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This paper provides results in the area of the analytical derivation of the efficient set of a mean-variance portfolio selection problem that has more than three criteria. By "analytical" we mean derived by formula as opposed to being computed by algorithm. By "more than three criteria", we mean that beyond the mean and variance of regular portfolio selection, the problems addressed have two or more additional linear objectives. The additional objectives might include sustainability, dividend yield, liquidity, and R&D as extra objectives like these are being seen with greater frequency. While not all multiple criteria portfolio selection problems lend themselves to an analytical derivation, a certain class does and the problems in this class are covered by the mathematics of this paper. [ABSTRACT FROM AUTHOR]

This paper proposes an innovative statistical method to measure the impact of the class/school on student achievements in multiple subjects. We propose a semiparametric model for a bivariate response variable with random coefficients, that are assumed to follow a discrete distribution with an unknown number of support points, together with an Expectation-Maximization algorithm—called BSPEM algorithm—to estimate its parameters. In the case study, we apply the BSPEM algorithm to data about Italian middle schools, considering students nested within classes, and we identify subpopulations of classes, standing on their effects on student achievements in reading and mathematics. The proposed model is extremely informative in exploring the correlation between multiple class effects, which are typical of the educational production function. The estimated class effects on reading and mathematics student achievements are then explained in terms of various class and school level characteristics selected by means of a LASSO regression. [ABSTRACT FROM AUTHOR]

Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions $$p \ge 3$$ . In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity $$O\left( n^{p-1}\log (n)\right) $$ , while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth. [ABSTRACT FROM AUTHOR]

The objective of this paper is analyzing to which extent the multiverse hypothesis provides a real explanation of the peculiarities of the laws and constants in our universe. First we argue in favor of the thesis that all multiverses except Tegmark's 'mathematical multiverse' are too small to explain the fine tuning, so that they merely shift the problem up one level. But the 'mathematical multiverse" is surely too large. To prove this assessment, we have performed a number of experiments with cellular automata of complex behavior, which can be considered as universes in the mathematical multiverse. The analogy between what happens in some automata (in particular Conway's 'Game of Life") and the real world is very strong. But if the results of our experiments can be extrapolated to our universe, we should expect to inhabit-in the context of the multiverse-a world in which at least some of the laws and constants of nature should show a certain time dependence. Actually, the probability of our existence in a world such as ours would be mathematically equal to zero. In consequence, the results presented in this paper can be considered as an inkling that the hypothesis of the multiverse, whatever its type, does not offer an adequate explanation for the peculiarities of the physical laws in our world. [ABSTRACT FROM AUTHOR]

In this paper, we deepen the theoretical study of the geometric structure of a balanced complex polytope (b.c.p.), which is the generalization of a real centrally symmetric polytope to the complex space. We also propose a constructive algorithm for the representation of its facets in terms of their associated linear functionals. The b.c.p.s are used, for example, as a tool for the computation of the joint spectral radius of families of matrices. For the representation of real polytopes, there exist well-known algorithms such as, for example, the Beneath–Beyond method. Our purpose is to modify and adapt this method to the complex case by exploiting the geometric features of the b.c.p. However, due to the significant increase in the difficulty of the problem when passing from the real to the complex case, in this paper, we confine ourselves to examine the two-dimensional case. We also propose an algorithm for the computation of the norm the unit ball of which is a b.c.p. [ABSTRACT FROM AUTHOR]

Simulation and verification of numerically controlled (NC) manufacturing processes require efficient visualization and analysis of the swept volume generated by the motion of freeform NC tools along complex 3D paths. State-of-the-art methods are either based on approximation techniques (thus lacking the level of accuracy required in NC manufacturing) or are based on analytical solutions with high computational complexity, which are not suitable for real-time applications. In addition, until recently, modeling the self-intersection of a generated volume was thought to be obstructed by seemingly complex mathematics. This paper proposes solving the sweeping problem by using the sweep-envelope differential equation (SEDE). This method has advantages over other methods in terms of low computational complexity and high accuracy. Moreover, this method includes efficient tools for self-intersection detection and modeling. In this paper, we present an enhanced self-intersection algorithm and apply the SEDE algorithm on a ball-end cutter that is swept along non-intersecting and self-intersecting cutter paths. [ABSTRACT FROM AUTHOR]

Symbolic model checkers use BDDs to represent arithmetic constraints over bounded integer variables. The size of such BDDs can in the worst case be exponential in the number and size (in bits) of the integer variables. In this paper we show how to construct linear-sized BDDs for linear integer arithmetic constraints. We present basic constructions for atomic equality and inequality constraints and generalize our complexity results for arbitrary linear arithmetic formulas. We also present three alternative ways of handling out-of-bounds transitions and discuss heterogeneous bounds on integer variables. We experimentally compare our approach to other BDD-based symbolic model checkers and demonstrate that the algorithms presented in this paper can be used to improve their performance significantly. [ABSTRACT FROM AUTHOR]

