Categorical Notions ofLayered Tropical Algebra and Geometry

2013 ◽  
pp. 191-234 ◽  
Author(s):  
Zur Izhakian ◽  
Manfred Knebusch ◽  
Louis Rowen
Keyword(s):  
Tropical Algebra

scholarly journals Independent Rainbow Domination Numbers of Generalized Petersen Graphs P(n,2) and P(n,3)

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 996
Author(s):  
Boštjan Gabrovšek ◽  
Aljoša Peperko ◽  
Janez Žerovnik
Keyword(s):  
Path Algebra ◽  
Generalized Petersen Graphs ◽  
Tropical Algebra ◽  
Rainbow Domination ◽  
Petersen Graphs ◽  
Established Technique ◽  
Domination Numbers

We obtain new results on independent 2- and 3-rainbow domination numbers of generalized Petersen graphs P ( n , k ) for certain values of n , k ∈ N . By suitably adjusting and applying a well established technique of tropical algebra (path algebra) we obtain exact 2-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) and P ( n , 3 ) thus confirming a conjecture proposed by Shao et al. In addition, we compute exact 3-independent rainbow domination numbers of generalized Petersen graphs P ( n , 2 ) . The method used here is developed for rainbow domination and for Petersen graphs. However, with some natural modifications, the method used can be applied to other domination type invariants, and to many other classes of graphs including grids and tori.

Modified tropical algebra based median filter for removing salt and pepper noise in digital image

2019 ◽  
Vol 13 (14) ◽  
pp. 2790-2795
Author(s):  
Achmad Abdurrazzaq ◽  
Ismail Mohd ◽  
Ahmad Kadri Junoh ◽  
Zainab Yahya
Keyword(s):  
Digital Image ◽  
Median Filter ◽  
Salt And Pepper Noise ◽  
Tropical Algebra ◽  
Salt And Pepper

Tropical algebra based framework for error propagation analysis in systolic arrays

2013 ◽  
Vol 225 ◽  
pp. 512-525 ◽  
Author(s):  
Vladimir Ciric ◽  
Aleksandar Cvetković ◽  
Vladimir Simić ◽  
Ivan Milentijević
Keyword(s):  
Error Propagation ◽  
Systolic Arrays ◽  
Tropical Algebra ◽  
Propagation Analysis ◽  
Error Propagation Analysis

scholarly journals The Rényi Entropies Operate in Positive Semifields

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 780 ◽  
Author(s):  
Francisco J. Valverde-Albacete ◽  
Carmen Peláez-Moreno
Keyword(s):  
Computational Intelligence ◽  
Viterbi Algorithm ◽  
Morphological Processing ◽  
Information Function ◽  
Tropical Algebra ◽  
Idempotent Analysis ◽  
Standard Product ◽  
Rényi Entropies ◽  
Domain Transformation ◽  
Renyi Entropies

We set out to demonstrate that the Rényi entropies are better thought of as operating in a type of non-linear semiring called a positive semifield. We show how the Rényi’s postulates lead to Pap’s g-calculus where the functions carrying out the domain transformation are Rényi’s information function and its inverse. In its turn, Pap’s g-calculus under Rényi’s information function transforms the set of positive reals into a family of semirings where “standard” product has been transformed into sum and “standard” sum into a power-emphasized sum. Consequently, the transformed product has an inverse whence the structure is actually that of a positive semifield. Instances of this construction lead to idempotent analysis and tropical algebra as well as to less exotic structures. We conjecture that this is one of the reasons why tropical algebra procedures, like the Viterbi algorithm of dynamic programming, morphological processing, or neural networks are so successful in computational intelligence applications. But also, why there seem to exist so many computational intelligence procedures to deal with “information” at large.

On the numerical range in tropical algebra

2018 ◽  
Vol 67 (10) ◽  
pp. 1985-1998 ◽  
Author(s):  
Hanieh Tavakolipour ◽  
Fatemeh Shakeri
Keyword(s):  
Numerical Range ◽  
Tropical Algebra

scholarly journals Symmetric polynomials in tropical algebra semirings

2019 ◽  
Vol 93 ◽  
pp. 100-119 ◽  
Author(s):  
Sara Kališnik ◽  
Davorin Lešnik
Keyword(s):  
Symmetric Polynomials ◽  
Tropical Algebra

scholarly journals Generation of summand absorbing submodules

2020 ◽  
pp. 2150201
Author(s):  
Zur Izhakian ◽  
Manfred Knebusch ◽  
Louis Rowen
Keyword(s):  
Krull Dimension ◽  
Sums Of Squares ◽  
Finitely Generated ◽  
Tropical Algebra ◽  
Idempotent Semirings

An [Formula: see text]-module [Formula: see text] over a semiring [Formula: see text] lacks zero sums (LZS) if [Formula: see text] implies [Formula: see text]. More generally, a submodule [Formula: see text] of [Formula: see text] is “summand absorbing” (SA), if, for all [Formula: see text], [Formula: see text] These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.

scholarly journals Temporal State Machines: Using Temporal Memory to Stitch Time-based Graph Computations

2021 ◽  
Vol 17 (3) ◽  
pp. 1-27
Author(s):  
Advait Madhavan ◽  
Matthew W. Daniels ◽  
Mark D. Stiles
Keyword(s):  
Ad Hoc ◽  
Arrival Time ◽  
State Machine ◽  
Temporal Memory ◽  
Performance Improvements ◽  
Tropical Algebra ◽  
Methodical Approach ◽  
Systematic Exploration ◽  
And Performance ◽  
High Level

Race logic, an arrival-time-coded logic family, has demonstrated energy and performance improvements for applications ranging from dynamic programming to machine learning. However, the various ad hoc mappings of algorithms into hardware rely on researcher ingenuity and result in custom architectures that are difficult to systematize. We propose to associate race logic with the mathematical field of tropical algebra, enabling a more methodical approach toward building temporal circuits. This association between the mathematical primitives of tropical algebra and generalized race logic computations guides the design of temporally coded tropical circuits. It also serves as a framework for expressing high-level timing-based algorithms. This abstraction, when combined with temporal memory, allows for the systematic exploration of race logic–based temporal architectures by making it possible to partition feed-forward computations into stages and organize them into a state machine. We leverage analog memristor-based temporal memories to design such a state machine that operates purely on time-coded wavefronts. We implement a version of Dijkstra’s algorithm to evaluate this temporal state machine. This demonstration shows the promise of expanding the expressibility of temporal computing to enable it to deliver significant energy and throughput advantages.

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