A Stable Fast Time-Stepping Method for Fractional Integral and Derivative Operators

A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length \(\varDelta T\) and a history part, where the local part is approximated by the direct convolution method and the history part is approximated by a fast memory-saving method. The fast method has \(O(n_0+\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))\) active memory and \(O(n_0n_T+ (n_T-n_0)\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))\) operations, where \(L=\log (n_T-n_0)\), \(n_0={\varDelta T}/\tau ,n_T=T/\tau \), \(\tau \) is the stepsize, T is the final time, and \({q}_{\alpha }{(N_{\ell })}\) is the number of quadrature points used in the truncated Laguerre–Gauss (LG) quadrature. The error bound of the present fast method is analyzed. It is shown that the error from the truncated LG quadrature is independent of the stepsize, and can be made arbitrarily small by choosing suitable parameters that are given explicitly. Numerical examples are presented to verify the effectiveness of the current fast method.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

Mutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$

A way of constructing special entangled basis with fixed Schmidt number 2 (SEB2) in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) is proposed, and the conditions mutually unbiased SEB2s (MUSEB2s) satisfy are discussed. In addition, a very easy way of constructing MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\) is presented. We first establish the concrete construction of SEB2 and MUSEB2s in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4}\) and \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{8}\), respectively, and then generalize them into \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k\in z^+,3\not \mid k)\) and display the condition that MUSEB2s satisfy; we also give general form of two MUSEB2s as examples in \({\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}(k=2^l)\).     

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

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

     

Entanglement witness criteria of strong-<Emphasis Type="Italic">k</Emphasis>-separability for multipartite quantum states

Let \(H_{1}, H_{2},\ldots ,H_{n}\) be separable complex Hilbert spaces with \(\dim H_{i}\ge 2\) and \(n\ge 2\). Assume that \(\rho \) is a state in \(H=H_1\otimes H_2\otimes \cdots \otimes H_n\). \(\rho \) is called strong-k-separable \((2\le k\le n)\) if \(\rho \) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that \(\rho \) is not strong-k-separable if and only if there exist a k-division space \(H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}\) of H, a finite-rank linear elementary operator positive on product states \(\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})\) and a state \(\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})\), such that \(\mathrm {Tr}(W\rho )<0\), where \(W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}\) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.     

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete \(L^{\infty }(L^2)\) and \(L^2(H^1)\) errors are \(O(\triangle t^{2-\alpha }+h^2+H^3)\). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.     

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).

     

The Outer-connected Domination Number of Sierpiński-like Graphs

An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ? V satisfying the condition that, for each vertex v ? D, vertex v is adjacent to some vertex in D and the subgraph induced by V?D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by \(\tilde {\gamma }_{c}(G)\). In this paper, we determine \(\tilde {\gamma }_{c}(S(n,k))\), \(\tilde {\gamma }_{c}(S^{+}(n,k))\), \(\tilde {\gamma }_{c}(S^{++}(n,k))\), and \(\tilde {\gamma }_{c}(S_{n})\), where S(n, k), S ⁺(n, k), S ⁺⁺(n, k), and S _n are Sierpi\(\acute {\mathrm {n}}\)ski-like graphs.     

Master Lovas–Andai and equivalent formulas verifying the $$\frac{8}{33}$$ two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

