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Matroidal Structure of Skew Polynomial Rings with Application to Network Coding
[article]
2016
arXiv
pre-print
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This isometry allows one to use the F_q^m[x;σ]-matroid in a matroidal network coding application. ...
Over a finite field F_q^m, the evaluation of skew polynomials is intimately related to the evaluation of linearized polynomials. ...
The work of Felice Manganiello was supported by the Schweizerischer Nationalfonds (SNF), Switzerland. ...
arXiv:1610.07724v1
fatcat:3zelmu3dtngo3lldojkt3ffbeq
Computing Algebraic Matroids
[article]
2014
arXiv
pre-print
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The matroid has two natural decorations: a circuit polynomial attached to each circuit, and the degree of the projection map to each base, called the base degree. ...
An affine variety induces the structure of an algebraic matroid on the set of coordinates of the ambient space. ...
Output: List of bases of the matroid M(I) with base degree. circuits Input: Polynomial Ring, Ideal. Output: List of circuits of the matroid M(I). pCircuits Input: Polynomial Ring, Ideal. ...
arXiv:1403.8148v2
fatcat:mk35vjwvufdavgm3jixhkmhvmm
The complexity of matrix completion
2006
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
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This problem has important applications in the field of network coding. ...
This problem has deep connections to computational complexity and numerous important algorithmic applications. ...
Acknowledgments The authors would like to thank Swastik Kopparty, Kazuo Murota, David Woodruff, and the anonymous reviewers for many helpful comments and suggestions. ...
doi:10.1145/1109557.1109679
fatcat:oqhcbk3vjnd4xabxqbnyex6br4
Theory of supports for linear codes endowed with the sum-rank metric
[article]
2019
arXiv
pre-print
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In an appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given. ...
Although this metric has already shown to be of interest in several applications, not much is known about it. ...
The author also whishes to thank the anonymous reviewers for the valuable comments on this work. ...
arXiv:1808.02468v2
fatcat:lvyrfu2llffyleabgumzjcyxoa
Pfaffian Circuits
[article]
2010
arXiv
pre-print
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Holographic algorithms provide in general O(n^ω_p) time algorithms, where ω_p is the order of Pfaffian evaluation in the ring of interest (with 1.19 ≤ω_p ≤ 3 depending on the ring) and n is the number ...
We give examples (even before any change of basis) including efficient algorithms for certain lattice path problems and an O(n^ω_p) algorithm for evaluation of Tutte polynomials of lattice path matroids ...
It works over any semi-ring and exploits the tree structure to factor the problem, thus reducing a seemingly exponential problem to one which can be computed in polynomial time. ...
arXiv:1101.0129v1
fatcat:e24mjri75vdwvhjm6unwerxm6m
Algebraic Algorithms for Matching and Matroid Problems
2009
SIAM journal on computing (Print)
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For matroid intersection, our algorithm has running time O(nr ω−1 ) for matroids with n elements and rank r that satisfy some natural conditions. ...
For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω ) where n is the number of vertices and ω is the matrix multiplication exponent. ...
Additionally, the author thanks the anonymous referees for suggesting numerous improvements to the text of the paper and a concise proof of Theorem 4.1. ...
doi:10.1137/070684008
fatcat:ksxiwohxxvdvddo3xq4wabw2lq
Enumerative and Algebraic Combinatorics in the 1960's and 1970's
[article]
2021
arXiv
pre-print
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A partly autobiographical survey of the development of enumerative and algebraic combinatorics in the 1960's and 1970's. ...
I am grateful to Adriano Garsia, Ming-chang Kang, Igor Pak, Victor Reiner, Christophe Reutenauer and especially Curtis Greene for many helpful suggestions. ...
I proved polynomiality by showing that H n (r) is the Hilbert polynomial of a certain ring. ...
arXiv:2105.07884v3
fatcat:f332bkdpsjhprnjowgjceylvpi
Simplicial and cellular trees
[chapter]
2016
IMA Volumes in Mathematics and its Applications
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This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem ...
As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree -namely acyclicity and connectedness -can be generalized to arbitrary ...
Matroid structures such as cocircuits are useful in studying cellular trees (as in §5.2), so it is natural to ask what other matroid structures and invariants mean in the context of trees. ...
doi:10.1007/978-3-319-24298-9_28
fatcat:oh5q37h55jcmvady7lukhisolq
Fitness, Apprenticeship, and Polynomials
[article]
2016
arXiv
pre-print
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Combinatorics and concrete computations with polynomials (and theta functions) empowers young scholars in algebraic geometry, and it helps them to connect with the historic roots of their field. ...
