Linearity and iterator types for Gödel’s System

System Open image in new window is a linear λ-calculus with numbers and an iterator, which, although imposing linearity restrictions on terms, has all the computational power of Gödel’s System  Open image in new window . System Open image in new window owes its power to two features: the use of a closed reduction strategy (which permits the construction of an iterator on an open function, but only iterates the function after it becomes closed), and the use of a liberal typing rule for iterators based on iterative types. In this paper, we study these new types, and show how they relate to intersection types. We also give a sound and complete type reconstruction algorithm for System  Open image in new window .     

A methodology for verifying SysML requirements using activity diagrams

Designing complex and critical systems needs a methodology to ensure the correctness of their specifications. Within an overall approach which considers the validation of SysML designs, this paper proposes a methodology for verifying SysML requirements on activity diagrams. The objective is to define a complete process to formalize and verify SysML functional requirements related to activity diagrams. Our contributions lie, first, in the definition of AcTRL (Activity Temporal Requirement Language), a new language for the formalization of functional requirements at SysML level. Second, in the proposed verification methodology which is guided by the Open image in new window verify Open image in new window relationships between SysML requirements and activity diagrams. The verification is enabled by formalizing SysML activities with hierarchical coloured Petri nets (HCPNs) and by automatically translating SysML requirements expressed on AcTRL into temporal logic. Our methodology takes into account the hierarchical structure of SysML activities and their relations with SysML requirements to provide a modular and incremental verification. A case study for a ticket vending machine is presented to illustrate the different steps and the benefits of the proposed methodology.     

Herring–Flicker coupling and thermal quantum correlations in bipartite system

In this paper, we study thermal quantum correlations as quantum discord and entanglement in bipartite system imposed by external magnetic field with Herring–Flicker coupling, i.e., \(J(R)=1.642 e^{-2 R} R^{5/2}+O(R^{2}e^{-2R})\). The Herring–Flicker coupling strength is the function of R, which is the distance between spins and systems carry XXX Heisenberg interaction. By tuning the coupling distance R, temperature and magnetic field quantum correlations can be scaled in the bipartite system. We find the long sustainable behavior of quantum discord in comparison with entanglement over the coupling distance R. We also investigate the situations, where entanglement totally dies but quantum discord exists in the system.     

Fermat,Euler, Wilson - Three Case Studies in Number Theory

We report on computer assisted proofs of three theorems from Number Theory, viz. Fermat’s Little Theorem, Euler’s generalization of Fermat’s statement and Wilson’s Theorem. Common to the formal proofs is that permutation of certain number lists has to be proved, which causes the main effort in the development. We give a short survey of the Open image in new window   system used in this experiment and illustrate the proofs before presenting them formally. We also discuss alternative solutions, report on the required effort and conclude with some experiences gained from this experiment.     

Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

?ukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems ?3,..., Open image in new window ,..., \(\hbox {L}_{\omega }\); (2) the 4-valued logic ? he defined in the last years of his career. Unfortunately, all these systems contain “?ukasiewicz type (modal) paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following ?ukasiewicz’s strategy for defining truth-functional modal logics. The systems we define lack “?ukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics.     

Uniformly Convergent Cubic Nonconforming Element For Darcy–Stokes Problem

In this paper we construct a cubic element named DSC33 for the Darcy–Stokes problem of three-dimensional space. The finite element space \({{\varvec{V}}}_{h}\) for velocity is Open image in new window -conforming, i.e., the normal component of a function in \({{\varvec{V}}}_{h}\) is continuous across the element boundaries, meanwhile the tangential component of a function in \({{\varvec{V}}}_{h}\) is averagely continuous across the element boundaries, hence \({{\varvec{V}}}_{h}\) is \({{\varvec{H}}}^{1}\)-average conforming. We prove that this element is uniformly convergent with respect to the perturbation constant \(\varepsilon \) for the Darcy–Stokes problem. In addition, we construct a discrete de Rham complex corresponding to DSC33 element. The finite element spaces in the discrete de Rham complex can be applied to some singular perturbation problems.     

Thermal quantum correlation and entanglement in the Bose–Hubbard Hamiltonian

We study a two-qutrit system which is described by the Bose–Hubbard Hamiltonian with two external magnetic fields. The entanglement (through the negativity) and quantum correlation (through the geometric discord) between the qutrits are calculated as functions of the magnetic field (B), the temperature (T), the linear and nonlinear coupling constants among two spins (J and K). Then, we compare the effect of these parameters on entanglement and quantum correlation of this system. For some values of system parameters, we show that the negativity is zero while, the geometric discord is nonzero. Moreover, we investigate the effect of finite external magnetic fields direction on these measures. This study leads to some new and interesting results as well.     