We begin by investigating relationships between two forms of Hilbert–Schmidt two-rebit and two-qubit “separability functions”—those recently advanced by Lovas and Andai (J Phys A Math Theor 50(29):295303, 2017), and those earlier presented by Slater (J Phys A 40(47):14279, 2007). In the Lovas–Andai framework, the independent variable \(\varepsilon \in [0,1]\) is the ratio \(\sigma (V)\) of the singular values of the \(2 \times 2\) matrix \(V=D_2^{1/2} D_1^{-1/2}\) formed from the two \(2 \times 2\) diagonal blocks (\(D_1, D_2\)) of a \(4 \times 4\) density matrix \(D= \left||\rho _{ij}\right||\). In the Slater setting, the independent variable \(\mu \) is the diagonal-entry ratio \(\sqrt{\frac{\rho _{11} \rho _ {44}}{\rho _ {22} \rho _ {33}}}\)—with, of central importance, \(\mu =\varepsilon \) or \(\mu =\frac{1}{\varepsilon }\) when both \(D_1\) and \(D_2\) are themselves diagonal. Lovas and Andai established that their two-rebit “separability function” \(\tilde{\chi }_1 (\varepsilon )\) (\(\approx \varepsilon \)) yields the previously conjectured Hilbert–Schmidt separability probability of \(\frac{29}{64}\). We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts, \(\tilde{\chi _2}(\varepsilon ) =\frac{1}{3} \varepsilon ^2 \left( 4-\varepsilon ^2\right) \), \(\tilde{\chi _4}(\varepsilon ) =\frac{1}{35} \varepsilon ^4 \left( 15 \varepsilon ^4-64 \varepsilon ^2+84\right) \) and \(\tilde{\chi _8} (\varepsilon )= \frac{1}{1287}\varepsilon ^8 \left( 1155 \varepsilon ^8-7680 \varepsilon ^6+20160 \varepsilon ^4-25088 \varepsilon ^2+12740\right) \). These immediately lead to predictions of Hilbert–Schmidt separability/PPT-probabilities of \(\frac{8}{33}\), \(\frac{26}{323}\) and \(\frac{44482}{4091349}\), in full agreement with those of the “concise formula” (Slater in J Phys A 46:445302, 2013), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas–Andai “master formula,” \(\tilde{\chi _d}(\varepsilon )= \frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left( -\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right) }{\Gamma \left( \frac{d}{2}+1\right) ^2}\), encompassing both even and odd values of d. Remarkably, we are able to obtain the \(\tilde{\chi _d}(\varepsilon )\) formulas, \(d=1,2,4\), applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal \(D_1\) and \(D_2\), but also an additional pair of nullified entries. Nullification of a further pair still leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of \(1-\frac{256}{27 \pi ^2}\) is obtained based on the operator monotone function \(\sqrt{x}\), with the use of \(\tilde{\chi _2}(\varepsilon )\).     

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system \(\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)\). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of \((R,+)\) instead of \(\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}\). Particularly, if \(d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}\) where \(p_1, \ldots , p_s\) are distinct primes and \(3\le p_1^{a_1}\le \cdots \le p_s^{a_s}\), we present \(p_1^{a_1}-1\) MUMEB’s in \(\mathbb {C}^d\otimes \mathbb {C}^d\) by taking \(R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}\), direct sum of finite fields (Theorem 3.3).     

A Linearized Local Conservative Mixed Finite Element Method for Poisson–Nernst–Planck Equations

In this paper, a linearized local conservative mixed finite element method is proposed and analyzed for Poisson–Nernst–Planck (PNP) equations, where the mass fluxes and the potential flux are introduced as new vector-valued variables to equations of ionic concentrations (Nernst–Planck equations) and equation of the electrostatic potential (Poisson equation), respectively. These flux variables are crucial to PNP equations on determining the Debye layer and computing the electric current in an accurate fashion. The Raviart–Thomas mixed finite element is employed for the spatial discretization, while the backward Euler scheme with linearization is adopted for the temporal discretization and decoupling nonlinear terms, thus three linear equations are separately solved at each time step. The proposed method is more efficient in practice, and locally preserves the mass conservation. By deriving the boundedness of numerical solutions in certain strong norms, an unconditionally optimal error analysis is obtained for all six unknowns: the concentrations p and n, the mass fluxes \({{\varvec{J}}}_p=\nabla p + p {\varvec{\sigma }}\) and \({{\varvec{J}}}_n=\nabla n - n {\varvec{\sigma }}\), the potential \(\psi \) and the potential flux \({\varvec{\sigma }}= \nabla \psi \) in \(L^{\infty }(L^2)\) norm. Numerical experiments are carried out to demonstrate the efficiency and to validate the convergence theorem of the proposed method.     

