Denote by W (A) the numerical range of a bounded linear operator A, and [A, B] = AB −BA the Lie product of two operators A and B. Let H, K be complex Hilbert spaces of dimension ≥ 2 and Φ : B(H) → B(K) be a map whose range contains all operators of rank ≤ 1. It is shown that Φ satisfies that W ([Φ(A), Φ(B)]) = W ([A, B]) for any A, B ∈ B(H) if and only if dim H = dim K, there exist ε ∈ {1,−1}, ...

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. A direct “sieving” technique is used to establish this conjecture and establish the range of validity. Unlike the approach of Bringmann and Dousse, the technique readily yields the analogous result for Dyson’s partition rank and all ...

Abstract. We derive and study a Gauss-Newton method for computing a symmetric low-rank product XXT, where X ∈ Rn×k for k < n, that is the closest to a given symmetric matrix A ∈ Rn×n in Frobenius norm. When A = BTB (or BBT), this problem essentially reduces to finding a truncated singular value decomposition of B. Our Gauss-Newton method, which has a particularly simple form, shares the same or...

Stampfli has shown that for a given T £ B(H) there exists a K £ C(H) so that o(T + K) = ow(T). An analogous result holds for the essential numerical range We(T). A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator T £ B(H) if o(T + K) = o„(T) and W(T + K)= We(T). Theorem. For each block diagonal operator T, there exists a compact operator K wh...

Motivated by applications in quantum information and quantum control, a new type of C-numerical range, the relative C-numerical range denoted W K (C, A), is introduced. It arises upon replacing the unitary group U (N) in the definition of the classical C-numerical range by any of its compact and connected subgroups K ⊂ U (N). The geometric properties of the relative C-numerical range are analys...

Given a matrix A ∈ Rm×n (n vectors in m dimensions), we consider the problem of selecting a submatrix (subset of the columns) with maximum volume. The motivation to study such a problem is that if A can be approximately reconstructed from a small number k of its columns (A has “numerical” rank k), then any set of k independent columns of A should suffice to reconstruct A. However, numerical sta...

Let ? be a valuation of arbitrary rank on the polynomial ring K[x] with coefficients in field K. We prove comparison theorems between MacLane–Vaquié key polynomials for valuations ??? and abstract ?. Also, some results invariants associated to limit are obtained. In particular, if char(K)=0, we show that all unbounded continuous families augmentations have numerical character equal one.

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.