Quantum correlations of identical particles subject to classical environmental noise

In this work, we propose a measure for the quantum discord of indistinguishable particles, based on the definition of entanglement of particles given in Wiseman and Vaccaro (Phys Rev Lett 91:097902, 2003. doi: 10.1103/PhysRevLett.91.097902). This discord of particles is then used to evaluate the quantum correlations in a system of two identical bosons (fermions), where the particles perform a quantum random walk described by the Hubbard Hamiltonian in a 1D lattice. The dynamics of the particles is either unperturbed or subject to a classical environmental noise—such as random telegraph, pink or brown noise. The observed results are consistent with those for the entanglement of particles, and we observe that on-site interaction between particles have an important protective effect on correlations against the decoherence of the system.     

On quasi-interpretations of grammar forms

The notions of a grammar form and its interpretations were introduced to describe families of structurally related grammars. Basically, a grammar formG is a (context-free) grammar serving as a master grammar and the interpretation operator defines a family of grammars, each structurally related toG. In this paper, a new operator yielding a family of grammars, is introduced as a variant of . There are two major results. The first is that and commute. The second is that for each grammar formG, the collection of all families of grammars ,G′ in , is finite. Expressed otherwise, the second result is that for each grammar formG there is only a bounded number of grammar forms in (G) no two of which are strongly equivalent.     

A Bell inequality for a class of multilocal ring networks

Quantum networks with independent sources of entanglement (hidden variables) and nodes that execute joint quantum measurements can create strong quantum correlations spanning the breadth of the network. Understanding of these correlations has to the present been limited to standard Bell experiments with one source of shared randomness, bilocal arrangements having two local sources of shared randomness, and multilocal networks with tree topologies. We introduce here a class of quantum networks with ring topologies comprised of subsystems each with its own internally shared source of randomness. We prove a Bell inequality for these networks, and to demonstrate violations of this inequality, we focus on ring networks with three-qubit subsystems. Three qubits are capable of two non-equivalent types of entanglement, GHZ and W-type. For rings of any number N of three-qubit subsystems, our inequality is violated when the subsystems are each internally GHZ-entangled. This violation is consistently stronger when N is even. This quantitative even-odd difference for GHZ entanglement becomes extreme in the case of W-type entanglement. When the ring size N is even, the presence of W-type entanglement is successfully detected; when N is odd, the inequality consistently fails to detect its presence.     

Geometric quantum discord under noisy environment

In this work, we mainly analyze the dynamics of geometric quantum discord under a common dissipating environment. Our results indicate that geometric quantum discord is generated when the initial state is a product state. The geometric quantum discord increases from zero to a stable value with the increasing time, and the variations of stable values depend on the system size. For different initial product states, geometric quantum discord has some different behaviors in contrast with entanglement. For initial maximally entangled state, it is shown that geometric quantum discord decays with the increasing dissipated time. It is found that for EPR state, entanglement is more robust than geometric quantum discord, which is a sharp contrast to the existing result that quantum discord is more robust than entanglement in noisy environments. However, for GHZ state and W state, geometric quantum discord is more stable than entanglement. By the comparison of quantum discord and entanglement, we find that a common dissipating environment brings complicated effects on quantum correlation, which may deepen our understanding of physical impacts of decohering environment on quantum correlation. In the end, we analyze the effects of collective dephasing noise and rotating noise to a class of two-qubit X states, and we find that quantum correlation is not altered by the collective noises.     

Wigner–Yanase skew information and entanglement generation in quantum measurement

The first step of quantum measurement procedure is known as premeasurement, during which correlation is established between the system and the measurement apparatus. Such correlation may be classical or nonclassical in nature. One compelling nonclassical correlation is entanglement, a useful resource for various quantum information theoretic protocols. Quantifying the amount of entanglement, generated during quantum measurement, therefore, seeks importance from practical ground, and this is the central issue of the present paper. Interestingly, for a two-level quantum system, we obtain that the amount of entanglement, measured in term of negativity, generated in premeasurement process can be quantified by two factors: skew information, which quantifies the uncertainty in the measurement of an observable not commuting with some conserved quantity of the system, and mixedness parameter of the system’s initial state.     

