We show that a Schrödinger operator Aδ,α with a δ-interaction of strength α supported on a bounded or unbounded C-hypersurface Σ ⊂ R, d ≥ 2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self-adjoint realization of the formal differential expression ...

Motivated by a spectral analysis of the generator completely positive trace-preserving semigroup, we analyze real functionalA,B∈Mn(C)→r(A,B)=12(〈[B,A],BA〉+〈[B,A⁎],BA⁎〉)∈R where 〈A,B〉:=tr(A⁎B) is Hilbert-Schmidt inner product, and [A,B]:=AB−BA commutator. In particular discuss upper lower bounds form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 ‖A‖ Frobenius norm. We prove that optimal are given c±=1±22. If A r...

In this paper, we gives an upper bound estimation of the spectral norm for matrices A and B such that the entries in the first row of n×n r-circulant matrix A = Circr(a1, a2, . . . , an) and n×n symmetric r-circulant matrix B = SCircr(a1, a2, . . . , an) are ai = Pi or ai = P 2 i or ai = Pi−1 or ai = P 2 i−1, where {Pi}i=0 is Padovan sequence. At the last section, some illustrative numerical ex...

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

let $varphi(z)=z^m, z in mathbb{u}$, for some positive integer $m$, and $c_varphi$ be the composition operator on the bergman space $mathcal{a}^2$ induced by $varphi$. in this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $c^*_varphi c_varphi, c_varphi c^*_varphi$ as well as self-commutator and anti-self-commutators of $c_...

We study a simple two step procedure for estimating sparse precision matrices from data with missing values, which is tractable in high-dimensions and does not require imputation of the missing values. We provide rates of convergence for this estimator in the spectral norm, Frobenius norm and element-wise `∞ norm. Simulation studies show that this estimator compares favorably with the EM algori...

We consider the set Gn,k of graphs of order n with the chromatic number k ≥ 2. In this note, we prove that in Gn,k the Turán graph Tn,k has the maximal spectral radius; and Pn if k = 2, Cn if k = 3 and n is odd, C1 n−1 if k = 3 and n is even, K (l) k if k ≥ 4 has the minimal spectral radius. Thus we answer a problem raised by Cao [D.S. Cao, Index function of graphs, J. East China Norm. Univ. Sc...

Let C(r) = [Cij ], r = 1, 2, . . . , R, be block m×m matrices where Cij (r) are nonnegative Ni ×Nj matrices for i, j = 1, 2, . . . , m. Let ‖ · ‖ be a consistent matrix norm. Denote for each r by B(r) = [‖Cij (r)‖] an m×m matrix. The relation of the spectral radii ρ( ∏R r=1 C(r)) and ρ( ∏R r=1 B(r)) is studied in this paper. It is shown with two proofs that ρ   R ∏ r=1 C(r)   ρ   R ∏ r=1 ...

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of FredholmVolterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L∞ norm and weighted L2-norm. The numerical examp...