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### To the Steinitz lemma in coordinate form

Sergey Sevast'janov, Wojciech Banaszczyk
1997 Discrete Mathematics
We consider an arbitrary finite family of vectors in m-dimensional space with sum zero and the absolute value of any vector's coordinate at most 1.  ...  We prove that there exists an order of vector summation for this family such that the absolute value of the first coordinate for all partial sums is at most 1 while those of other coordinates are bounded  ...  Does there exists a Steinitz bound C = (C1, C2 ..... C,,) with C1 and C2 dependent neither on n nor on m? Or in a more general form for the first v coordinates.  ...

### A vector-sum theorem in two-dimensional space

I. Bárány, V. S. Grinberg
1985 Periodica Mathematica Hungarica
Given a finite set X of vectors from the uni{~ ball of the max norm in the twodimensional space whose sum is zero, it is always possible to write X = {x 1 .... , Xn} k in such a way that the fix'st coordinates  ...  of each partial sum x-xl lie in [--l, 1 ] and the 1 second coordinates lie in [--C, C] where C is a universal constant.  ...  To do so we apply Lemma 3 for the first coordinates. This shows that the first coordinates of each partial sum lie in [--1, 1].  ...

### Regular hypergraphs, Gordon's lemma, Steinitz' lemma and invariant theory

N Alon, K.A Berman
1986 Journal of combinatorial theory. Series A
STEINITZ' LEMMA AND UPPER BOUNDS FOR D(n), D(n,k) Steinitz' lemma asserts that every sequence of m vectors of norm < 1 in R" whose sum is the zero vector, can be rearranged such that all initial sums will  ...  We conjecture that in fact D(n, k) <n'("), where c(k) depends only on k. 1 The method used in the proofs of the last two propositions enables us to obtain an effective version of Gordon's lemma (Lemma  ...

### Vectors in a Box [article]

Kevin Buchin, Jiří Matoušek, Robin A. Moser, Dömötör Pálvölgyi
2009 arXiv   pre-print
Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of tau(d) <= d^d+o(d), and based on a construction of Alon and Vu, whose main idea goes back  ...  The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.  ...  Acknowledgment We would like to thank Tibor Szabó for raising the problem at the GWOP'09 workshop, Sanjeeb Dash for prompt answers to our questions, and Patrick Traxler for useful discussions.  ...

### Steinitz Representations of Polyhedra and the Colin de Verdière Number

László Lovász
2001 Journal of combinatorial theory. Series B (Print)
We show that the Steinitz representations of 3-connected planar graphs are correspond, in a well described way, to Colin de Verdière matrices of such graphs.  ...  I am grateful to Bob Connelly, Oded Schramm and Günter Ziegler for their valuable remarks on the topic of this paper.  ...  By lemma 1, u belongs to C f ∩ C g , but to no other cone C h , and thus C f and C g must share a face, proving that f g ∈ E * . This completes the proof of Claim 2 and of lemma 5.  ...

### On the Power of Linear Dependencies [chapter]

Imre Bárány
2008 Bolyai Society Mathematical Studies
Simple as they may be, linear dependencies have proved very useful in many ways.  ...  In this survey several geometric applications of linear dependencies are discussed, focusing on rearrangements of sums and on sums with ±1 signs.  ...  This is what Steinitz proved in  . The smallest constant the Steinitz lemma holds with is a number, to be denoted by S(B), that depends only on the unit ball B.  ...

### Vectors in a box

Kevin Buchin, Jiří Matoušek, Robin A. Moser, Dömötör Pálvölgyi
2011 Mathematical programming
Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of τ (d) ≤ d d+o (d) , and based on a construction of Alon and Vũ, whose main idea goes back  ...  The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.  ...  use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.  ...

### Small Grid Embeddings of 3-Polytopes

Ares Ribó Mor, Günter Rote, André Schulz
2010 Discrete & Computational Geometry
We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face.  ...  If the graph contains a triangle we can bound the integer coordinates by O(2^4.82n). If the graph contains a quadrilateral we can bound the integer coordinates by O(2^5.46n).  ...  Since L BI = (L IB ) T , the matrixL is symmetric and thereforeω ij =ω ji holds. To show that the expression F x =Lx B has the form stated in the lemma we have to check that all row sums inL equal 0.  ...

