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14285810
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D-alanine—D-alanine ligase
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Enzyme belonging to the ligase family
In enzymology, a D-alanine—D-alanine ligase (EC 6.3.2.4) is an enzyme that catalyzes the chemical reaction
ATP + 2 D-alanine formula_0 ADP + phosphate + D-alanyl-D-alanine
Thus, the two substrates of this enzyme are ATP and D-alanine, whereas its 3 products are ADP, phosphate, and D-alanyl-D-alanine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is D-alanine:D-alanine ligase (ADP-forming). Other names in common use include alanine:alanine ligase (ADP-forming), and alanylalanine synthetase. This enzyme participates in d-alanine metabolism and peptidoglycan biosynthesis. Phosphinate and D-cycloserine are known to inhibit this enzyme.
The N-terminal region of the D-alanine—D-alanine ligase is thought to be involved in substrate binding, while the C-terminus is thought to be a catalytic domain.
Structural studies.
As of late 2007, 8 structures have been solved for this class of enzymes, with PDB accession codes 1EHI, 1IOV, 1IOW, 2DLN, 2FB9, 2I80, 2I87, and 2I8C.
References.
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https://en.wikipedia.org/wiki?curid=14285810
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14285821
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D-alanine—poly(phosphoribitol) ligase
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In enzymology, a D-alanine—poly(phosphoribitol) ligase (EC 6.1.1.13) is an enzyme that catalyzes the chemical reaction
ATP + D-alanine + poly(ribitol phosphate) formula_0 AMP + diphosphate + O-D-alanyl-poly(ribitol phosphate)
The 3 substrates of this enzyme are ATP, D-alanine, and poly(ribitol phosphate), whereas its 3 products are AMP, diphosphate, and O-D-alanyl-poly(ribitol phosphate).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is D-alanine:poly(phosphoribitol) ligase (AMP-forming). Other names in common use include D-alanyl-poly(phosphoribitol) synthetase, D-alanine: membrane acceptor ligase, D-alanine-D-alanyl carrier protein ligase, D-alanine-membrane acceptor ligase, and D-alanine-activating enzyme. This enzyme participates in d-alanine metabolism.
References.
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https://en.wikipedia.org/wiki?curid=14285821
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14285830
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D-aspartate ligase
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Class of enzymes
In enzymology, a D-aspartate ligase (EC 6.3.1.12) is an enzyme that catalyzes the chemical reaction
ATP + D-aspartate + [beta-GlcNAc-(1->4)-Mur2Ac(oyl-L-Ala-gamma-D-Glu-L-Lys-D-Ala-D- Ala)]n formula_0 [beta-GlcNAc-(1->4)-Mur2Ac(oyl-L-Ala-gamma-D-Glu-6-N-(beta-D-Asp)-L- Lys-D-Ala-D-Ala)]n + ADP + phosphate
The 4 substrates of this enzyme are ATP, D-aspartate, beta-GlcNAc-(1->4)-Mur2Ac(oyl-L-Ala-gamma-D-Glu-L-Lys-D-Ala-D-, and Ala)]n, whereas its 4 products are beta-GlcNAc-(1->4)-Mur2Ac(oyl-L-Ala-gamma-D-Glu-6-N-(beta-D-Asp)-L-, Lys-D-Ala-D-Ala)]n, ADP, and phosphate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthases). The systematic name of this enzyme class is D-aspartate:[beta-GlcNAc-(1->4)-Mur2Ac(oyl-L-Ala-gamma-D-Glu-L-Lys-D -Ala-D-Ala)]n ligase (ADP-forming). Other names in common use include Aslfm, UDP-MurNAc-pentapeptide:D-aspartate ligase, and D-aspartic acid-activating enzyme.
References.
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https://en.wikipedia.org/wiki?curid=14285830
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14285841
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Dethiobiotin synthase
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In enzymology, a dethiobiotin synthase (EC 6.3.3.3) is an enzyme that catalyzes the chemical reaction
ATP + 7,8-diaminononanoate + CO2 formula_0 ADP + phosphate + dethiobiotin
The 3 substrates of this enzyme are ATP, 7,8-diaminononanoate, and CO2, whereas its 3 products are ADP, phosphate, and dethiobiotin.
This enzyme belongs to the family of ligases, specifically the cyclo-ligases, which form carbon-nitrogen bonds. The systematic name of this enzyme class is 7,8-diaminononanoate:carbon-dioxide cyclo-ligase (ADP-forming). This enzyme is also called desthiobiotin synthase. This enzyme participates in biotin metabolism.
Structural studies.
As of late 2007, 14 structures have been solved for this class of enzymes, with PDB accession codes 1A82, 1BS1, 1BYI, 1DAD, 1DAE, 1DAF, 1DAG, 1DAH, 1DAI, 1DAK, 1DAM, 1DBS, 1DTS, and 2QMO.
References.
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https://en.wikipedia.org/wiki?curid=14285841
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14285852
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Dicarboxylate—CoA ligase
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In enzymology, a dicarboxylate—CoA ligase (EC 6.2.1.23) is an enzyme that catalyzes the chemical reaction
ATP + an alphaomega-dicarboxylic acid + CoA formula_0 AMP + diphosphate + an omega-carboxyacyl-CoA
The 3 substrates of this enzyme are ATP, alphaomega-dicarboxylic acid, and CoA, whereas its 3 products are AMP, diphosphate, and omega-carboxyacyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is omega-dicarboxylate:CoA ligase (AMP-forming). Other names in common use include carboxylyl-CoA synthetase, and dicarboxylyl-CoA synthetase.
References.
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14285856
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Dihydrofolate synthase
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Class of enzymes
In enzymology, a dihydrofolate synthase (EC 6.3.2.12) is an enzyme that catalyzes the chemical reaction
ATP + 7,8-dihydropteroate + L-glutamate formula_0 ADP + phosphate + 7,8-dihydropteroylglutamate
The 3 substrates of this enzyme are ATP, 7,8-dihydropteroate, and L-glutamate, whereas its 3 products are ADP, phosphate, and 7,8-dihydropteroylglutamate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is 7,8-dihydropteroate:L-glutamate ligase (ADP-forming). Other names in common use include dihydrofolate synthetase, 7,8-dihydrofolate synthetase, H2-folate synthetase, 7,8-dihydropteroate:L-glutamate ligase (ADP), dihydrofolate synthetase-folylpolyglutamate synthetase, folylpoly-(gamma-glutamate) synthetase-dihydrofolate synthase, FHFS, FHFS/FPGS, dihydropteroate:L-glutamate ligase (ADP-forming), and DHFS. This enzyme participates in folate biosynthesis.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1W78, 1W7K, and 2BMB.
References.
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14285865
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Diphthine—ammonia ligase
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Class of enzymes
In enzymology, a diphthine—ammonia ligase (EC 6.3.1.14, "diphthamide synthase", "diphthamide synthetase") is an enzyme that catalyzes the chemical reaction
ATP + diphthine + NH3 formula_0 ADP + phosphate + diphthamide
The 3 substrates of this enzyme are ATP, diphthine, and NH3, whereas its 3 products are ADP, phosphate, and diphthamide.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is diphthine:ammonia ligase (ADP-forming). Other names in common use include diphthamide synthase, and diphthamide synthetase.
References.
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14285876
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Formate—dihydrofolate ligase
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In enzymology, a formate—dihydrofolate ligase (EC 6.3.4.17) is an enzyme that catalyzes the chemical reaction
ATP + formate + dihydrofolate formula_0 ADP + phosphate + 10-formyldihydrofolate
The 3 substrates of this enzyme are ATP, formate, and dihydrofolate, whereas its 3 products are ADP, phosphate, and 10-formyldihydrofolate.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. The systematic name of this enzyme class is formate:dihydrofolate ligase (ADP-forming).
References.
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Further reading.
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14285885
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Formate–tetrahydrofolate ligase
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In enzymology, a formate—tetrahydrofolate ligase (EC 6.3.4.3) is an enzyme that catalyzes the chemical reaction
ATP + formate + tetrahydrofolate formula_0 ADP + phosphate + 10-formyltetrahydrofolate
The 3 substrates of this enzyme are ATP, formate, and tetrahydrofolate, whereas its 3 products are ADP, phosphate, and 10-formyltetrahydrofolate.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. This enzyme participates in glyoxylate and dicarboxylate metabolism and one carbon pool by folate.
In eukaryotes the FTHFS activity is expressed by a multifunctional enzyme, C-1-tetrahydrofolate synthase (C1-THF synthase), which also catalyses the dehydrogenase and cyclohydrolase activities. Two forms of C1-THF syntheses are known, one is located in the mitochondrial matrix, while the second one is cytoplasmic. In both forms the FTHFS domain consists of about 600 amino acid residues and is located in the C-terminal section of C1-THF synthase. In prokaryotes FTHFS activity is expressed by a monofunctional homotetrameric enzyme of about 560 amino acid residues.
Nomenclature.
The systematic name of this enzyme class is formate:tetrahydrofolate ligase (ADP-forming). Other names in common use include:
Examples.
Human genes encoding formate-tetrahydrofolate ligases include:
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1EG7, 1FP7, and 1FPM.
The crystal structure of N(10)-formyltetrahydrofolate synthetase from "Moorella thermoacetica" shows that the subunit is composed of three domains organised around three mixed beta-sheets. There are two cavities between adjacent domains. One of them was identified as the nucleotide binding site by homology modelling. The large domain contains a seven-stranded beta-sheet surrounded by helices on both sides. The second domain contains a five-stranded beta-sheet with two alpha-helices packed on one side while the other two are a wall of the active site cavity. The third domain contains a four-stranded beta-sheet forming a half-barrel. The concave side is covered by two helices while the convex side is another wall of the large cavity. Arg 97 is likely involved in formyl phosphate binding. The tetrameric molecule is relatively flat with the shape of the letter X, and the active sites are located at the end of the subunits far from the subunit interface.
Related enzymes.
The reverse reaction converting 10-formyltetrahydrofolate to tetrahydrofolate is performed by formyltetrahydrofolate dehydrogenase.
References.
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14285895
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Gamma-glutamylhistamine synthase
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Class of enzymes
In enzymology, a γ-glutamylhistamine synthase (EC 6.3.2.18) is an enzyme that catalyzes the chemical reaction
ATP + -glutamate + histamine formula_0 products of ATP breakdown + "N"α-γ--glutamylhistamine
The 3 substrates of this enzyme are ATP, L-glutamate, and histamine, whereas its two products are products of ATP breakdown and Nalpha-gamma-L-glutamylhistamine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is L-glutamate:histamine ligase. Other names in common use include gamma-glutaminylhistamine synthetase, and gamma-GHA synthetase.
References.
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14285905
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Geranoyl-CoA carboxylase
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In enzymology, a geranoyl-CoA carboxylase (EC 6.4.1.5) is an enzyme that catalyzes the chemical reaction
ATP + geranoyl-CoA + HCO3- formula_0 ADP + phosphate + 3-(4-methylpent-3-en-1-yl)pent-2-enedioyl-CoA
The 3 substrates of this enzyme are ATP, geranoyl-CoA, and HCO3-, whereas its 3 products are ADP, phosphate, and 3-(4-methylpent-3-en-1-yl)pent-2-enedioyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon–carbon bonds. The systematic name of this enzyme class is geranoyl-CoA:carbon-dioxide ligase (ADP-forming). Other names in common use include geranoyl coenzyme A carboxylase, and geranyl-CoA carboxylase. It employs one cofactor, biotin.
References.
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14285917
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Glutamate–cysteine ligase
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Enzyme in glutathione biosynthesis
Glutamate–cysteine ligase (GCL) EC 6.3.2.2), previously known as γ-glutamylcysteine synthetase (GCS), is the first enzyme of the cellular glutathione (GSH) biosynthetic pathway that catalyzes the chemical reaction:
-glutamate + -cysteine + ATP formula_0 γ-glutamyl cysteine + ADP + Pi
GSH, and by extension GCL, is critical to cell survival. Nearly every eukaryotic cell, from plants to yeast to humans, expresses a form of the GCL protein for the purpose of synthesizing GSH. To further highlight the critical nature of this enzyme, genetic knockdown of GCL results in embryonic lethality. Furthermore, dysregulation of GCL enzymatic function and activity is known to be involved in the vast majority of human diseases, such as diabetes, Parkinson's disease, Alzheimer's disease, COPD, HIV/AIDS, and cancer. This typically involves impaired function leading to decreased GSH biosynthesis, reduced cellular antioxidant capacity, and the induction of oxidative stress. However, in cancer, GCL expression and activity is enhanced, which serves to both support the high level of cell proliferation and confer resistance to many chemotherapeutic agents.
Function.
Glutamate cysteine ligase (GCL) catalyzes the first and rate-limiting step in the production of the cellular antioxidant glutathione (GSH), involving the ATP-dependent condensation of cysteine and glutamate to form the dipeptide gamma-glutamylcysteine (γ-GC). This peptide coupling is unique in that it occurs between the amino moiety of the cysteine and the terminal carboxylic acid of the glutamate side chain (hence the name gamma-glutamyl cysteine). This peptide bond is resistant to cleavage by cellular peptidases and requires a specialized enzyme, gamma-glutamyl transpeptidase (γGT), to metabolize γ-GC and GSH into its constituent amino acids.
GCL enzymatic activity generally dictates cellular GSH levels and GSH biosynthetic capacity. GCL enzymatic activity is influenced by numerous factors, including cellular expression of the GCL subunit proteins, access to substrates (cysteine is typically limiting in the production of γ-GC), the degree of negative feedback inhibition by GSH, and functionally relevant post-translational modifications to specific sites on the GCL subunits. Given its status as the rate-limiting enzyme in GSH biosynthesis, changes in GCL activity directly equate to changes in cellular GSH biosynthetic capacity. Therefore, therapeutic strategies to alter GSH production have focused on this enzyme.
Regulation.
In keeping with its critical importance in maintaining life, GCL is subject to a multi-level regulation of its expression, function, and activity. GCL expression is regulated at the transcriptional (transcription of the GCLC and GCLM DNA to make mRNA), posttranscriptional (the stability of the mRNA over time), translational (processing of the mRNA into protein), and posttranslational levels (involving modifications to the existing proteins). Although baseline constitutive expression is required to maintain cell viability, expression of the GCL subunits is also inducible in response to oxidative stress, GSH depletion, and exposure to toxic chemicals, with the Nrf2, AP-1, and NF-κB transcription factors regulating the inducible and constitutive expression of both subunits
In terms of enzyme functional regulation, GSH itself acts as a feedback inhibitor of GCL activity. Under normal physiologic substrate concentrations, the GCLC monomer alone may synthesize gamma-glutamylcysteine; however, the normal physiologic levels of GSH (estimated at around 5 mM) far exceeds the GSH Ki for GCLC, suggesting that only the GCL holoenzyme is functional under baseline conditions. However, during oxidative stress or toxic insults that can result in the depletion of cellular GSH or its oxidation to glutathione disulfide (GSSG), the function of any monomeric GCLC in the cell is likely to become quite important. In support of this hypothesis, mice lacking expression of the GCLM subunit due to genetic knockdown exhibit low levels of tissue GSH (~10–20% of the normal level), which is roughly the level of the GSH Ki for monomeric GCLC.
Structure.
Animal glutamate–cysteine ligase.
Animal glutamate cysteine ligase (GCL) is a heterodimeric enzyme composed of two protein subunits that are coded by independent genes located on separate chromosomes:
In the majority of cells and tissues, the expression of GCLM protein is lower than GCLC and GCLM is therefore limiting in the formation of the holoenzyme complex. Thus, the sum total of cellular GCL activity is equal to the activity of the holoenzyme + the activity of the remaining monomeric GCLC. composed of a catalytic and a modulatory subunit. The catalytic subunit is necessary and sufficient for all GCL enzymatic activity, whereas the modulatory subunit increases the catalytic efficiency of the enzyme. Mice lacking the catalytic subunit (i.e., lacking all "de novo" GSH synthesis) die before birth. Mice lacking the modulatory subunit demonstrate no obvious phenotype, but exhibit marked decrease in GSH and increased sensitivity to toxic insults.
Plant glutamate cysteine ligase.
The plant glutamate cysteine ligase is a redox-sensitive homodimeric enzyme, conserved in the plant kingdom. In an oxidizing environment, intermolecular disulfide bridges are formed and the enzyme switches to the dimeric active state. The midpoint potential of the critical cysteine pair is -318 mV. In addition to the redox-dependent control, the plant GCL enzyme is feedback inhibited by glutathione. GCL is exclusively located in plastids, and glutathione synthetase (GS) is dual-targeted to plastids and cytosol, thus GSH and gamma-glutamylcysteine are exported from the plastids. Studies also shown that restricting GCL activity to the cytosol or glutathione biosynthesis to the plastids is sufficient for normal plant development and stress tolerance. Both glutathione biosynthesis enzymes are essential in plants; knock-outs of GCL and GS are lethal to embryo and seedling, respectively.
As of late 2007, six structures have been solved for this class of enzymes, with PDB accession codes 1V4G, 1VA6, 2D32, 2D33, 2GWC, and 2GWD.
References.
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https://en.wikipedia.org/wiki?curid=14285917
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14285924
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Glutamate—ethylamine ligase
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Class of enzymes
In enzymology, a glutamate—ethylamine ligase (EC 6.3.1.6) is an enzyme that catalyzes the chemical reaction
ATP + L-glutamate + ethylamine formula_0 ADP + phosphate + N5-ethyl-L-glutamine
The 3 substrates of this enzyme are ATP, L-glutamate, and ethylamine, whereas its 3 products are ADP, phosphate, and N5-ethyl-L-glutamine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthases). The systematic name of this enzyme class is L-glutamate:ethylamine ligase (ADP-forming). Other names in common use include N5-ethyl-L-glutamine synthetase, theanine synthetase, and N5-ethylglutamine synthetase.
References.
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14285931
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Glutamate—methylamine ligase
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In enzymology, a glutamate—methylamine ligase (EC 6.3.4.12) is an enzyme that catalyzes the chemical reaction
ATP + -glutamate + methylamine formula_0 ADP + phosphate + "N"5-methyl--glutamine
The 3 substrates of this enzyme are ATP, L-glutamate, and methylamine, whereas its 3 products are ADP, phosphate, and N5-methyl-L-glutamine.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. The systematic name of this enzyme class is L-glutamate:methylamine ligase (ADP-forming). This enzyme is also called gamma-glutamylmethylamide synthetase.
References.
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14285941
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Glutamate—putrescine ligase
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Class of enzymes
In enzymology, a glutamate-putrescine ligase (EC 6.3.1.11) is an enzyme that catalyzes the chemical reaction
ATP + L-glutamate + putrescine formula_0 ADP + phosphate + gamma-L-glutamylputrescine
The 3 substrates of this enzyme are ATP, L-glutamate, and putrescine, whereas its 3 products are ADP, phosphate, and gamma-L-glutamylputrescine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthases). The systematic name of this enzyme class is L-glutamate:putrescine ligase (ADP-forming). Other names in common use include gamma-glutamylputrescine synthetase, and YcjK. This enzyme participates in urea cycle and metabolism of amino groups.
References.
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14285948
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Glutamate—tRNA(Gln) ligase
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In enzymology, a glutamate—tRNAGln ligase (EC 6.1.1.24) is an enzyme that catalyzes the chemical reaction
ATP + L-glutamate + tRNAGlx formula_0 AMP + diphosphate + glutamyl-tRNAGlx
The 3 substrates of this enzyme are ATP, L-glutamate, and tRNAGlx, whereas its 3 products are AMP, diphosphate, and glutamyl-tRNAGlx.
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-glutamate:tRNAGlx ligase (AMP-forming). This enzyme is also called glutamyl-tRNA synthetase. This enzyme participates in glutamate metabolism.
References.
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14285954
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Glutamate—tRNA ligase
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In enzymology, a glutamate—tRNA ligase (EC 6.1.1.17) is an enzyme that catalyzes the chemical reaction
ATP + L-glutamate + tRNAGlu formula_0 AMP + diphosphate + L-glutamyl-tRNAGlu
The 3 substrates of this enzyme are ATP, L-glutamate, and tRNA(Glu), whereas its 3 products are AMP, diphosphate, and L-glutamyl-tRNA(Glu).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-glutamate:tRNAGlu ligase (AMP-forming). Other names in common use include glutamyl-tRNA synthetase, glutamyl-transfer ribonucleate synthetase, glutamyl-transfer RNA synthetase, glutamyl-transfer ribonucleic acid synthetase, glutamate-tRNA synthetase, and glutamic acid translase. This enzyme participates in 3 metabolic pathways: glutamate metabolism, porphyrin and chlorophyll metabolism, and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 16 structures have been solved for this class of enzymes, with PDB accession codes 1FYJ, 1G59, 1J09, 1N75, 1N77, 1N78, 2CFO, 2CUZ, 2CV0, 2CV1, 2CV2, 2DXI, 2HRA, 2HRK, 2HSM, and 2O5R.
References.
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14285960
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Glutamine—tRNA ligase
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Glutamine—tRNA ligase or glutaminyl-tRNA synthetase (GlnRS) is an aminoacyl-tRNA synthetase (aaRS or ARS), also called tRNA-ligase. is an enzyme that attaches the amino acid glutamine onto its cognate tRNA.
This enzyme participates in glutamate metabolism and aminoacyl-trna biosynthesis.
The human gene for glutaminyl-tRNA synthetase is "QARS".
Catalyzed reaction.
Glutamine—tRNA ligase (EC 6.1.1.18) is an enzyme that catalyzes the chemical reaction
ATP + L-glutamine + tRNAGln formula_0 AMP + diphosphate + L-glutaminyl-tRNAGln
The 3 substrates of this enzyme are ATP, L-glutamine, and tRNAGln, whereas its 3 products are AMP, diphosphate, and L-glutaminyl-tRNAGln. The cycle of aminoacylation reaction is shown in the figure.
Nomenclature.
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-glutamine:tRNAGln ligase (AMP-forming). Glutaminyl-tRNA synthetase or GlnRS is the primary name in use in the scientific literature. Other names that have been reported are:
Evolution.
In the eukaryotic cytoplasm and in some bacteria such as "E. coli," glutaminyl-tRNA synthetase catalyzes glutamine-tRNAGln formation. However a two-step formation process is necessary for its formation in all archaebacteria and most eubacteria as well as most eukaryotic organelles. In these cases, a glutamyl-tRNA synthetase first mis-aminoacylates tRNAGln with glutamate. Glutamine-tRNAGln is then formed by transamidation of the misacylated glutamate-tRNAGln by the glutaminyl-tRNA synthase (glutamine-hydrolysing) enzyme. It is believed that glutaminyl-tRNA synethetases have evolved from the glutamyl-tRNA synthetase enzyme.
Aminoacyl tRNA synthetases are divided into two major classes based on their active site structure: class I and II. Glutaminyl-tRNA synthetase belongs to the class-I aminoacyl-tRNA synthetase family.
Structure.
Of the glutaminyl-tRNA synthetases, the enzyme from "E. coli" is the most well studied structurally and biochemically. It is 553 amino acids long and is about 100Å long. At the N-terminus, it has its catalytic active site with a Rossmann di-nucleotide fold interacting with the 2'OH of the final nucleotide of tRNAGln (A76), while the C terminus interacts with the tRNA's anti-codon loop. The human human glutaminyl-tRNA synthetase structure at N-terminus contains a two tandem non-specific RNA binding regions, a catalytic domain, and two tandem anti-codon binding domains in the C-terminus.
The first crystal structure of a tRNA synthetase in complex with its cognate tRNA was that of the "E. coli" tRNA-Gln:GlnRS, determined in 1989 (PDB accession code (1GSG). This was also the first crystal structure of a non-viral protein:RNA complex. The purified enzyme was crystalized in complex with in vivo overexpressed tRNAGln.
As of late 2024, over 38 structures have been solved for this class of enzymes. Some of the PDB accession codes include 1EUQ, 1EUY, 1EXD, 1GSG, 1GTR, 1GTS, 1NYL, 1O0B, 1O0C, 1QRS, 1QRT, 1QRU, 1QTQ, 1ZJW, and 2HZ7. The "E. coli" glutaminyl-tRNA synethetase structure complexed with its cognate tRNA, tRNAGln is depicted in the figure (accession number 1EUG.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14285960
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14285968
|
Glutaminyl-tRNA synthase (glutamine-hydrolysing)
|
Glu-tRNAGln amidotransferase or glutaminyl-tRNA synthase (glutamine-hydrolysing) enzyme (EC 6.3.5.7) is an amidotransferase that catalyzes the conversion of the non-cognate amino acid glutamyl-tRNAGln to the cognate glutaminyl-tRNAGln.. It catalyzes the reaction:
ATP + glutamyl-tRNAGln + -glutamine formula_0 ADP + phosphate + glutaminyl-tRNAGln + -glutamate
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds carbon-nitrogen ligases with glutamine as amido-N-donor. The systematic name of this enzyme class is glutamyl-tRNAGln:L-glutamine amido-ligase (ADP-forming). This enzyme participates in glutamate metabolism and alanine and aspartate metabolism.
Function and evolutionary significance.
Most bacterial and all archaea genomes do not encode a glutaminyl-tRNA synthetase (GlnRS). Instead they first synthesize the attachment of an amino acid on the tRNAGln by first attaching a non-cognate glutamate to the tRNA. Then these organisms use the amidotransferase: glutaminyl-tRNA synthase (glutamine-hydrolysing) (EC 6.3.5.7) enzyme to convert the glutamate attached to tRNAGln to glutamine.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14285968
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14285977
|
Glutarate—CoA ligase
|
In enzymology, a glutarate—CoA ligase (EC 6.2.1.6) is an enzyme that catalyzes the chemical reaction
ATP + glutarate + CoA formula_0 ADP + phosphate + glutaryl-CoA
The 3 substrates of this enzyme are ATP, glutarate, and CoA, whereas its 3 products are ADP, phosphate, and glutaryl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is glutarate:CoA ligase (ADP-forming). Other names in common use include glutaryl-CoA synthetase, and glutaryl coenzyme A synthetase. This enzyme participates in fatty acid metabolism and lysine degradation.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14285977
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14285990
|
Glutathionylspermidine synthase
|
Class of enzymes
In enzymology, a glutathionylspermidine synthase (EC 6.3.1.8) is an enzyme that catalyzes the chemical reaction
glutathione + spermidine + ATP formula_0 glutathionylspermidine + ADP + phosphate
The 3 substrates of this enzyme are glutathione, spermidine, and ATP, whereas its 3 products are glutathionylspermidine, ADP, and phosphate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthases). The systematic name of this enzyme class is gamma-L-glutamyl-L-cysteinyl-glycine:spermidine ligase (ADP-forming) [spermidine is numbered so that atom N-1 is in the amino group of the aminopropyl part of the molecule]. This enzyme is also called glutathione:spermidine ligase (ADP-forming). This enzyme participates in glutathione metabolism. It employs one cofactor, magnesium.
Structural studies.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 2IO7, 2IO8, 2IO9, 2IOA, and 2IOB.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14285990
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14285996
|
Glycine—tRNA ligase
|
Protein-coding gene in the species Homo sapiens
Glycine—tRNA ligase also known as glycyl–tRNA synthetase is an enzyme that in humans is encoded by the "GARS1" gene.
Function.
This gene encodes glycyl-tRNA synthetase, one of the aminoacyl-tRNA synthetases that charge tRNAs with their cognate amino acids. The encoded enzyme is an (alpha)2 dimer which belongs to the class II family of tRNA synthetases.
Reaction.
In enzymology, a glycine—tRNA ligase (EC 6.1.1.14) is an enzyme that catalyzes the chemical reaction
ATP + glycine + tRNAGly formula_0 AMP + diphosphate + glycyl-tRNAGly
The 3 substrates of this enzyme are ATP, glycine, and tRNA(Gly), whereas its 3 products are AMP, diphosphate, and glycyl-tRNA(Gly).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is glycine:tRNAGly ligase (AMP-forming). Other names in common use include glycyl-tRNA synthetase, glycyl-transfer ribonucleate synthetase, glycyl-transfer RNA synthetase, glycyl-transfer ribonucleic acid synthetase, and glycyl translase. This enzyme participates in glycine, serine and threonine metabolism and aminoacyl-trna biosynthesis.
Interactions.
Glycyl-tRNA synthetase has been shown to interact with EEF1D. Mutant forms of the protein associated with peripheral nerve disease have been shown to aberrantly bind to the transmembrane receptor proteins neuropilin 1 and Trk receptors A-C.
Clinical relevance.
Glycyl-tRNA synthetase has been shown to be a target of autoantibodies in the human autoimmune diseases, polymyositis or dermatomyositis.
The peripheral nerve diseases Charcot-Marie-Tooth disease type 2D (CMT2D) and distal spinal muscular atrophy type V (dSMA-V) have been liked to dominant mutations in "GARS". CMT2D usually manifests during the teenage years, and results in muscle weakness predominantly in the hands and feet. Two mouse models of CMT2D have been used to better understand the disease, identifying that the disorder is caused by a toxic gain-of-function of the mutant glycine-tRNA ligase protein. The CMT2D mice display peripheral nerve axon degeneration and defective development and function> of the neuromuscular junction.
References.
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Further reading.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14285996
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14286
|
Holographic principle
|
Physics inside a bounded region is fully captured by physics at the boundary of the region
The holographic principle is a property of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region – such as a light-like boundary like a gravitational horizon. First proposed by Gerard 't Hooft, it was given a precise string theoretic interpretation by Leonard Susskind, who combined his ideas with previous ones of 't Hooft and Charles Thorn. Leonard Susskind said, "The three-dimensional world of ordinary experience—the universe filled with galaxies, stars, planets, houses, boulders, and people—is a hologram, an image of reality coded on a distant two-dimensional surface." As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. The prime example of holography is the AdS/CFT correspondence.
The holographic principle was inspired by the Bekenstein bound of black hole thermodynamics, which conjectures that the maximum entropy in any region scales with the radius squared, rather than cubed as might be expected. In the case of a black hole, the insight was that the information content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory.
However, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law (radius squared), hence in principle larger than those of a black hole. These are the so-called "Wheeler's bags of gold". The existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood.
High-level summary.
The physical universe is widely seen to be composed of "matter" and "energy". In his 2003 article published in Scientific American magazine, Jacob Bekenstein speculatively summarized a current trend started by John Archibald Wheeler, which suggests scientists may "regard the physical world as made of information, with energy and matter as incidentals". Bekenstein asks "Could we, as William Blake memorably penned, 'see a world in a grain of sand', or is that idea no more than 'poetic license'?", referring to the holographic principle.
Unexpected connection.
Bekenstein's topical overview "A Tale of Two Entropies" describes potentially profound implications of Wheeler's trend, in part by noting a previously unexpected connection between the world of information theory and classical physics. This connection was first described shortly after the seminal 1948 papers of American applied mathematician Claude Shannon introduced today's most widely used measure of information content, now known as Shannon entropy. As an objective measure of the quantity of information, Shannon entropy has been enormously useful, as the design of all modern communications and data storage devices, from cellular phones to modems to hard disk drives and DVDs, rely on Shannon entropy.
In thermodynamics (the branch of physics dealing with heat), entropy is popularly described as a measure of the "disorder" in a physical system of matter and energy. In 1877, Austrian physicist Ludwig Boltzmann described it more precisely in terms of the number of distinct microscopic states that the particles composing a macroscopic "chunk" of matter could be in, while still "looking" like the same macroscopic "chunk". As an example, for the air in a room, its thermodynamic entropy would equal the logarithm of the count of all the ways that the individual gas molecules could be distributed in the room, and all the ways they could be moving.
Energy, matter, and information equivalence.
Shannon's efforts to find a way to quantify the information contained in, for example, a telegraph message, led him unexpectedly to a formula with the same form as Boltzmann's. In an article in the August 2003 issue of "Scientific American" titled "Information in the Holographic Universe", Bekenstein summarizes that "Thermodynamic entropy and Shannon entropy are conceptually equivalent: the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information one would need to implement any particular arrangement" of matter and energy. The only salient difference between the thermodynamic entropy of physics and Shannon's entropy of information is in the units of measure; the former is expressed in units of energy divided by temperature, the latter in essentially dimensionless "bits" of information.
The holographic principle states that the entropy of ordinary mass (not just black holes) is also proportional to surface area and not volume; that volume itself is illusory and the universe is really a hologram which is isomorphic to the information "inscribed" on the surface of its boundary.
The AdS/CFT correspondence.
The anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality (after ref.) or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles.
The duality represents a major advance in our understanding of string theory and quantum gravity. This is because it provides a non-perturbative formulation of string theory with certain boundary conditions and because it is the most successful realization of the holographic principle.
It also provides a powerful toolkit for studying strongly coupled quantum field theories. Much of the usefulness of the duality results from the fact that it is a strong-weak duality: when the fields of the quantum field theory are strongly interacting, the ones in the gravitational theory are weakly interacting and thus more mathematically tractable. This fact has been used to study many aspects of nuclear and condensed matter physics by translating problems in those subjects into more mathematically tractable problems in string theory.
The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997. Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov, and by Edward Witten. By 2015, Maldacena's article had over 10,000 citations, becoming the most highly cited article in the field of high energy physics.
Black hole entropy.
An object with relatively high entropy is microscopically random, like a hot gas. A known configuration of classical fields has zero entropy: there is nothing random about electric and magnetic fields, or gravitational waves. Since black holes are exact solutions of Einstein's equations, they were thought not to have any entropy either.
But Jacob Bekenstein noted that this leads to a violation of the second law of thermodynamics. If one throws a hot gas with entropy into a black hole, once it crosses the event horizon, the entropy would disappear. The random properties of the gas would no longer be seen once the black hole had absorbed the gas and settled down. One way of salvaging the second law is if black holes are in fact random objects with an entropy that increases by an amount greater than the entropy of the consumed gas.
Given a fixed volume, a black hole whose event horizon encompasses that volume should be the object with the highest amount of entropy. Otherwise, suppose we have something with a larger entropy, then by throwing more mass into that something, we obtain a black hole with less entropy, violating the second law.
In a sphere of radius "R", the entropy in a relativistic gas increases as the energy increases. The only known limit is gravitational; when there is too much energy the gas collapses into a black hole. Bekenstein used this to put an upper bound on the entropy in a region of space, and the bound was proportional to the area of the region. He concluded that the black hole entropy is directly proportional to the area of the event horizon. Gravitational time dilation causes time, from the perspective of a remote observer, to stop at the event horizon. Due to the natural limit on maximum speed of motion, this prevents falling objects from crossing the event horizon no matter how close they get to it. Since any change in quantum state requires time to flow, all objects and their quantum information state stay imprinted on the event horizon. Bekenstein concluded that from the perspective of any remote observer, the black hole entropy is directly proportional to the area of the event horizon.
Stephen Hawking had shown earlier that the total horizon area of a collection of black holes always increases with time. The horizon is a boundary defined by light-like geodesics; it is those light rays that are just barely unable to escape. If neighboring geodesics start moving toward each other they eventually collide, at which point their extension is inside the black hole. So the geodesics are always moving apart, and the number of geodesics which generate the boundary, the area of the horizon, always increases. Hawking's result was called the second law of black hole thermodynamics, by analogy with the law of entropy increase.