In many applications of circuit design and synthesis, it is natural and in some instances essential to manipulate logic functions and model circuits using word-level representations and arithmetic operations in contrast to bit-level representations and logic operations. This paper reviews linear word-level structures and formulates their properties for combinational circuit modeling. The paper addresses the following problem: given a library of gates with their corresponding word-level representations such as linear arithmetic expressions or respective graph structures, find a word-level model of an arbitrary combinational circuit/netlist using that library of gates and minimizing memory allocation and time delay requirements. We present a comprehensive study on linearization assuming various circuit processing strategies. In particular, we develop a new approach to manipulate linear word-level representations by means of cascades. The practical applicability of linear structures and developed algorithms is strengthen by considering the problem of timing analysis. All this is supported by the experimental study on benchmark circuits. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of fast and efficient indexing techniques for sequences evolving in non-Euclidean spaces. This problem has several applications in the areas of human activity analysis, where there is a need to perform fast search, and recognition in very high dimensional spaces. The problem is made more challenging when representations such as landmarks, contours, and human skeletons etc. are naturally studied in a non-Euclidean setting where even simple operations are much more computationally intensive than their Euclidean counterparts. We propose a geometry and data adaptive symbolic framework that is shown to enable the deployment of fast and accurate algorithms for activity recognition, dynamic texture recognition, motif discovery. Toward this end, we present generalizations of key concepts of piece-wise aggregation and symbolic approximation for the case of non-Euclidean manifolds. We show that one can replace expensive geodesic computations with much faster symbolic computations with little loss of accuracy in activity recognition and discovery applications. The framework is general enough to work across both Euclidean and non-Euclidean spaces, depending on appropriate feature representations without compromising on the ultra-low bandwidth, high speed and high accuracy. The proposed methods are ideally suited for real-time systems and low complexity scenarios. [ABSTRACT FROM AUTHOR]

This paper presents a generalized version of the classic projective reconstruction theorem which helps to choose or assess depth constraints for projective depth estimation algorithms. The theorem shows that projective reconstruction is possible under a much weaker constraint than requiring all estimated projective depths to be nonzero. This result enables us to present classes of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera-point configuration. It also completely specifies the possible wrong configurations allowed by other constraints. We demonstrate the application of the theorem by analysing several constraints used in the literature, as well as presenting new constraints with desirable properties. We mention some of the implications of our results on iterative depth estimation algorithms and projective reconstruction via rank minimization. Our theory is verified by running experiments on both synthetic and real data. [ABSTRACT FROM AUTHOR]

Origami, the ancient Japanese art of paper folding, is mathematically deeper than it looks. Delving into its complexities allows the construction of elaborate and useful structures from simple, flat templates. [ABSTRACT FROM AUTHOR]

In the paper,we study algebras having n bilinearmultiplication operations [InlineMediaObject not available: see fulltext.]: A× A → A, s = 1, ..., n, such that ( a[InlineMediaObject not available: see fulltext.] b) [InlineMediaObject not available: see fulltext.] c = a[InlineMediaObject not available: see fulltext.] ( b[InlineMediaObject not available: see fulltext.] c), s, r = 1,..., n, a, b, c ∈ A. The radical of such an algebra is defined as the intersection of the annihilators of irreducible A-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is s-regular for some operation [InlineMediaObject not available: see fulltext.]. This implies that the quotient algebra by the radical is semisimple. If an n-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian n-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional n-tuple algebra A, over an algebraically closed field, which is a simple A-module. [ABSTRACT FROM AUTHOR]

The problem of enumerating and denumerating words generated by Dyck grammars arises in the work of compilers for high-level programming languages and a number of other applications. The present paper proposes an algorithm for the fast enumeration and denumeration of words of Dyck languages; the complexity of this algorithm per one symbol of enumerated words is O(log n log log n) bit operations, provided that the Schönhage-Strassen multiplication and division algorithmis used. The well-knownmethods applied earlier possess complexity O( n) per one symbol of enumerated words. The construction of the proposed algorithm is based on the Ryabko method. [ABSTRACT FROM AUTHOR]

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem ( Math. Program., doi:, ). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported. [ABSTRACT FROM AUTHOR]

With the advent use of sophisticated and high-cost machines coupled with higher labor costs, concurrent optimization of machining process parameters and tolerance allocation plays a vital role in producing the parts economically. In this paper, an effort is made to concurrently optimize the manufacturing cost of piston and cylinder components by optimizing the operating parameters of the machining processes. Design of experiments (DoE) is adopted to investigate systematically the machining process parameters that influence product quality. In addition, tolerance plays a vital role in assembly of parts in manufacturing industries. For the selected piston and cylinder component, improvements efforts are made to reduce the total manufacturing cost of the components. By making use of central composite rotatable design method, a module of DoE, a mathematical model is developed for predicting the standard deviation of the tolerance achieved by grinding process. This mathematical model, which gives 93.3% accuracy, is used to calculate the quality loss cost. The intent of concurrent optimization problem is to minimize total manufacturing cost and quality loss function. Genetic algorithm is followed for optimizing the parameters. The results prove that there is a considerable reduction in manufacturing cost without violating the required tolerance, cutting force, and power. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical programming based approach for the analysis of elastoplastic softening frames in the presence of geometric nonlinearity. Arbitrarily large deformations, albeit within a small strain regime, can be accommodated, if necessary. For the sake of efficiency, and without undue loss of accuracy, the algorithm processes the nonholonomic (path-dependent) problem in a stepwise holonomic (path-independent) fashion. This analysis capability has been used to investigate what level or order of geometric nonlinearity needs to be adopted to obtain sufficiently accurate results for practical frames. Two examples are provided for illustrative purposes. [ABSTRACT FROM AUTHOR]

