Functions of perturbed n-tuples of commuting self-adjoint operators

Let (_A1,…,_An)$(A_1,\dots ,A_n)$ and (_B1,…,_Bn)$(B_1,\dots ,B_n)$ be n-tuples of commuting self-adjoint operators on Hilbert space. For functions f   on Rⁿ${\mathbb{R}}^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of f(_A1,…,_An)−f(_B1,…,_Bn)$f(A_1,\dots ,A_n)-f(B_1,\dots ,B_n)$ in terms of the corresponding norms of _Aj−_Bj$A_j-B_j$, 1?j?n$1?j?n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for n?3$n?3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary n  . We also get sharp estimates for quasicommutators f(_A1,…,_An)R−Rf(_B1,…,_Bn)$f(A_1,\dots ,A_n)R-Rf(B_1,\dots ,B_n)$ in terms of norms of _AjR−R_Bj$A_{j}R-R{B}_j$, 1?j?n$1?j?n$, for a bounded linear operator R.     

Uniform hypergraphs containing no grids

A hypergraph is called an r×r$r\times r$grid   if it is isomorphic to a pattern of r$r$ horizontal and r$r$ vertical lines, i.e., a family of sets {_A1,…,_Ar,_B1,…,_Br}$\{A_1,\dots ,A_r,B_1,\dots ,B_r\}$ such that _Ai∩_Aj=_Bi∩_Bj=0?$A_i\cap A_j=B_i\cap B_j=0?$ for 1≤i<j≤r$1\le i<j\le r$ and |_Ai∩_Bj|=1$|A_i\cap B_j|=1$ for 1≤i,j≤r$1\le i,j\le r$. Three sets _C1,_C2,_C3$C_1,C_2,C_3$ form a triangle   if they pairwise intersect in three distinct singletons, |_C1∩_C2|=|_C2∩_C3|=|_C3∩_C1|=1$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$, _C1∩_C2≠_C1∩_C3$C_1\cap C_2\ne C_1\cap C_3$. A hypergraph is linear  , if |E∩F|≤1$|E\cap F|\le 1$ holds for every pair of edges E≠F$E\ne F$.     

Efficient rare-event simulation for perpetuities

We consider perpetuities of the form

D=_B1exp(_Y1)+_B2exp(_Y1+_Y2)+?,$D=B_{1}exp\left(Y_1\right)+B_{2}exp(Y_1+Y_2)+?\text{,}$

where the _Yj$Y_j$’s and _Bj$B_j$’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the _Yj$Y_j$’s satisfy the so-called Cramér condition with associated root _θ∗∈(0,∞)${\theta }_{\ast }\in (0,\infty )$ and that the tails of the _Bj$B_j$’s are appropriately behaved so that D$D$ is regularly varying with index _θ∗${\theta }_{\ast }$. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the _Yj$Y_j$’s according to _θ∗${\theta }_{\ast }$ fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.     

Parallel machine scheduling with nested processing set restrictions

We consider the problem of scheduling a set of n independent jobs on m parallel machines, where each job can only be scheduled on a subset of machines called its processing set. The machines are linearly ordered, and the processing set of job j   is given by two machine indexes _aj$a_j$ and _bj$b_j$; i.e., job j   can only be scheduled on machines _aj,_aj+1,…,_bj$a_j\text{,}{a}_j+1\text{,}\dots \text{,}{b}_j$. Two distinct processing sets are either nested or disjoint. Preemption is not allowed. Our goal is to minimize the makespan. It is known that the problem is strongly NP-hard and that there is a list-type algorithm with a worst-case bound of 2-1/m$2-1/m$. In this paper we give an improved algorithm with a worst-case bound of 7/4. For two and three machines, the algorithm gives a better worst-case bound of 5/4 and 3/2, respectively.     

Completely positive interpolations of compact,trace-class and Schatten-p class operators

Extending Li and Poon's results on interpolation problems for matrices, we give characterizations of the existence of a completely positive linear map _Φcp${\Phi }_{\mathrm{cp}}$ between compact (or Schatten-p class) operators sending a particular operator A to another B. It is shown that such a map exists if a multiple of the numerical range of A contains the numerical range of B. Given two commutative families of compact (or Schatten-p   class) operators {_Aα}$\{A_{\alpha }\}$ and {_Bα}$\{B_{\alpha }\}$, we provide sufficient and necessary conditions to ensure that we can choose a completely positive interpolation _Φcp${\Phi }_{\mathrm{cp}}$ to preserve trace and/or approximate units such that _Φcp(_Aα)=_Bα${\Phi }_{\mathrm{cp}}(A_{\alpha })=B_{\alpha }$ for all α.     

A Note on n-Perinormal Operators

The study of operators satisfying

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

is of significant interest. Does

σ_ja(T)=σa(T)${\sigma }_{ja}\left(T\right)={\sigma }_a\left(T\right)$

for n-perinormal operator

T∈B(H)?$T\in B\left(H\right)?$

This question was raised by Mecheri and Braha [Oper. Matrices 6 (2012), 725–734]. In the note we construct a counterexample to this question and obtain the following result: if T is a n-perinormal operator in B(H), then

σ_ja(T)\{0}=σa(T)\{0}.${\sigma }_{ja}\left(T\right)\backslash \left\{0\right\}={\sigma }_a\left(T\right)\backslash \left\{0\right\}.$

We also consider tensor product of n-perinormal operators.     