At first, Hawking did not take the analogy too seriously. He argued that the black hole must have zero temperature, since black holes do not radiate and therefore cannot be in thermal equilibrium with any black body of positive temperature. Then he discovered that black holes do radiate. When heat is added to a thermal system, the change in entropy is the increase in mass–energy divided by temperature:
formula_0
If black holes have a finite entropy, they should also have a finite temperature. In particular, they would come to equilibrium with a thermal gas of photons. This means that black holes would not only absorb photons, but they would also have to emit them in the right amount to maintain detailed balance.
Time-independent solutions to field equations do not emit radiation, because a time-independent background conserves energy. Based on this principle, Hawking set out to show that black holes do not radiate. But, to his surprise, a careful analysis convinced him that they do, and in just the right way to come to equilibrium with a gas at a finite temperature. Hawking's calculation fixed the constant of proportionality at 1/4; the entropy of a black hole is one quarter its horizon area in Planck units.
The entropy is proportional to the logarithm of the number of microstates, the enumerated ways a system can be configured microscopically while leaving the macroscopic description unchanged. Black hole entropy is deeply puzzling – it says that the logarithm of the number of states of a black hole is proportional to the area of the horizon, not the volume in the interior.
Later, Raphael Bousso came up with a covariant version of the bound based upon null sheets.
Black hole information paradox.
Hawking's calculation suggested that the radiation which black holes emit is not related in any way to the matter that they absorb. The outgoing light rays start exactly at the edge of the black hole and spend a long time near the horizon, while the infalling matter only reaches the horizon much later. The infalling and outgoing mass/energy interact only when they cross. It is implausible that the outgoing state would be completely determined by some tiny residual scattering.
Hawking interpreted this to mean that when black holes absorb some photons in a pure state described by a wave function, they re-emit new photons in a thermal mixed state described by a density matrix. This would mean that quantum mechanics would have to be modified because, in quantum mechanics, states which are superpositions with probability amplitudes never become states which are probabilistic mixtures of different possibilities.
Troubled by this paradox, Gerard 't Hooft analyzed the emission of Hawking radiation in more detail. He noted that when Hawking radiation escapes, there is a way in which incoming particles can modify the outgoing particles. Their gravitational field would deform the horizon of the black hole, and the deformed horizon could produce different outgoing particles than the undeformed horizon. When a particle falls into a black hole, it is boosted relative to an outside observer, and its gravitational field assumes a universal form. 't Hooft showed that this field makes a logarithmic tent-pole shaped bump on the horizon of a black hole, and like a shadow, the bump is an alternative description of the particle's location and mass. For a four-dimensional spherical uncharged black hole, the deformation of the horizon is similar to the type of deformation which describes the emission and absorption of particles on a string-theory world sheet. Since the deformations on the surface are the only imprint of the incoming particle, and since these deformations would have to completely determine the outgoing particles, 't Hooft believed that the correct description of the black hole would be by some form of string theory.
This idea was made more precise by Leonard Susskind, who had also been developing holography, largely independently. Susskind argued that the oscillation of the horizon of a black hole is a complete description of both the infalling and outgoing matter, because the world-sheet theory of string theory was just such a holographic description. While short strings have zero entropy, he could identify long highly excited string states with ordinary black holes. This was a deep advance because it revealed that strings have a classical interpretation in terms of black holes.
This work showed that the black hole information paradox is resolved when quantum gravity is described in an unusual string-theoretic way assuming the string-theoretical description is complete, unambiguous and non-redundant. The space-time in quantum gravity would emerge as an effective description of the theory of oscillations of a lower-dimensional black-hole horizon, and suggest that any black hole with appropriate properties, not just strings, would serve as a basis for a description of string theory.
In 1995, Susskind, along with collaborators Tom Banks, Willy Fischler, and Stephen Shenker, presented a formulation of the new M-theory using a holographic description in terms of charged point black holes, the D0 branes of type IIA string theory. The matrix theory they proposed was first suggested as a description of two branes in eleven-dimensional supergravity by Bernard de Wit, Jens Hoppe, and Hermann Nicolai. The later authors reinterpreted the same matrix models as a description of the dynamics of point black holes in particular limits. Holography allowed them to conclude that the dynamics of these black holes give a complete non-perturbative formulation of M-theory. In 1997, Juan Maldacena gave the first holographic descriptions of a higher-dimensional object, the 3+1-dimensional type IIB membrane, which resolved a long-standing problem of finding a string description which describes a gauge theory. These developments simultaneously explained how string theory is related to some forms of supersymmetric quantum field theories.
Limit on information density.
Information content is defined as the logarithm of the reciprocal of the probability that a system is in a specific microstate, and the information entropy of a system is the expected value of the system's information content. This definition of entropy is equivalent to the standard Gibbs entropy used in classical physics. Applying this definition to a physical system leads to the conclusion that, for a given energy in a given volume, there is an upper limit to the density of information (the Bekenstein bound) about the whereabouts of all the particles which compose matter in that volume. In particular, a given volume has an upper limit of information it can contain, at which it will collapse into a black hole.
This suggests that matter itself cannot be subdivided infinitely many times and there must be an ultimate level of fundamental particles. As the degrees of freedom of a particle are the product of all the degrees of freedom of its sub-particles, were a particle to have infinite subdivisions into lower-level particles, the degrees of freedom of the original particle would be infinite, violating the maximal limit of entropy density. The holographic principle thus implies that the subdivisions must stop at some level.
The most rigorous realization of the holographic principle is the AdS/CFT correspondence by Juan Maldacena. However, J. David Brown and Marc Henneaux had rigorously proved already in 1986, that the asymptotic symmetry of 2+1 dimensional gravity gives rise to a Virasoro algebra, whose corresponding quantum theory is a 2-dimensional conformal field theory.
Experimental tests.
The Fermilab physicist Craig Hogan claims that the holographic principle would imply quantum fluctuations in spatial position that would lead to apparent background noise or "holographic noise" measurable at gravitational wave detectors, in particular GEO 600. However these claims have not been widely accepted, or cited, among quantum gravity researchers and appear to be in direct conflict with string theory calculations.
Analyses in 2011 of measurements of gamma ray burst GRB 041219A in 2004 by the INTEGRAL space observatory launched in 2002 by the European Space Agency shows that Craig Hogan's noise is absent down to a scale of 10−48 meters, as opposed to the scale of 10−35 meters predicted by Hogan, and the scale of 10−16 meters found in measurements of the GEO 600 instrument. Research continued at Fermilab under Hogan as of 2013.
Jacob Bekenstein claimed to have found a way to test the holographic principle with a tabletop photon experiment.
See also.
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Notes.
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References.
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[
{
"math_id": 0,
"text": "\n{\\rm d}S = \\frac{{\\rm\\delta }M \\ c^2}{T}.\n"
}
] |
https://en.wikipedia.org/wiki?curid=14286
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14286018
|
Histidine—tRNA ligase
|
In enzymology, a histidine-tRNA ligase (EC 6.1.1.21) is an enzyme that catalyzes the chemical reaction
ATP + L-histidine + tRNAHis formula_0 AMP + diphosphate + L-histidyl-tRNAHis
The 3 substrates of this enzyme are ATP, L-histidine, and tRNA(His), whereas its 3 products are AMP, diphosphate, and L-histidyl-tRNA(His).
This enzyme participates in histidine metabolism and aminoacyl-trna biosynthesis.
Nomenclature.
Histidine—tRNA ligase belongs to the family of ligase enzymes, specifically those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-histidine:tRNAHis ligase (AMP-forming). Other names in common use include histidyl-tRNA synthetase, histidyl-transfer ribonucleate synthetase, and histidine translase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14286018
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14286028
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Homoglutathione synthase
|
Class of enzymes
In enzymology, a homoglutathione synthase (EC 6.3.2.23) is an enzyme that catalyzes the chemical reaction
ATP + γ--glutamyl--cysteine + β-alanine formula_0 ADP + phosphate + γ-glutamyl--cysteinyl-β-alanine
The 3 substrates of this enzyme are ATP, gamma-L-glutamyl-L-cysteine, and beta-alanine, whereas its 3 products are ADP, phosphate, and gamma-L-glutamyl-L-cysteinyl-beta-alanine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is gamma-L-glutamyl-L-cysteine:beta-alanine ligase (ADP-forming). Other names in common use include homoglutathione synthetase, and beta-alanine specific hGSH synthetase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14286028
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14286032
|
Hydrogenobyrinic acid a,c-diamide synthase (glutamine-hydrolysing)
|
In enzymology, a hydrogenobyrinic acid "a,c"-diamide synthase (glutamine-hydrolysing) (EC 6.3.5.9) is an enzyme that catalyzes the chemical reaction
2 ATP + hydrogenobyrinic acid + 2 -glutamine + 2 H2O formula_0 2 ADP + 2 phosphate + hydrogenobyrinic acid "a,c"-diamide + 2 -glutamate
The four substrates of this enzyme are ATP, hydrogenobyrinic acid, L-glutamine, and H2O; its four products are ADP, phosphate, hydrogenobyrinic acid a,c-diamide, and L-glutamate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds carbon-nitrogen ligases with glutamine as amido-N-donor. The systematic name of this enzyme class is hydrogenobyrinic-acid:L-glutamine amido-ligase (AMP-forming). This enzyme is also called CobB and is part of the biosynthetic pathway to cobalamin (vitamin B12) in aerobic bacteria.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
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https://en.wikipedia.org/wiki?curid=14286032
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14286039
|
Imidazoleacetate—phosphoribosyldiphosphate ligase
|
In enzymology, an imidazoleacetate—phosphoribosyldiphosphate ligase (EC 6.3.4.8) is an enzyme that catalyzes a chemical reaction
ATP + imidazole-4-acetate + 5-phosphoribosyl diphosphate formula_0 ADP + phosphate + 1-(5-phosphoribosyl)imidazole-4-acetate + diphosphate
The 3 substrates of this enzyme are ATP, imidazole-4-acetate, and 5-phosphoribosyl diphosphate, whereas its 4 products are ADP, phosphate, 1-(5-phosphoribosyl)imidazole-4-acetate, and diphosphate.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. The systematic name of this enzyme class is imidazoleacetate:5-phosphoribosyl-diphosphate ligase (ADP- and diphosphate-forming). This enzyme is also called 5-phosphoribosylimidazoleacetate synthetase. This enzyme participates in histidine metabolism.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14286039
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14286046
|
Indoleacetate—lysine synthetase
|
Class of enzymes
In enzymology, an indoleacetate—lysine synthetase (EC 6.3.2.20) is an enzyme that catalyzes the chemical reaction
ATP + (indol-3-yl)acetate + L-lysine formula_0 ADP + phosphate + N6-[(indol-3-yl)acetyl]-L-lysine
The 3 substrates of this enzyme are ATP, (indol-3-yl)acetate, and L-lysine, whereas its 3 products are ADP, phosphate, and N6-[(indol-3-yl)acetyl]-L-lysine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is (indol-3-yl)acetate:L-lysine ligase (ADP-forming). This enzyme is also called indoleacetate:L-lysine ligase (ADP-forming).
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14286046
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14286051
|
Isoleucine—tRNA ligase
|
In enzymology, an isoleucine—tRNA ligase (EC 6.1.1.5) is an enzyme that catalyzes the chemical reaction
ATP + L-isoleucine + tRNAIle formula_0 AMP + diphosphate + L-isoleucyl-tRNAIle
The 3 substrates of this enzyme are ATP, L-isoleucine, and tRNA(Ile), whereas its 3 products are AMP, diphosphate, and L-isoleucyl-tRNA(Ile).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-isoleucine:tRNAIle ligase (AMP-forming). Other names in common use include isoleucyl-tRNA synthetase, isoleucyl-transfer ribonucleate synthetase, isoleucyl-transfer RNA synthetase, isoleucine-transfer RNA ligase, isoleucine-tRNA synthetase, and isoleucine translase. This enzyme participates in valine, leucine and isoleucine biosynthesis and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1FFY, 1JZQ, 1JZS, 1QU2, 1QU3, 1UDZ, 1UE0, 1WK8, 1WNY, and 1WNZ.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14286051
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14286058
|
L-amino-acid alpha-ligase
|
Class of enzymes
In enzymology, an L-amino-acid alpha-ligase (EC 6.3.2.28) is an enzyme that catalyzes the chemical reaction
ATP + an L-amino acid + an L-amino acid formula_0 ADP + phosphate + L-aminoacyl-L-amino acid
Thus, the two substrates of this enzyme are ATP and L-amino acid, whereas its 3 products are ADP, phosphate, and L-aminoacyl-L-amino acid.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is L-amino acid:L-amino acid ligase (ADP-forming). Other names in common use include L-amino acid alpha-ligase, bacilysin synthetase, YwfE, and L-amino acid ligase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286058
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14286066
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Leucine—tRNA ligase
|
In enzymology, a leucine—tRNA ligase (EC 6.1.1.4) is an enzyme that catalyzes the chemical reaction
ATP + L-leucine + tRNALeu formula_0 AMP + diphosphate + L-leucyl-tRNALeu
The 3 substrates of this enzyme are ATP, L-leucine, and tRNA(Leu), whereas its 3 products are AMP, diphosphate, and L-leucyl-tRNA(Leu).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-leucine:tRNALeu ligase (AMP-forming). Other names in common use include leucyl-tRNA synthetase, leucyl-transfer ribonucleate synthetase, leucyl-transfer RNA synthetase, leucyl-transfer ribonucleic acid synthetase, leucine-tRNA synthetase, and leucine translase. This enzyme participates in valine, leucine and isoleucine biosynthesis and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 1WKB, 1WZ2, 2AJG, 2AJH, and 2AJI.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286066
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14286073
|
Long-chain-fatty-acid—(acyl-carrier-protein) ligase
|
In enzymology, a long-chain-fatty-acid—[acyl-carrier-protein] ligase (EC 6.2.1.20) is an enzyme that catalyzes the chemical reaction
ATP + an acid + [acyl-carrier-protein] formula_0 AMP + diphosphate + acyl-[acyl-carrier-protein]
The 3 substrates of this enzyme are ATP, acid, and acyl-carrier-protein, whereas its 3 products are AMP, diphosphate, and acyl-[acyl-carrier-protein].
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is long-chain-fatty-acid:[acyl-carrier-protein] ligase (AMP-forming). Other names in common use include acyl-[acyl-carrier-protein] synthetase, acyl-[acyl carrier protein] synthetase, acyl-ACP synthetase, acyl-[acyl-carrier-protein]synthetase, stearoyl-ACP synthetase, and acyl-acyl carrier protein synthetase. This enzyme participates in fatty acid metabolism.
References.
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[
{
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"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286073
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14286079
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Long-chain-fatty-acid—luciferin-component ligase
|
In enzymology, a long-chain-fatty-acid—luciferin-component ligase (EC 6.2.1.19) is an enzyme that catalyzes the chemical reaction
ATP + an acid + protein formula_0 AMP + diphosphate + an acyl-protein thioester
The 3 substrates of this enzyme are ATP, acid, and protein, whereas its 3 products are AMP, diphosphate, and acyl-protein thioester.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is long-chain-fatty-acid:protein ligase (AMP-forming). This enzyme is also called acyl-protein synthetase.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286079
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14286088
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Lysine—tRNA ligase
|
In enzymology, a lysine—tRNA ligase (EC 6.1.1.6) is an enzyme that catalyzes the chemical reaction
ATP + L-lysine + tRNALys formula_0 AMP + diphosphate + L-lysyl-tRNALys
The 3 substrates of this enzyme are ATP, L-lysine, and tRNA(Lys), whereas its 3 products are AMP, diphosphate, and L-lysyl-tRNA(Lys).
This enzyme participates in 3 metabolic pathways: lysine biosynthesis, aminoacyl-trna biosynthesis, and amyotrophic lateral sclerosis (als).
Nomenclature.
This enzyme belongs to the family of ligases, to be specific, those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-lysine:tRNALys ligase (AMP-forming). Other names in common use include lysyl-tRNA synthetase, lysyl-transfer ribonucleate synthetase, lysyl-transfer RNA synthetase, L-lysine-transfer RNA ligase, lysine-tRNA synthetase, and lysine translase.
References.
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Further reading.
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[
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https://en.wikipedia.org/wiki?curid=14286088
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14286098
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Lysine—tRNA(Pyl) ligase
|
In enzymology, a lysine-tRNAPyl ligase (EC 6.1.1.25) is an enzyme that catalyzes the chemical reaction
ATP + L-lysine + tRNAPyl formula_0 AMP + diphosphate + L-lysyl-tRNAPyl
The 3 substrates of this enzyme are ATP, L-lysine, and tRNA(Pyl), whereas its 3 products are AMP, diphosphate, and L-lysyl-tRNA(Pyl).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-lysine:tRNAPyl ligase (AMP-forming).
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286098
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14286121
|
Malate—CoA ligase
|
In enzymology, a malate—CoA ligase (EC 6.2.1.9) is an enzyme that catalyzes the chemical reaction
ATP + malate + CoA formula_0 ADP + phosphate + malyl-CoA
The 3 substrates of this enzyme are ATP, malate, and CoA, whereas its 3 products are ADP, phosphate, and malyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is malate:CoA ligase (ADP-forming). Other names in common use include malyl-CoA synthetase, malyl coenzyme A synthetase, and malate thiokinase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
References.
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[
{
"math_id": 0,
"text": "\\rightleftharpoons"
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https://en.wikipedia.org/wiki?curid=14286121
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14286126
|
Methionine—tRNA ligase
|
In enzymology, a methionine—tRNA ligase (EC 6.1.1.10) is an enzyme that catalyzes the chemical reaction
ATP + L-methionine + tRNAMet formula_0 AMP + diphosphate + L-methionyl-tRNAMet
The 3 substrates of this enzyme are ATP, L-methionine, and tRNA(Met), whereas its 3 products are AMP, diphosphate, and L-methionyl-tRNA(Met).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-methionine:tRNAMet ligase (AMP-forming). Other names in common use include methionyl-tRNA synthetase, methionyl-transfer ribonucleic acid synthetase, methionyl-transfer ribonucleate synthetase, methionyl-transfer RNA synthetase, methionine translase, and MetRS. This enzyme participates in 3 metabolic pathways: methionine metabolism, selenoamino acid metabolism, and aminoacyl-trna biosynthesis.
Role in oxidative stress.
During oxidative stress, methionine—tRNA ligase might be phosphorylated, which results in promiscuity of this enzyme, where it aminoacylates methionine to various non-Met tRNAs. This in turn leads to substitution of amino acids in proteins with methionine, which helps relieve oxidative stress in the cell.
Structural studies.
As of late 2007, 21 structures have been solved for this class of enzymes, with PDB accession codes 1A8H, 1F4L, 1MEA, 1MED, 1MKH, 1P7P, 1PFU, 1PFV, 1PFW, 1PFY, 1PG0, 1PG2, 1QQT, 1RQG, 1WOY, 2CSX, 2CT8, 2D54, 2D5B, 2DJV, and 2HSN.
References.
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Further reading.
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[
{
"math_id": 0,
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https://en.wikipedia.org/wiki?curid=14286126
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14286131
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ACV synthetase
|
Class of enzymes
ACV synthetase (ACVS, -δ-(α-aminoadipoyl)--cysteinyl--valine synthetase, "N"-(5-amino-5-carboxypentanoyl)--cysteinyl--valine synthase, EC 6.3.2.26) is an enzyme that catalyzes the chemical reaction
3 ATP + -2-aminohexanedioate + -cysteine + -valine + H2O formula_0 3 AMP + 3 PPi + "N"-[-5-amino-5-carboxypentanoyl]--cysteinyl--valine
The five substrates of this enzyme are ATP, -2-aminohexanedioate, -cysteine, -valine, and H2O, whereas its three products are AMP, diphosphate, and "N"-[-5-amino-5-carboxypentanoyl]--cysteinyl--valine.
ACVS is an example of a nonribosomal peptide synthetase (NRPS). It participates in penicillin and cephalosporin biosyntheses.
References.
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https://en.wikipedia.org/wiki?curid=14286131
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14286146
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NAD+ synthase
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Enzyme
In enzymology, a NAD+ synthetase (EC 6.3.1.5) is an enzyme that catalyzes the chemical reaction
ATP + deamido-NAD+ + NH3 formula_0 AMP + diphosphate + NAD+
The 3 substrates of this enzyme are ATP, deamido-NAD+, and NH3, whereas its 3 products are AMP, diphosphate, and NAD+.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthetase). The systematic name of this enzyme class is deamido-NAD+:ammonia ligase (AMP-forming). Other names in common use include NAD+ synthetase, NAD+ synthetase, nicotinamide adenine dinucleotide synthetase, and diphosphopyridine nucleotide synthetase. This enzyme participates in nicotinate and nicotinamide metabolism and nitrogen metabolism.
Structural studies.
As of late 2007, 11 structures have been solved for this class of enzymes, with PDB accession codes 1WXE, 1WXF, 1WXG, 1WXH, 1WXI, 1XNG, 1XNH, 2E18, 2PZ8, 2PZA, and 2PZB.
References.
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https://en.wikipedia.org/wiki?curid=14286146
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14286156
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NAD+ synthase (glutamine-hydrolysing)
|
In enzymology, a NAD+ synthase (glutamine-hydrolysing) (EC 6.3.5.1) is an enzyme that catalyzes the chemical reaction
ATP + deamido-NAD+ + -glutamine + H2O formula_0 AMP + diphosphate + NAD+ + L-glutamate. In eukaryotes, this enzyme contains a glutaminase domain related to nitrilase.
The substrates of this enzyme are ATP, deamido-NAD+, L-glutamine, and H2O, whereas its 4 products are AMP, diphosphate, NAD+, and glutamate
This enzyme participates in glutamate metabolism and nicotinate and nicotinamide metabolism.
Nomenclature.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds carbon-nitrogen ligases with glutamine as amido-N-donor. The systematic name of this enzyme class is deamido-NAD+:L-glutamine amido-ligase (AMP-forming).
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14286156
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14286171
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Oxalate—CoA ligase
|
In enzymology, an oxalate—CoA ligase (EC 6.2.1.8) is an enzyme that catalyzes the chemical reaction
ATP + oxalate + CoA formula_0 AMP + diphosphate + oxalyl-CoA
The 3 substrates of this enzyme are ATP, oxalate, and coenzyme A (CoA), whereas its 3 products are AMP, diphosphate, and oxalyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is oxalate:CoA ligase (AMP-forming). Other names in common use include oxalyl-CoA synthetase, and oxalyl coenzyme A synthetase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
Organisms with Oxalate-CoA Ligases include:
Arabidopsis thaliana
Saccharomyces cerevisiae
References.
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https://en.wikipedia.org/wiki?curid=14286171
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14286185
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Pantoate—beta-alanine ligase
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Class of enzymes
In enzymology, a pantoate—β-alanine ligase (EC 6.3.2.1) is an enzyme that catalyzes the chemical reaction
ATP + (R)-pantoate + β-alanine formula_0 AMP + diphosphate + (R)-pantothenate
The 3 substrates of this enzyme are ATP, (R)-pantoate, and beta-alanine, whereas its 3 products are AMP, diphosphate, and (R)-pantothenate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is (R)-pantoate:beta-alanine ligase (AMP-forming). Other names in common use include pantothenate synthetase, pantoate activating enzyme, pantoic-activating enzyme, and D-pantoate:beta-alanine ligase (AMP-forming). This enzyme participates in beta-alanine metabolism and pantothenate and CoA biosynthesis.
Structural studies.
As of late 2007, 15 structures have been solved for this class of enzymes, with PDB accession codes 1IHO, 1MOP, 1N2B, 1N2E, 1N2G, 1N2H, 1N2I, 1N2J, 1N2O, 1UFV, 1V8F, 2A7X, 2A84, 2A86, and 2A88.
References.
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https://en.wikipedia.org/wiki?curid=14286185
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14286193
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Phenylacetate—CoA ligase
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In enzymology, a phenylacetate—CoA ligase is an enzyme (EC 6.2.1.30) that catalyzes the chemical reaction
ATP + phenylacetate + CoA formula_0 AMP + diphosphate + phenylacetyl-CoA
The 3 substrates of this enzyme are ATP, phenylacetate, and CoA. Its 3 products are AMP, diphosphate, and phenylacetyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is phenylacetate:CoA ligase (AMP-forming). Other names in common use include phenylacetyl-CoA ligase, PA-CoA ligase, and phenylacetyl-CoA ligase (AMP-forming). This enzyme participates in tyrosine metabolism and phenylalanine metabolism.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14286193
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14286201
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Phenylalanine—tRNA ligase
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In enzymology, a phenylalanine—tRNA ligase (EC 6.1.1.20) is an enzyme that catalyzes the chemical reaction
ATP + L-phenylalanine + tRNAPhe formula_0 AMP + diphosphate + L-phenylalanyl-tRNAPhe
The 3 substrates of this enzyme are ATP, L-phenylalanine, and tRNAPhe, whereas its 3 products are AMP, diphosphate, and L-phenylalanyl-tRNAPhe.
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-phenylalanine:tRNAPhe ligase (AMP-forming). Other names in common use include phenylalanyl-tRNA synthetase, phenylalanyl-transfer ribonucleate synthetase, phenylalanine-tRNA synthetase, phenylalanyl-transfer RNA synthetase, phenylalanyl-tRNA ligase, phenylalanyl-transfer RNA ligase, L-phenylalanyl-tRNA synthetase, and phenylalanine translase. This enzyme participates in phenylalanine, tyrosine and tryptophan biosynthesis and aminoacyl-tRNA biosynthesis.
Phenylalanine-tRNA synthetase (PheRS) is known to be among the most complex enzymes of the aaRS (Aminoacyl-tRNA synthetase) family. Bacterial and mitochondrial PheRSs share a ferredoxin-fold anticodon binding (FDX-ACB) domain, which represents a canonical double split alpha+beta motif having no insertions. The FDX-ACB domain displays a typical RNA recognition fold (RRM) formed by the four-stranded antiparallel beta sheet, with two helices packed against it.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1B70, 1B7Y, 1EIY, 1JJC, 1PYS, 2AKW, 2ALY, 2AMC, 2CXI, and 2IY5.
References.
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https://en.wikipedia.org/wiki?curid=14286201
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14286205
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Phosphopantothenate—cysteine ligase
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Mammalian protein found in Homo sapiens
In enzymology, a phosphopantothenate—cysteine ligase also known as phosphopantothenoylcysteine synthetase (PPCS) is an enzyme (EC 6.3.2.5) that catalyzes the chemical reaction which constitutes the second of five steps involved in the conversion of pantothenate to Coenzyme A. The reaction is:
NTP + (R)-4'-phosphopantothenate + L-cysteine formula_0 NMP + diphosphate + N-[(R)-4'-phosphopantothenoyl]-L-cysteine
The nucleoside triphosphate (NTP) involved in the reaction varies from species to species. Phosphopantothenate—cysteine ligase from the bacterium "Escherichia coli" uses cytidine triphosphate (CTP) as an energy donor, whilst the human isoform uses adenosine triphosphate (ATP).
Nomenclature.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide syntheses). The systematic name of this enzyme class is (R)-4'-phosphopantothenate:L-cysteine ligase. This enzyme is also called phosphopantothenoylcysteine synthetase.
Gene.
Phosphopantothenoylcysteine synthetase in humans is encoded by the PPCS gene.
Protein structure.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 1P9O, 1U7U, 1U7W, 1U7Z, and 1U80.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14286205
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14286213
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Phosphoribosylamine—glycine ligase
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Phosphoribosylamine—glycine ligase, also known as glycinamide ribonucleotide synthetase (GARS), (EC 6.3.4.13) is an enzyme that catalyzes the chemical reaction
ATP + 5-phospho--ribosylamine + glycine formula_0 ADP + phosphate + N1-(5-phospho--ribosyl)glycinamide
which is the second step in purine biosynthesis.
The 3 substrates of this enzyme are ATP, 5-phospho-D-ribosylamine, and glycine, whereas its 3 products are ADP, phosphate, and N1-(5-phospho-D-ribosyl)glycinamide.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds.
In bacteria, GARS is a monofunctional enzyme (encoded by the purD gene). The purD genes often contain PurD RNA motif in their 5' UTR. In yeast, GARS is part of a bifunctional enzyme (encoded by the ADE5/7 gene) in conjunction with phosphoribosylformylglycinamidine cyclo-ligase (AIRS). In higher eukaryotes, including humans, GARS is part of a trifunctional enzyme in conjunction with AIRS and with phosphoribosylglycinamide formyltransferase (GART), forming GARS-AIRS-GART.
Nomenclature.
The systematic name of this enzyme class is 5-phospho-D-ribosylamine:glycine ligase (ADP-forming). Other names in common use include:
Mechanism.
GARS operates via an ordered, sequential mechanism. 5-phospho-D-ribosylamine (PRA) binds first, then ATP, and finally glycine. Phosphate is released first, followed by ADP and GAR. The oxygen in the ribose ring of PRA is important in substrate binding, likely due to favorable energetics from hydrogen bonding and the ring conformation it confers. In addition, the phosphate group of GAR has been implicated in GARS substrate recognition.
The reaction starts with the oxygen of glycine acting as a nucleophile to attack the γ-phosphorus of ATP. Then, the nitrogen of PRA attacks the carbonyl carbon in the intermediate, and phosphate leaves, forming GAR.
Structural studies.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1GSO, 1VKZ, and 2QK4. The overall structure of the enzyme, based on crystallization from "E. coli", consists of 16 alpha helices which connect to 20 beta strands by turns and loops. There are four main domains: N, A, B, and C. Each domain has a central beta sheet with an alpha helix on at least one side. The N, A, and C domains are clustered together, while the B domain is slightly separated from the others and connected to them by two hinge regions. The active site is between the NAC group and the B domain. The A and B domains appear to facilitate ATP binding, while the N and C domains confer substrate specificity. The N domain is very similar to that of glycinamide ribonucleotide transformylase. Although the orientation of the B domains varies, the structure of GARS is very similar across organisms. Furthermore, the gene has been sequenced in many organisms, and "E. coli" shows between 41 and 52% identity with the GARS sequences of "B. subtilis", "S. cerevisiae", "D. melanogaster", and "D. pseudobscura". Human GARS-AIRS-GART has been shown to be most similar to that of mice, chimpanzees, and cows. Among the amino acids that are identical in "B. subtilis", "S. cerevisiae", "D. melanogaster", and "D. pseudobscura", almost a third are glycine and proline, which suggests that they play an important role in proper folding of the protein. In addition to similar structure across species, GARS as a whole has a very similar structure to D-alanine:D-alanine ligase, biotin carboxylase, and glutathione synthetase. All of these enzymes have an ATP binding domain classified as ATP-grasp domains.
Disease relevance.
In humans, the gene that codes for GARS-AIRS-GART is on chromosome 21, and individuals with Down Syndrome have higher purine levels, which has been correlated with mental retardation. Thus, studies have been conducted to investigate its involvement in Down Syndrome. It has been found that GARS is expressed for longer in individuals with Down Syndrome than in unaffected individuals. In unaffected individuals, GARS is highly expressed in the cerebellum before birth but is barely expressed by three weeks after birth. In individuals with Down Syndrome, GARS expression continues until at least seven weeks after birth. This suggests that GARS may be a main contributor to the development of Down Syndrome. However, so far no mutations to GARS have been identified that could change its function and cause Down Syndrome related mental retardation.
References.
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https://en.wikipedia.org/wiki?curid=14286213
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14286218
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Phosphoribosylaminoimidazolesuccinocarboxamide synthase
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Class of enzymes
In molecular biology, the protein domain SAICAR synthase is an enzyme which catalyses a reaction to create SAICAR. In enzymology, this enzyme is also known as phosphoribosylaminoimidazolesuccinocarboxamide synthase (EC 6.3.2.6). It is an enzyme that catalyzes the chemical reaction
ATP + 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxylate + L-aspartate formula_0 ADP + phosphate + (S)-2-[5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxamido]succinate
The 3 substrates of this enzyme are ATP, 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxylate, and L-aspartate, whereas its 3 products are ADP, phosphate, and (S)-2-[5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxamido]succinate.
This enzyme belongs to the family of ligases, to be specific those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is 5-amino-1-(5-phospho-D-ribosyl)imidazole-4-carboxylate:L-aspartate ligase (ADP-forming). This enzyme participates in purine metabolism.
This particular protein family is of huge importance as it is found in all three domains of life. It is the seventh step in the pathway of purine biosynthesis. Purines are vital to all cells as they are involved in energy metabolism and DNA synthesis. Furthermore, they are of specific interest to scientific researchers as the study of the purine biosynthesis pathway could lead to the development of chemotherapeutic drugs. This is because most cancers lack a salvage pathway for adenine nucleotides and rely entirely on the SAICAR pathway.
Protein domain.
This protein domain is found in eukaryotes, bacteria and archaea. It is vital for living organisms since it catalyses a step in the purine biosynthesis pathway which aids energy metabolism and DNA synthesis.
Protein domain function.
In bacteria and plants this protein domain only catalyses the synthesis of SAICAR. However, in mammals it also catalyses phosphoribosylaminoimidazole carboxylase (AIRC) activity.
Protein domain structure.
This particular protein is an octamer made up of 8 identical subunits. Each monomer consists of a central domain and a C-terminal alpha helix. The central domain consists
of a five-stranded parallel beta sheet flanked by three alpha helices
one side of the sheet and two alpha helices on the other, forming a three-layer (alpha beta alpha) sandwich.
Structural studies.
As of late 2007, 10 structures have been solved for this class of enzymes, with PDB accession codes 1A48, 1KUT, 1OBD, 1OBG, 2CNQ, 2CNU, 2CNV, 2GQR, 2GQS, and 2H31.
References.
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https://en.wikipedia.org/wiki?curid=14286218
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14286236
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Phosphoribosylformylglycinamidine synthase
|
In enzymology, a phosphoribosylformylglycinamidine synthase (EC 6.3.5.3) is an enzyme that catalyzes the chemical reaction
ATP + "N"2-formyl-"N"1-(5-phospho--ribosyl)glycinamide + -glutamine + H2O formula_0 ADP + phosphate + 2-(formamido)-"N"1-(5-phospho--ribosyl)acetamidine + -glutamate
The 4 substrates of this enzyme are ATP, N2-formyl-N1-(5-phospho-D-ribosyl)glycinamide, L-glutamine, and H2O, whereas its 4 products are ADP, phosphate, 2-(formamido)-N1-(5-phospho-D-ribosyl)acetamidine, and L-glutamate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds carbon-nitrogen ligases with glutamine as amido-N-donor. The systematic name of this enzyme class is N2-formyl-N1-(5-phospho-D-ribosyl)glycinamide:L-glutamine amido-ligase (ADP-forming). Other names in common use include phosphoribosylformylglycinamidine synthetase, formylglycinamide ribonucloetide amidotransferase, phosphoribosylformylglycineamidine synthetase, FGAM synthetase, FGAR amidotransferase, 5'-phosphoribosylformylglycinamide:L-glutamine amido-ligase, (ADP-forming), 2-N-formyl-1-N-(5-phospho-D-ribosyl)glycinamide:L-glutamine, and amido-ligase (ADP-forming).
It is known as ADE6 in "Saccharomyces cerevisiae" (budding yeast) genetics.
Structural studies.
As of late 2007, 8 structures have been solved for this class of enzymes, with PDB accession codes 1T3T, 1VK3, 1VQ3, 2HRU, 2HRY, 2HS0, 2HS3, and 2HS4.
