A desired state can not be found with certainty for Grover��s algorithm in a possible three-dimensional complex subspace

Using an accurate method, we prove that no matter what the initial superposition may be, neither a superposition of desired states nor a unique desired state can be found with certainty in a possible three-dimensional complex subspace, provided that the deflection angle Φ is not exactly equal to zero. By this method, we derive such a result that, if N is sufficiently large (where N denotes the total number of the desired and undesired states in an unsorted database), then corresponding to the case of identical rotation angles, the maximum success probability of finding a unique desired state is approximately equal to cos² Φ for any given F ? [0,p/2){\Phi\in\left[0,\pi/2\right)}.     

The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

We prove asymptotically optimal bounds on the Gaussian noise sensitivity and Gaussian surface area of degree-d polynomial threshold functions. In particular, we show that for f a degree-d polynomial threshold function that the Gaussian noise sensitivity of f with parameter e{\epsilon} is at most \fracdarcsin(?{2e-e²})p{\frac{d\arcsin\left(\sqrt{2\epsilon-\epsilon^2}\right)}{\pi}} . This bound translates into an optimal bound on the Gaussian surface area of such functions, namely that the Gaussian surface area is at most \fracd?{2p}{\frac{d}{\sqrt{2\pi}}} . Finally, we note that the later result implies bounds on the runtime of agnostic learning algorithms for polynomial threshold functions.     

Distributed discovery of large near-cliques

Given an undirected graph and 0 ￡ e ￡ 1{0\le\epsilon\le1}, a set of nodes is called an e{\epsilon}-near clique if all but an e{\epsilon} fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size e{\epsilon}-near clique if there exists an e³{\epsilon^3}-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n ^−Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) ^α ) for some a ? (0,1){\alpha \in (0,1)}. Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.     

Attractive energy contribution to nanoconfined fluids behavior: the normal pressure tensor

The aim of our research is to demonstrate the role of attractive intermolecular potential energy on normal pressure tensor of confined molecular fluids inside nanoslit pores of two structureless purely repulsive parallel walls in xy plane at z = 0 and z = H, in equilibrium with a bulk homogeneous fluid at the same temperature and at a uniform density. To achieve this we have derived the perturbation theory version of the normal pressure tensor of confined inhomogeneous fluids in nanoslit pores:

Entanglement and Berry phase in a 9 × 9 Yang–Baxter system

A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang–Baxterization approach, we obtain a unitary solution \breveR(q,j₁,j₂){\breve{R}(\theta,\varphi_{1},\varphi_{2})} of Yang–Baxter equation. It is shown that any pure two-qutrit entangled states can be generated via the universal \breveR{\breve{R}}-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the \breveR{\breve{R}}-matrix, and Berry phase of the Yang–Baxter system is investigated. Specifically, for j₁ = j₂{\varphi_{1}\,{=}\,\varphi_{2}}, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.     

The sudden death of entanglement in constructed Yang–Baxter systems

Some Hamiltonians are constructed from the unitary \checkR_i,i+1(q, j){\check{R}_{i,i+1}(\theta, \varphi)}-matrices, where θ and j{\varphi} are time-independent parameters. We show that the entanglement sudden death (ESD) can happen in these closed Yang–Baxter systems. It is found that the ESD is not only sensitive to the initial condition, but also has a great connection with different Yang–Baxter systems. Especially, we find that the meaningful parameter j{\varphi} has a great influence on ESD.     

Entanglement and Yangian in a {V^{\otimes 3}} Yang-Baxter system

In this paper, a 8 × 8 unitary Yang-Baxter matrix \breveR₁₂₃(q₁,q₂,f){\breve{R}_{123}(\theta_{1},\theta_{2},\phi)} acting on the triple tensor product space, which is a solution of the Yang-Baxter Equation for three qubits, is presented. Then quantum entanglement and the Berry phase of the Yang-Baxter system are studied. The Yangian generators, which can be viewed as the shift operators, are investigated in detail. And it is worth mentioning that the Yangian operators we constructed are independent of choice of basis.     

