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.     

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 \(J(u_h)\).     

Fast arithmetic in algorithmic self-assembly

In this paper we consider the time complexity of adding two n-bit numbers together within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum of two n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output sum is encoded in the final, terminal assembly of the system. We show that this problem, along with multiplication, has a worst case lower bound of \(\varOmega ( \sqrt{n} )\) in 2D assembly, and \(\varOmega (\root 3 \of {n})\) in 3D assembly. We further design algorithms for both 2D and 3D that meet this bound with worst case run times of \(O(\sqrt{n})\) and \(O(\root 3 \of {n})\) respectively, which beats the previous best known upper bound of O(n). Finally, we consider average case complexity of addition over uniformly distributed n-bit strings and show how we can achieve \(O(\log n)\) average case time with a simultaneous \(O(\sqrt{n})\) worst case run time in 2D. As additional evidence for the speed of our algorithms, we implement our algorithms, along with the simpler O(n) time algorithm, into a probabilistic run-time simulator and compare the timing results.     

Generalized Ramsey numbers through adiabatic quantum optimization

Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers r(G, H), the emergent order is characterized by graphs G and H. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers \(r({\mathscr {T}}_{m},{\mathscr {T}}_{n})\) for trees of order \(m,n = 6,7,8\), most of which were previously unknown.     

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.     

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.     

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.     

The panpositionable panconnectedness of crossed cubes

In many parallel and distributed multiprocessor systems, the processors are connected based on different types of interconnection networks. The topological structure of an interconnection network is typically modeled as a graph. Among the many kinds of network topologies, the crossed cube is one of the most popular. In this paper, we investigate the panpositionable panconnectedness problem with respect to the crossed cube. A graph G is r-panpositionably panconnected if for any three distinct vertices x, y, z of G and for any integer \(l_1\) satisfying \(r \le l_1 \le |V(G)| - r - 1\), there exists a path \(P = [x, P_1, y, P_2, z]\) in G such that (i) \(P_1\) joins x and y with \(l(P_1) = l_1\) and (ii) \(P_2\) joins y and z with \(l(P_2) = l_2\) for any integer \(l_2\) satisfying \(r \le l_2 \le |V(G)| - l_1 - 1\), where |V(G)| is the total number of vertices in G and \(l(P_1)\) (respectively, \(l(P_2)\)) is the length of path \(P_1\) (respectively, \(P_2\)). By mathematical induction, we demonstrate that the n-dimensional crossed cube \(CQ_n\) is n-panpositionably panconnected. This result indicates that the path embedding of joining x and z with a mediate vertex y in \(CQ_n\) is extremely flexible. Moreover, applying our result, crossed cube problems such as panpositionable pancyclicity, panpositionably Hamiltonian connectedness, and panpositionable Hamiltonicity can be solved.     

A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics

The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from \(O(N^{3})\) required by traditional methods to O(Nlog\(^{2}N)\) and the memory requirement from \(O(N^{2})\) to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.     

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 \(r\ge 11\).     

Generalized Tweakable Even-Mansour Cipher and Its Applications

This paper describes a generalized tweakable blockcipher HPH (Hash-Permutation-Hash), which is based on a public random permutation P and a family of almost-XOR-universal hash functions \( \mathcal{H}={\left\{ HK\right\}}_{K\in \mathcal{K}} \) as a tweak and key schedule, and defined as y = HPH_K((t₁, t₂), x) = P(x ⊕ H_K(t₁)) ⊕ H_K(t₂), where K is a key randomly chosen from a key space \( \mathcal{K} \), (t₁, t₂) is a tweak chosen from a valid tweak space \( \mathcal{T} \), x is a plaintext, and y is a ciphertext. We prove that HPH is a secure strong tweakable pseudorandom permutation (STPRP) by using H-coefficients technique. Then we focus on the security of HPH against multi-key and related-key attacks. We prove that HPH achieves both multi-key STPRP security and related-key STPRP security. HPH can be extended to wide applications. It can be directly applied to authentication and authenticated encryption modes. We apply HPH to PMAC1 and OPP, provide an improved authentication mode HPMAC and a new authenticated encryption mode OPH, and prove that the two modes achieve single-key security, multi-key security, and related-key security.     

Occurrence indices of elements in linear recurrence sequences over primary residue rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring \(\mathbb{Z}_{p^n } \). We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over \(\mathbb{Z}_{p^n } \), we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over ?₄ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.     

Faster search by lackadaisical quantum walk

In the typical model, a discrete-time coined quantum walk searching the 2D grid for a marked vertex achieves a success probability of \(O(1/\log N)\) in \(O(\sqrt{N \log N})\) steps, which with amplitude amplification yields an overall runtime of \(O(\sqrt{N} \log N)\). We show that making the quantum walk lackadaisical or lazy by adding a self-loop of weight 4 / N to each vertex speeds up the search, causing the success probability to reach a constant near 1 in \(O(\sqrt{N \log N})\) steps, thus yielding an \(O(\sqrt{\log N})\) improvement over the typical, loopless algorithm. This improved runtime matches the best known quantum algorithms for this search problem. Our results are based on numerical simulations since the algorithm is not an instance of the abstract search algorithm.     

A locking-free stabilized mixed finite element method for linear elasticity: the high order case

In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\)-order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\)-norm and for the displacement in \(L^2\)-norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.     

An interval extension of SMS method for computing weighted Moore–Penrose inverse

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore–Penrose inverse \(A^{\dagger }_{MN}\) along with its rigorous error bounds is proposed for given full rank \(m \times n\) complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing \(A^{\dagger }_{MN}\), this method generates a sequence of complex interval matrices each enclosing \(A^{\dagger }_{MN}\) and converging to it. A new method is developed for constructing initial complex interval matrix containing \(A^{\dagger }_{MN}\). Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where \(l \ge 2\). A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.     

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.     

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.     

New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

Quantum algorithm to solve function inversion with time–space trade-off

In general, it is a difficult problem to solve the inverse of any function. With the inverse implication operation, we present a quantum algorithm for solving the inversion of function via using time–space trade-off in this paper. The details are as follows. Let function \(f(x)=y\) have k solutions, where \(x\in \{0, 1\}^{n}, y\in \{0, 1\}^{m}\) for any integers n, m. We show that an iterative algorithm can be used to solve the inverse of function f(x) with successful probability \(1-\left( 1-\frac{k}{2^{n}}\right) ^{L}\) for \(L\in Z^{+}\). The space complexity of proposed quantum iterative algorithm is O(Ln), where L is the number of iterations. The paper concludes that, via using time–space trade-off strategy, we improve the successful probability of algorithm.