Regulation.
This enzyme participates in purine metabolism. Oncogenic and physiological signals lead to the ERK-dependent PFAS phosphorylation at the T619 site, stimulating de novo purine synthesis flux. In addition, ERK-mediated PFAS phosphorylation is required for cell and tumor growth.
References.
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https://en.wikipedia.org/wiki?curid=14286236
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14286244
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Phytanate—CoA ligase
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In enzymology, a phytanate—CoA ligase (EC 6.2.1.24) is an enzyme that catalyzes the chemical reaction
ATP + phytanate + CoA formula_0 AMP + diphosphate + phytanoyl-CoA
The 3 substrates of this enzyme are ATP, phytanate, and CoA, whereas its 3 products are AMP, diphosphate, and phytanoyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is phytanate:CoA ligase (AMP-forming). This enzyme is also called phytanoyl-CoA ligase.
References.
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[
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https://en.wikipedia.org/wiki?curid=14286244
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14286252
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Proline—tRNA ligase
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In enzymology, a proline—tRNA ligase (EC 6.1.1.15) is an enzyme that catalyzes the chemical reaction
ATP + L-proline + tRNAPro formula_0 AMP + diphosphate + L-prolyl-tRNAPro
The 3 substrates of this enzyme are ATP, L-proline, and tRNA(Pro), whereas its 3 products are AMP, diphosphate, and L-prolyl-tRNA(Pro).
This enzyme participates in arginine and proline metabolism and aminoacyl-trna biosynthesis.
Nomenclature.
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-proline:tRNAPro ligase (AMP-forming). Other names in common use include prolyl-tRNA synthetase, prolyl-transferRNA synthetase, prolyl-transfer ribonucleate synthetase, proline translase, prolyl-transfer ribonucleic acid synthetase, prolyl-s-RNA synthetase, and prolinyl-tRNA ligase.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14286252
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14286260
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Propionate—CoA ligase
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In enzymology, a propionate—CoA ligase (EC 6.2.1.17) is an enzyme that catalyzes the chemical reaction
ATP + propanoate + CoA formula_0 AMP + diphosphate + propanoyl-CoA
The 3 substrates of this enzyme are ATP, propanoate, and CoA, whereas its 3 products are AMP, diphosphate, and propanoyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is propanoate:CoA ligase (AMP-forming). This enzyme is also called propionyl-CoA synthetase. This enzyme participates in propanoate metabolism.
References.
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https://en.wikipedia.org/wiki?curid=14286260
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14286267
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Ribose-5-phosphate—ammonia ligase
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In enzymology, a ribose-5-phosphate—ammonia ligase (EC 6.3.4.7) is an enzyme that catalyzes the chemical reaction
ATP + ribose 5-phosphate + NH3 formula_0 ADP + phosphate + 5-phosphoribosylamine
The 3 substrates of this enzyme are ATP, ribose 5-phosphate, and NH3, whereas its 3 products are ADP, phosphate, and 5-phosphoribosylamine.
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. The systematic name of this enzyme class is ribose-5-phosphate:ammonia ligase (ADP-forming). This enzyme participates in purine metabolism.
References.
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https://en.wikipedia.org/wiki?curid=14286267
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14286274
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RNA-3'-phosphate cyclase
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In enzymology, a RNA-3′-phosphate cyclase (EC 6.5.1.4) is an enzyme that catalyzes the chemical reaction
ATP + RNA 3'-terminal-phosphate formula_0 AMP + diphosphate + RNA terminal-2',3'-cyclic-phosphate
Thus, the two substrates of this enzyme are ATP and RNA 3'-terminal-phosphate, whereas its 3 products are AMP, diphosphate, and RNA terminal-2',3'-cyclic-phosphate.
This enzyme belongs to the family of ligases, specifically those forming phosphoric-ester bonds. The systematic name of this enzyme class is RNA-3'-phosphate:RNA ligase (cyclizing, AMP-forming). This enzyme is also called RNA cyclase.
Structural studies.
As of 2010, three structures have been solved for this class of enzymes, with PDB accession codes 1QMH and 1QMI, (un-adenylated) and 3KGD (adenylated).
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14286274
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14286279
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RNA ligase (ATP)
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In enzymology, an RNA ligase (ATP) (EC 6.5.1.3) is an enzyme that catalyzes the chemical reaction "ATP + (ribonucleotide)n + (ribonucleotide)m formula_0 AMP + diphosphate + (ribonucleotide)n+m". Their three substrates are ATP, (ribonucleotide)n and (ribonucleotide)m, and their three products are AMP, diphosphate and (ribonucleotide)n+m.
These enzymes belong to the family of ligases, specifically those forming phosphoric-ester bonds. The systematic name of this enzyme class is poly(ribonucleotide):poly(ribonucleotide) ligase (AMP-forming). Other names in common use include polyribonucleotide synthase (ATP), RNA ligase, polyribonucleotide ligase and ribonucleic ligase.
As of late 2007, two structures have been solved for this class of enzymes, with PDB accession codes 1VDX and 2C5U.
References.
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[
{
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https://en.wikipedia.org/wiki?curid=14286279
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14286292
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Serine—tRNA ligase
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In enzymology, a serine—tRNA ligase (EC 6.1.1.11) is an enzyme that catalyzes the chemical reaction
ATP + L-serine + tRNASer formula_0 AMP + diphosphate + L-seryl-tRNASer
The 3 substrates of this enzyme are ATP, L-serine, and tRNA(Ser), whereas its 3 products are AMP, diphosphate, and L-seryl-tRNA(Ser).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-serine:tRNASer ligase (AMP-forming). Other names in common use include seryl-tRNA synthetase, SerRS, seryl-transfer ribonucleate synthetase, seryl-transfer RNA synthetase, seryl-transfer ribonucleic acid synthetase, and serine translase. This enzyme participates in glycine, serine and threonine metabolism and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 13 structures have been solved for this class of enzymes, with PDB accession codes 1SER, 1SES, 1SET, 1SRY, 1WLE, 2CIM, 2CJ9, 2CJA, 2CJB, 2DQ0, 2DQ1, 2DQ2, and 2DQ3.
References.
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https://en.wikipedia.org/wiki?curid=14286292
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14286303
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Succinate—CoA ligase (ADP-forming)
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In enzymology, a succinate-CoA ligase (ADP-forming) (EC 6.2.1.5) is an enzyme that catalyzes the chemical reaction
ATP + succinate + CoA formula_0 ADP + phosphate + succinyl-CoA
The 3 substrates of this enzyme are ATP, succinate, and CoA, whereas its 3 products are ADP, phosphate, and succinyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is succinate:CoA ligase (ADP-forming). Other names in common use include succinyl-CoA synthetase (ADP-forming), succinic thiokinase, succinate thiokinase, succinyl-CoA synthetase, succinyl coenzyme A synthetase (adenosine diphosphate-forming), succinyl coenzyme A synthetase, A-STK (adenin nucleotide-linked succinate thiokinase), STK, and A-SCS. This enzyme participates in 4 metabolic pathways: Citric acid cycle, propanoate metabolism, c5-branched dibasic acid metabolism, and reductive carboxylate cycle (CO2 fixation).
Structural studies.
As of late 2007, 12 structures have been solved for this class of enzymes, with PDB accession codes 1CQI, 1CQJ, 1JKJ, 1JLL, 1OI7, 1SCU, 2NU6, 2NU7, 2NU8, 2NU9, 2NUA, and 2SCU.
References.
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https://en.wikipedia.org/wiki?curid=14286303
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14286308
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Succinate—CoA ligase (GDP-forming)
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In enzymology, a succinate—CoA ligase (GDP-forming) (EC 6.2.1.4) is an enzyme that catalyzes the chemical reaction
GTP + succinate + CoA formula_0 GDP + phosphate + succinyl-CoA
The 3 substrates of this enzyme are GTP, succinate, and CoA, whereas its 3 products are GDP, phosphate, and succinyl-CoA.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is succinate:CoA ligase (GDP-forming). Other names in common use include succinyl-CoA synthetase (GDP-forming), succinyl coenzyme A synthetase (guanosine diphosphate-forming), succinate thiokinase, succinic thiokinase, succinyl coenzyme A synthetase, succinate-phosphorylating enzyme, P-enzyme, SCS, G-STK, succinyl coenzyme A synthetase (GDP-forming), succinyl CoA synthetase, and succinyl coenzyme A synthetase. This enzyme participates in the citric acid cycle and propanoate metabolism.
Structural studies.
As of late 2007, 6 structures have been solved for this class of enzymes, with PDB accession codes 1EUC, 1EUD, 2FP4, 2FPG, 2FPI, and 2FPP.
References.
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14286313
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Tetrahydrofolate synthase
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Class of enzymes
In enzymology, a tetrahydrofolate synthase (EC 6.3.2.17) is an enzyme that catalyzes the chemical reaction
ATP + tetrahydropteroyl-[gamma-Glu]n + L-glutamate formula_0 ADP + phosphate + tetrahydropteroyl-[gamma-Glu]n+1
The 3 substrates of this enzyme are ATP, tetrahydropteroyl-[gamma-Glu]n, and L-glutamate, whereas its 3 products are ADP, phosphate, and tetrahydropteroyl-[gamma-Glu]n+1.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is tetrahydropteroyl-gamma-polyglutamate:L-glutamate gamma-ligase (ADP-forming). This enzyme participates in folate biosynthesis.
Structural studies.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1FGS, 1JBV, 1JBW, 2GC5, 2GC6, 2GCA, and 2GCB.
References.
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https://en.wikipedia.org/wiki?curid=14286313
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14286321
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Threonine—tRNA ligase
|
In enzymology, a threonine-tRNA ligase (EC 6.1.1.3) is an enzyme that catalyzes the chemical reaction
ATP + L-threonine + tRNA(Thr) formula_0 AMP + diphosphate + L-threonyl-tRNA(Thr)
The three substrates of this enzyme are ATP, L-threonine, and threonine-specific transfer RNA [tRNA(Thr)], whereas its three products are AMP, diphosphate, and L-threonyl-tRNA(Thr).
The systematic name of this enzyme class is L-threonine:tRNAThr ligase (AMP-forming). Other names in common use include threonyl-tRNA synthetase, threonyl-transfer ribonucleate synthetase, threonyl-transfer RNA synthetase, threonyl-transfer ribonucleic acid synthetase, threonyl ribonucleic synthetase, threonine-transfer ribonucleate synthetase, threonine translase, threonyl-tRNA synthetase, and TARS.
Threonine—tRNA ligase (TARS) belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in tRNA and related compounds. More precisely, it belongs to the family of the aminoacyl-tRNA synthetases. These latter enzymes link amino acids to their cognate transfer RNAs (tRNA) in aminoacylation reactions that establish the connection between a specific amino acid and a nucleotide triplet anticodon embedded in the tRNA. During their long evolution, some of these enzymes have acquired additional functions, including roles in RNA splicing, RNA trafficking, transcriptional regulation, translational regulation, and cell signaling.
Structural studies.
As of late 2007, 17 structures have been solved for this class of enzymes, with PDB accession codes 1EVK, 1EVL, 1FYF, 1KOG, 1NYQ, 1NYR, 1QF6, 1TJE, 1TKE, 1TKG, 1TKY, 1WWT, 1Y2Q, 2HKZ, 2HL0, 2HL1, and 2HL2.
Translational regulation.
Threonyl-tRNA synthetase (TARS) from "Escherichia coli" is encoded by the "thrS" gene. It is a homodimeric enzyme that aminoacylates tRNA(Thr) with the amino acid threonine. In addition, TARS has the ability to bind to its own messenger RNA (mRNA) immediately upstream of the AUG start codon, to inhibit its translation by competing with ribosome binding, and thus to negatively regulate the expression of its own gene. The cis-acting region responsible for the control, called operator, can be folded into four distinct domains. Each of domains 2 and 4 can be folded in a stem and loop structure that mimics the anticodon arm of "E. coli" tRNA(Thr). Mutagenesis and biochemical experiments have shown that the two anticodon-like domains of the operator bind to the two tRNA(Thr) anticodon recognition sites (one per subunit) of the dimeric TARS in a quasi-symmetrical manner.
The crystal structures of (i) TARS complexed with two tRNA(Thr) molecules, and (ii) TARS complexed with two isolated domains 2, have confirmed that TARS recognition is primarily governed by similar base-specific interactions between the anticodon loop of tRNA(Thr) and the loop of the operator domain 2. The same amino acids interact with the CGU anticodon sequence of tRNA(Thr) and the analogous residues in domain 2.
References.
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14286326
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Trans-feruloyl-CoA synthase
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In enzymology, a "trans"-feruloyl—CoA synthase (EC 6.2.1.34) is an enzyme that catalyzes the chemical reaction
ferulic acid + CoASH + ATP formula_0 trans-feruloyl-CoA + products of ATP breakdown
The 3 substrates of this enzyme are ferulic acid, CoASH, and ATP, whereas its two products are trans-feruloyl-CoA and products of ATP breakdown.
This enzyme belongs to the family of ligases, specifically those forming carbon-sulfur bonds as acid-thiol ligases. The systematic name of this enzyme class is trans-ferulate:CoASH ligase (ATP-hydrolysing). This enzyme is also called trans-feruloyl-CoA synthetase.
References.
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14286334
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Trypanothione synthase
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Class of enzymes
In enzymology, a trypanothione synthase (EC 6.3.1.9) is an enzyme that catalyzes the chemical reaction
glutathione + glutathionylspermidine + ATP formula_0 N1,N8-bis(glutathionyl)spermidine + ADP + phosphate
The 3 substrates of this enzyme are glutathione, glutathionylspermidine, and ATP, whereas its 3 products are N1,N8-bis(glutathionyl)spermidine, ADP, and phosphate.
This reaction is especially important for protozoa in the order kinetoplastida as the molecule of N1,N8-bis(glutathionyl)spermidine, also known as trypanothione, is homologous to the function of glutathione in most other prokaryotic and eukaryotic cells. This means that it is a key intermediate in maintaining thiol redox within the cell and defending against harmful oxidative effects in such protozoa.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-ammonia (or amine) ligases (amide synthases). The systematic name of this enzyme class is glutathionylspermidine:glutathione ligase (ADP-forming).
Structure.
The active bifunctional enzyme of trypanothione synthase is found as a 74.4 KDa monomer consisting of 652 residues with two catalytic domains. Its C-terminal domain is a synthetase and has an ATP-grasp family fold that is usually found in carbon-nitrogen ligases. The N-terminal domain is a cysteine, histidine-dependent aminohydrolase amidase. Structurally the synthetase and amidase domains are bound together by three residues of Glu-650-Asp-651-Glu-652 through hydrogen bonding and salt bridge interactions with basic side chains in order for the protein to properly fold. These three residues also block the catalytic Cys-59 in the amidase domain.
It is currently known that the synthetase active site is shaped in the fashion of a triangular cavity that binds the three substrates such that the end of each molecule is nestled in a vertex of the triangle. The particular residues of Arg-553 and Arg-613 have been found to key for synthetic function, however further research into the structure of trypanothione synthase must be done in order to fully understand the enzyme's active sites.
Function.
The main function of trypanothione synthase is to use the free energy generated from ATP hydrolysis to conjugate glutathione and spermidine to form the intermediate of glutathionylspermidine and then the final product of trypanothione. It also catalyzes the reverse reaction as well, albeit at a much lower rate. Under conditions found to be the optimum for both the forward and backwards reactions, trypanothione synthase from trypanosoma cruzi was found to have an amidase activity that was only about 1% of the forwards synthetase activity. This low activity can be explained by the blocking of the catalytic Cys in the amidase active site in order for the protein to properly fold.
In parasitic kinetoplastids trypanothione synthase activity is key to survival. Due to the need for trypanothione in order to defend against oxidative stresses, and maintain thiol and ribonucleotide metabolism. It was observed that induced knockout of trypanothione synthase through RNA interference caused a reduced growth rate of twofold among trypanosoma brucei due to the immediate disruption of flux through thiol redox. In other similar experiments which observed cell death after knocking out trypanothione synthase, it was shown that after two hours of being exposed to hydrogen peroxide in order to mimic the oxidant attack of phagocytes, the cells which did not contain working trypanothione synthase had a much higher death rate than wild type T. brucei.
Mechanism.
The current believed mechanism for synthetase activity is that first glutathione and Mg2+-ATP bind to the enzyme in a ternary complex where glutathione becomes activated by ATP and forms glutathionyl phosphate. ADP then leaves the active site and the activated phosphate still bound to the enzyme in what is equivalent to a substituted enzyme reacts with glutathionylspermidine to form trypanothione.
Regulation.
The regulation of trypanothione synthase is currently thought to be driven by conformational changes caused by allosteric interactions as the enzyme must regulate the relative levels of spermidine, glutathionylspermidine, glutathione and trypanothione in the cell. Evidence for this regulation is that the residues which allow the synthase domain to block the amidase active site are highly conserved among different species of kinetoplastids, indicating that they are key in the enzyme's function and that the binding of certain substrates might cause conformational shifts that would open up the amidase active site.
Clinical Significance.
Many diseases such as Human African trypanosomiasis, Nagana disease in cattle, and Chagas disease are caused by kinetoplastid parasites. Such diseases infect an estimated 15 to 20 million people per year worldwide and kill 100000 to 150000 of those infected. Current treatments for these diseases were generally made almost 100 years ago and in that time many of the parasites have developed resistance, in addition, many of the original treatments are highly toxic. Targeting trypanothione synthase could be a novel way of preventing and curing these diseases through disruption of the parasites' metabolism. Scientists believe that the thiol metabolic pathway is an especially good target for anti-parasitic drug production as trypanothione based thiol redox is absent in humans and it is thought that thiol redox is key in the mechanisms some parasites have in order to obtain drug resistance .
References.
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14286343
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Tryptophan—tRNA ligase
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In enzymology, a tryptophan-tRNA ligase (EC 6.1.1.2) is an enzyme that catalyzes the chemical reaction
ATP + L-tryptophan + tRNATrp formula_0 AMP + diphosphate + L-tryptophyl-tRNATrp
The 3 substrates of this enzyme are ATP, L-tryptophan, and tRNA(Trp), whereas its 3 products are AMP, diphosphate, and L-tryptophyl-tRNATrp.
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-tryptophan:tRNATrp ligase (AMP-forming). Other names in common use include tryptophanyl-tRNA synthetase, L-tryptophan-tRNATrp ligase (AMP-forming), tryptophanyl-transfer ribonucleate synthetase, tryptophanyl-transfer ribonucleic acid synthetase, tryptophanyl-transfer RNA synthetase, tryptophanyl ribonucleic synthetase, tryptophanyl-transfer ribonucleic synthetase, tryptophanyl-tRNA synthase, tryptophan translase, and TrpRS. This enzyme participates in tryptophan metabolism and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 21 structures have been solved for this class of enzymes, with PDB accession codes 1D2R, 1I6K, 1I6L, 1I6M, 1M83, 1MAU, 1MAW, 1MB2, 1O5T, 1R6T, 1R6U, 1ULH, 1YIA, 1YID, 2A4M, 2AKE, 2AZX, 2DR2, 2G36, 2IP1, and 2OV4.
References.
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https://en.wikipedia.org/wiki?curid=14286343
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14286349
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Tubulin—tyrosine ligase
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Class of enzymes
In enzymology, a tubulin—tyrosine ligase (EC 6.3.2.25) is an enzyme that catalyzes the chemical reaction
ATP + detyrosinated α-tubulin + -tyrosine formula_0 α-tubulin + ADP + phosphate
The 3 substrates of this enzyme are ATP, detyrosinated alpha-tubulin, and L-tyrosine, whereas its 3 products are alpha-tubulin, ADP, and phosphate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is alpha-tubulin:L-tyrosine ligase (ADP-forming).
References.
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{
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https://en.wikipedia.org/wiki?curid=14286349
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14286357
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Tyrosine—arginine ligase
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Class of enzymes
In enzymology, a tyrosine—arginine ligase (EC 6.3.2.24) is an enzyme that catalyzes the chemical reaction
ATP + -tyrosine + -arginine formula_0 AMP + diphosphate + -tyrosyl--arginine
The 3 substrates of this enzyme are ATP, L-tyrosine, and L-arginine, whereas its 3 products are AMP, diphosphate, and L-tyrosyl-L-arginine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is L-tyrosine:L-arginine ligase (AMP-forming). Other names in common use include tyrosyl-arginine synthase, kyotorphin synthase, kyotorphin-synthesizing enzyme, and kyotorphin synthetase.
References.
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https://en.wikipedia.org/wiki?curid=14286357
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14286366
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Tyrosine—tRNA ligase
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Tyrosine—tRNA ligase (EC 6.1.1.1), also known as tyrosyl-tRNA synthetase is an enzyme that is encoded by the gene YARS. Tyrosine—tRNA ligase catalyzes the chemical reaction
ATP + L-tyrosine + tRNA(Tyr) formula_0 AMP + diphosphate + L-tyrosyl-tRNA(Tyr)
The three substrates of this enzyme are ATP, L-tyrosine, and a tyrosine-specific transfer RNA [tRNA(Tyr) or tRNATyr], whereas its three products are AMP, diphosphate, and L-tyrosyl-tRNA(Tyr).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in tRNA and related compounds. More specifically, it belongs to the family of the aminoacyl-tRNA synthetases. These latter enzymes link amino acids to their cognate transfer RNAs (tRNA) in aminoacylation reactions that establish the connection between a specific amino acid and a nucleotide triplet anticodon embedded in the tRNA. Therefore, they are the enzymes that translate the genetic code "in vivo". The 20 enzymes, corresponding to the 20 natural amino acids, are divided into two classes of 10 enzymes each. This division is defined by the unique architectures associated with the catalytic domains and by signature sequences specific to each class.
Structural studies.
As of late 2007, 34 structures have been solved for this class of enzymes, with PDB accession codes
The tyrosyl-tRNA synthetases (YARS) are either homodimers or monomers with a pseudo-dimeric structure. Each subunit or pseudo-subunit comprises an N-terminal domain which has: (i) about 230 amino acid residues; (ii) the mononucleotide binding fold (also known as Rossmann fold) of the class I aminoacyl-tRNA synthetases; (iii) an idiosynchratic insertion between the two halves of the fold (known as Connective Peptide 1 or CP1); (iv) the two signature sequences HIGH and KMSKS of the class I aminoacyl-tRNA synthetases. The N-terminal domain contains the catalytic site of the enzyme. The C-terminal moiety of the YARSs varies in sequence, length and organization and is involved in the recognition of the tRNA anticodon.
Eubacteria.
Tyrosyl-tRNA synthetase from "Bacillus stearothermophilus" was the first synthetase whose crystal structure has been solved at high resolution (2.3 Å), alone or in complex with tyrosine, tyrosyl-adenylate or tyrosinyl-adenylate.(P. Brick 1989) The structures of the "Staphylococcus aureus" YARS and of a truncated version of "Escherichia coli" YARS have also been solved. A structural model of the complex between "B. sterothermophilus" YARS and tRNA(Tyr) was constructed using extensive mutagenesis data on both YARS and tRNATyr and found consistent with the crystal structure of the complex between YARS and tRNA(Tyr) from "Thermus thermophilus", which was subsequently solved at 2.9 Å resolution.
The C-terminal moiety of the eubacterial YARSs comprises two domains: (i) a proximal α-helical domain (known as Anticodon Binding Domain or α-ACB) of about 100 amino acids; (ii) a distal domain (known as S4-like) that shares high homology with the C-terminal domain of ribosomal protein S4. The S4-like domain was disordered in the crystal structure of "B. stearothermophilus" YARS. However, biochemical and NMR experiments have shown that the S4-like domain is folded in solution, and that its structure is similar to that in the crystal structure of the "T. thermophilus" YARS. Mutagenesis experiments have shown that the flexibility of the peptide that links the α-ACB and S4-like domains is responsible for the disorder of the latter in the structure and that elements of sequence in this linker peptide are essential for the binding of tRNA(Tyr) by YARS and its aminoacylation with tyrosine. TyrRSs from eubacterial species are divided into two subgroups according to variation in their C-terminal moiety.
Archaea and lower eukaryotes.
The crystal structures of several archaeal tyrosyl-tRNA synthetases are available. The crystal structure of the complex between YARS from "Methanococcus jannaschii", tRNA(Tyr) and L-tyrosine has been solved at 1.95 Å resolution. The crystal structures of the YARSs from "Archeoglobus fulgidus", "Pyrococcus horikoshii" and "Aeropyrum pernix" have also been solved at high resolution.(M. Kuratani 2006) The C-terminal moieties of the archaeal YARSs contain only one domain. This domain is different from the α-ACB domain of eubacteria; it shares strong homology with the C-terminal domain of the tryptophanyl-tRNA synthetases and was therefore named C-W/Y domain. It is present in all eukarya. The structure of the complex between YARS from "Saccharomyces cerevisiae", tRNA(Tyr) and an analog of tysosyl-adenylate has been solved at 2.4 Å resolution. The YARS from this lower eukaryote has an organization which is similar to that of the archaeal YARSs.
"Homo sapiens" cytoplasm.
The human YARS has a C-terminal moiety that include a proximal C-W/Y domain and a distal domain which is not found in the YARSs of lower eukaryotes, archaea or eubacteria, and is a homolog of endothelial monocyte-activating polypeptide II (EMAP II, a mammalian cytokine). Although full-length, native YARS has no cell-signaling activity, the enzyme is secreted during apoptosis in cell culture and can be cleaved with an extracellular enzyme such as leukocyte elastase. The two released fragments, an N-terminal mini-YARS and a C-terminal EMAP II-like C-terminal domain, are active cytokines. The structure of mini-YARS has been solved at 1.18 Å resolution. It has an N-terminal Rossmann-fold domain and a C-terminal C-W/Y domain, similar to those of other YARSs.
"Homo sapiens" mitochondria.
The mitochondrial tyrosyl-tRNA synthetases (mt-YARSs) and in particular "H. sapiens" mt-YARS, likely originate from a YARS of eubacterial origin. Their C-terminal moiety includes both α-ACB and S4-like domains like the eubacterial YARSs and share a low sequence identity with their cytosolic relatives. The crystal structure of a complex between a recombinant "H. sapiens" mt-YARS, devoid of the S4-like domain, and an analog of tyrosyl-adenylate has been solved at 2.2 Å resolution.
"Neurospora crassa" mitochondria.
The mitochondrial (mt) tyrosyl-tRNA synthetase of "Neurospora crassa", which is encoded by the nuclear gene cyt-18, is a bifunctional enzyme that catalyzes the aminoacylation of mt-tRNA(Tyr) and promotes the splicing of the mitochondrial group I introns. The crystal structure of a C-terminally truncated "N. crassa" mt-YARS that functions in splicing group I introns, has been determined at 1.95 Å resolution. Its Rossmann-fold domain and intermediate α-ACB domain superimpose on those of eubacterial YARSs, except for an additional N-terminal extension and three small insertions. The structure of the complex between a group I intron ribozyme and the splicing-active, carboxy-terminally truncated mt-YARS has been solved at 4.5 Å resolution. The structure shows that the group I intron binds across the two subunits of the homodimeric protein with a newly evolved RNA-binding surface distinct from that which binds tRNA(Tyr). This RNA binding surface provides an extended scaffold for the phosphodiester backbone of the conserved catalytic core of the intron RNA, allowing the protein to promote the splicing of a wide variety of group I introns. The group I intron-binding surface includes three small insertions and additional structural adaptations relative to non-splicing eubacterial YARSs, indicating a multistep adaptation for splicing function.
"Plasmodium falciparum".
The structure of the complex between "Plasmodium falciparum" tyrosyl-tRNA synthetase (Pf-YARS) and tyrosyl-adenylate at 2.2 Å resolution, shows that the overall fold of Pf-YARS is typical of class I synthetases. It comprises an N-terminal catalytic domain (residues 18–260) and an anticodon-binding domain (residues 261–370). The polypeptide loop that includes the KMSKS motif, is highly ordered and close to the bound substrate at the active site. Pf-YARS contains the ELR motif, which is present in "H. sapiens" mini-YARS and chemokines. Pf-YARS is expressed in all asexual parasite stages (rings, trophozoites and schizonts) and is exported to the host erythrocyte cytosol, from where it is released into blood plasma on iRBC rupture. Using its ELR peptide motif, Pf-YARS specifically binds to and internalizes into host macrophages, leading to enhanced secretion of the pro-inflammatory cytokines TnF-α and IL-6. The interaction between Pf-YARS and macrophages augments expression of adherence-linked host endothelial receptors ICAm-1 and VCAm-1.
Mimivirus.
"Acanthamoeba polyphaga" mimivirus is the largest known DNA virus. It genome encodes four aminoacyl-tRNA synthetases: RARS, CARS, MARS, and YARS. The crystal structure of the mimivirus tyrosyl-tRNA synthetase in complex with tyrosinol has been solved at 2.2 Å resolution. The mimiviral YARS exhibits the typical fold and active-site organization of archaeal-type YARSs, with an N-terminal Rossmann-fold catalytic domain, an anticodon binding domain, and no extra C-terminal domain. It presents a unique dimeric conformation and significant differences in its anticodon binding site, when compared with the YARSs from other organisms.
"Leishmania major".
The single YARS gene that is present in the genomes of trypanosomatids, codes for a protein that has twice the length of tyrosyl-tRBA synthetase from other organisms. Each half of the double-length YARS contains a catalytic domain and an anticodon-binding domain; however, the two halves retain only 17% sequence identity to each other. Crystal structures of "Leishmania major" YARS at 3.0 Å resolution show that the two halves of a single molecule form a pseudo-dimer that resembles the canonical YARS dimer. The C-terminal copy of the catalytic domain has lost the catalytically important HIGH and KMSKS motifs, characteristic of class I aminoacyl-tRNA synthetases. Thus, the pseudo-dimer contains only one functional active site (contributed by the N-terminal half) and only one functional anticodon recognition site (contributed by the C-terminal half). Thus, the "L. major" YARS pseudo-dimer is inherently asymmetric.
Roles of the subunits and domains.
The N-terminal domain of tyrosyl-tRNA synthetase provides the chemical groups necessary for converting the substrates tyrosine and ATP into a reactive intermediate, tyrosyl-adenylate (the first step of the aminoacylation reaction) and for transferring the amino-acid moiety from tyrosyl-adenylate to the 3'OH-CCA terminus of the cognate tRNA(Tyr) (the second step of the aminoacylation reaction). The other domains are responsible (i) for the recognition of the anticodon bases of the cognate tRNA(Tyr); (ii) for the binding of the long variable arm of tRNA(Tyr) in eubacteria; and (iii) for unrelated functions such as cytokine activity.
Recognition of tRNA(Tyr).
The tRNA(Tyr) molecule has an L-shaped structure. Its recognition involves both subunits of the tyrosyl-tRNA synthetase dimer. The acceptor arm of tRNA(Tyr) interacts with the catalytic domain of one YARS monomer whereas the anticodon arm interacts with the C-terminal moiety of the other monomer. In most YARS structures, the monomers are related to each other by a twofold rotational symmetry. Moreover, all available crystal structures of complexes between YARS and tRNA(Tyr) are also planar, with symmetrical conformations of the two monomers in the dimer and with two tRNA(Tyr) molecules simultaneously interacting with one YARS dimer. However, kinetic studies of tyrosine activation and tRNA(Tyr) charging have revealed an anticooperative behavior of the TyrRS dimer in solution: each TyrRS dimer binds and tyrosylates only one tRNA(Tyr) molecule at a time. Thus, only one of the two sites is active at any given time.
The presence of base pair Gua1:Cyt72 in the acceptor stem of tRNA(Tyr) from eubacteria and of base pair Cyt1-Gua72 in tRNA(Tyr) from archaea and eukaryotes results in a species specific recognition of tRNATyr by tyrosyl-tRNA synthetase. This characteristic of the recognition between YARS and tRNA(Tyr) has been used to obtain aminoacyl-tRNA synthetases that can specifically charge non-sense suppressor derivatives of tRNA(Tyr) with unnatural aminoacids in vivo without interfering with the normal process of translation in the cell.
Both tyrosyl-tRNA synthetases and tryptophanyl-tRNA synthetases belong to Class I of the aminoacyl-tRNA synthetases, both are dimers and both have a class II mode of tRNA recognition, i.e. they interact with their cognate tRNAs from the variable loop and major groove side of the acceptor stem. This is in strong contrast to the other class I enzymes, which are monomeric and approach their cognate tRNA from minor groove side of the acceptor stem.
Folding and stability.
The unfolding reaction and stability of tyrosyl-tRNA synthetase from "Bacillus stearothermophilus" have been studied under equilibrium conditions. This homodimeric enzyme is highly stable with a variation of free energy upon unfolding equal to 41 ± 1 kcal/mol. It unfolds through a compact monomeric intermediate. About one-third of the global energy of stabilization comes from the association between the two subunits, and one-third come from the secondary and tertiary interactions stabilizing each of the two molecules of the monomeric intermediate. Both mutations within the dimer interface and mutations distal to the interface can destabilize the association between the subunits. These experiments have shown in particular that the monomer of YARS is enzymatically inactive.
References.
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Further reading.
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https://en.wikipedia.org/wiki?curid=14286366
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14286371
|
Ubiquitin—calmodulin ligase
|
Class of enzymes
In enzymology, an ubiquitin-calmodulin ligase (EC 6.3.2.21) is an enzyme that catalyzes the chemical reaction
n ATP + calmodulin + n ubiquitin formula_0 n AMP + n diphosphate + (ubiquitin)n-calmodulin
The 3 substrates of this enzyme are ATP, calmodulin, and ubiquitin, whereas its 3 products are AMP, diphosphate, and (ubiquitin)n-calmodulin.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is calmodulin:ubiquitin ligase (AMP-forming). Other names in common use include ubiquityl-calmodulin synthase, ubiquitin-calmodulin synthetase, ubiquityl-calmodulin synthetase, and uCaM-synthetase.
References.
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{
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https://en.wikipedia.org/wiki?curid=14286371
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14286388
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UDP-N-acetylmuramate—L-alanine ligase
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Class of enzymes
In enzymology, a UDP-"N"-acetylmuramate—-alanine ligase (EC 6.3.2.8) is an enzyme that catalyzes the chemical reaction
ATP + UDP-N-acetylmuramate + L-alanine formula_0 ADP + phosphate + UDP-N-acetylmuramoyl-L-alanine
The 3 substrates of this enzyme are ATP, UDP-N-acetylmuramate, and L-alanine, whereas its 3 products are ADP, phosphate, and UDP-N-acetylmuramoyl-L-alanine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is UDP-N-acetylmuramate:L-alanine ligase (ADP-forming). Other names in common use include MurC synthetase, UDP-N-acetylmuramoyl-L-alanine synthetase, uridine diphospho-N-acetylmuramoylalanine synthetase, UDP-N-acetylmuramoylalanine synthetase, L-alanine-adding enzyme, UDP-acetylmuramyl-L-alanine synthetase, UDPMurNAc-L-alanine synthetase, L-Ala ligase, uridine diphosphate N-acetylmuramate:L-alanine ligase, uridine 5'-diphosphate-N-acetylmuramyl-L-alanine synthetase, uridine-diphosphate-N-acetylmuramate:L-alanine ligase, UDP-MurNAc:L-alanine ligase, alanine-adding enzyme, and UDP-N-acetylmuramyl:L-alanine ligase. This enzyme participates in d-glutamine and d-glutamate metabolism and peptidoglycan biosynthesis.
