selberg class.

The Ramanujan journal·2021
Same author

Diophantine equations in separated variables.

Periodica mathematica Hungarica·2019
Same author

Extremal problems for topological indices in combinatorial chemistry.

Journal of computational biology : a journal of computational molecular cell biology·2005

Related Experiment Video

Updated: Jan 22, 2026

An Orbital Shaking Culture of Mammalian Cells in O-shaped Vessels to Produce Uniform Aggregates
05:40

An Orbital Shaking Culture of Mammalian Cells in O-shaped Vessels to Produce Uniform Aggregates

Published on: January 7, 2019

10.1K

On Weyl products and uniform distribution modulo one.

Christoph Aistleitner1, Gerhard Larcher2, Friedrich Pillichshammer2

  • 11Institute for Analysis and Number Theory, Graz University of Technology, Graz, Austria.

Monatshefte Fur Mathematik
|July 2, 2019
PubMed
Summary

This study analyzes trigonometric products using evenly distributed points in the unit interval. We establish new bounds for these products, improving upon prior research and exploring special sequences.

Keywords:
Kronecker sequenceStar-discrepancyTrigonometric productvan der Corput sequence

More Related Videos

Fabrication of Uniform Nanoscale Cavities via Silicon Direct Wafer Bonding
10:32

Fabrication of Uniform Nanoscale Cavities via Silicon Direct Wafer Bonding

Published on: January 9, 2014

10.2K
Xylem Water Distribution in Woody Plants Visualized with a Cryo-scanning Electron Microscope
10:47

Xylem Water Distribution in Woody Plants Visualized with a Cryo-scanning Electron Microscope

Published on: June 20, 2019

9.2K

Related Experiment Videos

Last Updated: Jan 22, 2026

An Orbital Shaking Culture of Mammalian Cells in O-shaped Vessels to Produce Uniform Aggregates
05:40

An Orbital Shaking Culture of Mammalian Cells in O-shaped Vessels to Produce Uniform Aggregates

Published on: January 7, 2019

10.1K
Fabrication of Uniform Nanoscale Cavities via Silicon Direct Wafer Bonding
10:32

Fabrication of Uniform Nanoscale Cavities via Silicon Direct Wafer Bonding

Published on: January 9, 2014

10.2K
Xylem Water Distribution in Woody Plants Visualized with a Cryo-scanning Electron Microscope
10:47

Xylem Water Distribution in Woody Plants Visualized with a Cryo-scanning Electron Microscope

Published on: June 20, 2019

9.2K

Area of Science:

  • Number Theory
  • Harmonic Analysis
  • Numerical Integration

Background:

  • Trigonometric products are fundamental in various mathematical fields.
  • Understanding their asymptotic behavior is crucial for analyzing sequences and distributions.
  • Previous work by Hlawka provided initial bounds for such products.

Purpose of the Study:

  • To investigate the asymptotic behavior of trigonometric products involving sequences of points.
  • To establish new, improved lower and upper bounds for these products.
  • To examine the behavior of these products for specific types of sequences, namely Kronecker and van der Corput sequences.

Main Methods:

  • Analysis of trigonometric products of the form $\prod_{k=1}^{N} 2\sin(\pi x_k)$ as $N \to \infty$.
  • Utilizing the concept of star-discrepancy to derive bounds for the trigonometric products.
  • Examining specific cases involving Kronecker and van der Corput sequences.

Main Results:

  • Derived matching lower and upper bounds for the trigonometric products.
  • These bounds are expressed in terms of the star-discrepancy of the underlying point sets.
  • The results improve upon the bounds previously established by Hlawka.

Conclusions:

  • The star-discrepancy effectively characterizes the asymptotic behavior of these trigonometric products.
  • The study provides a refined understanding of trigonometric products for uniformly distributed sequences.
  • Probabilistic analogues of the main results are also considered, suggesting broader applicability.