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1
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Monodromy, differential equations and the Jacobian conjecture

100%
Friedland S.
Annales Polonici Mathematici
|
1999
|
tom 72
|
nr 3
219-249
EN
We study certain problems on polynomial mappings related to the Jacobian conjecture.
2
Content available remote

Some finitely generated modules and cohomologies and the Jacobian conjecture

100%
Friedland S.
Annales Polonici Mathematici
|
2001
|
tom 76
|
nr 1-2
101-112
EN
We show that the plane Jacobian conjecture is equivalent to finite generatedness of certain modules.
3
Content available remote

Equality in Wielandt’s eigenvalue inequality

100%
Friedland S.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
In this paper we give necessary and sufficient conditions for the equality case in Wielandt’s eigenvalue inequality.
4
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Nonnegative definite hermitian matrices with increasing principal minors

100%
Friedland S.
Special Matrices
|
2013
|
tom 1
1-2
EN
A nonnegative definite hermitian m × m matrix A≠0 has increasing principal minors if det A[I] ≤ det A[J] for I⊂J, where det A[I] is the principal minor of A based on rows and columns in the set I ⊆ {1,...,m}. For m > 1 we show A has increasing principal minors if and only if A−1 exists and its diagonal entries are less or equal to 1.
5
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Some approximation problems in semi-algebraic geometry

64%
Friedland S., Stawiska M.
Banach Center Publications
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2015
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tom 107
|
nr 1
133-147
EN
In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set C in the space ℝⁿ endowed with a semi-algebraic norm ν. Under additional assumptions on ν we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to C. For C irreducible algebraic we study the critical point correspondence and introduce the ν-distance degree, generalizing the notion developed by other authors for the Euclidean norm. We discuss separately the case of the $ℓ^p$ norm (p > 1).
6
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Generalized interval exchanges and the 2–3 conjecture

64%
Friedland S., Weiss B.
Open Mathematics
|
2005
|
tom 3
|
nr 3
412-429
EN
We introduce the notion of a generalized interval exchange $$\phi _\mathcal{A} $$ induced by a measurable k-partition $$\mathcal{A} = \left\{ {A_1 ,...,A_k } \right\}$$ of [0,1). $$\phi _\mathcal{A} $$ can be viewed as the corresponding restriction of a nondecreasing function $$f_\mathcal{A} $$ on ℝ with $$f_\mathcal{A} (0) = 0, f_\mathcal{A} (k) = 1$$ . A is called λ-dense if λ(A i∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that $$f_\mathcal{A} \circ f_\mathcal{B} = f_\mathcal{B} \circ f_\mathcal{A} $$ . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which $$f_\mathcal{A} $$ and $$f_\mathcal{B} $$ commute.
7
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A simple spectral algorithm for recovering planted partitions

51%
Cole S., Friedland S., Reyzin L.
Special Matrices
|
2017
|
tom 5
|
nr 1
139-157
EN
In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k “clusters,” each of size s. Edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least (√n). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph’s adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Θ (√n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.
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