Structural studies.
As of late 2007, 6 structures have been solved for this class of enzymes, with PDB accession codes 1GQQ, 1GQY, 1J6U, 1P31, 1P3D, and 2F00.
References.
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{
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https://en.wikipedia.org/wiki?curid=14286388
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14286395
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UDP-N-acetylmuramoyl-L-alanine—D-glutamate ligase
|
Class of enzymes
In enzymology, a UDP-"N"-acetylmuramoyl--alanine—-glutamate ligase (EC 6.3.2.9) is an enzyme that catalyzes the chemical reaction
ATP + UDP-N-acetylmuramoyl-L-alanine + D-glutamate formula_0 ADP + phosphate + UDP-N-acetylmuramoyl-L-alanyl-D-glutamate
The 3 substrates of this enzyme are ATP, UDP-N-acetylmuramoyl-L-alanine, and D-glutamate, whereas its 3 products are ADP, phosphate, and UDP-N-acetylmuramoyl-L-alanyl-D-glutamate.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is UDP-N-acetylmuramoyl-L-alanine:D-glutamate ligase (ADP-forming). Other names in common use include MurD synthetase, UDP-N-acetylmuramoyl-L-alanyl-D-glutamate synthetase, uridine diphospho-N-acetylmuramoylalanyl-D-glutamate synthetase, D-glutamate-adding enzyme, D-glutamate ligase, UDP-Mur-NAC-L-Ala:D-Glu ligase, UDP-N-acetylmuramoyl-L-alanine:glutamate ligase (ADP-forming), and UDP-N-acetylmuramoylalanine-D-glutamate ligase. This enzyme participates in d-glutamine and d-glutamate metabolism and peptidoglycan biosynthesis.
Structural studies.
As of late 2007, 9 structures have been solved for this class of enzymes, with PDB accession codes 1E0D, 1EEH, 1UAG, 2JFF, 2JFG, 2JFH, 2UAG, 3UAG, and 4UAG.
References.
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https://en.wikipedia.org/wiki?curid=14286395
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14286401
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UDP-N-acetylmuramoyl-L-alanyl-D-glutamate—L-lysine ligase
|
Class of enzymes
In enzymology, a UDP-"N"-acetylmuramoyl--alanyl--glutamate—-lysine ligase (EC 6.3.2.7) is an enzyme that catalyzes the chemical reaction
ATP + UDP-N-acetylmuramoyl-L-alanyl-D-glutamate + L-lysine formula_0 ADP + phosphate + UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine
The 3 substrates of this enzyme are ATP, UDP-N-acetylmuramoyl-L-alanyl-D-glutamate, and L-lysine, whereas its 3 products are ADP, phosphate, and UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases). The systematic name of this enzyme class is UDP-N-acetylmuramoyl-L-alanyl-D-glutamate:L-lysine gamma-ligase (ADP-forming). Other names in common use include MurE synthetase, UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine synthetase, uridine diphospho-N-acetylmuramoylalanyl-D-glutamyllysine, synthetase, and UPD-MurNAc-L-Ala-D-Glu:L-Lys ligase. This enzyme participates in peptidoglycan biosynthesis.
References.
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https://en.wikipedia.org/wiki?curid=14286401
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14286411
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UDP-N-acetylmuramoyl-tripeptide—D-alanyl-D-alanine ligase
|
Class of enzymes
In enzymology, a UDP-"N"-acetylmuramoyl-tripeptide—-alanyl--alanine ligase (EC 6.3.2.10) is an enzyme that catalyzes the chemical reaction
ATP + UDP-N-acetylmuramoyl-L-alanyl-gamma-D-glutamyl-L-lysine + D-alanyl-D-alanine formula_0 ADP + phosphate + UDP-N-acetylmuramoyl-L-alanyl-gamma-D-glutamyl-L-lysyl-D-alanyl-D- alanine
The 3 substrates of this enzyme are ATP, UDP-N-acetylmuramoyl-L-alanyl-gamma-D-glutamyl-L-lysine, and D-alanyl-D-alanine, whereas its 4 products are ADP, phosphate, UDP-N-acetylmuramoyl-L-alanyl-gamma-D-glutamyl-L-lysyl-D-alanyl-D-alanine.
This enzyme belongs to the family of ligases, specifically those forming carbon-nitrogen bonds as acid-D-amino-acid ligases (peptide synthases).
Nomenclature.
The systematic name of this enzyme class is UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysine:D-alanyl-D-alanine ligase (ADP-forming). Other names in common use include MurF synthetase, UDP-N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysyl-D-alanyl-D-alanine, synthetase, UDP-N-acetylmuramoylalanyl-D-glutamyl-lysine-D-alanyl-D-alanine, ligase, uridine diphosphoacetylmuramoylpentapeptide synthetase, UDPacetylmuramoylpentapeptide synthetase, and UDP-MurNAc-L-Ala-D-Glu-L-Lys:D-Ala-D-Ala ligase. This enzyme participates in lysine biosynthesis and peptidoglycan biosynthesis.
References.
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https://en.wikipedia.org/wiki?curid=14286411
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14286421
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Urea carboxylase
|
Class of enzymes
In enzymology, a urea carboxylase (EC 6.3.4.6) is an enzyme that catalyzes the chemical reaction
ATP + urea + HCO3- formula_0 ADP + phosphate + urea-1-carboxylate
The 3 substrates of this enzyme are ATP, urea, and HCO3-, whereas its 3 products are ADP, phosphate, and urea-1-carboxylate (allophanate).
This enzyme belongs to the family of ligases, specifically those forming generic carbon-nitrogen bonds. The systematic name of this enzyme class is urea:carbon-dioxide ligase (ADP-forming). This enzyme participates in urea cycle and metabolism of amino groups. It employs one cofactor, biotin.
References.
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https://en.wikipedia.org/wiki?curid=14286421
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14286432
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Valine—tRNA ligase
|
In enzymology, a valine—tRNA ligase (EC 6.1.1.9) is an enzyme that catalyzes the chemical reaction
ATP + -valine + tRNAVal formula_0 AMP + diphosphate + -valyl-tRNAVal
The 3 substrates of this enzyme are ATP, L-valine, and tRNA(Val), whereas its 3 products are AMP, diphosphate, and L-valyl-tRNA(Val).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is L-valine:tRNAVal ligase (AMP-forming). Other names in common use include valyl-tRNA synthetase, valyl-transfer ribonucleate synthetase, valyl-transfer RNA synthetase, valyl-transfer ribonucleic acid synthetase, valine transfer ribonucleate ligase, and valine translase. This enzyme participates in valine, leucine and isoleucine biosynthesis and aminoacyl-trna biosynthesis.
Structural studies.
As of late 2007, 5 structures have been solved for this class of enzymes, with PDB accession codes 1GAX, 1IVS, 1IYW, 1WK9, and 1WKA.
References.
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https://en.wikipedia.org/wiki?curid=14286432
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14289175
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Nonhypotenuse number
|
In mathematics, a nonhypotenuse number is a natural number whose square "cannot" be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number "cannot" form the hypotenuse of a right angle triangle with integer sides.
The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is "not" a nonhypotenuse number as 52 equals 32 + 42.
The first fifty nonhypotenuse numbers are:
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84 (sequence in the OEIS)
Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value "x" scales asymptotically with "x"/√log "x".
The nonhypotenuse numbers are those numbers that have no prime factors of the form 4"k"+1. Equivalently, they are the number that cannot be expressed in the form formula_0 where "K", "m", and "n" are all positive integers. A number whose prime factors are not all of the form 4"k"+1 cannot be the hypotenuse of a "primitive" integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle.
The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first formula_1 square numbers using only formula_2 additions.
References.
<templatestyles src="Reflist/styles.css" />
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[
{
"math_id": 0,
"text": "K(m^2+n^2)"
},
{
"math_id": 1,
"text": "n"
},
{
"math_id": 2,
"text": "n+o(n)"
}
] |
https://en.wikipedia.org/wiki?curid=14289175
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14293177
|
Federal University of ABC
|
Public federal university in São Paulo, Brazil
Federal University of ABC (, UFABC) is a Brazilian federal public institution of higher learning based in Santo André and São Bernardo do Campo, municipalities belonging to the ABC region, both in the state of São Paulo.
UFABC is the only federal university in Brazil with 100% of its professors holding PhDs and, for the second consecutive year in 2011, emerged as the only university in Brazil with impact factor in scientific publications above the world average according to SCImago Institutions Rankings. The institution was evaluated by the General Course Index (IGC) of the Ministry of Education (MEC) as the best university in the State of São Paulo, being rated as the 1st in the ranking of undergraduate courses among all universities in Brazil. The IGC takes into account in its assessment factors such as infrastructure, faculty and graduates' scores in the National Student Performance Exam (ENADE). It occupies the 1st place among Brazilian universities in the "Internationalization" item in the University Ranking of the Folha de São Paulo newspaper.
The chairman of the committee that formulated the proposal of the university was Luiz Bevilacqua, who became its second rector.
History and expansion.
In 2004, the Ministry of Education sent Bill 3.962 / 2004 to the National Congress, which provided for the creation of the Federal University of ABC Foundation. This law was sanctioned by the President of the Republic Luiz Inácio Lula da Silva and published in the Federal Official Gazette (D.O.U.) of 27 July 2005, under No. 11,145 and dated 26 July 2005.
UFABC aims to integrate several campuses in the Greater ABC region. The current campuses are based in the municipalities of Santo André (headquarters) and São Bernardo do Campo.
Deans.
The first rector of UFABC was the ex-rector of the State University of Campinas (UNICAMP), professor Hermano Tavares (PhD in Automation at the Université de Toulouse), whose term lasted from 2005 to 2007.
Tavares was succeeded by the former president of the Brazilian Space Agency (AEB), professor Luiz Bevilacqua (PhD in Applied Mechanics at Stanford University and Grand Cross of the National Order of Scientific Merit), who had been his vice-chancellor and presided over the university implantation committee.
For health reasons, Bevilacqua left office in mid-2008, being succeeded, pro tempore, by the former president of the Brazilian Society of Physics, Professor Adalberto Fazzio (PhD in Renewable Energies by the National Renewable Energy Laboratory and Great Cross of the National Order of Scientific Merit).
In January 2010, President Lula appointed the former Dean of Research at the State University of Campinas (UNICAMP) and Full Professor at the University of São Paulo (USP), Professor Helio Waldman (PhD in Electrical Engineering from Stanford University and Commander of the National Order of Scientific Merit) to exercise the position of rector of UFABC for the next four years. Professor Waldman had obtained the unanimous preference of the Electoral College in the composition of the triple list and won the Consultation to the Community, obtaining the majority of votes in all categories of voters (students, teachers and administrative technicians).
Undergraduate teaching.
Unlike many universities in Brazil, where the curriculum has a fixed order, at UFABC, the student can decide the order in which the subjects will be taken, without prerequisites, only recommendations. That means that anyone can take any subject, given that the class isn't full. Also the year is divided in three four-months periods, instead of two semesters.
At the end of each period, the student chooses the classes they want to take. On each class, the students are listed by their CR (which is UFABC's version of the GPA) and the first ones will attend that class on the next period. However, if the class is from a specific course, the criteria becomes the CP (a number ranging from 0 to 1 that indicates how much of that specific course was completed by that student). All disciplines have an associated "credit", which represents the number of classes per two weeks.
After their basic university education, the student has several options: they can enter the job market with a bachelor's degree in Science and Technology or may continue at the university and attend one or two years in a bachelor's program specifically or eight forms of engineering. There is also the possibility of holding a master's degree and moving to master's degree programs in other national and international programs.
The graduate of an interdisciplinary bachelor's program of a university may enroll in up to two of 21 options for undergraduate courses as their basic training, divided into bachelor's degrees and.
The UFABC was the first institution of higher education in Brazil to adopt the interdisciplinary bachelor's degree.
Disciplines.
There are three types of disciplines at UFABC, mandatory, limited option and free. The type of any given discipline is different for each course, meaning that the same discipline might be of one type on a course, but of other type on another. Mandatory disciplines are the ones required by a course in order to finish it. For computer science students, "Databases" discipline required, so for the computer science course, it is mandatory. Limited option disciplines are meant to enhance the student's knowledge on a certain field of study. Molecular evolution, for example, is a limited option discipline for Biology students. Free disciplines are all disciplines that don't fit in either one of the previous categories. LIBRAS is a free discipline on all non-licentiate courses.
Evaluation.
The fact that the curriculum has no fixed order introduces some disadvantages, for example, one professor may apply harder exams than the others, creating a disparity between grades on the same discipline. Some, however, are "unified", meaning that the same exams and evaluations will be applied to all students, preventing that disparity. Excluding Quantum Physics and Structure of Matter, all Physics related disciplines are unified.
Different disciplines have different evaluation systems. One may have three, two, one or no exams at all (in this case, the evaluation is made via a project developed by the student). There may also be a student project and one or two exams.
Students who get a lower than C grade have the option to do one last exam, which is sometimes referred to as "recovery exam", that can increase their grade up to a C. If one gets an F and doesn't do well on the recovery exam, they will have to redo the discipline.
Coefficients.
Source:
At UFABC, all undergrad students have four coefficients. They are the formula_0, the formula_1, the formula_2 and the formula_3.
CR (""Coeficiente de Rendimento", orYield Coefficient).
It represents how well the student is doing at the disciplines. As mentioned, it is the equivalent of the GPA. It can be calculated by the following formula:formula_4
Where formula_5 is the number of completed disciplines, formula_6 is the grade on that discipline, formula_7 is the discipline's credits and formula_8 is:
formula_9 and formula_10.
CA ("Coeficiente Acadêmico", or Academic Coefficient).
It is similar to the CR, however, if a student takes the same discipline multiple times, only the one with the higher grade will be accounted. It is calculated using the following formula:formula_11
Where formula_12 is the number of different disciplines taken by the student, formula_13 is the number of credits on that discipline formula_14is the greatest grade on that discipline. formula_15 is:
formula_9 and formula_10.
CPk ("Coeficiente de Progressão", or Progression Coefficient).
Represents the ratio between the approved disciplines' credits on a course formula_16 and the total of credits on that course. It grows as the student finishes disciplines according to their category (mandatory, limited option and free). When it reaches 1, it means that student has finished all the required credits on that course formula_16. Its formula is:
formula_17
Where:
Ik ("Coeficiente de Afinidade"", or Affinity Coefficient).
It combines the formula_24, the formula_0 and the number of periods a student has studied on the university. The calculation is as follows:
formula_25
Bachelor of Science and Technology (BC&T).
Bachelor of Science and Technology is a general course that teaches the foundation on mathematics, physics, biology, chemistry and computer science, while giving a slight introduction on human sciences. The idea behind that is to have students working with interdisciplinary subjects via a methodology that encourages an investigative approach to situations and stimulates scientific research and production.
Examples of subjects studied are:
The course takes a total of 2,400 hours of study to be completed.
Bachelor of Science and Humanities (BC&H).
The Bachelor of Science and Humanities is a course of general science. The initiation in Natural Science, Formal, Social, and Philosophy is through lectures with participation in research groups, under the supervision of a senior researcher.
The course also takes 2,400 hours to finish.
Post-BC&T and post-BC&H degrees.
After completing either the BC&T or the BC&H, the student can choose up to two specific degrees to pursue. Students that graduate on BC&T have the following options:
Students that graduate on BC&H, can choose between (all of them at São Bernardo's campus):
Research.
The UFABC is the only federal university in the Brazil that has 100% of professors with a doctoral degree. UFABC research in Nanotechnology has been recognised by CNPq. Researchers in Nanotechnology in UFABC were able to publish articles in the scientific journal "Nano Letters".
The university employs professors who received the Capes awards for Best Thesis, researchers with the Commendation and the Grand Cross of the Brazilian Order of Scientific Merit, members of the Brazilian Academy of Sciences, and representatives of national scientific organizations.
In March 2007 the CAPES approved 10 of 11 applications for the opening of post-graduate studies (Master and Doctorate). The CAPES on average approves only one third of the applications referred to it.
UFABC participates in major research projects worldwide, among them the presence of the university in the largest scientific project of all time, the Large Hadron Collider (LHC) of the European Organization for Nuclear Research (CERN) that is operating the largest particle accelerator ever built and also in the DZero Experiment of the Fermi National Accelerator Laboratory (Fermilab) of the United States Department of Energy that conducts research on the basic structure of matter in the second-largest particle accelerator in the world. The university has a presence in the first Brazilian space mission to deep space, the Mission Aster ("Missão Aster"), of the Brazilian Space Agency, which provides for the launching of a space probe in 2015.
UFABC has research completed or in progress with research institutions such as the COPPE-UFRJ, the National Institute for Space Research, the Brazilian Space Agency, the Angra Nuclear Power Plant, the National Education and Research Network (RNP), UNICAMP, Unifesp, USP, UFSCar, the Brazilian National Synchrotron Light Laboratory (LNLS), the Institute for Advanced Studies of the Brazilian General Command for Aerospace Technology (CTA-IEAv), the Eletronuclear, the Institute for Research on Stem Cells (IPCTRON), the Brazilian National Laboratory for Scientific Computing (LNCC), the Brazilian Navy Technology Center (CTMSP), the National Institute of Science and Technology for Optical Communications (FOTONICOM), the ICMC-USP, the Brazilian Society of Biophysics, UFRGS, UNIFEI and UNIFENAS.
The university collaborates with research institutes as the Max Planck Institute (Germany), the Paris-Sud 11 University (France), the National Acoustic Laboratories (Australia) and the Pierre Auger Observatory (Argentina), the National Institute for Materials Science (Japan), the University of Texas System (USA), the Universidade do Algarve (Portugal) and the Universidade de Coimbra (Portugal).
Undergraduate research.
For science projects the institution created the PDPD program ("Researching Since The First Day" or "Pesquisando Desde o Primeiro Dia" in Portuguese), where first-year students participate in research projects funded by the institution and supervised by doctors.
Besides this program, students from other university years participate in undergraduate research programs with funding from organs such as the Institutional Program of Scientific Initiation, PIC-UFABC and PIBIC-UFABC, financed with resources from the UFABC. There is also the PIBIC-CNPq, financed by CNPq and PIBIC-Affirmative Action a pilot program launched and funded in UFABC by CNPq that provides scholarships for research by undergraduate students who entered the university system by affirmative actions (social quota). Students can also obtain research grants from organs such as Fapesp.
The university has instituted a Scholarship Assistance - Events, which finances the participation of its students in extracurricular activities such as presentation of scientific work (e.g. work of graduate students approved at international conferences) and going to technological events such as the Campus Party.
Infrastructure for research.
The campus of Santo André has equipment such as Atomic force microscopy and Tunneling, Circular dichroism, Elemental Analysis, Electron Paramagnetic Resonance, Atomic Absorption Spectroscopy of High Resolution, Ultraviolet-visible spectroscopy, Gas chromatograph, Atomic Emission Spectroscopy, Infrared spectroscopy, Potentiostat / Galvanostat, Centrifuge Refrigerated Superspeed, Absorption Spectroscopy and Atomic Emission For Multicomponent Analysis, Gas Chromatography-Mass Detector, Liquid Chromatography System, Modular Electrochemical Microscope, Spectroscopy of Fluorescence, Nuclear Magnetic Resonance, Dynamic Mechanical Analyzer, Differential scanning calorimetry, Thermogravimetric Analysis, Optical microscope, High performance liquid chromatography Coupled With Mass spectrometry With Mass Detector, Scanning electron microscope, X-ray crystallography, wind tunnel and supersonic wind tunnel.
In addition, UFABC has two networks of high-performance computers. The Chromo UFABC went into operation in June 2007 and is a project financed by FAPESP and CNPq, and the Bull Novascale UFABC is a project funded by CAPES.
Library.
The library website provides services such as queries to the collection, and renewal of loans of books. Also available are a list of acquisitions in recent months and the articles published on scientific journals by researchers of the UFABC. In addition, the library website is integrated to Twitter via Twitter @AnyWhere allowing interaction among the users of the system.
Postgraduate education.
The UFABC has courses of postgraduate education (Master or PhD) approved by the Capes. They are:
High school.
Since 2010 UFABC has offered a preparatory course for the ENEM, free for public school students in the ABC Region, as part of an extension project of the university. In this course, classes are held Monday through Friday by students from the undergraduate institution; these are complemented with cultural activities and lectures by professors from UFABC on Saturdays. Some of the students who teach classes are scholarship of the Pro-rectory of University Extension while the majority are student volunteers.
Selection of students.
The first three classes of UFABC were formed through the selection processes via Vestibular. The students enrolled in the years 2007 to 2009.
From 2010 the selection of new students is done with the ENEM. classes. The Bachelor of Science and Technology of UFABC was the most sought-after course in the country in the "Unified Selecting System" (SISU) of Ministry of Education in 2010.
According to statistics collected by the institution, the Bachelor of Science and Technology (BC&T) had a demand of 13,758 applicants competing for 1,500 places (candidate/vacancy ratio of 9.2, with a 10,9% acceptance rate), while the Bachelor of Science and Humanities (BC&H) has 2320 registered for 200 places (candidate/vacancy ratio of 11.6, with an 8,6% acceptance rate).
There is a reservation of 50% of seats of all its courses in both periods (morning and evening) for candidates who attended high school in public schools, as well as those self-declared indigenous and African descent.
Infrastructure.
The Federal University of ABC has modern facilities in the city of Santo André (SP) and facilities under construction in São Bernardo do Campo (SP).
Centers and cores.
The university has abolished the system of "departments" and "schools" that exist in other universities. In UFABC, all professors are bound to one of three major Centres Interdisciplinary:
Besides these, professors may be associated with the Interdisciplinary Centers linked to the rectory of UFABC:
There are organs like the President's Office, Office of International Affairs, Center for Information Technology and Center for Technological Innovation, in addition to several secretaries and numerous research groups.
Campus of Santo André.
Santo André's Campus of the UFABC has six blocks, a total of 96409.03 square meters of built area in the headquarters campus of the university. The design of the campus of Santo André was by Libeskindllovet Architects S/S Ltda. and the construction work was in charge of the Constructor Augusto Velloso SA.
Besides these main blocks located between the Avenida dos Estados and Avenida Santa Adelia, in Santo André city, the UFABC has a building with teaching laboratory, computer rooms and classrooms on Avenida Altântica, in Valparaíso neighborhood, and an administrative headquarters at Rua Catequese, in Jardim neighborhood, both in Santo André. The university has acquired a large piece of land to the other side of the Avenida dos Estados (across from the current headquarters) to build new research laboratories and support to engineering.
Campus of São Bernardo do Campo.
On 25 August 2009 President Lula, accompanied by first lady Marisa Leticia, the Minister of Education, Fernando Haddad, the mayor of Sao Bernardo do Campo, Luiz Marinho, the deputy mayor, Frank Aguiar, and the Senator of the state of São Paulo, Aloisio Mercadante, laid the foundation stone of the campus of UFABC the city of São Bernardo do Campo. The campus of São Bernardo do Campo UFABC has six blocks.
The project is designed by Benno Perelmutter Architecture and Planning S/C Ltda.
Besides these six blocks, UFABC has a building with teaching laboratory, computer rooms and classrooms located in downtown São Bernardo do Campo, in the former headquarters of the College Salete named Block Sigma.
University-enterprise integration.
The Board of Education and Research (Consep) of UFABC published in 2010 a resolution asking for corporations and industries interested in signing agreements of internship to enable the students to form placements in these firms. More than 100 companies signed the agreement with UFABC in 2010. Among such companies are names like IBM, Itaú, Nestlé, Globo Organizations, General Motors (GM), Ford, Siemens, Honda, Sun Microsystems, Pirelli, BASF, and Accenture.
Sport.
Students can play basketball, chess, handball, futsal, volleyball, swimming, table tennis, soccer, rugby and martial arts.
Students take part in competitions with universities throughout Brazil. In the Tournament "Semana da Asa" that occurs in the CTA, annually organized by the ITA, the athletes of UFABC won 3rd place overall in the first year of participation in 2008, and 1st place overall in the second year of participation in 2009;
In 2010, the Engenharíadas tournament brought together engineering colleges), the University Games Paulistanos (organized by the city of São Paulo), the Liga Deportiva Universitaria Paulista (major league sports university in the country) and the Cup of the University of São Carlos, the TUSCA, (organized by UFSCar and USP-São Carlos).
Academic Athletic Association September XI.
On 11 September 2006, the first class entered the Federal University of ABC. In 2007 a group of students began the Athletic UFABC, starting the Academic Athletic Association XI September (Portuguese: Associação Atlética Acadêmica XI de Setembro), AXIS, whose name was chosen in homage to the days of the first class.
AXIS is now a student representation recognized by UFABC formed by an elected board and that represents all athletes of the university. The Athletics (AXIS) of UFABC supports sports at the university, and conducts the selection of athletes for the leagues in which they participate.
The campuses of UFABC have sports fields, swimming pools, gyms and sports courts. Furthermore, AXIS has agreements with SESI (Social Service of Industry) of Santo André and with Santo André's Club Bochófilo for the use of sport facilities of these sites by the students of the institution.
AXIS works closely with the DCE (Central Directory of Students) of UFABC organizing parties, sponsoring the travel of athletes, performing the manufacture and sale of customized sports equipment (such as shirts, kimono, swimming caps, etc.), and providing training and transportation for the athletes to participate in tournaments and doing extra activities such as mountaineering, abseiling and internal tournaments.
Groups linked to Athletic's UFABC are Cheerleaders, the Infantry (responsible for the UFABC's fanfare in sports tournaments) and the Martial Arts Group, who train practitioners of Karate, Kung Fu, Tae Kwon Do and other martial arts.
Student representation.
The DCE (Central Directory of Students) represents the graduate students. Its members are chosen in an annual election in which all undergraduates can vote. The Directory of UFABC was established in 2008, from the former Academic Center, and helps students' stay, by helping with their problems, monitoring projects that relate to academic life, as discussions in ConsEP, ConsUni and CEU.
The UFABC has student organizations representing groups of students. Some of these are officially recognized by the university while others rely on the coordination of support from professors and education cores, including:
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "CR"
},
{
"math_id": 1,
"text": "CA"
},
{
"math_id": 2,
"text": "CP_k"
},
{
"math_id": 3,
"text": "I_k\n"
},
{
"math_id": 4,
"text": "CR = \\textstyle \\frac{\\textstyle \\sum_{k=1}^{NC}C_i \\cdot f(N_i) \\displaystyle}{\\textstyle \\sum_{k=1}^{NC}C_i \\displaystyle}\\displaystyle"
},
{
"math_id": 5,
"text": "NC"
},
{
"math_id": 6,
"text": "N_i"
},
{
"math_id": 7,
"text": "C_i\n"
},
{
"math_id": 8,
"text": "f(N_i)"
},
{
"math_id": 9,
"text": "f(A) = 4, f(B) = 3, f(C) = 2, f(D) = 1"
},
{
"math_id": 10,
"text": "f(F) = 0"
},
{
"math_id": 11,
"text": "CA = \\frac{\\sum_{k=1}^{ND} CR_i \\cdot f(MC_i)}{\\sum_{k=1}^{ND} CR_i}"
},
{
"math_id": 12,
"text": "ND"
},
{
"math_id": 13,
"text": "CR_i"
},
{
"math_id": 14,
"text": "MC_i"
},
{
"math_id": 15,
"text": "f(MC_i)"
},
{
"math_id": 16,
"text": "k"
},
{
"math_id": 17,
"text": "CP_k = \\frac{n^k_{man} + min[(N^k_{lim} + N^k_{free}), n^k_{lim} + min(n^k_{free},N^k_{free})]}{NC_k}"
},
{
"math_id": 18,
"text": "n^k_{man}\n"
},
{
"math_id": 19,
"text": "n^k_{lim}\n"
},
{
"math_id": 20,
"text": "n^k_{free}\n"
},
{
"math_id": 21,
"text": "N^k_{man}\n"
},
{
"math_id": 22,
"text": "N^k_{lim}\n"
},
{
"math_id": 23,
"text": "N^k_{free}\n"
},
{
"math_id": 24,
"text": "CP_k\n"
},
{
"math_id": 25,
"text": "I_k = (0,07 \\cdot CR) + (0,63 \\cdot CPk) + (0,005 \\cdot T)\n"
}
] |
https://en.wikipedia.org/wiki?curid=14293177
|
14294
|
Hausdorff dimension
|
Invariant measure of fractal dimension
In mathematics, Hausdorff dimension is a measure of "roughness", or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the "Hausdorff–Besicovitch dimension."
More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined. The dimension is drawn from the extended real numbers, formula_0, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an "n"-dimensional inner product space equals "n". This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
Intuition.
The intuitive concept of dimension of a geometric object "X" is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and "continuously", so that a one-dimensional object completely fills up a higher-dimensional object.
Every space-filling curve hits some points multiple times and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is the greatest integer "n" such that in every covering of "X" by small open balls there is at least one point where "n" + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension "n" = 1.
But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number "N"("r") of balls of radius at most "r" required to cover "X" completely. When "r" is very small, "N"("r") grows polynomially with 1/"r". For a sufficiently well-behaved "X", the Hausdorff dimension is the unique number "d" such that N("r") grows as 1/"rd" as "r" approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value "d" is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.
For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benoit Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.
Formal definition.
The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed:
Let formula_1 be a metric space. If formula_2 and formula_3,
formula_4
where the infimum is taken over all countable covers formula_5 of formula_6. The Hausdorff d-dimensional outer measure is then defined as formula_7, and the restriction of the mapping to measurable sets justifies it as a measure, called the formula_8-dimensional Hausdorff Measure.
Hausdorff dimension.
The Hausdorff dimension formula_9 of formula_1 is defined by
formula_10
This is the same as the supremum of the set of formula_3 such that the formula_8-dimensional Hausdorff measure of formula_1 is infinite (except that when this latter set of numbers formula_8 is empty the Hausdorff dimension is zero).
Hausdorff content.
The formula_8-dimensional unlimited Hausdorff content of formula_6 is defined by
formula_11
In other words, formula_12 has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes (Here, we use the standard convention that formula_13). The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree.
Properties of Hausdorff dimension.
Hausdorff dimension and inductive dimension.
Let "X" be an arbitrary separable metric space. There is a topological notion of inductive dimension for "X" which is defined recursively. It is always an integer (or +∞) and is denoted dimind("X").
Theorem. Suppose "X" is non-empty. Then
formula_17
Moreover,
formula_18
where "Y" ranges over metric spaces homeomorphic to "X". In other words, "X" and "Y" have the same underlying set of points and the metric "d""Y" of "Y" is topologically equivalent to "d""X".
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
Hausdorff dimension and Minkowski dimension.
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.
Hausdorff dimensions and Frostman measures.
If there is a measure μ defined on Borel subsets of a metric space "X" such that "μ"("X") > 0 and "μ"("B"("x", "r")) ≤ "rs" holds for some constant "s" > 0 and for every ball "B"("x", "r") in "X", then dimHaus("X") ≥ "s". A partial converse is provided by Frostman's lemma.
Behaviour under unions and products.
If formula_19 is a finite or countable union, then
formula_20
This can be verified directly from the definition.
If "X" and "Y" are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
formula_21
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when "X" and "Y" are Borel subsets of R"n", the Hausdorff dimension of "X" × "Y" is bounded from above by the Hausdorff dimension of "X" plus the upper packing dimension of "Y". These facts are discussed in Mattila (1995).
Self-similar sets.
Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set "E" is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ("E") = "E", although the exact definition is given below.
Theorem. Suppose
formula_22
are each a contraction mapping on R"n" with contraction constant "ri" < 1. Then there is a unique "non-empty" compact set "A" such that
formula_23
The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of R"n" with the Hausdorff distance.
The open set condition.
To determine the dimension of the self-similar set "A" (in certain cases), we need a technical condition called the "open set condition" (OSC) on the sequence of contractions ψ"i".
There is an open set "V" with compact closure, such that
formula_24
where the sets in union on the left are pairwise disjoint.
The open set condition is a separation condition that ensures the images ψ"i"("V") do not overlap "too much".
Theorem. Suppose the open set condition holds and each ψ"i" is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is "s" where "s" is the unique solution of
formula_25
The contraction coefficient of a similitude is the magnitude of the dilation.
In general, a set "E" which is carried onto itself by a mapping
formula_26
is self-similar if and only if the intersections satisfy the following condition:
formula_27
where "s" is the Hausdorff dimension of "E" and "Hs" denotes s-dimensional Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\overline{\\mathbb{R}}"
},
{
"math_id": 1,
"text": "X"
},
{
"math_id": 2,
"text": "S\\subset X"
},
{
"math_id": 3,
"text": "d\\in [0,\\infty)"
},
{
"math_id": 4,
"text": "H^d_\\delta(S)=\\inf\\left \\{\\sum_{i=1}^\\infty (\\operatorname{diam} U_i)^d: \\bigcup_{i=1}^\\infty U_i\\supseteq S, \\operatorname{diam} U_i<\\delta\\right \\},"
},
{
"math_id": 5,
"text": "U"
},
{
"math_id": 6,
"text": "S"
},
{
"math_id": 7,
"text": "\\mathcal{H}^d(S)=\\lim_{\\delta\\to 0}H^d_\\delta(S)"
},
{
"math_id": 8,
"text": "d"
},
{
"math_id": 9,
"text": "\\dim_{\\operatorname{H}}{(X)}"
},
{
"math_id": 10,
"text": "\\dim_{\\operatorname{H}}{(X)}:=\\inf\\{d\\ge 0: \\mathcal{H}^d(X)=0\\}."
},
{
"math_id": 11,
"text": "C_H^d(S):= H_\\infty^d(S) = \\inf\\left \\{ \\sum_{k=1}^\\infty (\\operatorname{diam} U_k)^d: \\bigcup_{k=1}^\\infty U_k\\supseteq S \\right \\}"
},
{
"math_id": 12,
"text": "C_H^d(S)"
},
{
"math_id": 13,
"text": "\\inf\\varnothing=\\infty"
},
{
"math_id": 14,
"text": "\\R^n"
},
{
"math_id": 15,
"text": "n"
},
{
"math_id": 16,
"text": "S^1"
},
{
"math_id": 17,
"text": " \\dim_{\\mathrm{Haus}}(X) \\geq \\dim_{\\operatorname{ind}}(X). "
},
{
"math_id": 18,
"text": " \\inf_Y \\dim_{\\operatorname{Haus}}(Y) =\\dim_{\\operatorname{ind}}(X), "
},
{
"math_id": 19,
"text": "X=\\bigcup_{i\\in I}X_i"
},
{
"math_id": 20,
"text": " \\dim_{\\operatorname{Haus}}(X) =\\sup_{i\\in I} \\dim_{\\operatorname{Haus}}(X_i)."
},
{
"math_id": 21,
"text": " \\dim_{\\operatorname{Haus}}(X\\times Y)\\ge \\dim_{\\operatorname{Haus}}(X)+ \\dim_{\\operatorname{Haus}}(Y)."
