Thwarting intercept-and-resend attack on Zhang’s quantum secret sharing using collective rotation noises

This work deliberately introduces collective-rotation noise into quantum states to prevent an intercept-resend attack on Zhang’s quantum secret sharing scheme over a collective-noise quantum channel (Zhang in Phys A 361:233–238, 2006). The noise recovering capability of the scheme remains intact. With this design, the quantum bit efficiency of the protocol is doubled when compared to Sun et al.’s improvement on Zhang’s scheme (Sun et al. in Opt Commun 283:181–183, 2010).     

All-Pairs Shortest Paths with Real Weights in <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis><Superscript>3</Superscript>/log <Emphasis Type="Italic">n</Emphasis>) Time

We describe an O(n ³/log n)-time algorithm for the all-pairs-shortest-paths problem for a real-weighted directed graph with n vertices. This slightly improves a series of previous, slightly subcubic algorithms by Fredman (SIAM J. Comput. 5:49–60, 1976), Takaoka (Inform. Process. Lett. 43:195–199, 1992), Dobosiewicz (Int. J. Comput. Math. 32:49–60, 1990), Han (Inform. Process. Lett. 91:245–250, 2004), Takaoka (Proc. 10th Int. Conf. Comput. Comb., Lect. Notes Comput. Sci., vol. 3106, pp. 278–289, Springer, 2004), and Zwick (Proc. 15th Int. Sympos. Algorithms and Computation, Lect. Notes Comput. Sci., vol. 3341, pp. 921–932, Springer, 2004). The new algorithm is surprisingly simple and different from previous ones. A preliminary version of this paper appeared in Proc. 9th Workshop Algorithms Data Struct. (WADS), Lect. Notes Comput. Sci., vol. 3608, pp. 318–324, Springer, 2005.     

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338–353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682–687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80–98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69–81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313–334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169–186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741–747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.      

State BL-algebras

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna (An algebraic approach to states on MV-algebras. In: Novák V (ed) Fuzzy logic 2, proceedings of the 5th EUSFLAT conference, September 11–14, Ostrava, vol II, pp 201–206, 2007; Int J Approx Reason 50:138–152, 2009) as an MV-algebra with internal state as a unary operation. Di Nola and Dvurečenskij (Ann Pure Appl Logic 161:161–173, 2009a; Math Slovaca 59:517–534, 2009b) gave a stronger version of a state MV-algebra. In the present paper, we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.     

New Results on the Most Significant Bit of Integer Multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Analyzing the limits of symbolic graph algorithms for the reachability problem Sawitzki (Proc. of LATIN, LNCS, vol. 3887, pp. 781–792, Springer, Berlin, 2006) has presented the first exponential lower bound on the π-OBDD size for the most significant bit of integer multiplication according to one predefined variable order π. Since the choice of the variable order is a main issue to obtain OBDDs of small size the investigation is continued. As a result a new upper bound method and the first non-trivial upper bound on the size of OBDDs according to an arbitrary variable order is presented. Furthermore, Sawitzki’s lower bound is improved.     

Orientation-Matching Minimization for Image Denoising and Inpainting

In this paper, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the two-step TV-Stokes algorithm (Rahman et al. in Scale space and variational methods in computer vision, pp. 473–482, Springer, Heidelberg, 2007; Tai et al. in Image processing based on partial differential equations, pp. 3–22, Springer, Heidelberg, 2006; Bertalmio et al. in Proc. conf. comp. vision pattern rec., pp. 355–362, 2001), a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation (PDE) for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting (AOS) scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.     

