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Rearrangement Inequalities s
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Rearrangement may refer to: Chemistry * Rearrangement reaction Mathematics * Rearrangement inequality * The Riemann rearrangement theorem, also called the Riemann series theorem ** see also Lévy–Steinitz theorem * A permutation of the terms of a conditionally convergent series Genetics * Chromosomal rearrangements, such as: ** Translocations ** Ring chromosomes ** Chromosomal inversion An inversion is a chromosome rearrangement in which a segment of a chromosome becomes inverted within its original position. An inversion occurs when a chromosome undergoes a two breaks within the chromosomal arm, and the segment between the two br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rearrangement Reaction
In organic chemistry, a rearrangement reaction is a broad class of organic reactions where the carbon skeleton of a molecule is rearranged to give a structural isomer of the original molecule. Often a substituent moves from one atom to another atom in the same molecule, hence these reactions are usually intramolecular. In the example below, the substituent R moves from carbon atom 1 to carbon atom 2: :\underset\ce\ce\underset\ce\ce Intermolecular rearrangements also take place. A rearrangement is not well represented by simple and discrete electron transfers (represented by curved arrows in organic chemistry texts). The actual mechanism of alkyl groups moving, as in Wagner-Meerwein rearrangement, probably involves transfer of the moving alkyl group fluidly along a bond, not ionic bond-breaking and forming. In pericyclic reactions, explanation by orbital interactions give a better picture than simple discrete electron transfers. It is, nevertheless, possible to draw the curv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rearrangement Inequality
In mathematics, the rearrangement inequality states that x_n y_1 + \cdots + x_1 y_n \leq x_ y_1 + \cdots + x_ y_n \leq x_1 y_1 + \cdots + x_n y_n for every choice of real numbers x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n and every permutation x_, \ldots, x_ of x_1, \ldots, x_n. If the numbers are different, meaning that x_1 < \cdots < x_n \quad \text \quad y_1 < \cdots < y_n, then the lower bound is attained only for the permutation which reverses the order, that is, for all and the upper bound is attained only for the identity, that is, for all Note that the rearrangement inequality makes no assumptions on the signs of the real numbers. Applications Many important inequalities can be proved by the rearrangement inequality, such as the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Series Theorem
In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or Divergent series, diverges. This implies that a series of real numbers is Absolute convergence, absolutely convergent if and only if it is Unconditional convergence, unconditionally convergent. As an example, the series 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ⋯ converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ⋯, which sums to infinity. Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lévy–Steinitz Theorem
In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in R''n'' can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method. In an expository article, Peter Rosenthal stated the theorem in the following way.. : The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a subspace (i.e., a set of the form ''v'' + ''M'', where ''v'' is a given vector and ''M'' is a linear subspace). See also *Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditionally Convergent Series
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Definition More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if \lim_\,\sum_^m a_n exists (as a finite real number, i.e. not \infty or -\infty), but \sum_^\infty \left, a_n\ = \infty. A classic example is the alternating harmonic series given by 1 - + - + - \cdots =\sum\limits_^\infty , which converges to \ln (2), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see ''Riemann series theorem''. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in R''n'' can converge. A typical conditionally convergent integral is that on the non-negative real axis of \sin (x^2) (see Fresnel integral). See also *Absolute convergen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromosomal Rearrangement
In genetics, a chromosomal rearrangement is a mutation that is a type of chromosome abnormality involving a change in the structure of the native chromosome. Such changes may involve several different classes of events, like deletions, duplications, inversions, and translocations. Usually, these events are caused by a breakage in the DNA double helices at two different locations, followed by a rejoining of the broken ends to produce a new chromosomal arrangement of genes, different from the gene order of the chromosomes before they were broken. Structural chromosomal abnormalities are estimated to occur in around 0.5% of newborn infants. Some chromosomal regions are more prone to rearrangement than others and thus are the source of genetic diseases and cancer. This instability is usually due to the propensity of these regions to misalign during DNA repair, exacerbated by defects of the appearance of replication proteins (like FEN1 or Pol δ) that ubiquitously affect the integri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromosomal Translocation
In genetics, chromosome translocation is a phenomenon that results in unusual rearrangement of chromosomes. This includes balanced and unbalanced translocation, with two main types: reciprocal-, and Robertsonian translocation. Reciprocal translocation is a chromosome abnormality caused by exchange of parts between non-homologous chromosomes. Two detached fragments of two different chromosomes are switched. Robertsonian translocation occurs when two non-homologous chromosomes get attached, meaning that given two healthy pairs of chromosomes, one of each pair "sticks" and blends together homogeneously. A gene fusion may be created when the translocation joins two otherwise-separated genes. It is detected on cytogenetics or a karyotype of affected cells. Translocations can be balanced (in an even exchange of material with no genetic information extra or missing, and ideally full functionality) or unbalanced (where the exchange of chromosome material is unequal resulting in extra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Chromosome
A ring chromosome is an aberrant chromosome whose ends have fused together to form a ring. Ring chromosomes were first discovered by Lilian Vaughan Morgan in 1926. A ring chromosome is denoted by the symbol ''r'' in human genetics and ''R'' in ''Drosophila'' genetics. Ring chromosomes may form in cells following genetic damage by mutagens like radiation, but they may also arise spontaneously during development. Formation In order for a chromosome to form a ring, both ends of the chromosome are usually missing, enabling the broken ends to fuse together. In rare cases, the telomeres at the ends of a chromosome fuse without any loss of genetic material, which results in a normal phenotype. Complex rearrangements, including segmental microdeletions and microduplications, have been seen in numerous ring chromosomes, providing important clues regarding the mechanisms of their formation. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
