No nexiste nce of n-qubit u nexte ndible product bases of size $$2^ n-5$$2 n-5

It is known that the n-qubit system has no unextendible product bases (UPBs) of cardinality \(2^n-1\), \(2^n-2\) and \(2^n-3\). On the other hand, the n-qubit UPBs of cardinality \(2^n-4\) exist for all \(n\ge 3\). We prove that they do not exist for cardinality \(2^n-5\).     

Construction of quantum caps in projective space <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, 4) and quantum codes of distance 4

Constructions of quantum caps in projective space PG(r, 4) by recursive methods and computer search are discussed. For each even n satisfying \(n\ge 282\) and each odd z satisfying \(z\ge 275\), a quantum n-cap and a quantum z-cap in \(PG(k-1, 4)\) with suitable k are constructed, and \([[n,n-2k,4]]\) and \([[z,z-2k,4]]\) quantum codes are derived from the constructed quantum n-cap and z-cap, respectively. For \(n\ge 282\) and \(n\ne 286\), 756 and 5040, or \(z\ge 275\), the results on the sizes of quantum caps and quantum codes are new, and all the obtained quantum codes are optimal codes according to the quantum Hamming bound. While constructing quantum caps, we also obtain many large caps in PG(r, 4) for \(r\ge 11\). These results concerning large caps provide improved lower bounds on the maximal sizes of caps in PG(r, 4) for \(r\ge 11\).     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

A parameterized complexity view on non-preemptively scheduling interval-constrained jobs: few machines,small looseness,and small slack

We study the problem of non-preemptively scheduling n jobs, each job j with a release time \(t_j\), a deadline \(d_j\), and a processing time \(p_j\), on m parallel identical machines. Cieliebak et al. (2004) considered the two constraints \(|d_j-t_j|\le \lambda {}p_j\) and \(|d_j-t_j|\le p_j +\sigma \) and showed the problem to be NP-hard for any \(\lambda >1\) and for any \(\sigma \ge 2\). We complement their results by parameterized complexity studies: we show that, for any \(\lambda >1\), the problem remains weakly NP-hard even for \(m=2\) and strongly W[1]-hard parameterized by m. We present a pseudo-polynomial-time algorithm for constant m and \(\lambda \) and a fixed-parameter tractability result for the parameter m combined with \(\sigma \).     

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.     

Close to linear space routing schemes

Let \(G=(V,E)\) be an unweighted undirected graph with n vertices and m edges, and let \(k>2\) be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance \(\varDelta \) from s, routes a message from s to t on a path whose length is \(O(k\varDelta +m^{1/k})\). The total space used by our routing scheme is \(mn^{O(1/\sqrt{\log n})}\), which is almost linear in the number of edges of the graph. We present also a routing scheme with \(n^{O(1/\sqrt{\log n})}\) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every \(v\in V\) is at most \(kn^{O(1/\sqrt{\log n})}deg(v)\), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein (2009), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.     

Local distinguishability of generalized Bell states

We investigate the distinguishability of orthogonal generalized Bell states (GBSs) in \(d\otimes d\) system by local operations and classical communication (LOCC), where d is a prime. We show that |S| is no more than \(d+1\) for any l GBSs, i.e., \(|S|\le d+1\), where S is maximal set which is composed of pairwise noncommuting pairs in \({\varDelta } U\). If \(|S|\le d\), then the l GBSs can be distinguished by LOCC according to our main Theorem. Compared with the results (Fan in Phys Rev Lett 92:177905, 2004; Tian et al. in Phys Rev A 92:042320, 2015), our result is more general. It can determine local distinguishability of \(l (> k)\) GBSs, where k is the number of GBSs in Fan’s and Tian’s results. Only for \(|S|=d+1\), we do not find the answer. We conjecture that any l GBSs cannot be distinguished by one-way LOCC if \(|S|=d+1\). If this conjecture is right, the problem about distinguishability of GBSs with one-way LOCC is completely solved in \(d\otimes d\).     