On the Planar Piecewise Quadratic 1-Center Problem

In this paper we introduce a minimax model unifying several classes of single facility planar center location problems. We assume that the transportation costs of the demand points to the serving facility are convex functions {Q _i }, i=1,…,n, of the planar distance used. Moreover, these functions, when properly transformed, give rise to piecewise quadratic functions of the coordinates of the facility location. In the continuous case, using results on LP-type models by Clarkson (J. ACM 42:488–499, 1995), Matoušek et al. (Algorithmica 16:498–516, 1996), and the derandomization technique in Chazelle and Matoušek (J. Algorithms 21:579–597, 1996), we claim that the model is solvable deterministically in linear time. We also show that in the separable case, one can get a direct O(nlog n) deterministic algorithm, based on Dyer (Proceedings of the 8th ACM Symposium on Computational Geometry, 1992), to find an optimal solution. In the discrete case, where the location of the center (server) is restricted to some prespecified finite set, we introduce deterministic subquadratic algorithms based on the general parametric approach of Megiddo (J. ACM 30:852–865, 1983), and on properties of upper envelopes of collections of quadratic arcs. We apply our methods to solve and improve the complexity of a number of other location problems in the literature, and solve some new models in linear or subquadratic time complexity.     

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

In this paper we continue the study, which was initiated in (Ben-Artzi et al. in Math. Model. Numer. Anal. 35(2):313–303, 2001; Fishelov et al. in Lecture Notes in Computer Science, vol. 2667, pp. 809–817, 2003; Ben-Artzi et al. in J. Comput. Phys. 205(2):640–664, 2005 and SIAM J. Numer. Anal. 44(5):1997–2024, 2006) of the numerical resolution of the pure streamfunction formulation of the time-dependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in the last three afore-mentioned articles, to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is actually obtained for several test-cases.     

Capabilities and Limits of Compact Error Resilience Methods for Algorithmic Self-Assembly

Winfree’s pioneering work led the foundations in the area of error-reduction in algorithmic self-assembly (Winfree and Bekbolatov in DNA Based Computers 9, LNCS, vol. 2943, pp. 126–144, [2004]), but the construction resulted in increase of the size of assembly. Reif et al. (Nanotechnol. Sci. Comput. 79–103, [2006]) contributed further in this area with compact error-resilient schemes that maintained the original size of the assemblies, but required certain restrictions on the Boolean functions to be used in the algorithmic self-assembly. It is a critical challenge to improve these compact error resilient schemes to incorporate arbitrary Boolean functions, and to determine how far these prior results can be extended under different degrees of restrictions on the Boolean functions. In this work we present a considerably more complete theory of compact error-resilient schemes for algorithmic self-assembly in two and three dimensions. In our error model, ε is defined to be the probability that there is a mismatch between the neighboring sides of two juxtaposed tiles and they still stay together in the equilibrium. This probability is independent of any other match or mismatch and hence we term this probabilistic model as the independent error model. In our model all the error analysis is performed under the assumption of kinetic equilibrium. First we consider two-dimensional algorithmic self-assembly. We present an error correction scheme for reduction of errors from ε to ε ² for arbitrary Boolean functions in two dimensional algorithmic self-assembly. Then we characterize the class of Boolean functions for which the error can be reduced from ε to ε ³, and present an error correction scheme that achieves this reduction. Then we prove ultimate limits on certain classes of compact error resilient schemes: in particular we show that they can not provide reduction of errors from ε to ε ⁴ is for any Boolean functions. Further, we develop the first provable compact error resilience schemes for three dimensional tiling self-assemblies. We also extend the work of Winfree on self-healing in two-dimensional self-assembly (Winfree in Nanotechnol. Sci. Comput. 55–78, [2006]) to obtain a self-healing tile set for three-dimensional self-assembly.     

A New Mapped Weighted Essentially Non-oscillatory Scheme

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Recently Henrick et al. (J. Comput. Phys. 207:542–567, 2005) noted that the fifth-order WENO method by Jiang and Shu (J. Comput. Phys. 126:202–228, 1996) is only third-order accurate near critical points of the smooth regions in general. Using a simple mapping function to the original weights in Jiang and Shu (J. Comput. Phys. 126:202–228, 1996), Henrick et al. developed a mapped WENO method to achieve the optimal order of accuracy near critical points. In this paper we study the mapped WENO scheme and find that, when it is used for solving the problems with discontinuities, the mapping function in Henrick et al. (J. Comput. Phys. 207:542–567, 2005) may amplify the effect from the non-smooth stencils and thus cause a potential loss of accuracy near discontinuities. This effect may be difficult to be observed for the fifth-order WENO method unless a long time simulation is desired. However, if the mapping function is applied to seventh-order WENO methods (Balsara and Shu in J. Comput. Phys. 160:405–452, 2000), the error can increase much faster so that it can be observed with a moderate output time. In this paper a new mapping function is proposed to overcome this potential loss of accuracy.     