Local Discontinuous Galerkin Method for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we develop local discontinuous Galerkin method for the two-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in \(L^{\infty }(0, T; L^{2})\) for concentration c, \(L^{2}(0, T; L^{2})\) for \(\nabla c\) and \(L^{\infty }(0, T; L^{2})\) for velocity \(\mathbf{u}\) are derived. The main techniques in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments are shown to demonstrate the theoretical results.     

A New Collocation Scheme Using Non-polynomial Basis Functions

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator \({\mathscr {L}}_\lambda \) being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.     

Quantum one-way permutation over the finite field of two elements

In quantum cryptography, a one-way permutation is a bounded unitary operator \(U:\mathcal {H} \rightarrow \mathcal {H}\) on a Hilbert space \(\mathcal {H}\) that is easy to compute on every input, but hard to invert given the image of a random input. Levin (Probl Inf Transm 39(1):92–103, 2003) has conjectured that the unitary transformation \(g(a,x)=(a,f(x)+ax)\), where f is any length-preserving function and \(a,x \in \hbox {GF}_{{2}^{\Vert x\Vert }}\), is an information-theoretically secure operator within a polynomial factor. Here, we show that Levin’s one-way permutation is provably secure because its output values are four maximally entangled two-qubit states, and whose probability of factoring them approaches zero faster than the multiplicative inverse of any positive polynomial poly(x) over the Boolean ring of all subsets of x. Our results demonstrate through well-known theorems that existence of classical one-way functions implies existence of a universal quantum one-way permutation that cannot be inverted in subexponential time in the worst case.     

Weighted Finite Fourier Transform Operator: Uniform Approximations of the Eigenfunctions,Eigenvalues Decay and Behaviour

In this paper, we investigate two uniform asymptotic approximations as well as some spectral properties of the eigenfunctions of the weighted finite Fourier transform operator, defined by \({\displaystyle {\mathcal {F}}_c^{(\alpha )} f(x)=\int _{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha }\, dy.}\) Here, \( c >0, \alpha \ge -1/2\) are two fixed real numbers. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs) and they are firstly introduced and studied in Wang and Zhang (Appl Comput Harmon Anal 29(3):303–329, 2010). The present study is motivated by the promising concrete applications of the GPSWFs in various scientific area such as numerical analysis, mathematical physics and signal processing. We should mention that these two uniform approximation results of the GPSWFs can be considered as generalizations of the results given in the joint work of one of us (Bonami and Karoui in Constr Approx 43(1):15–45, 2016). As it will be seen, these generalizations require some involved extra work, especially in the case where \(\alpha > 1/2.\) By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator \({\mathcal {F}}_c^{(\alpha )}\) in the case where \(0<\alpha < 3/2.\) Moreover, by computing the trace and an estimate of the norm of the operator \({\displaystyle {\mathcal {Q}}_c^{(\alpha )}=\frac{c}{2\pi } {\mathcal {F}}_c^{{(\alpha )}^*} \circ {\mathcal {F}}_c^{(\alpha )},}\) we give a lower bound for the counting number of the eigenvalues of \(Q_c^{(\alpha )},\) when \(c>>1.\) Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

Cubature formulae for nearly singular and highly oscillating integrals

The paper deals with the approximation of integrals of the type

$$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$

where \({\mathrm {D}}=[-\,1,1]^2\), f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\), \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed.

     

Constructive quantum scaling of unitary matrices

In this work, we present a method of decomposition of arbitrary unitary matrix \(U\in \mathbf {U}(2^k)\) into a product of single-qubit negator and controlled-\(\sqrt{\text{ NOT }}\) gates. Since the product results with negator matrix, which can be treated as complex analogue of bistochastic matrix, our method can be seen as complex analogue of Sinkhorn–Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively, and resulting circuit consists of \(O(4^k)\) entangling gates, which is proved to be optimal. An example of such transformation is presented.     