Pure state ‘really’ informationally complete with rank-1 POVM

What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space \(\mathcal {H}_d\)? The known result is that the number is no less than \(3d-2\). We show that this lower bound is not tight except for \(d=2\) or 4. Then we give an upper bound \(4d-3\). For \(d=2\), many rank-1 POVMs with four elements can determine any pure states in \(\mathcal {H}_2\). For \(d=3\), we show eight is the minimal number by construction. For \(d=4\), the minimal number is in the set of \(\{10,11,12,13\}\). We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in \(\mathcal {H}_4\). For any dimension d, we construct \(d+2k-2\) adaptive rank-1 positive operators for the reconstruction of any unknown pure state in \(\mathcal {H}_d\), where \(1\le k \le d\).     

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.     

An optimal adaptive stabilizing algorithm for two edge-disjoint paths problem

The problem of two edge-disjoint paths is to identify two paths \(Q_1\) and \(Q_2\) from source \(s \in V\) to target \(t \in V\) without any common arc in a directed connected graph \(G=(V, E)\). In this paper, we present an adaptive stabilizing algorithm for finding a pair of edge-disjoint paths from s to t in G in O(D) rounds with state-space complexity of \(O(log\; n)\) bits per process, where n is the number of nodes and D is the diameter of the graph. The proposed algorithm is optimal with respect to its time complexity, and the total length of the shortest paths. In addition, it can also be used to solve the problem for undirected graphs. Since the proposed algorithm is stabilizing, it does not require initialization and is capable of withstanding transient faults. We view a fault that perturbs the state of the system but not its program as a transient fault. In addition, the proposed algorithm is adaptive since it is capable of dealing with topology changes in the form of addition/removal of arcs and/or nodes as well as changes in the directions of arcs provided that two edge-disjoint paths between s and t exist after the topology change.     

Good and asymptotically good quantum codes derived from algebraic geometry

In this paper, we construct several new families of quantum codes with good parameters. These new quantum codes are derived from (classical) t-point (\(t\ge 1\)) algebraic geometry (AG) codes by applying the Calderbank–Shor–Steane (CSS) construction. More precisely, we construct two classical AG codes \(C_1\) and \(C_2\) such that \(C_1\subset C_2\), applying after the well-known CSS construction to \(C_1\) and \(C_2\). Many of these new codes have large minimum distances when compared with their code lengths as well as they also have small Singleton defects. As an example, we construct a family \({[[46, 2(t_2 - t_1), d]]}_{25}\) of quantum codes, where \(t_1 , t_2\) are positive integers such that \(1<t_1< t_2 < 23\) and \(d\ge \min \{ 46 - 2t_2 , 2t_1 - 2 \}\), of length \(n=46\), with minimum distance in the range \(2\le d\le 20\), having Singleton defect at most four. Additionally, by applying the CSS construction to sequences of t-point (classical) AG codes constructed in this paper, we generate sequences of asymptotically good quantum codes.     

Counter machines and crystallographic structures

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages \(\mathcal {DCL}_d,\,d=0,1,2,\ldots \), within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class \(\mathcal {DCL}_d\). An intersection of d languages in \(\mathcal {DCL}_1\) defines \(\mathcal {DCL}_d\). We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in \(\mathcal {DCL}_d\). The proof uses the following result: given a digraph \(\Delta \) and a group G, there is a unique digraph \(\Gamma \) such that \(G\le \mathrm{Aut}\,\Gamma ,\,G\) acts freely on the structure, and \(\Gamma /G \cong \Delta \).     

