Spectra of expansion graphs.
Nonnegativity of Schur complements of nonnegative idempotent matrices.
The maximum number of perfect matchings in graphs with a given degree sequence.
Extremal eigenvalue problems defined on conformal classes of compact Riemannian manifolds.
Equality in Wielandt’s eigenvalue inequality
In this paper we give necessary and sufficient conditions for the equality case in Wielandt’s eigenvalue inequality.
Nonnegative definite hermitian matrices with increasing principal minors
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.
Monodromy, differential equations and the Jacobian conjecture
We study certain problems on polynomial mappings related to the Jacobian conjecture.
Some finitely generated modules and cohomologies and the Jacobian conjecture
We show that the plane Jacobian conjecture is equivalent to finite generatedness of certain modules.
The Collatz-Wielandt quotient for pairs of nonnegative operators
In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and is the identity operator, then one version of this quotient is the spectral radius of . In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with...
Positive entries of stable matrices.
On the first eigenvalue of bipartite graphs.
Minimum rank of matrices described by a graph or pattern over the rational, real and complex numbers.
Generalized interval exchanges and the 2–3 conjecture
We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . 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 . We give necessary and sufficient conditions for this equality to hold. We show that...
Some approximation problems in semi-algebraic geometry
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...
The generalized Radon-Hurwitz numbers
A simple spectral algorithm for recovering planted partitions
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...
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