Improved Approximation Results for the Minimum Energy Broadcasting Problem

In this paper we present new results on the performance of the Minimum Spanning Tree heuristic for the Minimum Energy Broadcast Routing (MEBR) problem. We first prove that, for any number of dimensions d≥2, the approximation ratio of the heuristic does not increase when the power attenuation coefficient α, that is the exponent to which the coverage distance must be raised to give the emission power, grows. Moreover, we show that, for any fixed instance, as a limit for α going to infinity, the ratio tends to the lower bound of Clementi et al. (Proceedings of the 18th annual symposium on theoretical aspects of computer science (STACS), pp. 121–131, 2001), Wan et al. (Wirel. Netw. 8(6):607–617, 2002) given by the d-dimensional kissing number, thus closing the existing gap between the upper and the lower bound. We then introduce a new analysis allowing to establish a 7.45-approximation ratio for the 2-dimensional case, thus significantly decreasing the previously known 12 upper bound (Wan et al. in Wirel. Netw. 8(6):607–617, 2002) (actually corrected to 12.15 in Klasing et al. (Proceedings of the 3rd IFIP-TC6 international networking conference, pp. 866–877, 2004)). Finally, we extend our analysis to any number of dimensions d≥2 and any α≥d, obtaining a general approximation ratio of 3 ^d −1, again independent of α. The improvements of the approximation ratios are specifically significant in comparison with the lower bounds given by the kissing numbers, as these grow at least exponentially with respect to d. The research was partially funded by the European project COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL). Preliminary version of this paper appeared in Flammini et al. (Proceedings of ACM joint workshop on foundations of mobile computing (DIALM-POMC), pp. 85–91, 2004).     

Approximation Algorithms for Reconstructing the Duplication History of Tandem Repeats

Computing the duplication history of a tandem repeated region is an important problem in computational biology (Fitch in Genetics 86:623–644, 1977; Jaitly et al. in J. Comput. Syst. Sci. 65:494–507, 2002; Tang et al. in J. Comput. Biol. 9:429–446, 2002). In this paper, we design a polynomial-time approximation scheme (PTAS) for the case where the size of the duplication block is 1. Our PTAS is faster than the previously best PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002). For example, to achieve a ratio of 1.5, our PTAS takes O(n ⁵) time while the PTAS in Jaitly et al. (J. Comput. Syst. Sci. 65:494–507, 2002) takes O(n ¹¹) time. We also design a ratio-6 polynomial-time approximation algorithm for the case where the size of each duplication block is at most 2. This is the first polynomial-time approximation algorithm with a guaranteed ratio for this case. Part of work was done during a Z.-Z. Chen visit at City University of Hong Kong.     

A posteriori error analysis of time-dependent Stokes problem by Chorin-Temam scheme

We present a posteriori-residual analysis for the approximate time-dependent Stokes model Chorin-Temam projection scheme (Chorin in Math. Comput. 23:341–353, 1969; Temam in Arch. Ration. Mech. Appl. 33:377–385, 1969). Based on the multi-step approach introduced in Bergam et al. (Math. Comput. 74(251):1117–1138, 2004), we derive error estimators, with respect to both time and space approximations, related to diffusive and incompressible parts of Stokes equations. Using a conforming finite element discretization, we prove the equivalence between error and estimators under specific conditions.     

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.     

