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

2016
*
arXiv
*
pre-print

This isometry allows one

arXiv:1610.07724v1
fatcat:3zelmu3dtngo3lldojkt3ffbeq
*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. ...##
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Computing Algebraic Matroids
[article]

2014
*
arXiv
*
pre-print

The

arXiv:1403.8148v2
fatcat:mk35vjwvufdavgm3jixhkmhvmm
*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. ...##
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The complexity of matrix completion

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

This problem has important

doi:10.1145/1109557.1109679
fatcat:oqhcbk3vjnd4xabxqbnyex6br4
*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. ...##
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Theory of supports for linear codes endowed with the sum-rank metric
[article]

2019
*
arXiv
*
pre-print

In an appendix,

arXiv:1808.02468v2
fatcat:lvyrfu2llffyleabgumzjcyxoa
*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. ...##
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Pfaffian Circuits
[article]

2010
*
arXiv
*
pre-print

Holographic algorithms provide in general O(n^ω_p) time algorithms, where ω_p is the order

arXiv:1101.0129v1
fatcat:e24mjri75vdwvhjm6unwerxm6m
*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. ...##
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Algebraic Algorithms for Matching and Matroid Problems

2009
*
SIAM journal on computing (Print)
*

For

doi:10.1137/070684008
fatcat:ksxiwohxxvdvddo3xq4wabw2lq
*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. ...##
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Enumerative and Algebraic Combinatorics in the 1960's and 1970's
[article]

2021
*
arXiv
*
pre-print

A partly autobiographical survey

arXiv:2105.07884v3
fatcat:f332bkdpsjhprnjowgjceylvpi
*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*. ...##
###
Simplicial and cellular trees
[chapter]

2016
*
IMA Volumes in Mathematics and its Applications
*

This point

doi:10.1007/978-3-319-24298-9_28
fatcat:oh5q37h55jcmvady7lukhisolq
*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. ...##
###
Fitness, Apprenticeship, and Polynomials
[article]

2016
*
arXiv
*
pre-print

Combinatorics and concrete computations

arXiv:1612.03539v1
fatcat:okcmfrn4endp5itfit3jr42ud4
*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? ...##
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Simplicial and Cellular Trees
[article]

2015
*
arXiv
*
pre-print

This point

arXiv:1506.06819v1
fatcat:hjorlrfwljemnawlytoz57ppp4
*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. ...##
###
Fourier's Method of Linear Programming and Its Dual

1986
*
The American mathematical monthly
*

and

doi:10.2307/2322281
fatcat:yvhgyh2epbcwdoqdhuaopkcrue
*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*...##
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Quantum Complexity: restrictions on algorithms and architectures
[article]

2010
*
arXiv
*
pre-print

A dissertation submitted

arXiv:1005.1425v1
fatcat:iusngjiypfgxzlezpnz5yiyarq
*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*. ...##
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Representability of Matroids by c-Arrangements is Undecidable
[article]

2020
*
arXiv
*
pre-print

It follows that certain

arXiv:1912.06123v2
fatcat:gb2akvn2svfpfpvo3wem3uv5e4
*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. ...##
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On a Class of Meromorphic Functions Related to Cho-Kwon-Srivastava Operator

2012
*
ISRN Applied Mathematics
*

by means

doi:10.5402/2012/272678
fatcat:kn3oeobqmzh7tgbpns35eeyy4u
*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. ...##
###
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. ...

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