How the Hawking radiation affect quantum Fisher information of Dirac particles in the background of a Schwarzschild black hole

In this work, the effect of Hawking radiation on the quantum Fisher information (QFI) of Dirac particles is investigated in the background of a Schwarzschild black hole. Interestingly, it has been verified that the QFI with respect to the weight parameter \(\theta \) of a target state is always independent of the Hawking temperature T. This implies that if we encode the information on the weight parameter, then we can affirm that the corresponding accuracy of the parameter estimation will be immune to the Hawking effect. Besides, it reveals that the QFI with respect to the phase parameter \(\phi \) exhibits a decay behavior with the increase in the Hawking temperature T and converges to a nonzero value in the limit of infinite Hawking temperature T. Remarkably, it turns out that the function \(F_\phi \) on \(\theta =\pi \big /4\) symmetry was broken by the influence of the Hawking radiation. Finally, we generalize the case of a three-qubit system to a case of a N-qubit system, i.e., \(|\psi \rangle _{1,2,3,\ldots ,N} =(\cos \theta | 0 \rangle ^{\otimes N}+\sin \theta \mathrm{e}^{i\phi }| 1 \rangle ^{\otimes N})\) and obtain an interesting result: the number of particles in the initial state does not affect the QFI \(F_\theta \), nor the QFI \(F_\phi \). However, with the increasing number of particles located near the event horizon, \(F_\phi \) will be affected by Hawking radiation to a large extent, while \(F_\theta \) is still free from disturbance resulting from the Hawking effects.     

On some sets of difference sequences of fuzzy numbers

In this paper we define the sequence space w_F(f,p,\Updelta){w}_{{\mathcal{F}}}\left(f,p,\Updelta\right) which is called the space of strongly \Updelta p\Updelta p-Cesàro summable sequences with modulus f. Furthermore the fuzzy Δ-statistically pre-Cauchy sequence is defined and the necessary and sufficient conditions are given for a sequence of fuzzy numbers to be fuzzy \Updelta\Updelta-statistically pre-Cauchy and to be fuzzy \Updelta\Updelta-statistically convergent. Also some relations between w_F(f,p,\Updelta)w_{{\mathcal{F}}}(f,p,\Updelta) and S_F(\Updelta){S}_{{\mathcal{F}}}(\Updelta) are given.     

Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

Two generalized Wigner–Yanase skew information and their uncertainty relations

In this paper, we first define two generalized Wigner–Yanase skew information \(|K_{\rho ,\alpha }|(A)\) and \(|L_{\rho ,\alpha }|(A)\) for any non-Hermitian Hilbert–Schmidt operator A and a density operator \(\rho \) on a Hilbert space H and discuss some properties of them, respectively. We also introduce two related quantities \(|S_{\rho ,\alpha }|(A)\) and \(|T_{\rho ,\alpha }|(A)\). Then, we establish two uncertainty relations in terms of \(|W_{\rho ,\alpha }|(A)\) and \(|\widetilde{W}_{\rho ,\alpha }|(A)\), which read

$$\begin{aligned}&|W_{\rho ,\alpha }|(A)|W_{\rho ,\alpha }|(B)\ge \frac{1}{4}\left| \mathrm {tr}\left( \left[ \frac{\rho ^{\alpha }+\rho ^{1-\alpha }}{2} \right] ^{2}[A,B]^{0}\right) \right| ^{2},\\&\sqrt{|\widetilde{W}_{\rho ,\alpha }|(A)| \widetilde{W}_{\rho ,\alpha }|(B)}\ge \frac{1}{4} \left| \mathrm {tr}\left( \rho ^{2\alpha }[A,B]^{0}\right) \mathrm {tr} \left( \rho ^{2(1-\alpha )}[A,B]^{0}\right) \right| . \end{aligned}$$

     

Fuzzy sets and cut systems in a category of sets with similarity relations

Let \Upomega\Upomega be a complete residuated lattice. Let SetR(\Upomega){\mathbf{SetR}}(\Upomega) be the category of sets with similarity relations with values in \Upomega\Upomega (called \Upomega\Upomega-sets), which is an analogy of the category of classical sets with relations as morphisms. A fuzzy set in an \Upomega\Upomega-set in the category SetR(\Upomega){\mathbf{SetR}}(\Upomega) is a morphism from \Upomega\Upomega-set to a special \Upomega\Upomega-set (\Upomega,?),(\Upomega,\leftrightarrow), where ?\leftrightarrow is the biresiduation operation in \Upomega.\Upomega. In the paper, we prove that fuzzy sets in \Upomega\Upomega-sets in the category SetR(\Upomega){\mathbf{SetR}}(\Upomega) can be expressed equivalently as special cut systems (C_a)_{a ? \Upomega}.(C_{\alpha})_{\alpha\in\Upomega}.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Computational Divided Differencing and Divided-Difference Arithmetics

Tools for computational differentiation transform a program that computes a numerical function F(x) into a related program that computes F(x) (the derivative of F). This paper describes how techniques similar to those used in computational-differentiation tools can be used to implement other program transformations—in particular, a variety of transformations for computational divided differencing. The specific technical contributions of the paper are as follows:– It presents a program transformation that, given a numerical function F(x) defined by a program, creates a program that computes F[x ₀, x ₁], the first divided difference of F(x), where – It shows how computational first divided differencing generalizes computational differentiation.– It presents a second program transformation that permits the creation of higher-order divided differences of a numerical function defined by a program.– It shows how to extend these techniques to handle functions of several variables.The paper also discusses how computational divided-differencing techniques could lead to faster and/or more robust programs in scientific and graphics applications.Finally, the paper describes how computational divided differencing relates to the numerical-finite-differencing techniques that motivated Robert Paige's work on finite differencing of set-valued expressions in SETL programs.     