},
{
"math_id": 22,
"text": " \\psi_i: \\mathbf{R}^n \\rightarrow \\mathbf{R}^n, \\quad i=1, \\ldots , m "
},
{
"math_id": 23,
"text": " A = \\bigcup_{i=1}^m \\psi_i (A). "
},
{
"math_id": 24,
"text": " \\bigcup_{i=1}^m\\psi_i (V) \\subseteq V, "
},
{
"math_id": 25,
"text": " \\sum_{i=1}^m r_i^s = 1. "
},
{
"math_id": 26,
"text": " A \\mapsto \\psi(A) = \\bigcup_{i=1}^m \\psi_i(A) "
},
{
"math_id": 27,
"text": " H^s\\left(\\psi_i(E)\\cap \\psi_j(E)\\right) =0, "
}
] |
https://en.wikipedia.org/wiki?curid=14294
|
14295806
|
Rotating wheel space station
|
Space station concept
A rotating wheel space station, also known as a von Braun wheel, is a concept for a hypothetical wheel-shaped space station. Originally proposed by Konstantin Tsiolkovsky in 1903, the idea was expanded by Herman Potočnik in 1929.
Specifications.
This type of station rotates about its axis, creating an environment of artificial gravity. Occupants of the station would experience centrifugal acceleration, according to the following equation:
formula_0
where formula_1 is the angular velocity of the station, formula_2 is its radius, and formula_3 is linear acceleration at any point along its perimeter.
In theory, the station could be configured to simulate the gravitational acceleration of Earth (9.81 m/s2), allowing for human long stays in space without the drawbacks of microgravity.
History.
Both scientists and science fiction writers have thought about the concept of a rotating wheel space station since the beginning of the 20th century. Konstantin Tsiolkovsky wrote about using rotation to create an artificial gravity in space in 1903. Herman Potočnik introduced a spinning wheel station with a 30-meter diameter in his "Problem der Befahrung des Weltraums" ("The Problem of Space Travel"). He even suggested it be placed in a geostationary orbit.
In the 1950s, Wernher von Braun and Willy Ley, writing in "Colliers Magazine", updated the idea, in part as a way to stage spacecraft headed for Mars. They envisioned a rotating wheel with a diameter of 76 meters (250 feet). The 3-deck wheel would revolve at 3 RPM to provide artificial one-third gravity. It was envisaged as having a crew of 80.
In 1959, a NASA committee opined that such a space station was the next logical step after the Mercury program. The Stanford torus, proposed by NASA in 1975, is an enormous version of the same concept that could harbor an entire city.
NASA has not attempted to build a rotating wheel space station, for several reasons. First, such a station would be difficult to construct, given the limited lifting capability available to the United States and other spacefaring nations. Assembling such a station and pressurizing it would present formidable obstacles, which, although not beyond NASA's technical capability, would be beyond available budgets. Second, NASA considers the present space station, the International Space Station (ISS), to be valuable as a zero gravity laboratory, and its current microgravity environment was a conscious choice.
In the 2010s, NASA explored plans for a Nautilus X centrifuge demonstration project. If flown, this would add a centrifuge sleep quarters module to the ISS. This makes it possible to experiment with artificial gravity without destroying the usefulness of the ISS for zero g experiments. It could lead to deep space missions under full g in centrifuge sleeping quarters following the same approach.
In fiction.
Many fictional space stations and ships use a rotating design.
1936: In Alexander Belyaev's novel "KETs Star" a circular space station provides pseudo-gravity of about 0.1g by its rotation.
1958: The film "Queen of Outer Space" features a rotating space station that gets blown up.
1968: Arthur C. Clarke's novel "" was developed concurrently with Stanley Kubrick's film version of . In it, the rotating space station Space Station V provides artificial gravity and features prominently on the book's first-edition cover. The Jupiter mission spacecraft, "Discovery One", features a centrifuge for the crew living quarters that provides artificial gravity.
1968: In the six part Doctor Who TV serial "The Wheel in Space" the titular station is the main setting of the story.
1970: The novel Ringworld describes a very large, habitable structure, centered on a star.
1984: The Peter Hyams directed film "2010" features a battleship-size, Russian built spacecraft (designed by futurist artist Syd Mead), the "Leonov", which has a continuously rotating central section, providing an artificial gravity for the occupants.
1985: The novel "Ender's Game" features a multi-ringed station, called "Battle School," with varying levels of simulated gravity. As the characters ascend through the station towards the center, there is a noticeable decline in the feeling of gravity.
1994: The humans in the science fiction series "Babylon 5" live in an O'Neill cylinder station using rotating sections to provide artificial gravity. Earth Alliance space stations such as the Babylon series (hence the name of the series), transfer stations such as the one at Io near the main Sol system jump gate, and EarthForce Omega-Class destroyer spaceships made extensive use of rotating sections to lengthen deployment times and increase mission flexibility as the effects of zero gravity are no longer a concern.
1999: The Japanese manga and anime "Planetes" has its main story set in "The Seven," the 7th wheel orbital station, and a 9th is under construction by 2075. In the Zenon trilogy (', ' and ""), 13-year-old Zenon lives on a rotating space station owned by the fictional WyndComm from 2049 though 2054, but it is not designed in a way that would allow for artificial gravity through centripetal force.
2000: In the film Mission to Mars, "Mars II", a NASA spacecraft hastily repurposed for a recovery mission of humanity's first mission to Mars in 2020, features a rotating crew habitat whose artificial gravitational rotation was shut down using the ship's attitude control thrusters to allow emergency repairs to the hull following a micrometeoroid shower.
2001: In the video-game series "Halo" created by Bungie, a planet-sized ring is depicted that can harbor Earth"-"like fauna and environments by simulating gravity through its spinning.
2003: In the re-imagined series "Battlestar Galactica". Ragnar Anchorage is a three ringed weapons storage station, and the civilian ship "Zephyr" is a luxury liner featuring a ringed midsection.
2007: The "Presidium" sector of the Citadel space station in the "Mass Effect" series of video games comprises a rotating toroidal section connected to a docking ring, with five large "wards" radiating out from the central ring like a flower's petals. In addition, Arcturus Station, the human seat of government on the galactic stage (not shown in the games, but described in detail) is also mentioned as being a rotating Stanford torus.
2010: In the OVA "Mobile Suit Gundam Unicorn", the official residence for the prime minister of the Earth Federation "Laplace" was an example of Stanford torus.
2011: Most space stations in the "Expanse" series make use of artificial gravity by rotation, most notably Tycho Station. Even larger celestial objects like Ceres and Eros have been hollowed out and spun up to generate gravitational pull for their inhabitants.
2013: The Neill Blomkamp film "Elysium" has an enormous space station called Elysium (an open-roofed station in diameter, somewhere between a much-larger open-roofed Bishop Ring and a smaller, fully enclosed Stanford Torus.) The station in the movie supports a city and habitat for the privileged upper classes of Earth.
2014: A vessel very similar in design to the NASA-designed "Nautilus-X" was used in "Interstellar". The ship, known as the "Endurance", was used as a staging station also capable of interplanetary flight.
2014: Space stations in the video game "" (and its prequels) rotate to create artificial gravity.
2015: Thunderbird 5 in the ITV TV show "Thunderbirds Are Go" features a rotating gravity ring section on the space station which features a glass floor to observe the Earth below. The series is set in the year 2060.
2015: The NASA-designed "Hermes" in the film "The Martian" was capable of space travel to Mars.
2018: A planetarium movie "Mars 1001" shows a fictional mission to Mars employing a rotating spacecraft. Fallout 76 includes a ruined space station that has a rotating wheel on it in a location called The Crater.
2019: The video game "Outer Wilds" features multiple: the base game contains a rotating gravity wheel inside of a planet to maintain a gravitational pull within the planet's center. The 2021 DLC "Echoes of the Eye" features a planet-sized wheel-shaped space station that rotates to create artificial gravity.
2022: The Mandalorian is shown on a rotating ring with artificial gravity in the Book of Boba Fett.
2022: The season 3 premiere of For All Mankind, an Apple TV+ original series, depicts a space hotel with a rotating wheel for gravity generation which becomes important to the storyline after the rotating mechanism malfunctions.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "a = -\\omega^2 r"
},
{
"math_id": 1,
"text": "\\omega"
},
{
"math_id": 2,
"text": "r"
},
{
"math_id": 3,
"text": "a"
}
] |
https://en.wikipedia.org/wiki?curid=14295806
|
14296821
|
Global meteoric water line
|
The Global Meteoric Water Line (GMWL) describes the global annual average relationship between hydrogen and oxygen isotope (oxygen-18 [18O] and deuterium [2H]) ratios in natural meteoric waters. The GMWL was first developed in 1961 by Harmon Craig, and has subsequently been widely used to track water masses in environmental geochemistry and hydrogeology.
Development and definition of GMWL.
When working on the global annual average isotopic composition of 18O and 2H in meteoric water, geochemist Harmon Craig observed a correlation between these two isotopes, and subsequently developed and defined the equation for GMWL:
formula_0
Where δ18O and δ2H (aka δD) are the ratio of heavy to light isotopes (e.g. 18O/16O, 2H/1H).
The relationship of δ18O and δ2H in meteoric water is caused by mass dependent fractionation of oxygen and hydrogen isotopes between evaporation from ocean seawater and condensation from vapor. As oxygen isotopes (18, 16O) and hydrogen isotopes (2, 1H) have different masses, they behave differently in the evaporation and condensation processes, and thus result in the fractionation between 18O and 16O as well as 2H and 1H. Equilibrium fractionation causes the isotope ratios of δ18O and δ2H to vary between localities within the area. The fractionation processes can be influenced by a number of factors including: temperature, latitude, continentality, and most importantly, humidity.
Applications.
Craig observed that δ18O and δ2H isotopic composition of cold meteoric water from sea ice in the Arctic and Antarctica are much more negative than that in warm meteoric water from the tropic. A correlation between temperature (T) and "δ"18O was proposed later in the 1970s. Such correlation is then applied to study surface temperature change over time. The δ18O of ancient meteoric water, preserved in ice cores, can also be collected and applied to reconstruct paleoclimate.
A meteoric water line can be calculated for a given area, named as local meteoric water line (LMWL), and used as a baseline within that area. Local meteoric water line can differ from the global meteoric water line in slope and intercept. Such deviated slope and intercept is a result largely from humidity. In 1964, the concept of deuterium excess d (d = δ2H - 8δ18O) was proposed. Later, a parameter of deuterium excess as a function of humidity has been established, as such the isotopic composition in local meteoric water can be applied to trace local relative humidity, study local climate and used as a tracer of climate change.
In hydrogeology, the δ18O and δ2H of groundwater are often used to study the origin of groundwater and groundwater recharge.
It has been shown that, even taking into account the standard deviation related to instrumental errors and the natural variability of the amount-weighted precipitations, the LMWL calculated with the EIV (error in variable regression) method has no differences on the slope compared to classic OLSR (ordinary least square regression) or other regression methods. However, for certain purposes such as the evaluation of the shifts from the line of the geothermal waters, it would be more appropriate to calculate the so-called "prediction interval" or "error wings" related to LMWL.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\delta\\ce D = 8.0\\cdot\\delta^{18}\\ce O + 10\\ {}^{0\\!}\\!/\\!_{00}"
}
] |
https://en.wikipedia.org/wiki?curid=14296821
|
142999
|
Circle of latitude
|
Geographic notion
A circle of latitude or line of latitude on Earth is an abstract east–west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line.
Circles of latitude are often called parallels because they are parallel to each other; that is, planes that contain any of these circles never intersect each other. A location's position along a circle of latitude is given by its longitude. Circles of latitude are unlike circles of longitude, which are all great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the distance from the Equator increases. Their length can be calculated by a common sine or cosine function. For example, the 60th parallel north or south is half as long as the Equator (disregarding Earth's minor flattening by 0.335%), stemming from formula_0. On the Mercator projection or on the Gall-Peters projection, a circle of latitude is perpendicular to all meridians. On the ellipsoid or on spherical projection, all circles of latitude are rhumb lines, except the Equator.
The latitude of the circle is approximately the angle between the Equator and the circle, with the angle's vertex at Earth's centre. The Equator is at 0°, and the North Pole and South Pole are at 90° north and 90° south, respectively. The Equator is the longest circle of latitude and is the only circle of latitude which also is a great circle. As such, it is perpendicular to all meridians.
There are 89 integral (whole degree) circles of latitude between the Equator and the poles in each hemisphere, but these can be divided into more precise measurements of latitude, and are often represented as a decimal degree (e.g. 34.637° N) or with minutes and seconds (e.g. 22°14'26" S).
On a map, the circles of latitude may or may not be parallel, and their spacing may vary, depending on which projection is used to map the surface of the Earth onto a plane. On an equirectangular projection, centered on the equator, the circles of latitude are horizontal, parallel, and equally spaced. On other cylindrical and pseudocylindrical projections, the circles of latitude are horizontal and parallel, but may be spaced unevenly to give the map useful characteristics. For instance, on a Mercator projection the circles of latitude are more widely spaced near the poles to preserve local scales and shapes, while on a Gall–Peters projection the circles of latitude are spaced more closely near the poles so that comparisons of area will be accurate. On most non-cylindrical and non-pseudocylindrical projections, the circles of latitude are neither straight nor parallel.
Arcs of circles of latitude are sometimes used as boundaries between countries or regions where distinctive natural borders are lacking (such as in deserts), or when an artificial border is drawn as a "line on a map", which was made in massive scale during the 1884 Berlin Conference, regarding huge parts of the African continent. North American nations and states have also mostly been created by straight lines, which are often parts of circles of latitudes. For instance, the northern border of Colorado is at 41° N while the southern border is at 37° N. Roughly half the length of the border between the United States and Canada follows 49° N.
Major circles of latitude.
There are five major circles of latitude, listed below from north to south. The position of the Equator is fixed (90 degrees from Earth's axis of rotation) but the latitudes of the other circles depend on the tilt of this axis relative to the plane of Earth's orbit, and so are not perfectly fixed. The values below are for :
These circles of latitude, excluding the Equator, mark the divisions between the five principal geographical zones.
Equator.
The equator is the circle that is equidistant from the North Pole and South Pole. It divides the Earth into the Northern Hemisphere and the Southern Hemisphere. Of the parallels or circles of latitude, it is the longest, and the only 'great circle' (a circle on the surface of the Earth, centered on Earth's center). All the other parallels are smaller and centered only on Earth's axis.
Polar circles.
The Arctic Circle is the southernmost latitude in the Northern Hemisphere at which the Sun can remain continuously above or below the horizon for 24 hours (at the June and December solstices respectively). Similarly, the Antarctic Circle marks the northernmost latitude in the Southern Hemisphere at which the Sun can remain continuously above or below the horizon for 24 hours (at the December and June Solstices respectively).
The latitude of the polar circles is equal to 90° minus the Earth's axial tilt.
Tropical circles.
The Tropic of Cancer and Tropic of Capricorn mark the northernmost and southernmost latitudes at which the Sun may be seen directly overhead at the June solstice and December solstice respectively.
The latitude of the tropical circles is equal to the Earth's axial tilt.
Movement of the Tropical and Polar Circles.
By definition, the positions of the Tropic of Cancer, Tropic of Capricorn, Arctic Circle and Antarctic Circle all depend on the tilt of the Earth's axis relative to the plane of its orbit around the Sun (the "obliquity of the ecliptic"). If the Earth were "upright" (its axis at right angles to the orbital plane) there would be no Arctic, Antarctic, or Tropical circles: at the poles the Sun would always circle along the horizon, and at the equator the Sun would always rise due east, pass directly overhead, and set due west.
The positions of the Tropical and Polar Circles are not fixed because the axial tilt changes slowly – a complex motion determined by the superimposition of many different cycles (some of which are described below) with short to very long periods. At noon of January 1st 2000 AD, the mean value of the tilt was 23° 26′ 21.406″ (according to IAU 2006, theory P03), the corresponding value being 23° 26′ 10.633" at noon of January 1st 2023 AD.
The main long-term cycle causes the axial tilt to fluctuate between about 22.1° and 24.5° with a period of 41,000 years. Currently, the "average" value of the tilt is decreasing by about 0.468″ per year. As a result (approximately, and on average), the Tropical Circles are drifting towards the equator (and the Polar Circles towards the poles) by 15 m per year, and the area of the Tropics, defined astronomically, is decreasing by per year. (However, the tropical belt as defined based on atmospheric conditions is expanding due to global warming.)
The Earth's axial tilt has additional shorter-term variations due to nutation, of which the main term, with a period of 18.6 years, has an amplitude of 9.2″ (corresponding to almost 300 m north and south). There are many smaller terms, resulting in varying daily shifts of some metres in any direction.
Finally, the Earth's rotational axis is not exactly fixed in the Earth, but undergoes small fluctuations (on the order of 15 m) called polar motion, which have a small effect on the Tropics and Polar Circles and also on the Equator.
Short-term fluctuations over a matter of days do not directly affect the location of the extreme latitudes at which the Sun may appear directly overhead, or at which 24-hour day or night is possible, except when they actually occur at the time of the solstices. Rather, they cause a theoretical shifting of the parallels, that would occur if the given axis tilt were maintained throughout the year.
Other planets.
These circles of latitude can be defined on other planets with axial inclinations relative to their orbital planes. Objects such as Pluto with tilt angles greater than 45 degrees will have the tropic circles closer to the poles and the polar circles closer to the equator.
Other notable parallels.
A number of sub-national and international borders were intended to be defined by, or are approximated by, parallels. Parallels make convenient borders in the northern hemisphere because astronomic latitude can be roughly measured (to within a few tens of metres) by sighting the North Star.
Elevation.
Normally the circles of latitude are defined at zero elevation. Elevation has an effect on a location with respect to the plane formed by a circle of latitude. Since (in the geodetic system) altitude and depth are determined by the normal to the Earth's surface, locations sharing the same latitude—but having different elevations (i.e., lying along this normal)—no longer lie within this plane. Rather, all points sharing the same latitude—but of varying elevation and longitude—occupy the surface of a truncated cone formed by the rotation of this normal around the Earth's axis of rotation.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\cos(60^{\\circ}) = 0.5"
}
] |
https://en.wikipedia.org/wiki?curid=142999
|
143021
|
Centers of gravity in non-uniform fields
|
Center of gravity of a material body
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity. This is a very good approximation for smaller bodies near the surface of Earth, so there is no practical need to distinguish "center of gravity" from "center of mass" in most applications, such as engineering and medicine.
In a non-uniform field, gravitational effects such as potential energy, force, and torque can no longer be calculated using the center of mass alone. In particular, a non-uniform gravitational field can produce a torque on an object, even about an axis through the center of mass. The center of gravity seeks to explain this effect. Formally, a center of gravity is an application point of the resultant gravitational force on the body. Such a point may not exist, and if it exists, it is not unique. One can further define a unique center of gravity by approximating the field as either parallel or spherically symmetric.
The concept of a center of gravity as distinct from the center of mass is rarely used in applications, even in celestial mechanics, where non-uniform fields are important. Since the center of gravity depends on the external field, its motion is harder to determine than the motion of the center of mass. The common method to deal with gravitational torques is a field theory.
Center of mass.
One way to define the center of gravity of a body is as the unique point in the body if it exists, that satisfies the following requirement: There is no torque about the point for any positioning of the body in the field of force in which it is placed. This center of gravity exists only when the force is uniform, in which case it coincides with the center of mass. This approach dates back to Archimedes.
Centers of gravity in a field.
When a body is affected by a non-uniform external gravitational field, one can sometimes define a "center of gravity" relative to that field that will act as a point where the gravitational force is applied. Textbooks such as "The Feynman Lectures on Physics" characterize the center of gravity as a point about which there is no torque. In other words, the center of gravity is a point of application for the resultant force. Under this formulation, the center of gravity rcg is defined as a point that satisfies the equation
formula_0
where F and τ are the total force and torque on the body due to gravity.
One complication concerning rcg is that its defining equation is not generally solvable. If F and τ are not orthogonal, then there is no solution; the force of gravity does not have a resultant and cannot be replaced by a single force at any point. There are some important special cases where F and τ are guaranteed to be orthogonal, such as if all forces lie in a single plane or are aligned with a single point.
If the equation is solvable, there is another complication: its solutions are not unique. Instead, there are infinitely many solutions; the set of all solutions is known as the line of action of the force. This line is parallel to the weight F. In general, there is no way to choose a particular point as the unique center of gravity. A single point may still be chosen in some special cases, such as if the gravitational field is parallel or spherically symmetric. These cases are considered below.
Parallel fields.
Some of the inhomogeneity in a gravitational field may be modeled by a variable but parallel field: g(r) = "g"(r)n, where n is some constant unit vector. Although a non-uniform gravitational field cannot be exactly parallel, this approximation can be valid if the body is sufficiently small. The center of gravity may then be defined as a certain weighted average of the locations of the particles composing the body. Whereas the center of mass averages over the mass of each particle, the center of gravity averages over the weight of each particle:
formula_1
where wi is the (scalar) weight of the ith particle and W is the (scalar) total weight of all the particles. This equation always has a unique solution, and in the parallel-field approximation, it is compatible with the torque requirement.
A common illustration concerns the Moon in the field of the Earth. Using the weighted-average definition, the Moon has a center of gravity that is lower (closer to the Earth) than its center of mass, because its lower portion is more strongly influenced by the Earth's gravity. This eventually lead to the Moon always showing the same face, a phenomenon known as tidal locking.
Spherically symmetric fields.
If the external gravitational field is spherically symmetric, then it is equivalent to the field of a point mass M at the center of symmetry r. In this case, the center of gravity can be defined as the point at which the total force on the body is given by Newton's Law:
formula_2
where G is the gravitational constant and m is the mass of the body. As long as the total force is nonzero, this equation has a unique solution, and it satisfies the torque requirement. A convenient feature of this definition is that if the body is itself spherically symmetric, then rcg lies at its center of mass. In general, as the distance between r and the body increases, the center of gravity approaches the center of mass.
Another way to view this definition is to consider the gravitational field of the body; then rcg is the apparent source of gravitational attraction for an observer located at r. For this reason, rcg is sometimes referred to as the center of gravity of M "relative to the point" r.
Usage.
The centers of gravity defined above are not fixed points on the body; rather, they change as the position and orientation of the body changes. This characteristic makes the center of gravity difficult to work with, so the concept has little practical use.
When it is necessary to consider a gravitational torque, it is easier to represent gravity as a force acting at the center of mass, plus an orientation-dependent couple. The latter is best approached by treating the gravitational potential as a field.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{r}_\\mathrm{cg} \\times \\mathbf{F} = \\boldsymbol{\\tau},"
},
{
"math_id": 1,
"text": "\\mathbf{r}_\\mathrm{cg} = \\frac{1}{W} \\sum_i w_i \\mathbf{r}_i,"
},
{
"math_id": 2,
"text": "\\frac {GmM (\\mathbf{r}_\\mathrm{cg} - \\mathbf{r})} {|\\mathbf{r}_\\mathrm{cg} - \\mathbf{r}|^3} = \\mathbf{F},"
}
] |
https://en.wikipedia.org/wiki?curid=143021
|
143023
|
Equatorial bulge
|
Outward bulge around a planet's equator due to its rotation
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.
On Earth.
The planet Earth has a rather slight equatorial bulge; its equatorial diameter is about greater than its polar diameter, with a difference of about <templatestyles src="Fraction/styles.css" />1⁄298 of the equatorial diameter. If Earth were scaled down to a globe with an equatorial diameter of , that difference would be only . While too small to notice visually, that difference is still more than twice the largest deviations of the actual surface from the ellipsoid, including the tallest mountains and deepest oceanic trenches.
Earth's rotation also affects the sea level, the imaginary surface used as a reference frame from which to measure altitudes. This surface coincides with the mean water surface level in oceans, and is extrapolated over land by taking into account the local gravitational potential and the centrifugal force.
The difference of the radii is thus about . An observer standing at sea level on either pole, therefore, is closer to Earth's center than if standing at sea level on the Equator. As a result, the highest point on Earth, measured from the center and outwards, is the peak of Mount Chimborazo in Ecuador rather than Mount Everest. But since the ocean also bulges, like Earth and its atmosphere, Chimborazo is not as high above sea level as Everest is. Similarly the lowest point on Earth, measured from the center and outwards, is the Litke Deep in the Arctic Ocean rather than Challenger Deep in the Pacific Ocean. But since the ocean also flattens, like Earth and its atmosphere, Litke Deep is not as low below sea level as Challenger Deep is.
More precisely, Earth's surface is usually approximated by an ideal oblate ellipsoid, for the purposes of defining precisely the latitude and longitude grid for cartography, as well as the "center of the Earth". In the WGS-84 standard Earth ellipsoid, widely used for map-making and the GPS system, Earth's radius is assumed to be to the Equator and to either pole, meaning a difference of between the radii or between the diameters, and a relative flattening of 1/298.257223563. The ocean surface is much closer to this standard ellipsoid than the solid surface of Earth is.
The equilibrium as a balance of energies.
Gravity tends to contract a celestial body into a sphere, the shape for which all the mass is as close to the center of gravity as possible. Rotation causes a distortion from this spherical shape; a common measure of the distortion is the flattening (sometimes called ellipticity or oblateness), which can depend on a variety of factors including the size, angular velocity, density, and elasticity.
A way for one to get a feel for the type of equilibrium involved is to imagine someone seated in a spinning swivel chair and holding a weight in each hand; if the individual pulls the weights inward towards them, work is being done and their rotational kinetic energy increases. The increase in rotation rate is so strong that at the faster rotation rate the required centripetal force is larger than with the starting rotation rate.
Something analogous to this occurs in planet formation. Matter first coalesces into a slowly rotating disk-shaped distribution, and collisions and friction convert kinetic energy to heat, which allows the disk to self-gravitate into a very oblate spheroid.
As long as the proto-planet is still too oblate to be in equilibrium, the release of gravitational potential energy on contraction keeps driving the increase in rotational kinetic energy. As the contraction proceeds, the rotation rate keeps going up, hence the required force for further contraction keeps going up. There is a point where the increase of rotational kinetic energy on further contraction would be larger than the release of gravitational potential energy. The contraction process can only proceed up to that point, so it halts there.
As long as there is no equilibrium there can be violent convection, and as long as there is violent convection friction can convert kinetic energy to heat, draining rotational kinetic energy from the system. When the equilibrium state has been reached then large scale conversion of kinetic energy to heat ceases. In that sense the equilibrium state is the lowest state of energy that can be reached.
The Earth's rotation rate is still slowing down, though gradually, by about two thousandths of a second per rotation every 100 years. Estimates of how fast the Earth was rotating in the past vary, because it is not known exactly how the moon was formed. Estimates of the Earth's rotation 500 million years ago are around 20 modern hours per "day".
The Earth's rate of rotation is slowing down mainly because of tidal interactions with the Moon and the Sun. Since the solid parts of the Earth are ductile, the Earth's equatorial bulge has been decreasing in step with the decrease in the rate of rotation.
Effect on gravitational acceleration.
Because of a planet's rotation around its own axis, the gravitational acceleration is less at the equator than at the poles. In the 17th century, following the invention of the pendulum clock, French scientists found that clocks sent to French Guiana, on the northern coast of South America, ran slower than their exact counterparts in Paris. Measurements of the acceleration due to gravity at the equator must also take into account the planet's rotation. Any object that is stationary with respect to the surface of the Earth is actually following a circular trajectory, circumnavigating the Earth's axis. Pulling an object into such a circular trajectory requires a force. The acceleration that is required to circumnavigate the Earth's axis along the equator at one revolution per sidereal day is 0.0339 m/s2. Providing this acceleration decreases the effective gravitational acceleration. At the Equator, the effective gravitational acceleration is 9.7805 m/s2. This means that the true gravitational acceleration at the Equator must be 9.8144 m/s2 (9.7805 + 0.0339 = 9.8144).
At the poles, the gravitational acceleration is 9.8322 m/s2. The difference of 0.0178 m/s2 between the gravitational acceleration at the poles and the true gravitational acceleration at the Equator is because objects located on the Equator are about further away from the center of mass of the Earth than at the poles, which corresponds to a smaller gravitational acceleration.
In summary, there are two contributions to the fact that the effective gravitational acceleration is less strong at the equator than at the poles. About 70% of the difference is contributed by the fact that objects circumnavigate the Earth's axis, and about 30% is due to the non-spherical shape of the Earth.
The diagram illustrates that on all latitudes the effective gravitational acceleration is decreased by the requirement of providing a centripetal force; the decreasing effect is strongest on the Equator.
Effect on satellite orbits.
The fact that the Earth's gravitational field slightly deviates from being spherically symmetrical also affects the orbits of satellites through secular orbital precessions. They depend on the orientation of the Earth's symmetry axis in the inertial space, and, in the general case, affect "all" the Keplerian orbital elements with the exception of the semimajor axis. If the reference "z" axis of the coordinate system adopted is aligned along the Earth's symmetry axis, then only the longitude of the ascending node Ω, the argument of pericenter ω and the mean anomaly "M" undergo secular precessions.
Such perturbations, which were earlier used to map the Earth's gravitational field from space, may play a relevant disturbing role when satellites are used to make tests of general relativity because the much smaller relativistic effects are qualitatively indistinguishable from the oblateness-driven disturbances.
Formulation.
The flattening formula_0 for the equilibrium configuration of a self-gravitating spheroid, composed of uniform density incompressible fluid, rotating steadily about some fixed axis, for a small amount of flattening, is approximated by:
formula_1
where
A related quantity is the body's second dynamic form factor, "J"2:
formula_10
with "J"2
where
"ε"E is the central body's oblateness,
"R"E is central body's equatorial radius ( for Earth),
"ω"E is the central body's rotation rate ( for Earth),
"GM"E is the product of the universal constant of gravitation and the central body's mass ( for Earth).
Typical values.
Real flattening is smaller due to mass concentration in the center of celestial bodies.
References.
<templatestyles src="Reflist/styles.css" />
|
[
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"math_id": 0,
"text": "f"
},
{
"math_id": 1,
"text": "f = \\frac{a_e - a_p}{a} = \\frac{5}{4} \\frac{\\omega^2 a^3}{G M} = \\frac{15 \\pi}{4} \\frac{1}{G T^2 \\rho}"
},
{
"math_id": 2,
"text": "G"
},
{
"math_id": 3,
"text": "a"
},
{
"math_id": 4,
"text": "a_e \\approx a\\, (1 + \\tfrac{f}{3})"
},
{
"math_id": 5,
"text": "a_p \\approx a\\,(1 - \\tfrac{2f}{3})"
},
{
"math_id": 6,
"text": "T"
},
{
"math_id": 7,
"text": "\\omega = \\tfrac{2 \\pi}{T}"
},
{
"math_id": 8,
"text": "\\rho"
},
{
"math_id": 9,
"text": "M \\simeq \\tfrac{4}{3} \\pi \\rho a^3"
},
{
"math_id": 10,
"text": "J_2 = \\frac{2 \\varepsilon_\\mathrm{E}}{3} - \\frac{{R_\\mathrm{E}}^3 {\\omega_\\mathrm{E}}^2}{3 G M_\\mathrm{E}}"
}
] |
https://en.wikipedia.org/wiki?curid=143023
|
1430274
|
Hold-And-Modify
|
Display mode used in Commodore Amiga computers
Hold-And-Modify, usually abbreviated as HAM, is a display mode of the Commodore Amiga computer. It uses a highly unusual technique to express the color of pixels, allowing many more colors to appear on screen than would otherwise be possible. HAM mode was commonly used to display digitized photographs or video frames, bitmap art and occasionally animation. At the time of the Amiga's launch in 1985, this near-photorealistic display was unprecedented for a home computer and it was widely used to demonstrate the Amiga's graphical capability. However, HAM has significant technical limitations which prevent it from being used as a general purpose display mode.
Background.
The original Amiga chipset uses a planar display with a 12-bit RGB color space that produces 4096 possible colors.
The bitmap of the playfield was held in a section of main memory known as "chip RAM", which was shared between the display system and the main CPU. The display system usually used an indexed color system with a color palette.
The hardware contained 32 registers that could be set to any of the 4096 possible colors, and the image could access up to 32 values using 5 bits per pixel. The sixth available bit could be used by a display mode known as Extra Half-Brite which reduced the luminosity of that pixel by half, providing an easy way to produce shadowing effects.
Hold-And-Modify mode.
The Amiga chipset was designed using a HSV (hue, saturation and luminance) color space, as was common for early home computers and games consoles which relied on television sets for display. HSV maps more directly to the YUV colorspace used by NTSC and PAL color TVs, requiring simpler conversion electronics compared to RGB encoding.
Color television, when transmitted over an RF or composite video link, uses a much reduced chroma bandwidth (encoded as two color-difference components, rather than hue + saturation) compared to the third component, luma. This substantially reduces the memory and bandwidth needed for a given perceived fidelity of display, by storing and transmitting the luminance at full resolution, but chrominance at a relatively lower resolution - a technique shared with image compression techniques like JPEG and MPEG, as well as in other HSV/YUV based video modes such as the YJK encoding of the V9958 MSX-Video chip (first used in the MSX2+).
The variant of HSV encoding used in the original form of HAM allowed for prioritising the update of luminance information over hue and particularly saturation, switching between the three components as needed, compared to the more regular interleaving of full-resolution luma (formula_0) with individual half- or quarter-resolution chromas (formula_1 + formula_2) as used by later digital video standards. This offered considerable efficiency benefits over RGB.
As the Amiga design migrated from a games console to a more general purpose home computer, the video chipset was itself changed from HSV to the modern RGB color model, seemingly negating much of the benefit of HAM mode. Amiga project lead Jay Miner relates:
The final form of Hold-And-Modify was, hardware-wise, functionally the same as the original HSV concept, but instead of operating on those three descriptive components (mostly prioritising the V component), it modifies one of the three RGB color channels. HAM can be considered a lossy compression technique, similar in operation and efficiency to JPEG minus the DCT stage; in HAM6 mode, an effective 4096-color (12-bit) playfield is encoded in half the memory that would normally be required - and HAM8 reduces this still further, to roughly 40%. There is a however a payoff for this simplistic compression: a greater overall color fidelity is achieved at the expense of horizontal artifacts, caused by the inability to set any single pixel to an arbitrary 12- (or 18, 24) bit value. At the extreme, it can take three pixels to change from one color to another, reducing the effective resolution at that point from a "320-pixel" to approximately "106-pixel" mode, and causing smears and shadows to spread along a scanline to the right of a high contrast feature if the 16 available palette registers prove insufficient.
"Decompression" of the HAM encoded color space is achieved in realtime by the display hardware, as the graphics buffer data is being displayed. Each encoded pixel acts as either a normal index to the color palette registers, or as a command to directly alter the value held in the output DAC (somewhat like updating just one-third of the active palette register), and is immediately acted on as such as it passes through the chipset.
Usage.
When the Amiga was launched in 1985, HAM mode offered a significant advantage over competing systems. HAM allows display of all 4096 colors simultaneously, though with the aforementioned limitations. This pseudo-photorealistic display was unprecedented for a home computer of the time and allowed display of digitized photographs and rendered 3D images. In comparison, the then IBM-PC standard EGA allowed 16 on-screen colors from a palette of 64. EGA's successor VGA released in 1987 with its flagship games mode, Mode 13h, allowed 256 on-screen colors from 262,144. HAM mode was frequently used to demonstrate the Amiga's ability in store displays and trade presentations, since competing hardware could not match the color depth. Due to the limitations described above HAM was mainly used for display of static images and developers largely avoided its use with games or applications requiring animation.
