### Extremal eigenvalue problems defined on conformal classes of compact Riemannian manifolds.

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In this paper we give necessary and sufficient conditions for the equality case in Wielandt’s eigenvalue inequality.

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.

We study certain problems on polynomial mappings related to the Jacobian conjecture.

We show that the plane Jacobian conjecture is equivalent to finite generatedness of certain modules.

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators $A,B$ 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 $B$ is the identity operator, then one version of this quotient is the spectral radius of $A$. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with...

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 $${f}_{\mathcal{A}}\left(0\right)=0,{f}_{\mathcal{A}}\left(k\right)=1$$ . 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...

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...

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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