Narrow Results

We show that the matrix of chromatic joins and the Gram matrix of a Temperley-Lieb algebra are similar.

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert–Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every $(U,C)$-controlled Riesz basis ${\{f_k\}}_{k=1}^{\infty }$ is in the form ${\{U^{-1}CM{e}_k\}}_{k=1}^{\infty }$, where $M$ is a bijective operator on $H$. Furthermore, we propose an equivalent accessible condition to the sequence ${\{f_k\}}_{k=1}^{\infty }$ being a $(U,C)$-controlled Riesz basis.

Fusion frame is a generalization of frame, which can analyze signals by projecting them onto multidimensional subspaces. Controlled fusion frame as generalization of fusion frame, it can improve the numerical efficiency of iterative algorithms for inverting the fusion frame operators. In this paper, we first introduce the notion of controlled g-fusion frame, discuss several properties of controlled g-fusion Bessel sequence. Then, we present some sufficient conditions and some characterizations of controlled g-fusion frames. Finally, we study the sum of controlled g-fusion frames.

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.

A geometrical model object oriented based on declarative geometry is presented in this paper. This method is used to test measurement algorithms for complex system as for a human body. Geometrical objects are actually defined by a vector family. This family can be efficiently described by a Gram matrix. This kind of modeling, as can be a metric tensor, provide the advantage to propose a geometrical object invariant to translation, rotation and all symmetric transformation of the affine space. This work will be used to simulate Total Knee Arthroplasty surgery.