HAM mode was only used for gameplay in twelve games, starting with "Pioneer Plague" in 1988. Other HAM titles include "Knights of the Crystallion", "", "Overdrive (Infacto)", "Kang Fu", "AMRVoxel", "RTG", "Zdzislav: Hero Of The Galaxy 3D", "OloFight" and "Genetic Species".
With the introduction of the Advanced Graphics Architecture, a conventional planar image could have a palette of 256 colors, offering significantly higher color fidelity. The original HAM mode, with its limited color resolution, became far less attractive to users of an AGA machine, though it was still included for backward compatibility. The new HAM8 mode was far less useful to the AGA chipset than the HAM mode was to the original chipset, since the more straightforward indexed 256-color (as well as higher performance, planar 128- and 64-color) modes greatly increased the options to the artist without suffering from the drawbacks of HAM. A well-programmed "sliced"-palette mode could prove to be more useful than HAM8, with up to 256 unique colors per line - enough to directly define a distinct color for each pixel if a 256-pixel-wide video mode was defined, and in higher resolutions even a single 256-color palette for the entire screen, let alone each line, allowed much more effective and accurate simulation of higher color depths using dithering than could be achieved with only 32.
The original purpose of HAM, which was to allow more color resolution despite limited video buffer size and limited memory bandwidth, had become largely irrelevant thanks to the lifting of those limits. As more modern computers are inherently capable of high resolution truecolor displays without any special tricks, there is no longer any need for display techniques like HAM; as PC-style graphics cards offering modes such as 800x600 SVGA in hi-color (16 bpp, or 65536 directly-selectable colors) were already available for the Amiga in the dying days of the platform, it is unlikely that any further developments of the technique would have been bothered with had it survived to the present day.
Limitations.
HAM mode places restrictions on the value of adjacent pixels on each horizontal line of the playfield. In order to render two arbitrary colors adjacently, it may take up to two intermediary pixels to change to the intended color (if the red, green and blue components must all be modified). In the worst case this reduces the horizontal usable chroma resolution in half, from 320~360 pixels to 106~120. Even so, it compares favorably to contemporary video technologies like VHS that has a chroma resolution of around 40 "television lines", roughly equivalent to 80 pixels.
Displaying such images over a composite video connection provides some horizontal smoothing that minimizes color artifacts. But if an RGB monitor is used, artifacts becomes particularly noticeable in areas of sharp contrast (strong horizontal image gradients), where an undesirable multi-hued artifact or "fringe" may appear. Various rendering techniques were used to minimize the impact of "fringing" and HAM displays were often designed to incorporate subtle horizontal color gradients, avoiding vertical edges and contrasts.
Displaying a full color image in HAM mode requires some careful preprocessing. Because HAM can only modify one of the RGB components at a time, rapid color transitions along a scan line may be best achieved by using one of the preset color registers for these transitions. To render an arbitrary image, a programmer may choose to first examine the original image for the most noticeable of these transitions and then assign those colors to one of the registers, a technique known as adaptive palettes. However, with only 16 available registers in the original HAM mode, some loss in color fidelity is common.
Additionally, HAM mode does not easily permit arbitrary animation of the display. For example, if an arbitrary portion of the playfield is to be moved to another on-screen position, the Hold-and-Modify values may have to be recomputed on all source and target lines in order to display the image correctly (an operation not well-suited to animation). Specifically, if the left-most edge of the animated object contains any 'modify' pixels, or if the image immediately to the right of the object contains any 'modify' pixels, then those Hold-and-Modify values must be recomputed. An attempt to move an object around the screen (such as with the use of the blitter) will create noticeable fringing at the left and right borders of that image, unless the graphics are specially designed to avoid this. In order to avoid recomputing Hold-and-Modify values and circumvent fringing, the programmer would have to ensure the left-most pixel of every blitter object and the left-most pixel of every line of a scrolling playfield is a "set" pixel. The palette would have to be designed so that it incorporates every such left-most pixel. Alternatively, a HAM display can be animated by generating pixel values through procedural generation, though this is generally useful for synthetic images only, for example, the "rainbow" effects used in demos.
Note, however, that Hold-and-Modify only applies to playfield pixels. 128 pixels of sprite data (in DMA mode) per scanline are still available for placement on top of the HAM playfield.
Implementations.
Original Chip Set HAM mode (HAM6).
HAM6 mode, named for the 6 bits of data per pixel, was introduced with the Original Chip Set and was retained in the later Enhanced Chip Set and Advanced Graphics Architecture (AGA). HAM6 allows up to 4096 colors to be displayed simultaneously at resolutions from 320×200 to 360×576.
HAM6 encoding uses six bits per pixel: two bits for control and four bits for data. If the two control bits are both set to zero, the four remaining bits are used to index one of the 16 preset color registers, operating in the fashion of a normal indexed bitmap. The other three possible control bit patterns indicate that the color of the previous pixel (to the left) on the scanline should be used and the data bits should instead be used to modify the value of the red, green or blue component. Consequently, there are four possibilities:
HAM5.
A similar mode, HAM5, is also available where only 5 bits of data per pixel are used. The sixth bit is always zero, so only the blue color component can be modified. Because only the blue component can be modified without a SET command, the effect is limited to moderate increase of the number of yellow-blue color shades displayed.
This mode is not as flexible as HAM6 and not widely used.
On the AGA chipset, HAM5 no longer exists.
HAM4.
It's also possible to use HAM mode with 4 bitplanes. Practical use is limited, but this technique was used in demos.
HAM7.
It is possible to set up HAM mode with 7 bitplanes on OCS/ECS, but that will use only 4 bitplanes. This technique was demonstrated in the “HAM Eager” demo. On the AGA chipset, HAM7 no longer exists.
Sliced HAM mode (SHAM).
The Original Amiga Chipset included a support chip known as the "Copper" that handles interrupts and other timing and housekeeping duties independently of the CPU and the video system. Using the Copper, it is possible to modify chipset registers or interrupt the CPU at any display coordinate synchronously to the video output. This allows programmers to use either Copper-specific code assembled into a Copperlist or CPU code for video effects with very low overhead.
Using this technique, programmers developed the Sliced HAM or SHAM mode, also known as dynamic HAM. SHAM changes some or all color registers on selected scan lines to change the palette during display. This meant that every scan line can have its own set of 16 base colors. This removes some constraints caused by the limited palette, which can then be chosen per-line instead of per-image. The only downsides to this approach are that the Copperlist uses extra clock cycles of chip RAM for the register changes, that the image is not bitmap-only, and the added complexity of setting up the SHAM mode.
This technique is not limited to HAM, and was widely used with the machine's more conventional graphics modes as well. Dynamic HiRes uses a similar palette changing technique to produce 16 colors per line in the high resolution modes, whereas HAM is limited to low resolution but allows both 16 indexed colors as well as modifications of them.
The SHAM idea was deprecated when HAM8 was introduced with the AGA chipset, since even an unsliced HAM8 image has far more color resolution than a sliced HAM6 image. However, SHAM remains the best available HAM mode on those Amigas with the OCS or ECS chipsets.
Advanced Graphics Architecture HAM mode (HAM8).
With the release of the Advanced Graphics Architecture (AGA) in 1992, the original HAM mode was renamed "HAM6", and a new "HAM8" mode was introduced (the numbered suffix represents the bitplanes used by the respective HAM mode). With AGA, instead of 4 bits per color component, the Amiga now had up to 8 bits per color component, resulting in 16,777,216 possible colors (24-bit color space).
HAM8 operates in the same way as HAM6, using two "control" bits per pixel, but with six bits of data per pixel instead of four. The "set" operation selects from a palette of 64 colors instead of 16. The "modify" operation modifies the six most significant bits of either the red, green or blue color component - the two least significant bits of the color cannot be altered by this operation and remain as set by the most recent set operation.
Compared to HAM6, HAM8 can display many more on-screen colors. The maximum number of on-screen colors using HAM8 was widely reported to be 262,144 colors (18-bit RGB color space). In fact, the maximum number of unique on-screen colors can be greater than 262,144, depending on the two least significant bits of each color component in the 64 color palette. In theory, all 16.7 million colors could be displayed with a large enough screen and an appropriate base palette, but in practice the limitations in achieving full precision mean that the two least significant bits are typically ignored. In general, the perceived HAM8 color depth is roughly equivalent to a high color display.
The vertical display resolutions for HAM8 are the same as for HAM6. The horizontal resolution can be 320 (360 with overscan) as before, doubled to 640 (720 with overscan) or even quadrupled to 1280 pixels (1440 with overscan). The AGA chipset also introduced even higher resolutions for the traditional planar display modes. The total number of pixels in a HAM8 image cannot exceed 829,440 (1440×576) using PAL modes but can exceed 1,310,720 (1280×1024) using third-party display hardware (Indivision AGA flicker-fixer).
Like the original HAM mode, a HAM8 screen cannot display any arbitrary color at any arbitrary position, since every pixel relies on either a limited palette or relies on up to two color components of the previous pixel. As with the original HAM mode, designers may also choose to 'slice' the display (see above) in order to circumvent some of these restrictions.
HAM emulation.
HAM is unique to the Amiga and its distinct chipsets. To allow direct rendering of legacy images encoded in HAM format software-based HAM emulators have been developed which do not require the original display hardware. Pre-4.0 versions of AmigaOS can use HAM mode in the presence of the native Amiga chipset. AmigaOS 4.0 and up, designed for radically different hardware, provides HAM emulation for use on modern chunky graphics hardware. Dedicated Amiga emulators running on non-native hardware are able to display HAM mode by emulation of the display hardware. However, since no other computer architecture used the HAM technique, viewing a HAM image on any other architecture requires programmatic interpretation of the image file. Faithful software-based decoding will produce identical results, setting aside variations in color fidelity between display setups.
However, if the goal is merely to display a SHAM image on a non-Amiga platform, the required color values may be pre-calculated based on the palette entries that are programmed via the Copperlist, regardless of whether the palette is modified in the middle of a scanline. It is always possible to up-convert a HAM or SHAM image losslessly to a 32-bit palette.
Third-party HAM implementations.
A device produced by Black Belt known as HAM-E was able to produce images with HAM8 color depth at low horizontal resolution from an Amiga with an Original Chip Set.
The Amiga would be set up to produce high resolution images (640 pixels wide, 720 with overscan). This required the use of four bitplanes at 70 ns per pixel. The first few lines of the image encoded information to configure the HAM-E unit. Then each pair of pixels was encoded with information for the HAM-E unit, which converted the information into one 140 ns pixel (generating an image 320 pixels wide, or 360 with overscan, at a color depth of eight bitplanes). The quality of HAM-E was thus comparable to a low-resolution HAM8 image. The HAM-E technique exploited the fact that a high resolution image with four bitplanes delivers a third more memory bandwidth, and therefore a third more data, than a low resolution image with six bitplanes.
The HAM technique was also implemented on the HAM256 and HAM8x1 modes of ULAplus / HAM256 / HAM8x1 for the ZX Spectrum, where it provides the ability to display 256 colors on screen, by modifying a base 64 color palette.
The HAM very similar to the Amiga HAM8 is a part of the HGFX, a planar-based system, provided in the form of the FPGA extension of the original video circuity (ULA), for ZX Spectrum computers. Proposed and tested with the LnxSpectrum emulator as the HGFX/Q, realized in the eLeMeNt ZX computer, in 2021.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "Y"
},
{
"math_id": 1,
"text": "U"
},
{
"math_id": 2,
"text": "V"
}
] |
https://en.wikipedia.org/wiki?curid=1430274
|
1430494
|
Price-cap regulation
|
Form of market regulation
Price-cap regulation is a form of incentive regulation capping the prices that firms in a natural monopoly position may charge their customers. Designed in the 1980s by UK Treasury economist Stephen Littlechild, it has been applied to all privatised British network utilities. It is contrasted with both rate-of-return regulation, with utilities being permitted a set rate of return on capital, and with revenue-cap regulation, with total revenue being the regulated variable.
Functioning.
Price-cap regulation adjusts the operator's prices according to
Revenue cap regulation attempts to do the same thing but for revenue, rather than prices.
Price-cap regulation is sometimes called "CPI - X", (in the United Kingdom "RPI-X") after the basic formula employed to set price caps. This takes the rate of inflation, measured by the Consumer Price Index (UK Retail Prices Index, RPI) and subtracts expected efficiency savings formula_0. In the water industry, the formula is formula_1, where formula_2 is based on capital investment requirements.
The system is intended to provide incentives for efficiency savings, as any savings above the predicted rate formula_0 can be passed on to shareholders, at least until the price caps are next reviewed (usually every five years). A key part of the system is that the rate formula_0 is based not only a firm's past performance, but on the performance of other firms in the industry: formula_0 is intended to be a proxy for a competitive market, in industries which are natural monopolies.
Use.
Energy.
Notably, in 2018, the UK Government introduced a form of price cap regulation through a new
customers on standard variable tariffs. In August 2022, the energy price cap was raised to £3,549 which would have pushed 8.2 million people into fuel poverty in October 2022 until March 2023. However, in the event, as a political decision, the UK Government subsidised the supply of domestic electricity by reducing bills by £400 for each household, spread out over six months, and subsidising each unit of electricity as well.
Telecoms.
Price-cap regulation is no longer a uniquely British form of regulation. Particularly in the telecommunications industry, many Asian countries are implementing some form of price cap on their newly privatised operators. In addition, many US local exchange carriers are regulated by price-cap rather than rate-of-return regulation: in 2003, of the 73 companies reporting to Federal Communications Commission "Automated Reporting Management Information System" (ARMIS) database, 22 were regulated according to an RPI-X price cap (and a further 35 were subject to other retail price controls). In Australia, the preferred form of price regulation for utilities is the CPI-X regime.
Airports.
Many airports are local monopolies. To prevent them from abusing their market power, governments around the world regulate how much airports may charge to airlines. This price-cap regulation can follow a dual-till approach, in which the regulator considers aeronautical and non-aeronautical (commercial) revenue separately, or a single-till approach, in which the regulator considers all the airport's revenues when determining acceptable airport charges.
Under dual-till regulation, airports do not use profits derived from concession businesses to finance airport infrastructure, implying higher charges to airlines. The single-till approach typically leads to lower charges to airlines, which better controls airport market power.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "X"
},
{
"math_id": 1,
"text": "RPI-X+K"
},
{
"math_id": 2,
"text": "K"
}
] |
https://en.wikipedia.org/wiki?curid=1430494
|
1430548
|
Preon
|
Hypothetical subatomic particle
In particle physics, preons are hypothetical point particles, conceived of as sub-components of quarks and leptons. The word was coined by Jogesh Pati and Abdus Salam, in 1974. Interest in preon models peaked in the 1980s but has slowed, as the Standard Model of particle physics continues to describe physics mostly successfully, and no direct experimental evidence for lepton and quark compositeness has been found. Preons come in four varieties: plus, anti-plus, zero, and anti-zero. W bosons have six preons, and quarks and leptons have only three.
In the hadronic sector, some effects are considered anomalies within the Standard Model. For example, the proton spin puzzle, the EMC effect, the distributions of electric charges inside the nucleons, as found by Robert Hofstadter in 1956, and the ad hoc CKM matrix elements.
When the term "preon" was coined, it was primarily to explain the two families of spin- fermions: quarks and leptons. More recent preon models also account for spin-1 bosons, and are still called "preons". Each of the preon models postulates a set of fewer fundamental particles than those of the Standard Model, together with the rules governing how those fundamental particles combine and interact. Based on these rules, the preon models try to explain the Standard Model, often predicting small discrepancies with this model and generating new particles and certain phenomena which do not belong to the Standard Model.
Goals of preon models.
Preon research is motivated by the desire to:
Background.
Before the Standard Model was developed in the 1970s (the key elements of the Standard Model known as quarks were proposed by Murray Gell-Mann and George Zweig in 1964), physicists observed hundreds of different kinds of particles in particle accelerators. These were organized into relationships on their physical properties in a largely ad-hoc system of hierarchies, not entirely unlike the way taxonomy grouped animals based on their physical features. Not surprisingly, the huge number of particles was referred to as the "particle zoo".
The Standard Model, which is now the prevailing model of particle physics, dramatically simplified this picture by showing that most of the observed particles were mesons, which are combinations of two quarks, or baryons which are combinations of three quarks, plus a handful of other particles. The particles being seen in the ever-more-powerful accelerators were, according to the theory, typically nothing more than combinations of these quarks.
Comparisons of quarks, leptons, and bosons.
Within the Standard Model, there are several classes of particles. One of these, the quarks, has six types, of which there are three varieties in each (dubbed "colors", red, green, and blue, giving rise to quantum chromodynamics).
Additionally, there are six different types of what are known as leptons. Of these six leptons, there are three charged particles: the electron, muon, and tau. The neutrinos comprise the other three leptons, and each neutrino pairs with one of the three charged leptons.
In the Standard Model, there are also bosons, including the photons and gluons; W+, W−, and Z bosons; and the Higgs boson; and an open space left for the graviton. Almost all of these particles come in "left-handed" and "right-handed" versions (see " chirality"). The quarks, leptons, and W boson all have antiparticles with opposite electric charge (or in the case of the neutrinos, opposite weak isospin).
Unresolved problems with the Standard Model.
The Standard Model also has a number of problems which have not been entirely solved. In particular, no successful theory of gravitation based on a particle theory has yet been proposed. Although the Model assumes the existence of a graviton, all attempts to produce a consistent theory based on them have failed.
Kalman asserts that, according to the concept of atomism, fundamental building blocks of nature are indivisible bits of matter that are ungenerated and indestructible. Neither leptons nor quarks are truly indestructible, since some leptons can decay into other leptons, some quarks into other quarks. Thus, on fundamental grounds, quarks are not themselves fundamental building blocks, but must be composed of other, fundamental quantities—preons. Although the mass of each successive particle follows certain patterns, predictions of the rest mass of most particles cannot be made precisely, except for the masses of almost all baryons which have been modeled well by de Souza (2010).
The Standard Model also has problems predicting the large scale structure of the universe. For instance, the SM generally predicts equal amounts of matter and antimatter in the universe. A number of attempts have been made to "fix" this through a variety of mechanisms, but to date none have won widespread support. Likewise, basic adaptations of the Model suggest the presence of proton decay, which has not yet been observed.
Motivation for preon models.
Several models have been proposed in an attempt to provide a more fundamental explanation of the results in experimental and theoretical particle physics, using names such as "parton" or "preon" for the hypothetical basic particle constituents.
Preon theory is motivated by a desire to replicate in particle physics the achievements of the periodic table in Chemistry, which reduced 94 naturally occurring elements to combinations of just three building-blocks (proton, neutron, electron). Likewise, the Standard Model later organized the "particle zoo" of hadrons by reducing several dozen particles to combinations at a more fundamental level of (at first) just three quarks, consequently reducing the huge number of arbitrary constants in mid-twentieth-century particle physics prior to the Standard Model and quantum chromodynamics.
However, the particular preon model discussed below has attracted comparatively little interest among the particle physics community to date, in part because no evidence has been obtained so far in collider experiments to show that the fermions of the Standard Model are composite.
Attempts.
A number of physicists have attempted to develop a theory of "pre-quarks" (from which the name "preon" derives) in an effort to justify theoretically the many parts of the Standard Model that are known only through experimental data. Other names which have been used for these proposed fundamental particles (or particles intermediate between the most fundamental particles and those observed in the Standard Model) include "prequarks", "subquarks", "maons", "alphons", "quinks", "rishons", "tweedles", "helons", "haplons", "Y-particles", and "primons". "Preon" is the leading name in the physics community.
Efforts to develop a substructure date at least as far back as 1974 with a paper by Pati and Salam in "Physical Review".
Other attempts include a 1977 paper by Terazawa, Chikashige, and Akama, similar, but independent, 1979 papers by Ne'eman,
Harari,
and Shupe, a 1981 paper by Fritzsch and Mandelbaum,
and a 1992 book by D'Souza and Kalman. None of these have gained wide acceptance in the physics world. However, in a recent work
de Souza has shown that his model describes well all weak decays of hadrons according to selection rules dictated by a quantum number derived from his compositeness model. In his model leptons are elementary particles and each quark is composed of two "primons", and thus, all quarks are described by four "primons". Therefore, there is no need for the Standard Model Higgs boson and each quark mass is derived from the interaction between each pair of "primons" by means of three Higgs-like bosons.
In his 1989 Nobel Prize acceptance lecture, Hans Dehmelt described a most fundamental elementary particle, with definable properties, which he called the "cosmon", as the likely result of a long but finite chain of increasingly more elementary particles.
Composite Higgs.
Many preon models either do not account for the Higgs boson or rule it out, and propose that electro-weak symmetry is broken not by a scalar Higgs field but by composite preons. For example, Fredriksson preon theory does not need the Higgs boson, and explains the electro-weak breaking as the rearrangement of preons, rather than a Higgs-mediated field. In fact, the Fredriksson preon model and the de Souza model predict that the Standard Model Higgs boson does not exist.
Rishon model.
The "rishon model" (RM) is the earliest effort (1979) to develop a preon model to explain the phenomenon appearing in the Standard Model (SM) of particle physics. It was first developed by Haim Harari and Michael A. Shupe (independently of each other), and later expanded by Harari and his then-student Nathan Seiberg.
The model has two kinds of fundamental particles called rishons (ראשונים) (which means "First" in Hebrew). They are T ("Third" since it has an electric charge of ⅓ "e", or Tohu (תוהו) which means "Chaos") and V ("Vanishes", since it is electrically neutral, or Vohu which means "void"). All leptons and all flavours of quarks are three-rishon ordered triplets. These groups of three rishons have spin-½.
The Rishon model illustrates some of the typical efforts in the field. Many of the preon models theorize that the apparent imbalance of matter and antimatter in the universe is in fact illusory, with large quantities of preon-level antimatter confined within more complex structures.
Criticisms.
The mass paradox.
One preon model started as an internal paper at the Collider Detector at Fermilab (CDF) around 1994. The paper was written after an unexpected and inexplicable excess of jets with energies above 200 GeV were detected in the 1992–1993 running period. However, scattering experiments have shown that quarks and leptons are "point like" down to distance scales of less than 10−18 m (or <templatestyles src="Fraction/styles.css" />1⁄1000 of a proton diameter). The momentum uncertainty of a preon (of whatever mass) confined to a box of this size is about 200 GeV/c, which is 50,000 times larger than the (model dependent) rest mass of an up-quark, and 400,000 times larger than the rest mass of an electron.
Heisenberg's uncertainty principle states that formula_0 and thus anything confined to a box smaller than formula_1 would have a momentum uncertainty proportionally greater. Thus, the preon model proposed particles smaller than the elementary particles they make up, since the momentum uncertainty formula_2 should be greater than the particles themselves.
So the preon model represents a mass paradox: How could quarks or electrons be made of smaller particles that would have many orders of magnitude greater mass-energies arising from their enormous momenta? One way of resolving this paradox is to postulate a large binding force between preons that cancels their mass-energies.
Conflicts with observed physics.
Preon models propose additional unobserved forces or dynamics to account for the observed properties of elementary particles, which may have implications in conflict with observation. For example, now that the LHC's observation of a Higgs boson is confirmed, the observation contradicts the predictions of many preon models that excluded it.
Preon theories require quarks and leptons to have a finite size. It is possible that the Large Hadron Collider will observe this after it is upgraded to higher energies.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\operatorname{\\Delta} x \\cdot \\operatorname{\\Delta} p \\ge \\tfrac{1}{2}\\hbar"
},
{
"math_id": 1,
"text": "\\operatorname{\\Delta} x"
},
{
"math_id": 2,
"text": "\\operatorname{\\Delta} p"
}
] |
https://en.wikipedia.org/wiki?curid=1430548
|
1430697
|
Coefficient of multiple correlation
|
Statistical concept
In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables.
The coefficient of multiple correlation takes values between 0 and 1. Higher values indicate higher predictability of the dependent variable from the independent variables, with a value of 1 indicating that the predictions are exactly correct and a value of 0 indicating that no linear combination of the independent variables is a better predictor than is the fixed mean of the dependent variable.
The coefficient of multiple correlation is known as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been derived from a model-fitting procedure.
Definition.
The coefficient of multiple correlation, denoted "R", is a scalar that is defined as the Pearson correlation coefficient between the predicted and the actual values of the dependent variable in a linear regression model that includes an intercept.
Computation.
The square of the coefficient of multiple correlation can be computed using the vector formula_0 of correlations formula_1 between the predictor variables formula_2 (independent variables) and the target variable formula_3 (dependent variable), and the correlation matrix formula_4 of correlations between predictor variables. It is given by
formula_5
where formula_6 is the transpose of formula_7, and formula_8 is the inverse of the matrix
formula_9
If all the predictor variables are uncorrelated, the matrix formula_4 is the identity matrix and formula_10 simply equals formula_11, the sum of the squared correlations with the dependent variable. If the predictor variables are correlated among themselves, the inverse of the correlation matrix formula_4 accounts for this.
The squared coefficient of multiple correlation can also be computed as the fraction of variance of the dependent variable that is explained by the independent variables, which in turn is 1 minus the unexplained fraction. The unexplained fraction can be computed as the sum of squares of residuals—that is, the sum of the squares of the prediction errors—divided by the sum of squares of deviations of the values of the dependent variable from its expected value.
Properties.
With more than two variables being related to each other, the value of the coefficient of multiple correlation depends on the choice of dependent variable: a regression of formula_3 on formula_12 and formula_13 will in general have a different formula_14 than will a regression of formula_13 on formula_12 and formula_3. For example, suppose that in a particular sample the variable formula_13 is uncorrelated with both formula_12 and formula_3, while formula_12 and formula_3 are linearly related to each other. Then a regression of formula_13 on formula_3 and formula_12 will yield an formula_14 of zero, while a regression of formula_3 on formula_12 and formula_13 will yield a strictly positive formula_14. This follows since the correlation of formula_3 with its best predictor based on formula_12 and formula_13 is in all cases at least as large as the correlation of formula_3 with its best predictor based on formula_12 alone, and in this case with formula_13 providing no explanatory power it will be exactly as large.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{c} = {(r_{x_1 y}, r_{x_2 y},\\dots,r_{x_N y})}^\\top"
},
{
"math_id": 1,
"text": "r_{x_n y}"
},
{
"math_id": 2,
"text": "x_n"
},
{
"math_id": 3,
"text": "y"
},
{
"math_id": 4,
"text": "R_{xx}"
},
{
"math_id": 5,
"text": "R^2 = \\mathbf{c}^\\top R_{xx}^{-1}\\, \\mathbf{c},"
},
{
"math_id": 6,
"text": "\\mathbf{c}^\\top"
},
{
"math_id": 7,
"text": "\\mathbf{c}"
},
{
"math_id": 8,
"text": "R_{xx}^{-1}"
},
{
"math_id": 9,
"text": "R_{xx} = \\left(\\begin{array}{cccc}\n r_{x_1 x_1} & r_{x_1 x_2} & \\dots & r_{x_1 x_N} \\\\\n r_{x_2 x_1} & \\ddots & & \\vdots \\\\\n \\vdots & & \\ddots & \\\\\n r_{x_N x_1} & \\dots & & r_{x_N x_N}\n\\end{array}\\right)."
},
{
"math_id": 10,
"text": "R^2"
},
{
"math_id": 11,
"text": "\\mathbf{c}^\\top\\, \\mathbf{c}"
},
{
"math_id": 12,
"text": "x"
},
{
"math_id": 13,
"text": "z"
},
{
"math_id": 14,
"text": "R"
}
] |
https://en.wikipedia.org/wiki?curid=1430697
|
14307
|
Hall effect
|
Electromagnetic effect in physics
The Hall effect is the production of a potential difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879.
The "Hall coefficient" is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made, since its value depends on the type, number, and properties of the charge carriers that constitute the current.
Discovery.
Wires carrying current in a magnetic field experience a mechanical force perpendicular to both the current and magnetic field. André-Marie Ampère in the 1820s observed this underlying mechanism that led to the discovery of the Hall effect. However it was not until a solid mathematical basis for electromagnetism was systematized by James Clerk Maxwell's "On Physical Lines of Force" (published in 1861–1862) that details of the interaction between magnets and electric current could be understood.
Edwin Hall then explored the question of whether magnetic fields interacted with the conductors "or" the electric current, and reasoned that if the force was specifically acting on the current, it should crowd current to one side of the wire, producing a small measurable voltage. In 1879, he discovered this "Hall effect" while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland. Eighteen years before the electron was discovered, his measurements of the tiny effect produced in the apparatus he used were an experimental tour de force, published under the name "On a New Action of the Magnet on Electric Currents".
Hall effect within voids.
The term ordinary Hall effect can be used to distinguish the effect described in the introduction from a related effect which occurs across a void or hole in a semiconductor or metal plate when current is injected via contacts that lie on the boundary or edge of the void. The charge then flows outside the void, within the metal or semiconductor material. The effect becomes observable, in a perpendicular applied magnetic field, as a Hall voltage appearing on either side of a line connecting the current-contacts. It exhibits apparent sign reversal in comparison to the "ordinary" effect occurring in the simply connected specimen. It depends only on the current injected from within the void.
Hall effect superposition.
Superposition of these two forms of the effect, the ordinary and void effects, can also be realized. First imagine the "ordinary" configuration, a simply connected (void-less) thin rectangular homogeneous element with current-contacts on the (external) boundary. This develops a Hall voltage, in a perpendicular magnetic field. Next, imagine placing a rectangular void within this ordinary configuration, with current-contacts, as mentioned above, on the interior boundary of the void. (For simplicity, imagine the contacts on the boundary of the void lined up with the ordinary-configuration contacts on the exterior boundary.) In such a combined configuration, the two Hall effects may be realized and observed simultaneously in the same doubly connected device: A Hall effect on the external boundary that is proportional to the current injected only via the outer boundary, and an apparently sign-reversed Hall effect on the interior boundary that is proportional to the current injected only via the interior boundary. The superposition of multiple Hall effects may be realized by placing multiple voids within the Hall element, with current and voltage contacts on the boundary of each void.
Further "Hall effects" may have additional physical mechanisms but are built on these basics.
Theory.
The Hall effect is due to the nature of the current in a conductor. Current consists of the movement of many small charge carriers, typically electrons, holes, ions (see Electromigration) or all three. When a magnetic field is present, these charges experience a force, called the Lorentz force. When such a magnetic field is absent, the charges follow approximately straight paths between collisions with impurities, phonons, etc. However, when a magnetic field with a perpendicular component is applied, their paths between collisions are curved; thus, moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element, arising from a force that is perpendicular to both the straight path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electric potential is established for as long as the charge is flowing.
In classical electromagnetism electrons move in the opposite direction of the current "I" (by convention "current" describes a theoretical "hole flow"). In some metals and semiconductors it "appears" "holes" are actually flowing because the direction of the voltage is opposite to the derivation below.
For a simple metal where there is only one type of charge carrier (electrons), the Hall voltage "V"H can be derived by using the Lorentz force and seeing that, in the steady-state condition, charges are not moving in the "y"-axis direction. Thus, the magnetic force on each electron in the "y"-axis direction is cancelled by a "y"-axis electrical force due to the buildup of charges. The "vx" term is the drift velocity of the current which is assumed at this point to be holes by convention. The "vxBz" term is negative in the "y"-axis direction by the right hand rule.
formula_0
In steady state, F = 0, so 0 = "Ey" − "vxBz", where "Ey" is assigned in the direction of the "y"-axis, (and not with the arrow of the induced electric field "ξy" as in the image (pointing in the −"y" direction), which tells you where the field caused by the electrons is pointing).
In wires, electrons instead of holes are flowing, so "vx" → −"vx" and "q" → −"q". Also "Ey" = −. Substituting these changes gives
formula_1
The conventional "hole" current is in the negative direction of the electron current and the negative of the electrical charge which gives "Ix" = "ntw"(−"vx")(−"e") where "n" is charge carrier density, "tw" is the cross-sectional area, and −"e" is the charge of each electron. Solving for formula_2 and plugging into the above gives the Hall voltage:
formula_3
If the charge build up had been positive (as it appears in some metals and semiconductors), then the "V"H assigned in the image would have been negative (positive charge would have built up on the left side).
The Hall coefficient is defined as
formula_4
or
formula_5
where j is the current density of the carrier electrons, and "Ey" is the induced electric field. In SI units, this becomes
formula_6
(The units of "R"H are usually expressed as m3/C, or Ω·cm/G, or other variants.) As a result, the Hall effect is very useful as a means to measure either the carrier density or the magnetic field.
One very important feature of the Hall effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. In the diagram above, the Hall effect with a negative charge carrier (the electron) is presented. But consider the same magnetic field and current are applied but the current is carried inside the Hall effect device by a positive particle. The particle would of course have to be moving in the opposite direction of the electron in order for the current to be the same—down in the diagram, not up like the electron is. And thus, mnemonically speaking, your thumb in the , representing (conventional) current, would be pointing the "same" direction as before, because current is the same—an electron moving up is the same current as a positive charge moving down. And with the fingers (magnetic field) also being the same, interestingly "the charge carrier gets deflected to the left in the diagram regardless of whether it is positive or negative." But if positive carriers are deflected to the left, they would build a relatively "positive voltage" on the left whereas if negative carriers (namely electrons) are, they build up a negative voltage on the left as shown in the diagram. Thus for the same current and magnetic field, the electric polarity of the Hall voltage is dependent on the internal nature of the conductor and is useful to elucidate its inner workings.
This property of the Hall effect offered the first real proof that electric currents in most metals are carried by moving electrons, not by protons. It also showed that in some substances (especially p-type semiconductors), it is contrarily more appropriate to think of the current as positive "holes" moving rather than negative electrons. A common source of confusion with the Hall effect in such materials is that holes moving one way are really electrons moving the opposite way, so one expects the Hall voltage polarity to be the same as if electrons were the charge carriers as in most metals and n-type semiconductors. Yet we observe the opposite polarity of Hall voltage, indicating positive charge carriers. However, of course there are no actual positrons or other positive elementary particles carrying the charge in p-type semiconductors, hence the name "holes". In the same way as the oversimplistic picture of light in glass as photons being absorbed and re-emitted to explain refraction breaks down upon closer scrutiny, this apparent contradiction too can only be resolved by the modern quantum mechanical theory of quasiparticles wherein the collective quantized motion of multiple particles can, in a real physical sense, be considered to be a particle in its own right (albeit not an elementary one).
Unrelatedly, inhomogeneity in the conductive sample can result in a spurious sign of the Hall effect, even in ideal van der Pauw configuration of electrodes. For example, a Hall effect consistent with positive carriers was observed in evidently n-type semiconductors. Another source of artefact, in uniform materials, occurs when the sample's aspect ratio is not long enough: the full Hall voltage only develops far away from the current-introducing contacts, since at the contacts the transverse voltage is shorted out to zero.
Hall effect in semiconductors.
When a current-carrying semiconductor is kept in a magnetic field, the charge carriers of the semiconductor experience a force in a direction perpendicular to both the magnetic field and the current. At equilibrium, a voltage appears at the semiconductor edges.
The simple formula for the Hall coefficient given above is usually a good explanation when conduction is dominated by a single charge carrier. However, in semiconductors and many metals the theory is more complex, because in these materials conduction can involve significant, simultaneous contributions from both electrons and holes, which may be present in different concentrations and have different mobilities. For moderate magnetic fields the Hall coefficient is
formula_7
or equivalently
formula_8
with
formula_9
Here "n" is the electron concentration, "p" the hole concentration, "μ"e the electron mobility, "μ"h the hole mobility and "e" the elementary charge.
