English

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group \({\mathbb{F}_n}\) on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group \({\mathbb{Z}_2}\). We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by C^*-algebras is taken over by operator systems. As part of the techniques we treat C^*-envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a C^*-algebra. We analyze this in the context of the finite Fourier transform on the cyclic group. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d(x,y)< varepsilon between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

OpenCV is a widely used computer vision library written in C++. Basic Linear Algebraic Operations (BLAOP) involving matrices are at the heart of OpenCV. Though OpenCV provides ubiquity in the computer vision field, it runs slow when ported on embedded processors. Accelerating the LAOPs using a co-processor certainly helps improving the throughput. In this paper we present a floating point VLIW-Vector Co-processor Architecture with Vector Floating Point Datapath (VFPDP) and a 4-slot VLIW processor core to accelerate BLAOps achieving performance of two GFLOPS when run at 500MHz clock frequency. We also demonstrate a detailed mapping strategy of One sided Jacobi Singular Value Decomposition (OJSVD) algorithm onto the proposed architecture. The proposed architecture is designed using Verilog HDL and it is synthesized using Synopsis Design Compiler with 28nm TSMC target libraries. The clock period is set to 2ns and the timing constraints are met. Using the Altera's SOPC builder, an experimental system is created with the co-processor interfaced to the NIOS II soft processor and implemented in Cyclone IV FPGA. The OJSVD algorithm is ported onto both the standalone NIOS II processor based system and the system with the proposed co-processor. The results show that 15X performance improvement achieved with this co-processor.

In this second part, we thoroughly examine the types of higher-order asymptotic variation of a function obtained by all possible basic algebraic operations on higher-order varying functions. The pertinent proofs are somewhat demanding except when all the involved functions are regularly varying. Next, we give an exposition of three types of exponential variation with an exhaustive list of various asymptotic functional equations satisfied by these functions and detailed results concerning operations on them. Simple applications to integrals of a product and asymptotic behavior of sums are given. The paper concludes with applications of higher-order regular, rapid or exponential variation to asymptotic expansions for an expression of type f(x+r(x)).

With the development of cloud storage, data owners no longer physically possess their data and thus how to ensure the integrity of their outsourced data becomes a challenging task. Several protocols have been proposed to audit cloud storage, all of which rely mainly on data block tags to check data integrity. However, their block tag constructions employ cryptographic operations, which makes them computationally complex. In this paper, we investigate a secure cloud storage protocol based on the classic discrete logarithm problem. Our protocol generates data block tags with only basic algebraic operations, which brings substantial computation savings compared with previous work. We also strictly prove that the proposed protocol is secure under a definition which captures the real-world uses of cloud storage. In order to fit more application scenarios, we extend the proposed protocol to support data dynamics by employing an index vector and third-party public auditing by using a random masking number, both of which are efficient and provably secure. At last, theoretical analysis and experimental evaluation are provided to validate the superiority of the proposed protocol.

For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.

A program, written for the IBM 7090, which defines a model for a finite collection of associative algebras for the computer, is described and illustrated. The program contains a structure broad enough in scope to allow one to perform operations on such diverse mathematical concepts as differential equations, infinite series and differential forms in a simple yet comprehensive manner, while also serving in a foundation upon which a variety of higher level symbol manipulation languages can be developed. The development of the structure of this model is given. In particular, it is assumed that a family of not necessarily commutative algebras defined over a collection of commutative rings is given, where the algebras are algebras of modules generated by recursive sets. A brief discussion is given on how the vectors and scalars of these algebras should be represented, and how their basic algebraic operations (addition and multiplication) can be handled in a systematic manner without regard to the specific underlying scalar rings involved. Functional expressions are treated by developing a procedure for assigning an arbitrary list of operators to an algebra, and then assigning to each operator a list of assumptions that it is to satisfy. The latter is accomplished through the use of what are called "axiom" or "partial evaluation" functions. The resulting structure, which is called ALGEBRA, is shown to serve as a model for a family of associative algebras, where the resulting families of equivalence classes of representations for the vectors are uniquely representable. Various related notions, such as topological and decision considerations, are also briefly discussed. After having outlined the development of the system ALGEBRA, polynomial, exterior and series algebras are given as examples to illustrate how the model can be used. It is observed that a wide variety of mathematical data can be handled fairly simply, depending only on the formulation of the problem under consideration. The possibility of constructing various higher level symbol manipulation languages that have a mathematical structure, broad enough in scope to allow an analyst to manipulate various problems in a variety of ways, is also discussed. A programming language called FLAP has already been developed, demonstrating that such a language can indeed be studied and formulated in a consistent manner.