### Hamming spaces and locally matrix algebras [article]

Oksana Bezushchak, Bogdana Oliynyk
2020 arXiv   pre-print
We classify countable locally standard Hamming spaces and show that each of them can be realized as the Boolean algebra of idempotents of a Cartan subalgebra of a locally matrix algebra.  ...  Steinitz numbers A Steinitz number  is a infinite formal product of the form p∈P p rp , where P is the set of all primes, r p ∈ N ∪ {0, ∞} for all p ∈ P.  ...  In view of Lemma 2 it is sufficient to find a general Cartan subalgebra H such that for an arbitrary invertible element x ∈ A * either x ∈ H or x −1 Hx = H. Lemma 2.  ...

### Uniform distribution of the Steinitz invariants of quadratic and cubic extensions

Anthony C. Kable, David J. Wright
2006 Compositio Mathematica
As mentioned in the abstract, the full version of this result allows the extension K to be restricted by specifying its completion, up to isomorphism, at each of finitely-many places of k.  ...  It is shown that the Steinitz invariants of the cubic extensions of a number field are uniformly distributed in the class group when the cubic extensions are ordered by the ideal norm of their relative  ...  The first-named author realized the significance of the invariant S for the space of binary cubic forms whilst returning from a talk at the HAAR seminar at The Ohio State University.  ...

### Algorithmic Discrepancy Beyond Partial Coloring [article]

Nikhil Bansal, Shashwat Garg
2017 arXiv   pre-print
Similarly, for the Steinitz problem we give the first algorithm that matches the best known non-constructive bounds due to Banaszczyk [Banaszczyk 2012] in the ℓ_∞ case, and improves the previous algorithmic  ...  However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number of times, and the errors can add up in an adversarial way.  ...  Acknowledgments We would like to thank Yin-Tat Lee for sharing with us a simpler proof of the martingale concentration in [BDG16] ; and Daniel Dadush, Aleksandar Nikolov and Mohit Singh for insightful  ...

### Gauss-Lucas Theorems for Entire Functions on CM [article]

Marek Kanter
2012 arXiv   pre-print
Essential use is made of the Levy-Steinitz theorem for conditionally convergent real number series.  ...  Previous work in this area is mostly restricted to univariate entire functions (of genus no greater than one unless "realness" assumptions are made.)  ...  These authors consider vectors in R ∞ , so it is useful to write γ −r n in the form x n (r; γ)+ix n (r+p; γ), for 1 ≤ r ≤ p, where x n (k; γ) ∈ R, for 1 ≤ k ≤ 2p and i = √ −1.  ...

### The Davenport constant of a box

Alain Plagne, Salvatore Tringali
2015 Acta Arithmetica
In this paper, we mainly investigate the case when G is a power of Z and X is a box (i.e., a product of intervals of G).  ...  Given an additively written abelian group G and a set X⊆ G, we let B(X) denote the monoid of zero-sum sequences over X and D(X) the Davenport constant of B(X), namely the supremum of the positive integers  ...  Acknowledgments The authors are indebted to Alfred Geroldinger for raising a question which gave birth to this paper.  ...

### A General Version of the Nullstellensatz for Arbitrary Fields

Edisson Gallego, Juan D. Vélez C., Danny A. J. Gómez-Ramírez
2019 Open Mathematics
The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to the ideal in consideration.  ...  We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields.  ...  This is an elementary consequence of the Artin-Tate lemma and the Steinitz theorem (see for example  ). Now, let us assume that I is an arbitrary ideal.  ...

### Graphs of polyhedra; polyhedra as graphs

Branko Grünbaum
2007 Discrete Mathematics
Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail.  ...  In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation.  ...  Concerning Lemma 2.3, it may be of some interest to note that it has a structural similarity to a technique Steinitz used much earlier, in his dissertation  .  ...
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