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In organic chemistry, a cross-coupling reaction is a reaction where two different fragments are joined. Cross-couplings are a subset of the more general coupling reactions. Often cross-coupling reactions require metal catalysts. One important reaction type is this:
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements.
The most common type of coupling reaction is the cross coupling reaction. [1] [2] [3] Richard F. Heck, Ei-ichi Negishi, and Akira Suzuki were awarded the 2010 Nobel Prize in Chemistry for developing palladium-catalyzed cross coupling reactions. [4] [5] Broadly speaking, two types of coupling reactions are recognized:
Quizlet made its first acquisition in March 2021, with the purchase of Slader, which offered detailed explanations of textbook concepts and practice problems, and eventually incorporated it into its paid platform, Quizlet Plus. [20] [21] [22] In November 2022, Quizlet announced a new CEO, Lex Bayer, the former CEO of Starship Technologies. [23]
Many decarboxylative cross coupling reactions involve the breaking of sp 2 C–COOH and sp C–COOH bonds, therefore subsequent studies have attempted to enable cross coupling with sp 3 C carboxylic acids. One such reaction by Shang et al. described a palladium catalyzed cross coupling that enables the formation of functionalized pyridines ...
In cross-coupling reactions, the component reagents are called cross-coupling partners or simply coupling partners. These reagents can be further classified according to their nucleophilic vs electrophilic character: R-X + R'-Y → R-R' + XY. Typically the electrophilic coupling partner (R-X) is an aryl halide, but triflates are also used ...
Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points C, D, and every blue circle separates the two points. In geometry, Apollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa.
Convex hull (red) of a polygon (yellow). The usual set closure from topology is a closure operator. Other examples include the linear span of a subset of a vector space, the convex hull or affine hull of a subset of a vector space or the lower semicontinuous hull ¯ of a function : {}, where is e.g. a normed space, defined implicitly (¯) = ¯, where is the epigraph of a function .
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