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Matroidal Structure of Skew Polynomial Rings with Application to Network Coding [article]

Siyu Liu, Felice Manganiello, Frank R. Kschischang
2016 arXiv   pre-print
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]

Zvi Rosen
2014 arXiv   pre-print
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

Nicholas J. A. Harvey, David R. Karger, Sergey Yekhanin
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
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]

Umberto Martínez-Peñas
2019 arXiv   pre-print
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]

Jason Morton
2010 arXiv   pre-print
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

Nicholas J. A. Harvey
2009 SIAM journal on computing (Print)  
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]

Richard P. Stanley
2021 arXiv   pre-print
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]

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2016 IMA Volumes in Mathematics and its Applications  
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]

Bernd Sturmfels
2016 arXiv   pre-print
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]

Art M. Duval, Caroline J. Klivans, Jeremy L. Martin
2015 arXiv   pre-print
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

H. P. Williams
1986 The American mathematical monthly  
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]

Daniel James Shepherd
2010 arXiv   pre-print
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]

Lukas Kühne, Geva Yashfe
2020 arXiv   pre-print
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

F. Ghanim, M. Darus
2012 ISRN Applied Mathematics  
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  
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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