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Tropical Analysis
In the mathematical discipline of idempotent analysis, tropical analysis is the study of the tropical semiring. Applications The max tropical semiring can be used appropriately to determine marking times within a given Petri net and a vector filled with marking state at the beginning: -\infty (unit for max, tropical addition) means "never before", while 0 (unit for addition, tropical multiplication) is "no additional time". Tropical cryptography is cryptography based on the tropical semiring. Tropical geometry is an analog to algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ..., using the tropical semiring. References * Further reading * * See also * Lunar arithmetic External links MaxPlus algebraworking group, INRIA Rocquencourt {{Mathanalys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Analysis
In mathematical analysis, idempotent analysis is the study of idempotent semirings, such as the tropical semiring In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical .... The lack of an additive inverse in the semiring is compensated somewhat by the idempotent rule A \oplus A = A. References * {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Semiring
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name ''tropical'' is a reference to the Hungarian-born computer scientist Imre Simon, so named because he lived and worked in Brazil. Definition The ' (or or ) is the semiring (ℝ ∪ , ⊕, ⊗), with the operations: : x \oplus y = \min\, : x \otimes y = x + y. The operations ⊕ and ⊗ are referred to as ''tropical addition'' and ''tropical multiplication'' respectively. The unit for ⊕ is +∞, and the unit for ⊗ is 0. Similarly, the ' (or or or ) is the semiring (ℝ ∪ , ⊕, ⊗), with operations: : x \oplus y = \max\, : x \otimes y = x + y. The unit for ⊕ is −∞, and the unit for ⊗ is 0. The two semirings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petri Net
A Petri net, also known as a place/transition (PT) net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph that has two types of elements, places and transitions. Place elements are depicted as white circles and transition elements are depicted as rectangles. A place can contain any number of tokens, depicted as black circles. A transition is enabled if all places connected to it as inputs contain at least one token. Some sources state that Petri nets were invented in August 1939 by Carl Adam Petri—at the age of 13—for the purpose of describing chemical processes. Like industry standards such as UML activity diagrams, Business Process Model and Notation, and event-driven process chains, Petri nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike these standards, Petri nets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Cryptography
In tropical analysis, tropical cryptography refers to the study of a class of cryptographic protocols built upon tropical algebras. In many cases, tropical cryptographic schemes have arisen from adapting classical (non-tropical) schemes to instead rely on tropical algebras. The case for the use of tropical algebras in cryptography rests on at least two key features of tropical mathematics: in the tropical world, there is no classical multiplication (a computationally expensive operation), and the problem of solving systems of tropical polynomial equations has been shown to be NP-hard. Basic Definitions The key mathematical object at the heart of tropical cryptography is the tropical semiring (\mathbb \cup \,\oplus,\otimes) (also known as the min-plus algebra), or a generalization thereof. The operations are defined as follows for x,y \in \mathbb \cup \: x \oplus y = \min\ x \otimes y = x + y It is easily verified that with \infty as the additive identity, these binary operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So for example, the classical polynomial x^3 + 2xy + y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lunar Arithmetic
Lunar arithmetic, formerly called dismal arithmetic, is a version of arithmetic in which the addition and multiplication operations on digits are defined as the max and min operations. Thus, in lunar arithmetic, :2+7=\max\=7 and 2\times 7 = \min\=2. The lunar arithmetic operations on nonnegative multidigit numbers are performed as in usual arithmetic as illustrated in the following examples. The world of lunar arithmetic is restricted to the set of nonnegative integers. 976 + 348 ---- 978 (adding digits column-wise) 976 × 348 ---- 876 (multiplying the digits of 976 by 8) 444 (multiplying the digits of 976 by 4) 333 (multiplying the digits of 976 by 3) ------ 34876 (adding digits column-wise) The concept of lunar arithmetic was proposed by David Applegate, Marc LeBrun, and Neil Sloane. In the general definition of lunar arithmetic, one considers numbers expressed in an arbitrary base b and define lunar arithmetic operations as the max and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical Analysis
In the mathematical discipline of idempotent analysis, tropical analysis is the study of the tropical semiring. Applications The max tropical semiring can be used appropriately to determine marking times within a given Petri net and a vector filled with marking state at the beginning: -\infty (unit for max, tropical addition) means "never before", while 0 (unit for addition, tropical multiplication) is "no additional time". Tropical cryptography is cryptography based on the tropical semiring. Tropical geometry is an analog to algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ..., using the tropical semiring. References * Further reading * * See also * Lunar arithmetic External links MaxPlus algebraworking group, INRIA Rocquencourt {{Mathanalys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