Super-quantum correlation for SU(2) invariant state in $$4\otimes 2$$ system

We analytically evaluate the weak one-way deficit and super-quantum discord for a system composed of spin-3/2 and spin-1/2 subsystems possessing SU(2) symmetry. We also make a comparative study of the relationships among the quantum discord, one-way deficit, weak one-way deficit, and super-quantum discord for the SU(2) invariant state. It is shown that super-quantum discord via weak measurement is greater than that via von Neumann measurement. But weak one-way deficit is less than the one-way deficit. As a result, weak measurement do not always reveal more quantumness.     

On the approximate solution of first-kind integral equations of Volterra type

The present paper deals with the approximate solution of integral equations of the first kind, ( y)x∈I:=[a, b], where denotes a (linear) integral operator of Volterra (or Abel) type, and whereg∈C(I), withg(a)=0. The given functiong is approximated uniformly onI (or on a finite subsetZ?I by using certain weak Chebyshev systems onI which are obtained in a natural way. By the linearity of this yields an approximation to the exact solutiony onI. Questions of uniqueness and characterization of such approximating functions, as well as numerical aspects of the approximation problem are discussed.     

Improved Bounds on Quantum Learning Algorithms

In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples.

Enclosing the solution set of linear systems with inaccurate data by iterative methods based on incomplete LU-decompositions

We present a class of iterative methods to enclose the solution set by an interval vector;A is varying in ann×n intervalH-Matrix andb is varying in an interval vector . The algorithm taken into consideration generalizes an iterative method of Meijerink/van der Vorst based on an incompleteLU-decomposition of anM-MatrixA. Theorems concerning the feasibility of the algorithm, its rate of convergence and its quality of enclosure are given. Since the original method of Meijerink/van der Vorst is a special case of our algorithm we have thus shown its applicability to the larger class ofH-matrices. Furthermore we relate theR ₁-factor (as defined in Ortega/Rheinboldt [9]) of the original method to the underlying setP of indices.     

An <Emphasis Type="Italic">R</Emphasis> || <Emphasis Type="Italic">C</Emphasis><Subscript>max</Subscript> Quantum Scheduling Algorithm

Grover’s search algorithm can be applied to a wide range of problems; even problems not generally regarded as searching problems, can be reformulated to take advantage of quantum parallelism and entanglement, and lead to algorithms which show a square root speedup over their classical counterparts. In this paper, we discuss a systematic way to formulate such problems and give as an example a quantum scheduling algorithm for an R||C_max problem. R||C_max is representative for a class of scheduling problems whose goal is to find a schedule with the shortest completion time in an unrelated parallel machine environment. Given a deadline, or a range of deadlines, the algorithm presented in this paper allows us to determine if a solution to an R||C_max problem with N jobs and M machines exists, and if so, it provides the schedule. The time complexity of the quantum scheduling algorithm is while the complexity of its classical counterpart is .     

Efficient hyperentanglement concentration for <Emphasis Type="Italic">N</Emphasis>-particle Greenberger–Horne–Zeilinger state assisted by weak cross-Kerr nonlinearity

In this scheme, based on the weak cross-Kerr nonlinearity, an hyperconcentration protocol for the arbitrary partially hyperentangled N-particle Greenberger–Horne–Zeilinger (GHZ) state is presented. Considering the N photons initially in the nonmaximally hyperentangled GHZ state in which photons are entangled simultaneously in the polarization and the spatial-mode degrees of freedom, we can obtain the maximally hyperentangled N-particle GHZ state by the projection measurements on the additional photons. Numerical simulation demonstrates that by iterating the entanglement concentration process, we can improve the success probability of the scheme. Furthermore, we discuss the feasibility of the setups of the protocol, concluding that the present protocol is feasible with existing experimental technology. All these advantages make this scheme more efficient and more convenient in quantum communication.     

Spatial dependence of entanglement renormalization in <Emphasis Type="Italic">XY</Emphasis> model

In this article, a comparative study of the renormalization of entanglement in one-, two- and three-dimensional space and its relation with quantum phase transition (QPT) near the critical point is presented by implementing the quantum renormalization group (QRG) method using numerical techniques. Adopting the Kadanoff’s block approach, numerical results for the concurrence are obtained for the spin \({-}\)1/2 XY model in all the spatial dimensions. The results show similar qualitative behavior as we move from the lower to the higher dimensions in space, but the number of iterations reduces for achieving the QPT in the thermodynamic limit. We find that in the two-dimensional and three-dimensional spin \({-}\)1/2 XY model, maximum value of the concurrence reduce by the factor of 1 / n \((n=2,3)\) with reference to the maximum value of one-dimensional case. Moreover, we study the scaling behavior and the entanglement exponent. We compare the results for one-, two- and three-dimensional cases and illustrate how the system evolves near the critical point.