Modified multi-component gas transport formulation with phoretic effects

The understanding of multicomponent mass transport processes is essential for modeling and optimization of many systems, such as fuel cells. The understanding of individual species behavior becomes quite significant at micro-nano scales, where average mixture models are not very accurate. Also at micro-nano scales, additional phoretic transport is present due to strong local temperature and pressure gradients, which is discussed by Chakraborty and Durst (Phys Fluids 19(8):088104-01–088104-04, 2007). To account for multicomponent mass transport, recently proposed mass transport model by Kerkhof and Geboers (AIChE J 51(1):79–121, 2005), provides a way to look into individual components. This article presents extended multicomponent mass transport equations for micro-nano scales within continuum region. The Kerkhof–Geboers theory and the modifications suggested by Chakraborty and Durst (2007) have been combined together to form a new set of equations. An extensive order of magnitude analysis has been done on the modified equations. The application of new equations to different problem situations has also been discussed. It is shown that at very small length scales and for highly diffusive transport, the phoretic transport dominates the system, thus rendering the conventional equations erroneous.     

On Buckley��s approach to fuzzy estimation

In several works, Buckley (Soft Comput 9:512–518, 2005a; Soft Comput 9:769–775 2005b; Fuzzy statistics, Springer, Heidelberg, 2005c) have introduced and developed an approach to the estimation of unknown parameters in statistical models. In this paper, we introduce an improved method for the estimation of parameters for cases in which the Buckley’s approach presents some drawbacks, as for example when the underlying statistic has a non-symmetric distribution.     

The Steiner Ratio Conjecture of Gilbert-Pollak May Still Be Open

We offer evidence in the disproof of the continuity of the length of minimum inner spanning trees with respect to a parameter vector having a zero component. The continuity property is the key step of the proof of the conjecture in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Therefore the Steiner ratio conjecture proposed by Gilbert-Pollak (SIAM J. Appl. Math. 16(1):1–29, 1968) has not been proved yet. The Steiner ratio of a round sphere has been discussed in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) by assuming the validity of the conjecture on a Euclidean plane in Du and Hwang (Proc. Nat. Acad. Sci. U.S.A. 87:9464–9466, 1990; Algorithmica 7(1):121–135, 1992). Hence the results in Rubinstein and Weng (J. Comb. Optim. 1:67–78, 1997) have not been proved yet.     

Constrained mesh optimization on boundary

This paper presents a new mesh optimization approach aiming to improve the mesh quality on the boundary. The existing mesh untangling and smoothing algorithms (Vachal et al. in J Comput Phys 196: 627–644, 2004; Knupp in J Numer Methods Eng 48: 1165–1185, 2002), which have been proved to work well to interior mesh optimization, are enhanced by adding constrains of surface and curve shape functions that approximate the boundary geometry from the finite element mesh. The enhanced constrained optimization guarantees that the boundary nodes to be optimized always move on the approximated boundary. A dual-grid hexahedral meshing method is used to generate sample meshes for testing the proposed mesh optimization approach. As complementary treatments to the mesh optimization, appropriate mesh topology modifications, including buffering element insertion and local mesh refinement, are performed in order to eliminate concave and distorted elements on the boundary. Finally, the optimization results of some examples are given to demonstrate the effectivity of the proposed approach.     

Speed-Up Theorems in Type-2 Computations Using Oracle Turing Machines

A classic result known as the speed-up theorem in machine-independent complexity theory shows that there exist some computable functions that do not have best programs for them (Blum in J. ACM 14(2):322–336, 1967 and J. ACM 18(2):290–305, 1971). In this paper we lift this result into type-2 computations. Although the speed-up phenomenon is essentially inherited from type-1 computations, we observe that a direct application of the original proof to our type-2 speed-up theorem is problematic because the oracle queries can interfere with the speed of the programs and hence the cancellation strategy used in the original proof is no longer correct at type-2. We also argue that a type-2 analog of the operator speed-up theorem (Meyer and Fischer in J. Symb. Log. 37:55–68, 1972) does not hold, which suggests that this curious speed-up phenomenon disappears in higher-typed computations beyond type-2. The result of this paper adds one more piece of evidence to support the general type-2 complexity theory under the framework proposed in Li (Proceedings of the Third International Conference on Theoretical Computer Science, pp. 471–484, 2004 and Proceedings of Computability in Europe: Logical Approach to Computational Barriers, pp. 182–192, 2006) and Li and Royer (On type-2 complexity classes: Preliminary report, pp. 123–138, 2001) as a reasonable setup.