The inductive wedge product of positive currents

We define the wedge product of positive currents. Our main issue is about studying the convergence of dd^c_gj∧T$d{d}^c{g}_j\wedge T$ where _gj$g_j$ is a given sequence of plurisubharmonic functions and T is a positive current.     

A branch-and-check algorithm for minimizing the weighted number of late jobs on a single machine with release dates

In this paper we consider the scheduling problem of minimizing the weighted number of late jobs on a single machine (1|_rj|∑_wjUj)$\left(1|r_j|\sum w_j{U}_j\right)$. A branch-and-check algorithm is proposed, where a relaxed integer programming formulation is solved by branch-and-bound and infeasible solutions are cut off using infeasibility cuts. We suggest two ways to generate cuts. First, tightened “no-good” cuts are derived using a modification of the algorithm by Carlier (1982, EJOR, v.11, 42–47) which was developed for the problem of minimizing maximum lateness on a single machine. Secondly we show how to create cuts by using constraint propagation. The proposed algorithm is implemented in the Mosel modelling and optimization language. Computational experiments on instances with up to 140 jobs are reported. A comparison is presented with the exact approach of Péridy at al. (2003, EJOR, v.148, 591–603).     

Note on a Korovkin-type theorem

A new Korovkin-type theorem and its converse theorem are established. We compare our direct result with the Korovkin-type theorem given by Wang [15]. As applications we obtain quantitative estimates for q  -Bernstein-type operators which preserve the functions _e0(x)=1$e_0(x)=1$ and _ej(x)=x^j$e_j(x)=x^j$.     

A new heuristic for open shop total completion time problem

The m  -machine open shop problem to minimize the sum of the completion times is investigated in our paper. First, a heuristic algorithm, Shortest Processing Time Block, is presented to deal with problem _Om|n=km|∑_Cj$O_m|n=\mathit{km}|\sum C_j$, where k   is positive integer. And then, the heuristic is extended to solve the general problem _Om‖∑_Cj$O_m\Vert \sum C_j$. It is proved that the heuristic is asymptotically optimal when the job number goes to infinity. At the end of this paper, numerical experimentation results show the effectiveness of the heuristic algorithm.     

Piecewise linear output measures in DEA (third revision)

It is assumed in the standard DEA model that the aggregate output (input) is a pure linear function of each output (input). This means, for example, that if DMU _j1$j_1$ generates twice as much of an output as does another DMU _j2$j_2$, then the former is credited with having created twice as much value  . In many situations, however, linear pricing (_μryrj)$({\mu }_r{y}_{\mathit{rj}})$ may not adequately reflect differences in value created from one DMU to another. In this paper, a generalization of the DEA methodology is presented that incorporates piecewise linear functions of factors. We deal specifically with those situations where for certain outputs in an input-oriented model, the weight function f(_yrj)$f(y_{\mathit{rj}})$ is described by either a non-increasing or non-decreasing set of multipliers for larger amounts of the factor. We refer to such a variable r as exhibiting diminishing marginal value (DMV) or increasing marginal value   (IMV). The DMV/IMV phenomenon is common in many for-profit applications. For example, in the case that _yrj$y_{\mathit{rj}}$ is the amount of a consumer product r generated by DMU j  , and _μr${\mu }_r$ is the price of that product, it may well be that the market will force lower prices if greater amounts of that product are generated; discounts automatically lead to this DMV situation. Such a phenomenon can arise as well in not-for-profit settings, and we examine such a situation based on earlier work by Cook et al. [Cook, W.D., Roll, Y., Kazakov, A., 1990. A DEA model for measuring the relative efficiency of highway maintenance patrols. INFOR 28 (2), 113–124].     

On the least distance eigenvalue of a graph

Denote by D(G)=_(di,j)n×n$D(G)={(d_{i,j})}_{n\times n}$ the distance matrix of a connected graph G with n   vertices, where _dij$d_{ij}$ is equal to the distance between vertices _vi$v_i$ and _vj$v_j$ in G  . The least eigenvalue of D(G)$D(G)$ is called the least distance eigenvalue of G  , denoted by _λn${\lambda }_n$. In this paper, we determine all the graphs with _λn∈[−2.383,0]${\lambda }_n\in [-2.383,0]$.     

Matrix representations of truncated Toeplitz operators

In this paper we describe matrix representations of truncated Toeplitz operators on the model space _KB$K_B$, where B is an infinite Blaschke product satisfying some additional conditions. Our results are extensions of that obtained by Cima, Ross and Wogen in 2008.     

S-Noetherian properties on amalgamated algebras along an ideal

Let A and B   be commutative rings with identity, f:A→B$f:A\to B$ a ring homomorphism and J an ideal of B  . Then the subring A?^fJ:={(a,f(a)+j)|a∈A and j∈J}$A{?}^{f}J:=\{(a,f(a)+j)|a\in A\text{and}j\in J\}$ of A×B$A\times B$ is called the amalgamation of A with B along with J with respect to f. In this paper, we investigate a general concept of the Noetherian property, called the S  -Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A?^fJ$A{?}^{f}J$ for a multiplicative subset S   of A?^fJ$A{?}^{f}J$. As particular cases of the amalgamation, we also devote to study the transfers of the S  -Noetherian property to the constructions D+(_X1,…,_Xn)E[_X1,…,_Xn]$D+(X_1,\dots ,X_n)E[X_1,\dots ,X_n]$ and D+(_X1,…,_Xn)E?_X1,…,_Xn?$D+(X_1,\dots ,X_n)E?X_1,\dots ,X_n?$ and Nagata?s idealization.