For large applied fields the simpler expression analogous to that for a single carrier type holds.
Relationship with star formation.
Although it is well known that magnetic fields play an important role in star formation, research models indicate that Hall diffusion critically influences the dynamics of gravitational collapse that forms protostars.
Quantum Hall effect.
For a two-dimensional electron system which can be produced in a MOSFET, in the presence of large magnetic field strength and low temperature, one can observe the quantum Hall effect, in which the Hall conductance σ undergoes quantum Hall transitions to take on the quantized values.
Spin Hall effect.
The spin Hall effect consists in the spin accumulation on the lateral boundaries of a current-carrying sample. No magnetic field is needed. It was predicted by Mikhail Dyakonov and V. I. Perel in 1971 and observed experimentally more than 30 years later, both in semiconductors and in metals, at cryogenic as well as at room temperatures.
The quantity describing the strength of the Spin Hall effect is known as Spin Hall angle, and it is defined as:
formula_10
Where formula_11 is the spin current generated by the applied current density formula_12.
Quantum spin Hall effect.
For mercury telluride two dimensional quantum wells with strong spin-orbit coupling, in zero magnetic field, at low temperature, the quantum spin Hall effect has been observed in 2007.
Anomalous Hall effect.
In ferromagnetic materials (and paramagnetic materials in a magnetic field), the Hall resistivity includes an additional contribution, known as the anomalous Hall effect (or the extraordinary Hall effect), which depends directly on the magnetization of the material, and is often much larger than the ordinary Hall effect. (Note that this effect is "not" due to the contribution of the magnetization to the total magnetic field.) For example, in nickel, the anomalous Hall coefficient is about 100 times larger than the ordinary Hall coefficient near the Curie temperature, but the two are similar at very low temperatures. Although a well-recognized phenomenon, there is still debate about its origins in the various materials. The anomalous Hall effect can be either an "extrinsic" (disorder-related) effect due to spin-dependent scattering of the charge carriers, or an "intrinsic" effect which can be described in terms of the Berry phase effect in the crystal momentum space ("k"-space).
Hall effect in ionized gases.
The Hall effect in an ionized gas (plasma) is significantly different from the Hall effect in solids (where the Hall parameter is always much less than unity). In a plasma, the Hall parameter can take any value. The Hall parameter, "β", in a plasma is the ratio between the electron gyrofrequency, "Ω"e, and the electron-heavy particle collision frequency, ν:
formula_13
where
The Hall parameter value increases with the magnetic field strength.
Physically, the trajectories of electrons are curved by the Lorentz force. Nevertheless, when the Hall parameter is low, their motion between two encounters with heavy particles (neutral or ion) is almost linear. But if the Hall parameter is high, the electron movements are highly curved. The current density vector, J, is no longer collinear with the electric field vector, E. The two vectors J and E make the Hall angle, θ, which also gives the Hall parameter:
formula_14
Other Hall effects.
The Hall Effects family has expanded to encompass other quasi-particles in semiconductor nanostructures. Specifically, a set of Hall Effects has emerged based on excitons and exciton-polaritons n 2D materials and quantum wells.
Applications.
Hall sensors amplify and use the Hall effect for a variety of sensing applications.
Corbino effect.
The Corbino effect, named after its discoverer Orso Mario Corbino, is a phenomenon involving the Hall effect, but a disc-shaped metal sample is used in place of a rectangular one. Because of its shape the Corbino disc allows the observation of Hall effect–based magnetoresistance without the associated Hall voltage.
A radial current through a circular disc, subjected to a magnetic field perpendicular to the plane of the disc, produces a "circular" current through the disc.
The absence of the free transverse boundaries renders the interpretation of the Corbino effect simpler than that of the Hall effect.
References.
<templatestyles src="Reflist/styles.css" />
Further reading.
<templatestyles src="Refbegin/styles.css" />
|
[
{
"math_id": 0,
"text": "\\mathbf{F} = q\\bigl(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}\\bigl)"
},
{
"math_id": 1,
"text": "V_\\mathrm{H}= v_x B_z w"
},
{
"math_id": 2,
"text": "w"
},
{
"math_id": 3,
"text": "V_\\mathrm{H} = \\frac{I_x B_z}{n t e}"
},
{
"math_id": 4,
"text": "R_\\mathrm{H} = \\frac{E_y}{j_x B_z}"
},
{
"math_id": 5,
"text": "\\mathbf{E} = -R_\\mathrm{H}(\\mathbf{J}_c \\times \\mathbf{B})"
},
{
"math_id": 6,
"text": "R_\\mathrm{H} =\\frac{E_y}{j_x B}= \\frac{V_\\mathrm{H} t}{IB}=\\frac{1}{ne}."
},
{
"math_id": 7,
"text": "R_\\mathrm{H}=\\frac{p\\mu_\\mathrm{h}^2 - n\\mu_\\mathrm{e}^2}{e(p\\mu_\\mathrm{h} + n\\mu_\\mathrm{e})^2}"
},
{
"math_id": 8,
"text": "R_\\mathrm{H}=\\frac{p-nb^2}{e(p+nb)^2}"
},
{
"math_id": 9,
"text": "b=\\frac{\\mu_\\mathrm{e}}{\\mu_\\mathrm{h}}."
},
{
"math_id": 10,
"text": "\\theta_{SH}=\\frac{2e}{\\hbar}\\frac{|j_s|}{|j_e|}"
},
{
"math_id": 11,
"text": "j_s"
},
{
"math_id": 12,
"text": "j_e"
},
{
"math_id": 13,
"text": "\\beta=\\frac {\\Omega_\\mathrm{e}}{\\nu}=\\frac {eB}{m_\\mathrm{e}\\nu}"
},
{
"math_id": 14,
"text": "\\beta = \\tan(\\theta)."
}
] |
https://en.wikipedia.org/wiki?curid=14307
|
14308303
|
Monte Carlo method for photon transport
|
Modeling application
Modeling photon propagation with Monte Carlo methods is a flexible yet rigorous approach to simulate photon transport. In the method, local rules of photon transport are expressed as probability distributions which describe the step size of photon movement between sites of photon-matter interaction and the angles of deflection in a photon's trajectory when a scattering event occurs. This is equivalent to modeling photon transport analytically by the radiative transfer equation (RTE), which describes the motion of photons using a differential equation. However, closed-form solutions of the RTE are often not possible; for some geometries, the diffusion approximation can be used to simplify the RTE, although this, in turn, introduces many inaccuracies, especially near sources and boundaries. In contrast, Monte Carlo simulations can be made arbitrarily accurate by increasing the number of photons traced. For example, see the movie, where a Monte Carlo simulation of a pencil beam incident on a semi-infinite medium models both the initial ballistic photon flow and the later diffuse propagation.
The Monte Carlo method is necessarily statistical and therefore requires significant computation time to achieve precision. In addition Monte Carlo simulations can keep track of multiple physical quantities simultaneously, with any desired spatial and temporal resolution. This flexibility makes Monte Carlo modeling a powerful tool. Thus, while computationally inefficient, Monte Carlo methods are often considered the standard for simulated measurements of photon transport for many biomedical applications.
Biomedical applications of Monte Carlo methods.
Biomedical imaging.
The optical properties of biological tissue offer an approach to biomedical imaging. There are many endogenous contrasts, including absorption from blood and melanin and scattering from nerve cells and cancer cell nuclei. In addition, fluorescent probes can be targeted to many different tissues. Microscopy techniques (including confocal, two-photon, and optical coherence tomography) have the ability to image these properties with high spatial resolution, but, since they rely on ballistic photons, their depth penetration is limited to a few millimeters. Imaging deeper into tissues, where photons have been multiply scattered, requires a deeper understanding of the statistical behavior of large numbers of photons in such an environment. Monte Carlo methods provide a flexible framework that has been used by different techniques to reconstruct optical properties deep within tissue. A brief introduction to a few of these techniques is presented here.
Radiation therapy.
The goal of radiation therapy is to deliver energy, generally in the form of ionizing radiation, to cancerous tissue while sparing the surrounding normal tissue. Monte Carlo modeling is commonly employed in radiation therapy to determine the peripheral dose the patient will experience due to scattering, both from the patient tissue as well as scattering from collimation upstream in the linear accelerator.
Photodynamic therapy.
In Photodynamic therapy (PDT) light is used to activate chemotherapy agents. Due to the nature of PDT, it is useful to use Monte Carlo methods for modeling scattering and absorption in the tissue in order to ensure appropriate levels of light are delivered to activate chemotherapy agents.
Implementation of photon transport in a scattering medium.
Presented here is a model of a photon Monte Carlo method in a homogeneous infinite medium. The model is easily extended for multi-layered media, however. For an inhomogeneous medium, boundaries must be considered. In addition for a semi-infinite medium (in which photons are considered lost if they exit the top boundary), special consideration must be taken. For more information, please visit the links at the bottom of the page. We will solve the problem using an infinitely small point source (represented analytically as a Dirac delta function in space and time). Responses to arbitrary source geometries can be constructed using the method of Green's functions (or convolution, if enough spatial symmetry exists). The required parameters are the absorption coefficient, the scattering coefficient, and the scattering phase function. (If boundaries are considered the index of refraction for each medium must also be provided.) Time-resolved responses are found by keeping track of the total elapsed time of the photon's flight using the optical path length. Responses to sources with arbitrary time profiles can then be modeled through convolution in time.
In our simplified model we use the following variance reduction technique to reduce computational time. Instead of propagating photons individually, we create a photon packet with a specific weight (generally initialized as unity). As the photon interacts in the turbid medium, it will deposit weight due to absorption and the remaining weight will be scattered to other parts of the medium. Any number of variables can be logged along the way, depending on the interest of a particular application. Each photon packet will repeatedly undergo the following numbered steps until it is either terminated, reflected, or transmitted. The process is diagrammed in the schematic to the right. Any number of photon packets can be launched and modeled, until the resulting simulated measurements have the desired signal-to-noise ratio. Note that as Monte Carlo modeling is a statistical process involving random numbers, we will be using the variable ξ throughout as a pseudo-random number for many calculations.
Step 1: Launching a photon packet.
In our model, we are ignoring initial specular reflectance associated with entering a medium that is not refractive index matched. With this in mind, we simply need to set the initial position of the photon packet as well as the initial direction. It is convenient to use a global coordinate system. We will use three Cartesian coordinates to determine position, along with three direction cosines to determine the direction of propagation. The initial start conditions will vary based on application, however for a pencil beam initialized at the origin, we can set the initial position and direction cosines as follows (isotropic sources can easily be modeled by randomizing the initial direction of each packet):
formula_0
Step 2: Step size selection and photon packet movement.
The step size, "s", is the distance the photon packet travels between interaction sites. There are a variety of methods for step size selection. Below is a basic form of photon step size selection (derived using the inverse distribution method and the Beer–Lambert law) from which we use for our homogeneous model:
formula_1
where formula_2 is a random number and formula_3 is the total interaction coefficient (i.e., the sum of the absorption and scattering coefficients).
Once a step size is selected, the photon packet is propagated by a distance "s" in a direction defined by the direction cosines. This is easily accomplished by simply updating the coordinates as follows:
formula_4
Step 3: Absorption and scattering.
A portion of the photon weight is absorbed at each interaction site. This fraction of the weight is determined as follows:
formula_5
where formula_6 is the absorption coefficient.
The weight fraction can then be recorded in an array if an absorption distribution is of interest for the particular study. The weight of the photon packet must then be updated as follows:
formula_7
Following absorption, the photon packet is scattered. The weighted average of the cosine of the photon scattering angle is known as scattering anisotropy ("g"), which has a value between −1 and 1. If the optical anisotropy is 0, this generally indicates that the scattering is isotropic. If "g" approaches a value of 1 this indicates that the scattering is primarily in the forward direction. In order to determine the new direction of the photon packet (and hence the photon direction cosines), we need to know the scattering phase function. Often the Henyey-Greenstein phase function is used. Then the scattering angle, θ, is determined using the following formula.
formula_8
And, the polar angle "φ" is generally assumed to be uniformly distributed between 0 and formula_9. Based on this assumption, we can set:
formula_10
Based on these angles and the original direction cosines, we can find a new set of direction cosines. The new propagation direction can be represented in the global coordinate system as follows:
formula_11
For a special case
formula_12
use
formula_13
or
formula_14
use
formula_15
C-code:
/*********************** Indicatrix *********************
*New direction cosines after scattering by angle theta, fi.
* mux new=(sin(theta)*(mux*muz*cos(fi)-muy*sin(fi)))/sqrt(1-muz^2)+mux*cos(theta)
* muy new=(sin(theta)*(muy*muz*cos(fi)+mux*sin(fi)))/sqrt(1-muz^2)+muy*cos(theta)
* muz new= - sqrt(1-muz^2)*sin(theta)*cos(fi)+muz*cos(theta)
*Input:
* muxs,muys,muzs - direction cosine before collision
* mutheta, fi - cosine of polar angle and the azimuthal angle
*Output:
* muxd,muyd,muzd - direction cosine after collision
*/
void Indicatrix (double muxs, double muys, double muzs, double mutheta, double fi, double *muxd, double *muyd, double *muzd)
double costheta = mutheta;
double sintheta = sqrt(1.0-costheta*costheta); // sin(theta)
double sinfi = sin(fi);
double cosfi = cos(fi);
if (muzs == 1.0) {
*muxd = sintheta*cosfi;
*muyd = sintheta*sinfi;
*muzd = costheta;
} elseif (muzs == -1.0) {
*muxd = sintheta*cosfi;
*muyd = -sintheta*sinfi;
*muzd = -costheta;
} else {
double denom = sqrt(1.0-muzs*muzs);
double muzcosfi = muzs*cosfi;
*muxd = sintheta*(muxs*muzcosfi-muys*sinfi)/denom + muxs*costheta;
*muyd = sintheta*(muys*muzcosfi+muxs*sinfi)/denom + muys*costheta;
*muzd = -denom*sintheta*cosfi + muzs*costheta;
Step 4: Photon termination.
If a photon packet has experienced many interactions, for most applications the weight left in the packet is of little consequence. As a result, it is necessary to determine a means for terminating photon packets of sufficiently small weight. A simple method would use a threshold, and if the weight of the photon packet is below the threshold, the packet is considered dead. The aforementioned method is limited as it does not conserve energy. To keep total energy constant, a Russian roulette technique is often employed for photons below a certain weight threshold. This technique uses a roulette constant "m" to determine whether or not the photon will survive. The photon packet has one chance in "m" to survive, in which case it will be given a new weight of "mW" where "W" is the initial weight (this new weight, on average, conserves energy). All other times, the photon weight is set to 0 and the photon is terminated. This is expressed mathematically below:
formula_16
Graphics Processing Units (GPU) and fast Monte Carlo simulations of photon transport.
Monte Carlo simulation of photon migration in turbid media is a highly parallelizable problem, where a large number of photons are propagated independently, but according to identical rules and different random number sequences. The parallel nature of this special type of Monte Carlo simulation renders it highly suitable for execution on a graphics processing unit (GPU). The release of programmable GPUs started such a development, and since 2008 there have been a few reports on the use of GPU for high-speed Monte Carlo simulation of photon migration.
This basic approach can itself be parallelized by using multiple GPUs linked together. One example is the "GPU Cluster MCML," which can be downloaded from the authors' website (Monte Carlo Simulation of Light Transport in Multi-layered Turbid Media Based on GPU Clusters):
http://bmp.hust.edu.cn/GPU_Cluster/GPU_Cluster_MCML.HTM
Inline references.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\\begin{align}\n x & = 0 \\\\\n\\text{Position: }y & = 0 \\\\\n z & = 0 \\\\ \\\\\n \\mu_x & = 0 \\\\\n\\text{Direction cosines: } \\mu_y & = 0 \\\\\n \\mu_z & = 1\n\\end{align}\n"
},
{
"math_id": 1,
"text": "s = -\\frac{\\ln\\xi}{\\mu_t}"
},
{
"math_id": 2,
"text": "\\xi"
},
{
"math_id": 3,
"text": "{\\mu_t}"
},
{
"math_id": 4,
"text": "\n\\begin{align}\nx & \\leftarrow x + \\mu_x s \\\\\ny & \\leftarrow y + \\mu_y s \\\\\nz & \\leftarrow z + \\mu_z s\n\\end{align}\n"
},
{
"math_id": 5,
"text": "\\Delta W = \\frac{\\mu_a}{\\mu_t} W"
},
{
"math_id": 6,
"text": "{\\mu_a}"
},
{
"math_id": 7,
"text": "W \\leftarrow W - \\Delta W \\, "
},
{
"math_id": 8,
"text": "\\cos\\theta = \n\\begin{cases}\n\\frac{1}{2g} \\left[ 1 + g^2 - \\left(\\frac{1-g^2}{1-g+2g\\xi}\\right)^2\\right]&\\text{ if }g\\ne 0 \\\\\n1-2\\xi&\\text{ if }g= 0\n\\end{cases}\n"
},
{
"math_id": 9,
"text": " 2\\pi "
},
{
"math_id": 10,
"text": "\n\\varphi = 2\\pi\\xi\\frac{}{}\n"
},
{
"math_id": 11,
"text": "\n\\begin{align}\n\\mu'_x & = \\frac{\\sin\\theta(\\mu_x \\mu_z \\cos\\varphi - \\mu_y \\sin\\varphi)}{\\sqrt{1-\\mu_z^2}}+ \\mu_x \\cos\\theta \\\\\n\\mu'_y & = \\frac{\\sin\\theta(\\mu_y \\mu_z \\cos\\varphi + \\mu_x \\sin\\varphi)}{\\sqrt{1-\\mu_z^2}}+ \\mu_y \\cos\\theta \\\\\n\\mu'_z & = -\\sqrt{1-\\mu_z^2}\\sin\\theta\\cos\\varphi + \\mu_z\\cos\\theta \\\\\n\\end{align}\n"
},
{
"math_id": 12,
"text": "\n\\begin{align}\n\\mu_z=1\n\\end{align}\n"
},
{
"math_id": 13,
"text": "\n\\begin{align}\n\\mu'_x & = \\sin\\theta\\cos\\varphi \\\\\n\\mu'_y & = \\sin\\theta\\sin\\varphi \\\\\n\\mu'_z & = \\cos\\theta \\\\\n\\end{align}\n"
},
{
"math_id": 14,
"text": "\n\\begin{align}\n\\mu_z=-1\n\\end{align}\n"
},
{
"math_id": 15,
"text": "\n\\begin{align}\n\\mu'_x & = \\sin\\theta\\cos\\varphi \\\\\n\\mu'_y & = -\\sin\\theta\\sin\\varphi \\\\\n\\mu'_z & = -\\cos\\theta \\\\\n\\end{align}\n"
},
{
"math_id": 16,
"text": " \nW = \\begin{cases}\nmW&\\xi \\leq 1/m \\\\\n0&\\xi > 1/m\n\\end{cases}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14308303
|
143133
|
Center of pressure (fluid mechanics)
|
Point at which the resultant force of a pressure field acts on a body
In fluid mechanics, the center of pressure is the point on a body where a single force acting at that point can represent the total effect of the pressure field acting on the body.
The total force vector acting at the center of pressure is the surface integral of the pressure vector field across the surface of the body. The resultant force and center of pressure location produce an equivalent force and moment on the body as the original pressure field.
Pressure fields occur in both static and dynamic fluid mechanics. Specification of the center of pressure, the reference point from which the center of pressure is referenced, and the associated force vector allows the moment generated about any point to be computed by a translation from the reference point to the desired new point. It is common for the center of pressure to be located on the body, but in fluid flows it is possible for the pressure field to exert a moment on the body of such magnitude that the center of pressure is located outside the body.
Hydrostatic example (dam).
Since the forces of water on a dam are hydrostatic forces, they vary linearly with depth. The total force on the dam is then the integral of the pressure multiplied by the width of the dam as a function of the depth. The center of pressure is located at the centroid of the triangular shaped pressure field formula_0 from the top of the water line. The hydrostatic force and tipping moment on the dam about some point can be computed from the total force and center of pressure location relative to the point of interest.
Historical usage for sailboats.
Center of pressure is used in sailboat design to represent the position on a sail where the aerodynamic force is concentrated.
The relationship of the aerodynamic center of pressure on the sails to the hydrodynamic center of pressure (referred to as the center of lateral resistance) on the hull determines the behavior of the boat in the wind. This behavior is known as the "helm" and is either a weather helm or lee helm. A slight amount of weather helm is thought by some sailors to be a desirable situation, both from the standpoint of the "feel" of the helm, and the tendency of the boat to head slightly to windward in stronger gusts, to some extent self-feathering the sails. Other sailors disagree and prefer a neutral helm.
The fundamental cause of "helm", be it weather or lee, is the relationship of the center of pressure of the sail plan to the center of lateral resistance of the hull. If the center of pressure is astern of the center of lateral resistance, a weather helm, the tendency of the vessel is to want to turn into the wind.
If the situation is reversed, with the center of pressure forward of the center of lateral resistance of the hull, a "lee" helm will result, which is generally considered undesirable, if not dangerous. Too much of either helm is not good, since it forces the helmsman to hold the rudder deflected to counter it, thus inducing extra drag beyond what a vessel with neutral or minimal helm would experience.
Aircraft aerodynamics.
A stable configuration is desirable not only in sailing, but in aircraft design as well. Aircraft design therefore borrowed the term center of pressure. And like a sail, a rigid non-symmetrical airfoil not only produces lift, but a moment.
The center of pressure of an aircraft is the point where all of the aerodynamic pressure field may be represented by a single force vector with no moment. A similar idea is the aerodynamic center which is the point on an airfoil where the pitching moment produced by the aerodynamic forces is constant with angle of attack.
The aerodynamic center plays an important role in analysis of the longitudinal static stability of all flying machines. It is desirable that when the pitch angle and angle of attack of an aircraft are disturbed (by, for example wind shear/vertical gust) that the aircraft returns to its original trimmed pitch angle and angle of attack without a pilot or autopilot changing the control surface deflection. For an aircraft to return towards its trimmed attitude, without input from a pilot or autopilot, it must have positive longitudinal static stability.
Missile aerodynamics.
Missiles typically do not have a preferred plane or direction of maneuver and thus have symmetric airfoils. Since the center of pressure for symmetric airfoils is relatively constant for small angle of attack, missile engineers typically speak of the complete center of pressure of the entire vehicle for stability and control analysis. In missile analysis, the center of pressure is typically defined as the center of the additional pressure field due to a change in the angle of attack off of the trim angle of attack.
For unguided rockets the trim position is typically zero angle of attack and the center of pressure is defined to be the center of pressure of the resultant flow field on the entire vehicle resulting from a very small angle of attack (that is, the center of pressure is the limit as angle of attack goes to zero). For positive stability in missiles, the total vehicle center of pressure defined as given above must be further from the nose of the vehicle than the center of gravity. In missiles at lower angles of attack, the contributions to the center of pressure are dominated by the nose, wings, and fins. The normalized normal force coefficient derivative with respect to the angle of attack of each component multiplied by the location of the center of pressure can be used to compute a centroid representing the total center of pressure. The center of pressure of the added flow field is behind the center of gravity and the additional force "points" in the direction of the added angle of attack; this produces a moment that pushes the vehicle back to the trim position.
In guided missiles where the fins can be moved to trim the vehicles in different angles of attack, the center of pressure is the center of pressure of the flow field at that angle of attack for the undeflected fin position. This is the center of pressure of any small change in the angle of attack (as defined above). Once again for positive static stability, this definition of center of pressure requires that the center of pressure be further from the nose than the center of gravity. This ensures that any increased forces resulting from increased angle of attack results in increased restoring moment to drive the missile back to the trimmed position. In missile analysis, positive static margin implies that the complete vehicle makes a restoring moment for any angle of attack from the trim position.
Movement of center of pressure for aerodynamic fields.
The center of pressure on a symmetric airfoil typically lies close to 25% of the chord length behind the leading edge of the airfoil. (This is called the "quarter-chord point".) For a symmetric airfoil, as angle of attack and lift coefficient change, the center of pressure does not move. It remains around the quarter-chord point for angles of attack below the stalling angle of attack. The role of center of pressure in the control characterization of aircraft takes a different form than in missiles.
On a cambered airfoil the center of pressure does not occupy a fixed location. For a conventionally cambered airfoil, the center of pressure lies a little behind the quarter-chord point at maximum lift coefficient (large angle of attack), but as lift coefficient reduces (angle of attack reduces) the center of pressure moves toward the rear. When the lift coefficient is zero an airfoil is generating no lift but a conventionally cambered airfoil generates a nose-down pitching moment, so the location of the center of pressure is an infinite distance behind the airfoil.
For a reflex-cambered airfoil, the center of pressure lies a little ahead of the quarter-chord point at maximum lift coefficient (large angle of attack), but as lift coefficient reduces (angle of attack reduces) the center of pressure moves forward. When the lift coefficient is zero an airfoil is generating no lift but a reflex-cambered airfoil generates a nose-up pitching moment, so the location of the center of pressure is an infinite distance ahead of the airfoil. This direction of movement of the center of pressure on a reflex-cambered airfoil has a stabilising effect.
The way the center of pressure moves as lift coefficient changes makes it difficult to use the center of pressure in the mathematical analysis of longitudinal static stability of an aircraft. For this reason, it is much simpler to use the aerodynamic center when carrying out a mathematical analysis. The aerodynamic center occupies a fixed location on an airfoil, typically close to the quarter-chord point.
The aerodynamic center is the conceptual starting point for longitudinal stability. The horizontal stabilizer contributes extra stability and this allows the center of gravity to be a small distance aft of the aerodynamic center without the aircraft reaching neutral stability. The position of the center of gravity at which the aircraft has neutral stability is called the neutral point.
See also.
<templatestyles src="Div col/styles.css"/>
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\tfrac{2}{3}"
}
] |
https://en.wikipedia.org/wiki?curid=143133
|
1431342
|
Potential theory
|
Harmonic functions as solutions to Laplace's equation
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation.
There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.
Modern potential theory is also intimately connected with probability and the theory of Markov chains. In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even in the finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others.
Symmetry.
A useful starting point and organizing principle in the study of harmonic functions is a consideration of the symmetries of the Laplace equation. Although it is not a symmetry in the usual sense of the term, we can start with the observation that the Laplace equation is linear. This means that the fundamental object of study in potential theory is a linear space of functions. This observation will prove especially important when we consider function space approaches to the subject in a later section.
As for symmetry in the usual sense of the term, we may start with the theorem that the symmetries of the formula_0-dimensional Laplace equation are exactly the conformal symmetries of the formula_0-dimensional Euclidean space. This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups (such as the group of rotations or translations). Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series. By taking linear superpositions of these solutions, one can produce large classes of harmonic functions which can be shown to be dense in the space of all harmonic functions under suitable topologies.
Second, one can use conformal symmetry to understand such classical tricks and techniques for generating harmonic functions as the Kelvin transform and the method of images.
Third, one can use conformal transforms to map harmonic functions in one domain to harmonic functions in another domain. The most common instance of such a construction is to relate harmonic functions on a disk to harmonic functions on a half-plane.
Fourth, one can use conformal symmetry to extend harmonic functions to harmonic functions on conformally flat Riemannian manifolds. Perhaps the simplest such extension is to consider a harmonic function defined on the whole of Rn (with the possible exception of a discrete set of singular points) as a harmonic function on the formula_0-dimensional sphere. More complicated situations can also happen. For instance, one can obtain a higher-dimensional analog of Riemann surface theory by expressing a multi-valued harmonic function as a single-valued function on a branched cover of Rn or one can regard harmonic functions which are invariant under a discrete subgroup of the conformal group as functions on a multiply connected manifold or orbifold.
Two dimensions.
From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwarz's theorem, Morera's theorem, the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results.
Local behavior.
An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of level sets of harmonic functions. There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities.
Inequalities.
A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the maximum principle. Another important result is Liouville's theorem, which states the only bounded harmonic functions defined on the whole of Rn are, in fact, constant functions. In addition to these basic inequalities, one has Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant.
One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. These convergence theorems are used to prove the existence of harmonic functions with particular properties.
Spaces of harmonic functions.
Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a vector space. By defining suitable norms and/or inner products, one can exhibit sets of harmonic functions which form Hilbert or Banach spaces. In this fashion, one obtains such spaces as the Hardy space, Bloch space, Bergman space and Sobolev space.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "n"
}
] |
https://en.wikipedia.org/wiki?curid=1431342
|
143135
|
Parity (mathematics)
|
Property of being an even or odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 7, and 21 are odd numbers.
The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings.
Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even.
Definition.
An even number is an integer of the form
formula_0
where "k" is an integer; an odd number is an integer of the form
formula_1
An equivalent definition is that an even number is divisible by 2:
formula_2
and an odd number is not:
formula_3
The sets of even and odd numbers can be defined as following:
formula_4
formula_5
The set of "even" numbers is a prime ideal of formula_6 and the quotient ring formula_7 is the field with two elements. Parity can then be defined as the unique ring homomorphism from formula_6 to formula_7 where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below.
Properties.
The following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which is not true for normal integer arithmetic.
Multiplication.
By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the field with two elements.
Division.
The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even "nor" odd, since the concepts of even and odd apply only to integers. But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor.
History.
The ancient Greeks considered 1, the monad, to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 "The Education of Man" instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches the philosophical afterthought,
<templatestyles src="Template:Blockquote/styles.css" />It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form, the right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.
Higher mathematics.
Higher dimensions and more general classes of numbers.
Integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance, the face-centered cubic lattice and its higher-dimensional that is generalizations, the "Dn" lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to moving between squares of the same parity, whereas knights alternate parity between moves. This form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other.
The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus a finite even number, and odd otherwise.
Let "R" be a commutative ring and let "I" be an ideal of "R" whose index is 2. Elements of the coset formula_8 may be called even, while elements of the coset formula_9 may be called odd.
As an example, let "R" = Z(2) be the localization of Z at the prime ideal (2). Then an element of "R" is even or odd if and only if its numerator is so in Z.
Number theory.
The even numbers form an ideal in the ring of integers, but the odd numbers do not—this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.
All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.
Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 1018, but still no general proof has been found.
Group theory.
The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In Rubik's Cube, Megaminx, and other twisting puzzles, the moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the configuration space of these puzzles.
The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.
Analysis.
The parity of a function describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It is possible for a function to be neither odd nor even, and for the case "f"("x") = 0, to be both odd and even. The Taylor series of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number.
Combinatorial game theory.
In combinatorial game theory, an "evil number" is a number that has an even number of 1's in its binary representation, and an "odious number" is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game Kayles. The parity function maps a number to the number of 1's in its binary representation, modulo 2, so its value is zero for evil numbers and one for odious numbers. The Thue–Morse sequence, an infinite sequence of 0's and 1's, has a 0 in position "i" when "i" is evil, and a 1 in that position when "i" is odious.
Additional applications.
In information theory, a parity bit appended to a binary number provides the simplest form of error detecting code. If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing the number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected. Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value.
In wind instruments with a cylindrical bore and in effect closed at one end, such as the clarinet at the mouthpiece, the harmonics produced are odd multiples of the fundamental frequency. (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason, the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See harmonic series (music).
In some countries, house numberings are chosen so that the houses on one side of a street have even numbers and the houses on the other side have odd numbers. Similarly, among United States numbered highways, even numbers primarily indicate east–west highways while odd numbers primarily indicate north–south highways. Among airline flight numbers, even numbers typically identify eastbound or northbound flights, and odd numbers typically identify westbound or southbound flights.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "x = 2k"
},
{
"math_id": 1,
"text": "x = 2k +1."
},
{
"math_id": 2,
"text": "2 \\ | \\ x"
},
{
"math_id": 3,
"text": "2\\not| \\ x"
},
{
"math_id": 4,
"text": "\\{ 2k: k \\in \\mathbb{Z} \\}"
},
{
"math_id": 5,
"text": "\\{ 2k+1: k \\in \\mathbb{Z} \\}"
},
{
"math_id": 6,
"text": "\\mathbb{Z}"
},
{
"math_id": 7,
"text": "\\mathbb{Z}/2\\mathbb{Z}"
},
{
"math_id": 8,
"text": "0+I"
},
{
"math_id": 9,
"text": "1+I"
}
] |
https://en.wikipedia.org/wiki?curid=143135
|
14314098
|
Delta formation
|
Delta formation is a flight pattern where multiple flying objects will come together in a V in order to fly more efficiently. Each trailing object is positioned slightly higher than the one in front, and uses the air moved by the forward object to reduce wind resistance.
The delta formation is frequently used by birds to migrate over long distances, in airplanes, and in UAVs.
The most famous use of the delta formation is by the United States Air Force's demonstration squadron the Thunderbirds. The Thunderbirds will use six aircraft that come together, typically at the end of an air show and fly in tight formation.
Advantages and basic mechanics.
Flying in a delta formation can allow for a longer flight time with the same amount of energy. This saves time and resources and reduces the potential need to stop in dangerous territories.
When in flight, upwash is generated behind the wing by wingtip vortices, this is air that was diverted upwards to generate lift. In a delta formation, the trailing object follows closely behind and slightly above the lead. This allows the upwash from the lead to generate lift for the trailing object in flight. When using the excess lift from the lead the trailing object does not need to generate as much lift, leading to the increased efficiency. This can be used in a large delta formation allowing for increased efficiency at scale. Since the excess lift from the lead was already being generated, this does not require more energy from the lead, and uses energy that would otherwise be unused. This allows for some birds to use up to 30% less energy when they fly in a delta formation.
United States Air Force's Thunderbirds.
The delta formation was made famous by the United States Air Force's Air Demonstration Squadron, the Thunderbirds. When the Thunderbirds fly in a delta formation all six of the squadron's airplanes fly in a tight delta. In close formation the aircraft can have as little as 0.5 meters (1.5 feet) of separation between them.
In shows the team forms in the delta multiple times and planes on the edge of the formation break off for solo stunts while the central four stay together. Delta formation tricks are usually saved until the end of the show.
Super Delta.
In 2021 the Navy's Blue Angels and the Air Force's Thunderbirds teamed up with a "super delta" formation. In the super delta the Blue Angels form a typical delta formation in the center and are flanked by three Thunderbirds on each side. The teams were able to train together with the additional practice time allowed to the COVID-19 pandemic canceling air shows.
Practical use.
As flight controls become more and more automated, large scale formation flight is becoming more realistic. In order to save fuel many UAV manufacturers are experimenting with using the delta formations, and formations like it, in commercial applications.
UAVs.
In unmanned aerial vehicles (UAVs) many manufacturers are developing UAVs that can fly in a delta formation over long distances to increase range. In these formations, called swarms, the UAVs would launch from a distant base station, then fly together to a destination and separate, or individual units could separate from the swarm sooner. They then could fly back together in formation, or fly back individually.
UAV application.
In UAVs the efficiency improvement is combined with a decrease in the amount of humans needed to monitor large scare operations. Some of the use cases include:
UAV communication.
In a swarm, the UAVs have a predefined trajectory path, the lead UAV is designated as the source of information. There is a reference point behind the lead that all the UAVs orient themselves around. Using the location of the UAVs in front they calculate the most efficient position for themselves.