The hierarchical Petersen network: a new interconnection network with fixed degree

Network cost and fixed-degree characteristic for the graph are important factors to evaluate interconnection networks. In this paper, we propose hierarchical Petersen network (HPN) that is constructed in recursive and hierarchical structure based on a Petersen graph as a basic module. The degree of HPN(n) is 5, and HPN(n) has \(10^n\) nodes and \(2.5 \times 10^n\) edges. And we analyze its basic topological properties, routing algorithm, diameter, spanning tree, broadcasting algorithm and embedding. From the analysis, we prove that the diameter and network cost of HPN(n) are \(3\log _{10}N-1\) and \(15 \log _{10}N-1\), respectively, and it contains a spanning tree with the degree of 4. In addition, we propose link-disjoint one-to-all broadcasting algorithm and show that HPN(n) can be embedded into FP\(_k\) with expansion 1, dilation 2k and congestion 4. For most of the fixed-degree networks proposed, network cost and diameter require \(O(\sqrt{N})\) and the degree of the graph requires O(N). However, HPN(n) requires O(1) for the degree and \(O(\log _{10}N)\) for both diameter and network cost. As a result, the suggested interconnection network in this paper is superior to current fixed-degree and hierarchical networks in terms of network cost, diameter and the degree of the graph.     

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form \(P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2\), where \(k\ge 1\) is the degree of polynomials in the interior of the element T, \(j\ge 0\) is the degree of polynomials on the boundary of T, and \(\ell \ge 0\) is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.     

Two-Level Penalty Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics Equations

This work is concerned with the study of two-level penalty finite element method for the 2D/3D stationary incompressible magnetohydrodynamics equations. The new method is an interesting combination of the Newton iteration and two-level penalty finite element algorithm with two different finite element pairs \(P_{1}b\)-\(P_{1}\)-\(P_{1}b\) and \(P_{1}\)-\(P_{0}\)-\(P_{1}\). Moreover, the rigorous analysis of stability and error estimate for the proposed method are given. Numerical results verify the theoretical results and show the applicability and effectiveness of the presented scheme.     

Throughput maximization for speed scaling with agreeable deadlines

We study the following energy-efficient scheduling problem. We are given a set of n jobs which have to be scheduled by a single processor whose speed can be varied dynamically. Each job \(J_j\) is characterized by a processing requirement (work) \(p_j\), a release date \(r_j\), and a deadline \(d_j\). We are also given a budget of energy E which must not be exceeded and our objective is to maximize the throughput (i.e., the number of jobs which are completed on time). We show that the problem can be solved optimally via dynamic programming in \(O(n^4 \log n \log P)\) time when all jobs have the same release date, where P is the sum of the processing requirements of the jobs. For the more general case with agreeable deadlines where the jobs can be ordered so that, for every \(i < j\), it holds that \(r_i \le r_j\) and \(d_i \le d_j\), we propose an optimal dynamic programming algorithm which runs in \(O(n^6 \log n \log P)\) time. In addition, we consider the weighted case where every job \(J_j\) is also associated with a weight \(w_j\) and we are interested in maximizing the weighted throughput (i.e., the total weight of the jobs which are completed on time). For this case, we show that the problem becomes \(\mathcal{NP}\)-hard in the ordinary sense even when all jobs have the same release date and we propose a pseudo-polynomial time algorithm for agreeable instances.     

Adjoint-Based,Superconvergent Galerkin Approximations of Linear Functionals

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where \(J(\cdot )\) is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if \(u_h\) is the approximation given by the continuous Galerkin method with piecewise polynomials of degree \(k>0\), then, as a direct consequence of its property of Galerkin orthogonality, the functional \(J(u_h)\) converges to J(u) with a rate of order \(h^{2k}\). We show how to define approximations to J(u), with a computational effort about twice of that of computing \(J(u_h)\), which converge with a rate of order \(h^{4k}\). The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for \(k=1,2,3\) in one-space dimension and for \(k=1,2\) in two-space dimensions, which show that \(J(u_h)\) converges to J(u) with order \(h^{2k+1}\) and that the new approximations converges with order \(h^{4k}\). The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of \(J(u_h)\).     

Greatest Lower Bounds of System Failure Probability on a Time Interval Under Incomplete Information About the Distribution Function of Time to Failure

The authors solve problems of finding the greatest lower bounds for the probability F (\( \upsilon \)) - F (u),0< u < \( \upsilon \) < ∞, in the set of distribution functions F (x) of nonnegative random variables with unimodal density with mode m, u < m < \( \upsilon \), and fixed two first moments.