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Related Concept Videos

Linear time-invariant Systems01:23

Linear time-invariant Systems

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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
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meta-Directing Deactivators: –NO2, –CN, –CHO, –⁠CO2R, –COR, –CO2H01:13

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All meta-directing substituents are deactivating groups. These substituents withdraw electrons from the aromatic ring, making the ring less reactive toward electrophilic substitution. For example, the nitration of nitrobenzene is 100,000 times slower than that of benzene because of the deactivating effect of the nitro group. The first step in an electrophilic aromatic substitution is the addition of an electrophile to form a resonance-stabilized carbocation. The energy diagrams for...
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Secondary amines react with nitrous acid to form N-nitrosamines, as depicted in Figure 1. Nitrous acid, a weak and unstable acid, is formed in situ from an aqueous solution of sodium nitrite and strong acids, such as hydrochloric acid or sulfuric acid, in cold conditions. In the presence of an acid, the nitrous acid gets protonated. The subsequent loss of water results in the formation of the electrophile known as nitrosonium ion.
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SN2 Reaction: Kinetics02:14

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Kinetic Studies and Significance
In a chemical reaction, a relationship exists between the concentration of reactants and the rate at which the reaction proceeds. The study to measure this relationship is known as the kinetics of a chemical reaction. Kinetic studies are used to deduce the rate law of a chemical reaction, which provides information about the species involved during the transition state of the rate-determining step. Thus, kinetic studies help to derive the mechanism of a...
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SN2 Reaction: Mechanism02:27

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The kinetic studies of SN2 reactions suggest an essential feature of its mechanism: it is a single-step process without intermediates. Here, both the nucleophile and the substrate participate in the rate-determining step.
The presence of the more electronegative halogen in the substrate creates a polarized carbon-halide bond. The halide pulls the electron cloud generating an electrophilic center at the carbon atom. Thus, the carbon atom carries a partial positive charge while the halide has a...
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SN2 Reaction: Transition State02:26

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An SN2 reaction of an alkyl halide is a single-step process in which bond formation between the nucleophile and the substrate and bond breaking between the substrate and the halide occurs simultaneously through a transition state without forming an intermediate.
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Purification and Expansion of Mouse Invariant Natural Killer T Cells for in vitro and in vivo Studies
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SU(2) -invariant -instantons.

Jason D Lotay1, Goncalo Oliveira2

  • 11Department of Mathematics, University College London, London, WC1E 6BT UK.

Mathematische Annalen
|July 2, 2019
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Summary
This summary is machine-generated.

This study introduces -instantons with SU(2) -symmetry, detailing their theory, existence, and classification. We explore their behavior in specific settings, observing phenomena like bubbling and removable singularities.

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Area of Science:

  • Differential Geometry
  • Mathematical Physics

Background:

  • Instantons are crucial in gauge theories and geometry.
  • Understanding instanton behavior on manifolds with special holonomy is an active research area.

Purpose of the Study:

  • To systematically study -instantons with SU(2) -symmetry.
  • To establish foundational theory and provide existence, non-existence, and classification results.
  • To analyze instanton behavior on specific manifolds with distinct holonomy metrics.

Main Methods:

  • Development of foundational theoretical framework for -instantons.
  • Analysis of instanton properties on manifolds.
  • Investigation of limiting behaviors including bubbling and singularity removal.

Main Results:

  • Established existence, non-existence, and classification criteria for -instantons.
  • Demonstrated differing instanton behaviors on due to distinct holonomy metrics and volume growths.
  • Provided an explicit example of sequences exhibiting bubbling and removable singularities.

Conclusions:

  • The study provides a comprehensive framework for understanding -instantons.
  • The results highlight the impact of manifold geometry on instanton properties.
  • The observed phenomena offer insights into the limiting behavior of instantons.