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In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base b for the exponent and logarithm (b is a choice of

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rdfs:label	Log semiring (en)
rdfs:comment	In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base b for the exponent and logarithm (b is a choice of (en)
dct:subject	Tropical analysis Logarithms
Wikipage page ID	37563991 (xsd:integer)
Wikipage revision ID	1071403440 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Probability semiring Decibel Deformation theory Algebraic operation Invariant measure Positive real numbers Mathematical optimization Mathematics Maximum Pullback Quasi-arithmetic mean Min-plus semiring Logarithm Logarithmic scale Tropical analysis Max-plus semiring Lebesgue measure Log-likelihood LogSumExp Log probability Logarithmic mean Logarithms Conjugation (group theory) Isomorphic Softmax function Minimum Semiring Wojciech Szpankowski Gesine Reinert Semifield Sophie Schbath Tropical analysis Tropical semiring Tropicalization Valérie Berthé Exponentiate Extended real numbers Logarithmic measure Logarithmic unit
Link from a Wikipage to an external page	https://archive.org/details/appliedcombinato0000loth
sameAs	Log semiring Log semiring
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Efn dbt:Math dbt:Mvar dbt:Notelist dbt:Refbegin dbt:Refend dbt:Reflist dbt:Sfn dbt:Short_description dbt:Slink dbt:Tmath
has abstract	In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base b for the exponent and logarithm (b is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base b < 1 is equivalent to using a negative sign and using the inverse 1/b > 1. If not qualified, the base is conventionally taken to be e or 1/e, which corresponds to e with a negative. The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity (max-plus semiring) or to zero (min-plus semiring), and thus can be viewed as a deformation ("quantization") of the tropical semiring. Notably, the addition operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see Decibel § Addition), log probability, or log-likelihoods. (en)
prov:wasDerivedFrom	wikipedia-en:Log_semiring?oldid=1071403440&ns=0
page length (characters) of wiki page	6484 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Log_semiring
is Link from a Wikipage to another Wikipage of	Positive real numbers Quasi-arithmetic mean Logarithm Logarithmic scale LogSumExp Logarithmic addition Logarithmic mean Extended real number line Finite-state transducer Softmax function Semiring Semifield
is Wikipage redirect of	Logarithmic addition
is foaf:primaryTopic of	wikipedia-en:Log_semiring

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