Abstract. We analyzed a great deal of high-resolution Synthetic Aperture Radar (SAR) data of dry-fallen intertidal flats in the German Wadden Sea with respect to the imaging of sediments, macrophytes, and mussels. TerraSAR-X and Radarsat-2 images of five test areas along the German North Sea coast acquired between 2008 and 2013 form the basis for the present investigation and are used to demonstrate that pairs of SAR images, if combined through basic algebraic operations, can already provide useful indicators for morphological changes and for bivalve (oyster and mussel) beds. Depending on the type of sediment, but also on the water level and on environmental conditions (wind speed) exposed sediments may show up on SAR imagery as areas of enhanced, or reduced, radar backscattering. The (multi-temporal) analysis of series of such images allows for the detection of mussel beds, and our results show evidence that also single-acquisition, multi-polarization SAR imagery can be used for that purpose.

Pure matrix states on operator systems

We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $\lambda$ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of $\lambda$-tensor product of $C^*$-algebras has also been discussed.

Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

This paper provides a general theory for characterizing and constructing a decoherence-free (DF) subsystem for an infinite dimensional linear open quantum system. The main idea is that, based on the Heisenberg picture of the dynamics rather than the commonly-taken Schrodinger picture, the notions of controllability and observability in control theory are employed to characterize a DF subsystem. A particularly useful result is a general if and only if condition for a linear system to have a DF component; this condition is used to demonstrate how to actually construct a DF dynamics in some specific examples. It is also shown that, as in the finite dimensional case, we are able to do coherent manipulation and preservation of a state of a DF subsystem.

Many system operators have issued grid code requirements to ensure a reliable and secure operation of power systems with high integration levels of wind generation. Then, wind farms are required to behave as much as possible like conventional power plants equipped with synchronous generators, providing thus active and reactive power regulation and fault ride through capability. In addition, it is necessary to assure that problems of small signal stability do not arise. This chapter deals with advanced control functionalities to improve the performance of double fed induction generators, regarding frequency control and fault ride through capability, and with robust tuning of classical power system stabilizers installed in double fed induction generators to provide damping as an ancillary service. Since fault ride through is a matter of major concern for many system operators, the control functionalities of a static compensator based solution to provide fault ride through capability of wind farms equipped with fixed speed induction generator s are also addressed. The effectiveness of the above mentioned advanced control functionalities is demonstrated through numerical simulations.

The circular Pythagorean fuzzy set is an expansion of the circular intuitionistic fuzzy set (CIFS), in which each component is represented by a circle. Nevertheless, even though CIFS improves the intuitionistic fuzzy set representation, it is still restricted to the inflexible intuitionistic fuzzy interpretation triangle (IFIT) space, where the square sum of membership and nonmembership in a circular Pythagorean fuzzy environment and the sum of membership and nonmembership in a circular intuitionistic fuzzy environment cannot exceed one. To overcome this restriction, we provide a fresh extension of the CIFS called the circular n,m -rung orthopair fuzzy set (C n,m -ROFS), which allows the IFIT region to be expanded or contracted while maintaining the features of CIFS. Consequently, decision makers can assess items over a wider and more flexible range when using a C n,m -ROFS, allowing for the making of more delicate decisions. In addition, we define several basic algebraic and arithmetic operations on C n,m -ROFS, such as intersection, union, multiplication, addition, and scalar multiplication, and we discuss their key characteristics together with some of the known relations over C n,m -ROFS. In addition, we present and study the new circular n,m -rung orthopair fuzzy weighted average/geometric aggregation operators and their properties. Further, a strategy for resolving multicriteria decision-making problems in a C n,m -ROF environment is provided. The suggested strategy is tested on two situations: the best teacher selection problem and the best school selection problem. To confirm and illustrate the efficacy of the suggested methodology, a comparative analysis with the intuitionistic fuzzy weighted average, intuitionistic fuzzy weighted geometric, q-rung orthopair fuzzy weighted averaging, q-rung orthopair fuzzy geometric averaging, circular PFWA max {{\rm{PFWA}}}_{{\rm{\max }}} , and circular PFWA min {{\rm{PFWA}}}_{{\rm{\min }}} operators approaches is also carried out. Ultimately, in the final section, there are discussions and ideas for future research.

The OPTIMA project aims to port and optimize industrial applications and a set of open-source libraries into two novel FPGA-populated HPC systems. Target applications are from the domains of robotics simulation, underground analysis and computational fluid dynamics (CFD), where data processing is based on differential equations, matrix-matrix and matrix-vector operations. Moreover, the OPTIMA OPen Source (OOPS) library will support basic linear algebraic operations, sparse matrix-vector arithmetic, as well as computer-aided engineering (CAE) solvers. The OPTIMA target platforms are JUMAX, an HPC system that couples an AMD Epyc Server with Maxeler FPGA-based Dataflow Engines (DFEs), and server class machines with Alveo FPGA cards installed. Experimental results show that performance on robotic simulation can be enhanced up to 1.2x, and CFD calculations up to 4.7x. Finally, BLAS L1 routines are improved up to 7x, with a performance-per-Watt ratio boost of more than 40x compared to multi-threaded software routines from the Intel Math Kernel Library (MKL) suite when executed on an Intel Xeon server-class machine.