Cost-efficient design of a quantum multiplier–accumulator unit

This paper proposes a cost-efficient quantum multiplier–accumulator unit. The paper also presents a fast multiplication algorithm and designs a novel quantum multiplier device based on the proposed algorithm with the optimum time complexity as multiplier is the major device of a multiplier–accumulator unit. We show that the proposed multiplication technique has time complexity \(O((3 {\hbox {log}}_{2}n)+1)\), whereas the best known existing technique has \(O(n{\hbox {log}}_{2} n)\), where n is the number of qubits. In addition, our design proposes three new quantum circuits: a circuit representing a quantum full-adder, a circuit known as quantum ANDing circuit, which performs the ANDing operation and a circuit presenting quantum accumulator. Moreover, the proposed quantum multiplier–accumulator unit is the first ever quantum multiplier–accumulator circuit in the literature till now, which has reduced garbage outputs and ancillary inputs to a great extent. The comparative study shows that the proposed quantum multiplier performs better than the existing multipliers in terms of depth, quantum gates, delays, area and power with the increasing number of qubits. Moreover, we design the proposed quantum multiplier–accumulator unit, which performs better than the existing ones in terms of hardware and delay complexities, e.g., the proposed (\(n\times n\))—qubit quantum multiplier–accumulator unit requires \(O(n^{2})\) hardware and \(O({\hbox {log}}_{2}n)\) delay complexities, whereas the best known existing quantum multiplier–accumulator unit requires \(O(n^{3})\) hardware and \(O((n-1)^{2} +1+n)\) delay complexities. In addition, the proposed design achieves an improvement of 13.04, 60.08 and 27.2% for \(4\times 4\), 7.87, 51.8 and 27.1% for \(8\times 8\), 4.24, 52.14 and 27% for \(16\times 16\), 2.19, 52.15 and 27.26% for \(32 \times 32\) and 0.78, 52.18 and 27.28% for \(128 \times 128\)-qubit multiplications over the best known existing approach in terms of number of quantum gates, ancillary inputs and garbage outputs, respectively. Moreover, on average, the proposed design gains an improvement of 5.62% in terms of area and power consumptions over the best known existing approach.     

A unified framework for two-grid methods for a class of nonlinear problems

In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence results indicate that the correct scaling between fine and coarse meshes is given by \(h={{\mathcal {O}}}(H^2)\) for all the nonlinear problems which can be written in and applied to the BRR framework. Next, a correction step can be added to the two-grid algorithm, which allows the choice \(h={\mathcal O}(H^3)\). On the other hand, the particular BRR framework with duality pairing, if it is applied to a semilinear problem, allows a higher order relation \(h={{\mathcal {O}}}(H^4)\). Furthermore, even the choice \(h={{\mathcal {O}}}(H^5)\) is possible with the correction step either on fine mesh or coarse mesh. In addition, elliptic problems with gradient nonlinearities and the Naiver–Stokes equations are considered to illustrate our unified theory. Finally, numerical experiments are conducted to confirm our theoretical findings. Numerical results indicate that the correction step used as a simple postprocessing enhances the solution accuracy, particularly for problems with layers.     

Symmetric triangular approximations of fuzzy numbers under a general condition and properties

We consider the set \(\mathcal {P}\) of real parameters associated to a fuzzy number, in a general form which includes the most important characteristics already introduced for fuzzy numbers. We find the set \(\mathcal {P}_{\mathrm{s}}\subset \mathcal {P}\) with the property that for any given fuzzy number there exists at least a symmetric triangular fuzzy number which preserves a fixed parameter \(p\in \mathcal {P}\). We compute the symmetric triangular approximation of a fuzzy number which preserves the parameter \(p\in \mathcal {P }_{\mathrm{s}}\). The uniqueness is an immediate consequence; therefore, an approximation operator is obtained. The properties of scale and translation invariance, additivity and continuity of this operator are studied. Some applications related with value and expected value, as important parameters, are given too.