The activities were structured into fitness, research and scholarship. ...
As in June Huh's July 18-22 course, determine the Chow ring of its matroid. 2. The coordinate ring of the Grassmannian Gr(3, 6) is a cluster algebra of finite type. What are the cluster variables? ...
arXiv:1612.03539v1
fatcat:okcmfrn4endp5itfit3jr42ud4
Simplicial and Cellular Trees
[article]
2015
arXiv
pre-print
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This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem ...
As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary ...
Matroid structures such as cocircuits are useful in studying cellular trees (as in §5.2), so it is natural to ask what other matroid structures and invariants mean in the context of trees. ...
arXiv:1506.06819v1
fatcat:hjorlrfwljemnawlytoz57ppp4
Fourier's Method of Linear Programming and Its Dual
1986
The American mathematical monthly
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and skew group rings, crossed products
16S36
Ordinary and skew polynomial rings and semigroup rings
[See also 20M25]
16S37
Quadratic and Koszul algebras
16S38
Rings arising from non-commutative ...
62P10 Applications to biology and medical sciences 62P12 Applications to environmental and related topics 62P15 Applications to psychology 62P20 Applications to economics [See also 91Bxx] 62P25 Applications ...
motions 83C45 Quantization of the gravitational field 83C47 Methods of quantum field theory [See also 81T20] 83C50 Electromagnetic fields 83C55 Macroscopic interaction of the gravitational field with ...
doi:10.2307/2322281
fatcat:yvhgyh2epbcwdoqdhuaopkcrue
Quantum Complexity: restrictions on algorithms and architectures
[article]
2010
arXiv
pre-print
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A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of Doctor of Philosophy (PhD) in the Faculty of Engineering, Department of Computer Science, July ...
RFW acknowledges support from the QUPRODIS network of the EU. ...
Weight enumerator polynomials Perhaps the main structural feature of a binary matroid is its weight enumerator polynomial. ...
arXiv:1005.1425v1
fatcat:iusngjiypfgxzlezpnz5yiyarq
Representability of Matroids by c-Arrangements is Undecidable
[article]
2020
arXiv
pre-print
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It follows that certain network coding problems are also undecidable. ...
In the proof, we introduce a generalized Dowling geometry to encode an instance of the uniform word problem for finite groups in matroids of rank three. ...
Multilinear matroids found applications to network coding capacity: In [ANLY00] , Ahlswede et al. introduced a model for network information flow problems. ...
arXiv:1912.06123v2
fatcat:gb2akvn2svfpfpvo3wem3uv5e4
On a Class of Meromorphic Functions Related to Cho-Kwon-Srivastava Operator
2012
ISRN Applied Mathematics
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by means of an operator given by Ghanim and Darus, in U={z:0<|z|<1}, and investigate a number of inclusion relationships of these classes. ...
We introduce new classes, Σn,λ*,μ,k(α,β,ρ) and Σn,λμ,k(α,β,ρ), of meromorphic functions defined by means of the Hadamard product of Cho-Kwon-Srivastava operator, and we define here a similar transformation ...
Gateva-Ivanova, Skew polynomial rings with binomial relations, J. Algebra, 185 (1996),
710-753. ...
doi:10.5402/2012/272678
fatcat:kn3oeobqmzh7tgbpns35eeyy4u
Index
1999
Linear Algebra and its Applications
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of the inertia of a parametric family of Bezoutians with application to some stability problems for bivariate polynomials 283 (1998) 221 Gemignani, L., see Bini, D.A. 284 (1998) 19 Genin, Y.V., Euclid ...
Pen Ä a, A matricial description of Neville elimination with applications to total positivity 202 (1994) 33 Gaøkowski, K., Matrix description of multivariable polynomials 234 (1996) 209 Gavrilyuk, I.P. ...
., On the nonexistence of rational ®rst integrals for systems of linear dierential equations 235 (1996) 107 Nummi, T., see Wang, S.-G. 289 (1999) 333 N uñez-Vald es, J., see G omez, J. ...
doi:10.1016/s0024-3795(99)00243-8
fatcat:xz5sbc3lzngydobg3ln6xp2cum
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