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Related Concept Videos

Improper Integrals: Infinite Intervals01:29

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An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
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When two or more physical quantities are linked by a single relationship, a change in one variable necessarily affects the others. This interdependence forms the basis of related rates analysis, which examines how different quantities change with respect to time. A classic physical example is an expanding balloon, where the size of the balloon changes continuously as air is added.For a hot air balloon, the inflated envelope is commonly idealized as a perfect sphere to simplify mathematical...
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Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
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An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the...
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Updated: Mar 21, 2026

Measurement of the Directional Information Flow in fNIRS-Hyperscanning Data using the Partial Wavelet Transform Coherence Method
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Inflationary spacetimes are incomplete in past directions.

Arvind Borde1, Alan H Guth, Alexander Vilenkin

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Many inflating spacetimes violate the weak energy condition. This study shows that even without this condition, inflating universes must have an incomplete past, requiring new physics beyond inflation to describe their origins.

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Area of Science:

  • Cosmology
  • General Relativity
  • Theoretical Physics

Background:

  • Singularity theorems in cosmology rely on the weak energy condition.
  • Many models of cosmic inflation are expected to violate this condition.

Purpose of the Study:

  • To investigate the past completeness of inflating spacetimes without assuming the weak energy condition.
  • To provide a kinematical argument for the incompleteness of inflationary models in the past.

Main Methods:

  • A simple kinematical argument is employed.
  • No energy conditions are assumed.
  • A bound is derived for the integral of the Hubble parameter along past-directed timelike or null geodesics.

Main Results:

  • Inflating or rapidly expanding cosmological models are shown to be incomplete in null and timelike past directions.
  • A specific bound on the integrated Hubble parameter is obtained.

Conclusions:

  • Inflationary models necessitate additional physics to account for the past boundary of the inflating region.
  • The standard singularity theorems may not apply to all inflating spacetimes due to the violation of energy conditions.