On the weakest failure detector ever

Many problems in distributed computing are impossible to solve when no information about process failures is available. It is common to ask what information about failures is necessary and sufficient to circumvent some specific impossibility, e.g., consensus, atomic commit, mutual exclusion, etc. This paper asks what information about failures is necessary to circumvent any impossibility and sufficient to circumvent some impossibility. In other words, what is the minimal yet non-trivial failure information. We present an abstraction, denoted U{\Upsilon} , that provides very little information about failures. In every run of the distributed system, U{\Upsilon} eventually informs the processes that some set of processes in the system cannot be the set of correct processes in that run. Although seemingly weak, for it might provide random information for an arbitrarily long period of time, and it eventually excludes only one set of processes (among many) that is not the set of correct processes in the current run, U{\Upsilon} still captures non-trivial failure information. We show that U{\Upsilon} is sufficient to circumvent the fundamental wait-free set-agreement impossibility. While doing so, (a) we disprove previous conjectures about the weakest failure detector to solve set-agreement and (b) we prove that solving set-agreement with registers is strictly weaker than solving n + 1-process consensus using n-process consensus. We show that U{\Upsilon} is the weakest stable non-trivial failure detector: any stable failure detector that circumvents some wait-free impossibility provides at least as much information about failures as U{\Upsilon} does. Our results are generalized, from the wait-free to the f-resilient case, through an abstraction U^f{\Upsilon^f} that we introduce and prove minimal to solve any problem that cannot be solved in an f-resilient manner, and yet sufficient to solve f-resilient f-set-agreement.     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in Wu et al. (Phys Rev E 93:032151, 2016), being called the time-tempered fractional Feynman–Kac equation named after Richard Feynman and Mark Kac who first considered the model describing the functional distribution of normal motion. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as

$$\begin{aligned} {^S\!}D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!=\!D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t)\!-\!\lambda ^\gamma G(x,p,t) \end{aligned}$$

with \(\widetilde{\lambda }=\lambda + pU(x),\, p=\rho +J\eta ,\, J=\sqrt{-1}\), where

$$\begin{aligned} D_t^{\gamma ,\widetilde{\lambda }} G(x,p,t) =\frac{1}{\varGamma (1-\gamma )} \left[ \frac{\partial }{\partial t}+\widetilde{\lambda } \right] \int _{0}^t{\left( t-z\right) ^{-\gamma }}e^{-\widetilde{\lambda }\cdot (t-z)}{G(x,p,z)}dz, \end{aligned}$$

and \(\lambda \ge 0\), \(0<\gamma <1\), \(\rho >0\), and \(\eta \) is a real number. The designed schemes are unconditionally stable and have the global truncation error \(\mathscr {O}(\tau ^2+h^2)\), being theoretically proved and numerically verified in complex space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the ‘physical’ equation (without artificial source term).

     

Lower estimation of approximation rate for neural networks

Let SFd and Πψ,n,d = { nj=1bjψ(ωj·x+θj) :bj,θj∈R,ωj∈Rd} be the set of periodic and Lebesgue’s square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SF d, Πψ,n,d) the deviation of the set SF d from the set Πψ,n,d. A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd, Πψ,n,d) is proved. That is, dist(SF d, Πψ,n,d) (nlogC2n)1/2 . T...     

Fuzzy n-ary hypergroups related to fuzzy points

In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called $(\in,\in \vee q)In this paper, we introduce and study a new sort of fuzzy n-ary sub-hypergroups of an n-ary hypergroup, called ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup. By using this new idea, we consider the ( ? , ? úq)(\in,\in\vee q)-fuzzy n-ary sub-hypergroup of a n-ary hypergroup. This newly defined ( ? , ? úq)(\in,\in \vee q)-fuzzy n-ary sub-hypergroup is a generalization of the usual fuzzy n-ary sub-hypergroup. Finally, we consider the concept of implication-based fuzzy n-ary sub-hypergroup in an n-ary hypergroup and discuss the relations between them, in particular, the implication operators in £\poundsukasiewicz system of continuous-valued logic are discussed.     

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Given a nonempty set of functions