Calculating optimum position in UAV formation.
formula_0
This system of equations can describe the optimum positioning of UAVs in a swarm. "In this configuration, the UAV can be considered as a point-mass system as where p = [px, py, pz]T stands for the position of UAV in the inertial coordinate system; "V" represents the air speed of UAV; y, x represent the flight path angle and the heading angle; "L", "T", "D" are the lift force, thrust force, and drag force, respectively; "g" denotes the gravity acceleration, α and σ are the attack angle and bank angle, respectively; Δv , Δy, Δx represent the ignored model items associated with the wind gradient and external disturbances. Denote vx = V cos(y) cos(x), vy = V cos(y) sin(x), vz = V sin y as the velocity in inertial coordinate system. For simplicity, the acceleration of air-speed is defined as , the climb rate and heading rate are defined as ωy and ωx, respectively. Then after conversion, the UAV swarm dynamics can be described approximately as the following second-order MAS"
Birds.
Birds use the delta formation to fly farther when they are flying as a group. In order to maximize the effect birds will switch the lead when they get tired to allow the flock to fly farther. Birds have evolved to maximize the formation and can sense the birds around them and can position themselves in the best possible spot. They also use the formation as a visual guide to help them stay together.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\n\\begin{cases} \\dot{P}x = V cos (y) * cos(x)\n\\\\ \\dot{P}y = V cos (y) * sin(x)\n\\\\ \\dot{P}z = V sin (y)\n\\\\ \\dot{V} = \\frac{Tcos(a-D)}{m - g * sin(y)} + \\Delta V\n\\\\ \\dot{y} = \\frac{\\frac{T sin(a) cos(\\sigma + L cos(\\sigma))}{mV - g cos(y)}}{V + \\Delta y} \n\\\\ \\dot{x} = \\left ( \\frac{T * sin(a) sin(\\sigma) + L sin(\\sigma) }{mV + \\Delta x} \\right ) \\end{cases}\n"
}
] |
https://en.wikipedia.org/wiki?curid=14314098
|
14315623
|
A-equivalence
|
Equivalence relation between map germs
In mathematics, formula_0-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
Let formula_1 and formula_2 be two manifolds, and let formula_3 be two smooth map germs. We say that formula_4 and formula_5 are formula_0-equivalent if there exist diffeomorphism germs formula_6 and formula_7 such that formula_8
In other words, two map germs are formula_0-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. formula_1) and the target (i.e. formula_2).
Let formula_9 denote the space of smooth map germs formula_10 Let formula_11 be the group of diffeomorphism germs formula_12 and
formula_13 be the group of diffeomorphism germs formula_14 The group formula_15 acts on formula_9 in the natural way: formula_16 Under this action we see that the map germs formula_3 are formula_0-equivalent if, and only if, formula_5 lies in the orbit of formula_4, i.e. formula_17 (or vice versa).
A map germ is called stable if its orbit under the action of formula_15 is open relative to the Whitney topology. Since formula_9 is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking formula_18-jets for every formula_18 and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of
these base sets.
Consider the orbit of some map germ formula_19 The map germ formula_4 is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs formula_20 for formula_21 are the infinite sequence formula_22 (formula_23), the infinite sequence formula_24 (formula_23), formula_25 formula_26 and formula_27
|
[
{
"math_id": 0,
"text": "\\mathcal{A}"
},
{
"math_id": 1,
"text": "M"
},
{
"math_id": 2,
"text": "N"
},
{
"math_id": 3,
"text": "f, g : (M,x) \\to (N,y)"
},
{
"math_id": 4,
"text": "f"
},
{
"math_id": 5,
"text": "g"
},
{
"math_id": 6,
"text": "\\phi : (M,x) \\to (M,x)"
},
{
"math_id": 7,
"text": "\\psi : (N,y) \\to (N,y)"
},
{
"math_id": 8,
"text": "\\psi \\circ f = g \\circ \\phi."
},
{
"math_id": 9,
"text": "\\Omega(M_x,N_y)"
},
{
"math_id": 10,
"text": "(M,x) \\to (N,y)."
},
{
"math_id": 11,
"text": "\\mbox{diff}(M_x)"
},
{
"math_id": 12,
"text": "(M,x) \\to (M,x)"
},
{
"math_id": 13,
"text": "\\mbox{diff}(N_y)"
},
{
"math_id": 14,
"text": "(N,y) \\to (N,y)."
},
{
"math_id": 15,
"text": " G := \\mbox{diff}(M_x) \\times \\mbox{diff}(N_y)"
},
{
"math_id": 16,
"text": " (\\phi,\\psi) \\cdot f = \\psi^{-1} \\circ f \\circ \\phi."
},
{
"math_id": 17,
"text": " g \\in \\mbox{orb}_G(f)"
},
{
"math_id": 18,
"text": "k"
},
{
"math_id": 19,
"text": "orb_G(f)."
},
{
"math_id": 20,
"text": "(\\mathbb{R}^n,0) \\to (\\mathbb{R},0)"
},
{
"math_id": 21,
"text": "1 \\le n \\le 3"
},
{
"math_id": 22,
"text": "A_k"
},
{
"math_id": 23,
"text": "k \\in \\mathbb{N}"
},
{
"math_id": 24,
"text": "D_{4+k}"
},
{
"math_id": 25,
"text": "E_6,"
},
{
"math_id": 26,
"text": "E_7,"
},
{
"math_id": 27,
"text": "E_8."
}
] |
https://en.wikipedia.org/wiki?curid=14315623
|
14317839
|
Minor allele frequency
|
Minor allele frequency (MAF) is the frequency at which the "second most common" allele occurs in a given population. They play a surprising role in heritability since MAF variants which occur only once, known as "singletons", drive an enormous amount of selection.
Single nucleotide polymorphisms (SNPs) with a minor allele frequency of 0.05 (5%) or greater were targeted by the HapMap project.
MAF is widely used in population genetics studies because it provides information to differentiate between common and rare variants in the population.
As an example, a 2015 study sequenced the whole genomes of Sardinian individuals. The authors classified the variants found in the study in three classes according to their MAF. It was observed that rare variants (MAF < 0.05) appeared more frequently in coding regions than common variants (MAF > 0.05) in this population.
Interpreting MAF data.
1. Introduce the reference of a SNP of interest, as an example: rs429358, in a database (dbSNP or other).
2. Find MAF/MinorAlleleCount link. MAF/MinorAlleleCount: C=0.1506/754 (1000 Genomes, where number of genomes sampled = N = 2504); where C is the minor allele for that particular locus; 0.1506 is the frequency of the C allele (MAF), i.e. 15% within the 1000 Genomes database; and 754 is the number of times this SNP has been observed in the population of the study. To find the number, note that formula_0, where formula_1 is to account for diploidy.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "0.1506 = \\frac{754}{2 \\times 2504}"
},
{
"math_id": 1,
"text": "2\\times"
}
] |
https://en.wikipedia.org/wiki?curid=14317839
|
14317956
|
Apparent oxygen utilisation
|
Method for measuring oxygen level change in water
In freshwater or marine systems apparent oxygen utilization (AOU) is the difference between oxygen gas solubility (i.e. the concentration at saturation) and the measured oxygen concentration in water with the same physical and chemical properties.
formula_0
General influences.
Differences in O2 solubility and measured concentration (AOU) typically occur when biological activity, ocean circulation, or ocean mixing act to change the ambient concentration of oxygen. For example, primary production liberates oxygen and increases its concentration, while respiration consumes it and decreases its concentration.
Consequently, the AOU of a water sample represents the sum of the biological activity that the sample has experienced since it was last in equilibrium with the atmosphere. In shallow water systems (e.g. lakes), the full water column is generally in close contact with the atmosphere, and oxygen concentrations are typically close to saturation, and AOU values are near zero. In deep water systems (e.g. oceans), water can be out of contact with the atmosphere for extremely long periods of time (years, decades, centuries) and large positive AOU values are typical. On occasion, where near-surface primary production has raised oxygen concentrations above saturation, negative AOU values are possible (i.e. oxygen has not been utilized to below saturation concentrations).
O2 trends and AOU in the ocean.
O2 concentrations in the ocean have decreased since the 1980s. Part of this decrease is due to increased ocean heat content (OHC) from global warming decreasing O2 solubility. As solubility in surface oceans decreases, O2 out gasses to the atmosphere. Increased AOU is likely also contributing to declining ocean O2 concentrations. Changes in AOU in the ocean could be caused by multiple forcings, such as changes in subduction rates, changes in water mass boundaries, initial O2 from water mass formation, biochemical consumption of O2, or changes in eddy mixing. Based on observations, global AOU increase seems to be linked to increasing OHC.
Spatial trends in AOU and O2.
Changes in AOU are not consistent across spatial domains. Notably, the Northern Hemisphere AOU spikes in the mid-1990s, while Southern Hemisphere AOU significantly decreases in the mid-2000s. This could be due to less dense data in the Southern Hemisphere. Since the 80s, regions of major O2 decrease (AOU increase) include the subpolar North Pacific, equatorial Atlantic, and eastern equatorial Pacific. Regions of O2 increase (AOU decrease) include the western subtropical North Pacific and eastern subpolar North Atlantic.
|
[
{
"math_id": 0,
"text": "AOU= [O_2 \\ solubility]-[O_2 \\ observed] "
}
] |
https://en.wikipedia.org/wiki?curid=14317956
|
14319966
|
Lexical density
|
Complexity of communication
Lexical density is a concept in computational linguistics that measures the structure and complexity of human communication in a language. Lexical density estimates the linguistic complexity in a written or spoken composition from the functional words (grammatical units) and content words (lexical units, lexemes). One method to calculate the lexical density is to compute the ratio of lexical items to the total number of words. Another method is to compute the ratio of lexical items to the number of higher structural items in a composition, such as the total number of clauses in the sentences.
The lexical density for an individual evolves with age, education, communication style, circumstances, unusual injuries or medical condition, and his or her creativity. The inherent structure of a human language and one's first language may impact the lexical density of the individual's writing and speaking style. Further, human communication in the written form is generally more lexically dense than in the spoken form after the early childhood stage. The lexical density impacts the readability of a composition and the ease with which the listener or reader can comprehend a communication. The lexical density may also impact the memorability and retention of a sentence and the message.
Discussion.
The lexical density is the proportion of content words (lexical items) in a given discourse. It can be measured either as the ratio of lexical items to total number of words, or as the ratio of lexical items to the number of higher structural items in the sentences (for example, clauses). A lexical item is typically the real content and it includes nouns, verbs, adjectives and adverbs. A grammatical item typically is the functional glue and thread that weaves the content and includes pronouns, conjunctions, prepositions, determiners, and certain classes of finite verbs and adverbs.
Lexical density is one of the methods used in discourse analysis as a descriptive parameter which varies with register and genre. There are many proposed methods for computing the lexical density of any composition or corpus. Lexical density may be determined as:
formula_0
Where:
formula_1 = the analysed text's lexical density
formula_2 = the number of lexical or grammatical tokens (nouns, adjectives, verbs, adverbs) in the analysed text
formula_3 = the number of all tokens (total number of words) in the analysed text
Ure lexical density.
Ure proposed the following formula in 1971 to compute the lexical density of a sentence:
Ld
* 100
Biber terms this ratio as "type-token ratio".
Halliday lexical density.
In 1985, Halliday revised the denominator of the Ure formula and proposed the following to compute the lexical density of a sentence:
Ld
* 100
In some formulations, the Halliday proposed lexical density is computed as a simple ratio, without the "100" multiplier.
Characteristics.
Lexical density measurements may vary for the same composition depending on how a "lexical item" is defined and which items are classified as lexical or as a grammatical item. Any adopted methodology when consistently applied across various compositions provides the lexical density of those compositions. Typically, the lexical density of a written composition is higher than a spoken composition. According to Ure, written forms of human communication in the English language typically have lexical densities above 40%, while spoken forms tend to have lexical densities below 40%. In a survey of historical texts by Michael Stubbs, the typical lexical density of fictional literature ranged between 40% and 54%, while non-fiction ranged between 40% and 65%.
The relation and intimacy between the participants of a particular communication impact the lexical density, states Ure, as do the circumstances prior to the start of communication for the same speaker or writer. The higher lexical density of written forms of communication, she proposed, is primarily because written forms of human communication involve greater preparation, reflection and revisions. Human discussions and conversations involving or anticipating feedback tend to be sparser and have lower lexical density. In contrast, state Stubbs and Biber, instructions, law enforcement orders, news read from screen prompts within the allotted time, and literature that authors expect will be available to the reader for re-reading tend to maximize lexical density. In surveys of lexical density of spoken and written materials across different European countries and age groups, Johansson and Strömqvist report that the lexical density of population groups were similar and depended on the morphological structure of the native language and within a country, the age groups sampled. The lexical density was highest for adults, while the variations estimated as lexical diversity, states Johansson, were higher for teenagers for the same age group (13-year-olds, 17-year-olds).
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": " L_d = (N_{\\mathrm{lex}}/N) \\times 100 "
},
{
"math_id": 1,
"text": "L_d"
},
{
"math_id": 2,
"text": "N_{\\mathrm{lex}}"
},
{
"math_id": 3,
"text": "N"
}
] |
https://en.wikipedia.org/wiki?curid=14319966
|
14320253
|
Carleman matrix
|
In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains.
Definition.
The Carleman matrix of an infinitely differentiable function formula_0 is defined as:
formula_1
so as to satisfy the (Taylor series) equation:
formula_2
For instance, the computation of formula_0 by
formula_3
simply amounts to the dot-product of row 1 of formula_4 with a column vector formula_5.
The entries of formula_6 in the next row give the 2nd power of formula_0:
formula_7
and also, in order to have the zeroth power of formula_0 in formula_6, we adopt the row 0 containing zeros everywhere except the first position, such that
formula_8
Thus, the dot product of formula_6 with the column vector formula_9 yields the column vector formula_10, i.e.,
formula_11
Generalization.
A generalization of the Carleman matrix of a function can be defined around any point, such as:
formula_12
or formula_13 where formula_14. This allows the matrix power to be related as:
formula_15
Another way to generalize it even further is think about a general series in the following way:
Let formula_16 be a series approximation of formula_17, where formula_18 is a basis of the space containing formula_17
Assuming that formula_18 is also a basis for formula_0, We can define formula_19, therefore we have formula_20, now we can prove that formula_21, if we assume that formula_18 is also a basis for formula_22 and formula_23.
Let formula_22 be such that formula_24 where formula_25.
Now
General Series.
formula_26
Comparing the first and the last term, and from formula_18 being a base for formula_0, formula_22 and formula_23 it follows that formula_27
Examples.
Rederive (Taylor) Carleman Matrix.
If we set formula_28 we have the Carleman matrix. Because
<br>formula_29<br>
then we know that the n-th coefficient formula_30 must be the nth-coefficient of the taylor series of formula_31. Therefore formula_32 <br>Therefore formula_33
<br>Which is the Carleman matrix given above. "(It's important to note that this is not an orthornormal basis)"
Carleman Matrix For Orthonormal Basis.
If formula_34 is an orthonormal basis for a Hilbert Space with a defined inner product formula_35, we can set formula_36 and formula_37 will be formula_38. Then
formula_39.
Carleman Matrix for Fourier Series.
If formula_40 we have the analogous for Fourier Series. Let formula_41 and formula_42 represent the carleman coefficient and matrix in the fourier basis. Because the basis is orthogonal, we have.
formula_43.
<br>Then, therefore, formula_44 which is
formula_45
Properties.
Carleman matrices satisfy the fundamental relationship
which makes the Carleman matrix "M" a (direct) representation of formula_0. Here the term formula_47 denotes the composition of functions formula_48.
Other properties include:
Examples.
The Carleman matrix of a constant is:
formula_53
The Carleman matrix of the identity function is:
formula_54
The Carleman matrix of a constant addition is:
formula_55
The Carleman matrix of the successor function is equivalent to the Binomial coefficient:
formula_56
formula_57
The Carleman matrix of the logarithm is related to the (signed) Stirling numbers of the first kind scaled by factorials:
formula_58
formula_59
The Carleman matrix of the logarithm is related to the (unsigned) Stirling numbers of the first kind scaled by factorials:
formula_60
formula_61
The Carleman matrix of the exponential function is related to the Stirling numbers of the second kind scaled by factorials:
formula_62
formula_63
The Carleman matrix of exponential functions is:
formula_64
formula_65
The Carleman matrix of a constant multiple is:
formula_66
The Carleman matrix of a linear function is:
formula_67
The Carleman matrix of a function formula_68 is:
formula_69
The Carleman matrix of a function formula_70 is:
formula_71
Related matrices.
The Bell matrix or the Jabotinsky matrix of a function formula_0 is defined as
formula_72
so as to satisfy the equation
formula_73
These matrices were developed in 1947 by Eri Jabotinsky to represent convolutions of polynomials. It is the transpose of the Carleman matrix and satisfy
formula_74 which makes the Bell matrix "B" an "anti-representation" of formula_0.
Notes.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "f(x)"
},
{
"math_id": 1,
"text": "M[f]_{jk} = \\frac{1}{k!}\\left[\\frac{d^k}{dx^k} (f(x))^j \\right]_{x=0} ~,"
},
{
"math_id": 2,
"text": "(f(x))^j = \\sum_{k=0}^{\\infty} M[f]_{jk} x^k."
},
{
"math_id": 3,
"text": "f(x) = \\sum_{k=0}^{\\infty} M[f]_{1,k} x^k. ~"
},
{
"math_id": 4,
"text": " M[f] "
},
{
"math_id": 5,
"text": "\\left[1,x,x^2,x^3,...\\right]^\\tau"
},
{
"math_id": 6,
"text": "M[f]"
},
{
"math_id": 7,
"text": "f(x)^2 = \\sum_{k=0}^{\\infty} M[f]_{2,k} x^k ~,"
},
{
"math_id": 8,
"text": "f(x)^0 = 1 = \\sum_{k=0}^{\\infty} M[f]_{0,k} x^k = 1+ \\sum_{k=1}^{\\infty} 0\\cdot x^k ~."
},
{
"math_id": 9,
"text": "\\begin{bmatrix}1,x,x^2,...\\end{bmatrix}^T"
},
{
"math_id": 10,
"text": "\\left[1,f(x),f(x)^2,...\\right]^T"
},
{
"math_id": 11,
"text": " M[f] \\begin{bmatrix}1\\\\x\\\\x^2\\\\ x^3\\\\\\vdots\\end{bmatrix} = \\begin{bmatrix} 1\\\\f(x)\\\\(f(x))^2\\\\(f(x))^3\\\\\\vdots\\end{bmatrix}. "
},
{
"math_id": 12,
"text": "M[f]_{x_0} = M_x[x - x_0]M[f]M_x[x + x_0]"
},
{
"math_id": 13,
"text": "M[f]_{x_0} = M[g]"
},
{
"math_id": 14,
"text": "g(x) = f(x + x_0) - x_0"
},
{
"math_id": 15,
"text": "(M[f]_{x_0})^n = M_x[x - x_0]M[f]^nM_x[x + x_0]"
},
{
"math_id": 16,
"text": "h(x) = \\sum_n c_n (h) \\cdot \\psi_n(x)"
},
{
"math_id": 17,
"text": "h(x)"
},
{
"math_id": 18,
"text": "\\{\\psi_n(x)\\}_n"
},
{
"math_id": 19,
"text": "G[f]_{mn} = c_n(\\psi_m \\circ f)"
},
{
"math_id": 20,
"text": "\\psi_m \\circ f = \\sum_n c_n (\\psi_m \\circ f) \\cdot \\psi_n = \\sum_n G[f]_{mn} \\cdot \\psi_n"
},
{
"math_id": 21,
"text": "G[g \\circ f] = G[g] \\cdot G[f]"
},
{
"math_id": 22,
"text": "g(x)"
},
{
"math_id": 23,
"text": "g(f(x))"
},
{
"math_id": 24,
"text": "\\psi_l \\circ g = \\sum_m G[g]_{lm} \\cdot \\psi_m"
},
{
"math_id": 25,
"text": "G[g]_{lm} = c_m (\\psi_l \\circ g)"
},
{
"math_id": 26,
"text": " \\begin{aligned} \\sum_n G[g \\circ f]_{ln} \\psi_n \n= \\psi_l \\circ (g \\circ f)\n&= (\\psi_l \\circ g) \\circ f\\\\\n&= \\sum_m G[g]_{lm} (\\psi_m \\circ f)\\\\\n&= \\sum_m G[g]_{lm} \\sum_n G[f]_{mn} \\psi_n\\\\\n&= \\sum_{n,m} G[g]_{lm} G[f]_{mn} \\psi_n\\\\\n&= \\sum_{n} (\\sum_m G[g]_{lm} G[f]_{mn}) \\psi_n\\end{aligned}"
},
{
"math_id": 27,
"text": "G[g \\circ f] =\\sum_m G[g]_{lm} G[f]_{mn}= G[g] \\cdot G[f]"
},
{
"math_id": 28,
"text": "\\psi_n(x) = x^n"
},
{
"math_id": 29,
"text": "h(x) = \\sum_n c_n (h) \\cdot \\psi_n(x) = \\sum_n c_n (h) \\cdot x^n"
},
{
"math_id": 30,
"text": "c_n(h)"
},
{
"math_id": 31,
"text": "h"
},
{
"math_id": 32,
"text": "c_n(h)=\\frac{1}{n!} h^{(n)}(0)"
},
{
"math_id": 33,
"text": " G[f]_{mn}=c_n(\\psi_m \\circ f)=c_n(f(x)^m)=\\frac{1}{n!}\\left[\\frac{d^n}{dx^n} (f(x))^m \\right]_{x=0}"
},
{
"math_id": 34,
"text": "\\{e_n(x)\\}_n"
},
{
"math_id": 35,
"text": "\\langle f,g \\rangle"
},
{
"math_id": 36,
"text": "\\psi_n = e_n"
},
{
"math_id": 37,
"text": "c_n (h)"
},
{
"math_id": 38,
"text": "{\\displaystyle \\langle h, e_n \\rangle } "
},
{
"math_id": 39,
"text": "G[f]_{mn}=c_n(e_m \\circ f)=\\langle e_m \\circ f,e_n\\rangle "
},
{
"math_id": 40,
"text": "e_n(x) = e^{i n x}"
},
{
"math_id": 41,
"text": "\\hat{c}_n"
},
{
"math_id": 42,
"text": "\\hat{G} "
},
{
"math_id": 43,
"text": "\\hat{c}_n (h) = \\langle h,e_n \\rangle = \\cfrac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\displaystyle h(x) \\cdot e^{-inx}dx"
},
{
"math_id": 44,
"text": "\\hat{G}[f]_{mn}=\\hat{c_n}(e_m \\circ f)=\\langle e_m \\circ f,e_n\\rangle "
},
{
"math_id": 45,
"text": "\\hat{G}[f]_{mn}=\\cfrac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\displaystyle e^{i m f(x)} \\cdot e^{-inx}dx "
},
{
"math_id": 46,
"text": "M[f \\circ g] = M[f]M[g] ~,"
},
{
"math_id": 47,
"text": "f \\circ g "
},
{
"math_id": 48,
"text": "f(g(x))"
},
{
"math_id": 49,
"text": "\\,M[f^n] = M[f]^n"
},
{
"math_id": 50,
"text": "\\,f^n"
},
{
"math_id": 51,
"text": "\\,M[f^{-1}] = M[f]^{-1}"
},
{
"math_id": 52,
"text": "\\,f^{-1}"
},
{
"math_id": 53,
"text": "M[a] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&0&0& \\cdots \\\\\na^2&0&0& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 54,
"text": "M_x[x] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&1&0& \\cdots \\\\\n0&0&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 55,
"text": "M_x[a + x] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&1&0& \\cdots \\\\\na^2&2a&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 56,
"text": "M_x[1 + x] = \\left(\\begin{array}{ccccc}\n1&0&0&0& \\cdots \\\\\n1&1&0&0& \\cdots \\\\\n1&2&1&0& \\cdots \\\\\n1&3&3&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 57,
"text": "M_x[1 + x]_{jk} = \\binom{j}{k}"
},
{
"math_id": 58,
"text": "M_x[\\log(1 + x)] = \\left(\\begin{array}{cccccc}\n1&0&0&0&0& \\cdots \\\\\n0&1&-\\frac{1}{2}&\\frac{1}{3}&-\\frac{1}{4}& \\cdots \\\\\n0&0&1&-1&\\frac{11}{12}& \\cdots \\\\\n0&0&0&1&-\\frac{3}{2}& \\cdots \\\\\n0&0&0&0&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 59,
"text": "M_x[\\log(1 + x)]_{jk} = s(k, j) \\frac{j!}{k!}"
},
{
"math_id": 60,
"text": "M_x[-\\log(1 - x)] = \\left(\\begin{array}{cccccc}\n1&0&0&0&0& \\cdots \\\\\n0&1&\\frac{1}{2}&\\frac{1}{3}&\\frac{1}{4}& \\cdots \\\\\n0&0&1&1&\\frac{11}{12}& \\cdots \\\\\n0&0&0&1&\\frac{3}{2}& \\cdots \\\\\n0&0&0&0&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 61,
"text": "M_x[-\\log(1 - x)]_{jk} = |s(k, j)| \\frac{j!}{k!}"
},
{
"math_id": 62,
"text": "M_x[\\exp(x) - 1] = \\left(\\begin{array}{cccccc}\n1&0&0&0&0& \\cdots \\\\\n0&1&\\frac{1}{2}&\\frac{1}{6}&\\frac{1}{24}& \\cdots \\\\\n0&0&1&1&\\frac{7}{12}& \\cdots \\\\\n0&0&0&1&\\frac{3}{2}& \\cdots \\\\\n0&0&0&0&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 63,
"text": "M_x[\\exp(x) - 1]_{jk} = S(k, j) \\frac{j!}{k!}"
},
{
"math_id": 64,
"text": "M_x[\\exp(a x)] = \\left(\\begin{array}{ccccc}\n1&0&0&0& \\cdots \\\\\n1&a&\\frac{a^2}{2}&\\frac{a^3}{6}& \\cdots \\\\\n1&2a&2a^2&\\frac{4a^3}{3}& \\cdots \\\\\n1&3a&\\frac{9a^2}{2}&\\frac{9a^3}{2}& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 65,
"text": "M_x[\\exp(a x)]_{jk} = \\frac{(j a)^k}{k!}"
},
{
"math_id": 66,
"text": "M_x[cx] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&c&0& \\cdots \\\\\n0&0&c^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 67,
"text": "M_x[a + cx] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&c&0& \\cdots \\\\\na^2&2ac&c^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 68,
"text": "f(x) = \\sum_{k=1}^{\\infty}f_k x^k"
},
{
"math_id": 69,
"text": "M[f] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&f_1&f_2& \\cdots \\\\\n0&0&f_1^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 70,
"text": "f(x) = \\sum_{k=0}^{\\infty}f_k x^k"
},
{
"math_id": 71,
"text": "M[f] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\nf_0&f_1&f_2& \\cdots \\\\\nf_0^2&2f_0f_1&f_1^2+2f_0f_2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)"
},
{
"math_id": 72,
"text": "B[f]_{jk} = \\frac{1}{j!}\\left[\\frac{d^j}{dx^j} (f(x))^k \\right]_{x=0} ~,"
},
{
"math_id": 73,
"text": "(f(x))^k = \\sum_{j=0}^{\\infty} B[f]_{jk} x^j ~,"
},
{
"math_id": 74,
"text": "B[f \\circ g] = B[g]B[f] ~,"
}
] |
https://en.wikipedia.org/wiki?curid=14320253
|
1432127
|
Mode (statistics)
|
Value that appears most often in a set of data
In statistics, the mode is the value that appears most often in a set of data values. If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e., "x"=argmax"x""i" P(X
"x""i")). In other words, it is the value that is most likely to be sampled.
Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.
The mode is not necessarily unique in a given discrete distribution since the probability mass function may take the same maximum value at several points "x"1, "x"2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.
A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value. When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution, so any peak is a mode. Such a continuous distribution is called multimodal (as opposed to unimodal).
In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.
Mode of a sample.
The mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] its mode is not unique. A dataset, in such a case, is said to be bimodal, while a set with more than two modes may be described as multimodal.
For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of the
intervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.
The following MATLAB (or Octave) code example computes the mode of a sample:
X = sort(x); % x is a column vector dataset
indices = find(diff([X; realmax]) > 0); % indices where repeated values change
[modeL,i] = max (diff([0; indices])); % longest persistence length of repeated values
mode = X(indices(i));
The algorithm requires as a first step to sort the sample in ascending order. It then computes the discrete derivative of the sorted list and finds the indices where this derivative is positive. Next it computes the discrete derivative of this set of indices, locating the maximum of this derivative of indices, and finally evaluates the sorted sample at the point where that maximum occurs, which corresponds to the last member of the stretch of repeated values.
Comparison of mean, median and mode.
Use.
Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.
Unlike median, the concept of mode makes sense for any random variable assuming values from a vector space, including the real numbers (a one-dimensional vector space) and the integers (which can be considered embedded in the reals). For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median and the centerpoint.
Uniqueness and definedness.
For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain pathological distributions (for example, the Cantor distribution) have no defined mode at all. For a finite data sample, the mode is one (or more) of the values in the sample.
Properties.
Assuming definedness, and for simplicity uniqueness, the following are some of the most interesting properties.
Example for a skewed distribution.
An example of a skewed distribution is personal wealth: Few people are very rich, but among those some are extremely rich. However, many are rather poor.
A well-known class of distributions that can be arbitrarily skewed is given by the log-normal distribution. It is obtained by transforming a random variable X having a normal distribution into random variable "Y"
"e""X". Then the logarithm of random variable Y is normally distributed, hence the name.
Taking the mean μ of X to be 0, the median of Y will be 1, independent of the standard deviation σ of X. This is so because X has a symmetric distribution, so its median is also 0. The transformation from X to Y is monotonic, and so we find the median "e"0
1 for Y.
When X has standard deviation σ = 0.25, the distribution of Y is weakly skewed. Using formulas for the log-normal distribution, we find:
formula_0
Indeed, the median is about one third on the way from mean to mode.
When X has a larger standard deviation, σ
1, the distribution of Y is strongly skewed. Now
formula_1
Here, Pearson's rule of thumb fails.
Van Zwet condition.
Van Zwet derived an inequality which provides sufficient conditions for this inequality to hold. The inequality
Mode ≤ Median ≤ Mean
holds if
F( Median - x ) + F( Median + x ) ≥ 1
for all x where F() is the cumulative distribution function of the distribution.
Unimodal distributions.
It can be shown for a unimodal distribution that the median formula_2 and the mean formula_3 lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other. In symbols,
formula_4
where formula_5 is the absolute value.
A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of each other:
formula_6
History.
The term mode originates with Karl Pearson in 1895.
Pearson uses the term "mode" interchangeably with "maximum-ordinate". In a footnote he says, "I have found it convenient to use the term "mode" for the abscissa corresponding to the ordinate of maximum frequency."
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\begin{array}{rlll}\n\\text{mean} & = e^{\\mu + \\sigma^2 / 2} & = e^{0 + 0.25^2 / 2} & \\approx 1.032 \\\\\n\\text{mode} & = e^{\\mu - \\sigma^2} & = e^{0 - 0.25^2} & \\approx 0.939 \\\\\n\\text{median} & = e^\\mu & = e^0 & = 1\n\\end{array}"
},
{
"math_id": 1,
"text": "\\begin{array}{rlll}\n\\text{mean} & = e^{\\mu + \\sigma^2 / 2} & = e^{0 + 1^2 / 2} & \\approx 1.649 \\\\\n\\text{mode} & = e^{\\mu - \\sigma^2} & = e^{0 - 1^2} & \\approx 0.368 \\\\\n\\text{median} & = e^\\mu & = e^0 & = 1\n\\end{array}"
},
{
"math_id": 2,
"text": "\\tilde{X}"
},
{
"math_id": 3,
"text": "\\bar{X}"
},
{
"math_id": 4,
"text": "\\frac{\\left|\\tilde{X} - \\bar{X}\\right|}{\\sigma} \\le (3/5)^{1/2}"
},
{
"math_id": 5,
"text": "|\\cdot|"
},
{
"math_id": 6,
"text": "\\frac{\\left|\\tilde{X} - \\mathrm{mode}\\right|}{\\sigma} \\le 3^{1/2}."
}
] |
https://en.wikipedia.org/wiki?curid=1432127
|
14322106
|
1,5-anhydro-D-fructose dehydratase
|
Class of enzymes
The enzyme 1,5-anhydro-D-fructose dehydratase (EC 4.2.1.111) catalyzes the chemical reaction
1,5-anhydro--fructose formula_0 1,5-anhydro-4-deoxy--"glycero"-hex-3-en-2-ulose + H2O
It catalyzes two steps in the anhydrofructose pathway process.
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is 1,5-anhydro--fructose hydro-lyase (ascopyrone-M-forming). Other names in common use include 1,5-anhydro--fructose 4-dehydratase, 1,5-anhydro--fructose hydrolyase, 1,5-anhydro--arabino-hex-2-ulose dehydratase, AFDH, AF dehydratase, and 1,5-anhydro--fructose hydro-lyase.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14322106
|
14322182
|
2-dehydro-3-deoxy-L-arabinonate dehydratase
|
Class of enzymes
The enzyme 2-dehydro-3-deoxy--arabinonate dehydratase (EC 4.2.1.43) catalyzes the chemical reaction
2-dehydro-3-deoxy--arabinonate formula_0 2,5-dioxopentanoate + H2O
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is 2-dehydro-3-deoxy--arabinonate hydro-lyase (2,5-dioxopentanoate-forming). Other names in common use include 2-keto-3-deoxy--arabinonate dehydratase, and 2-dehydro-3-deoxy--arabinonate hydro-lyase. This enzyme participates in ascorbate and aldarate metabolism.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14322182
|
14322196
|
2-hydroxyisoflavanone dehydratase
|
Class of enzymes
The enzyme 2-hydroxyisoflavanone dehydratase (EC 4.2.1.105) catalyzes the chemical reaction
2,7,4′-trihydroxyisoflavanone formula_0 daidzein + H2O
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is 2,7,4′-trihydroxyisoflavanone hydro-lyase (daidzein-forming). This enzyme is also called 2,7,4′-trihydroxyisoflavanone hydro-lyase. This enzyme participates in isoflavonoid biosynthesis.
The variant "GmHID1" from "Glycine max" converts 2-hydroxyisoflavone to isoflavones, mostly daidzein and genistein.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14322196
|
14322216
|
2-methylcitrate dehydratase
|
Class of enzymes
The enzyme 2-methylcitrate dehydratase (EC 4.2.1.79) catalyzes the chemical reaction
(2"S",3"S")-2-hydroxybutane-1,2,3-tricarboxylate formula_0 ("Z")-but-2-ene-1,2,3-tricarboxylate + H2O
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is (2"S",3"S")-2-hydroxybutane-1,2,3-tricarboxylate hydro-lyase [("Z")-but-2-ene-1,2,3-tricarboxylate-forming]. Other names in common use include 2-methylcitrate hydro-lyase, PrpD, and 2-hydroxybutane-1,2,3-tricarboxylate hydro-lyase. This enzyme participates in propanoate metabolism.
Structural studies.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 1SZQ.
References.
<templatestyles src="Reflist/styles.css" />
|
[
{
"math_id": 0,
"text": "\\rightleftharpoons"
}
] |
https://en.wikipedia.org/wiki?curid=14322216
|
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