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics. [ABSTRACT FROM AUTHOR]

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a supernode amalgamation algorithm which takes into account the characteristics of a hypermatrix data structure. The resulting frontal tree is then used to create a variable-sized partitioning of the hypermatrix. The sparse hypermatrix Cholesky factorization obtained runs slightly faster than the one which uses a fixed-sized partitioning. The algorithm also reduces data dependencies which limit exploitation of parallelism. [ABSTRACT FROM AUTHOR]

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti number, and the Euler–Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zeroth one. As a consequence we also obtain algorithms for computing semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in singly exponential time, which improves on the complexity of the previously published algorithms for this problem. [ABSTRACT FROM AUTHOR]

Multiprocessor task scheduling is an important and computationally difficult problem. A large number of algorithms were proposed which represent various tradeoffs between the quality of the solution and the computational complexity and scalability of the algorithm. Previous comparison studies have frequently operated with simplifying assumptions, such as independent tasks, artificially generated problems or the assumption of zero communication delay. In this paper, we propose a comparison study with realistic assumptions. Our target problems are two well known problems of linear algebra: LU decomposition and Gauss–Jordan elimination. Both algorithms are naturally parallelizable but have heavy data dependencies. The communication delay will be explicitly considered in the comparisons. In our study, we consider nine scheduling algorithms which are frequently used to the best of our knowledge: min–min, chaining, A*, genetic algorithms, simulated annealing, tabu search, HLFET, ISH, and DSH with task duplication. Based on experimental results, we present a detailed analysis of the scalability, advantages and disadvantages of each algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a new contact searching algorithm for large deformation mortar-based contact formulations. In this algorithm, a bounding volume hierarchy, defined in the context of a binary tree, is built for each contact surface based on the geometry of the surface. A global contact searching procedure based on these bounding volume trees is first performed to find all candidate contact element pairs, and then a local searching procedure is done to find all the mortar segments having contributions to the mortar integrals that define the contact formulation. The searching algorithm is shown to be very efficient and readily applicable to a variety of large sliding contact problems. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm which, for a given compact orientable irreducible boundary irreducible 3-manifold M, verifies whether M contains an essential orientable surface (possibly, with boundary), whose genus is at most N. The algorithm is based on Haken’s theory of normal surfaces, and on a trick suggested by Jaco and consisting in estimating the mean length of boundary curves in an unknown essential surface of a given genus in the given manifold. [ABSTRACT FROM AUTHOR]

In this paper, we consider algorithms for constructing n-terms approximations with nonnegative coefficients. The convergence theorem is proved for a “positive” analog of the Pure Greedy Algorithm. We establish a condition on the sequence of weakness coefficients which is sufficient for the convergence of the Positive Weak Greedy Algorithm. This condition is also necessary for the class of monotone sequences. [ABSTRACT FROM AUTHOR]

Consideration was given to control of a complex object whose motion obeys a multivariable nonlinear nonstationary mathematical model. Rigid constraints were imposed on the object’s dynamic precision. The paper considered computer-aided generation of the current equations of object motion with regard for the actuators which differ from subsystem to subsystem. The object is controlled adaptively with regard for the computer-based realization. The algorithms of control system operation that maintain the guaranteed precision of object motion were constructed. Conditions for problem solvability were formulated. The freeflying space robot was discussed by way of example. [ABSTRACT FROM AUTHOR]

This paper describes a parallel implementation of the Gröbner Bases algorithm and the Characteristic Sets method using a grid environment. The two algorithms, their parallelization, and grid-enabled implementations are presented. The performance of the implementations has been evaluated and the experiments have demonstrated considerable speedups. [ABSTRACT FROM AUTHOR]

In this paper, we propose a complete framework for 3D geometry modeling and processing that uses only fast geodesic computations. The basic building block for these techniques is a novel greedy algorithm to perform a uniform or adaptive remeshing of a triangulated surface. Our other contributions include a parameterization scheme based on barycentric coordinates, an intrinsic algorithm for computing geodesic centroidal tessellations, and a fast and robust method to flatten a genus-0 surface patch. On large meshes (more than 500,000 vertices), our techniques speed up computation by over one order of magnitude in comparison to classical remeshing and parameterization methods. Our methods are easy to implement and do not need multilevel solvers to handle complex models that may contain poorly shaped triangles. [ABSTRACT FROM AUTHOR]

We consider in this paper the class M k n of generalized Motzkin paths of length n, that is, lattice paths using steps (1,1), (1,−1), ( k,0), where k is a fixed positive integer, starting at the origin (0,0), running above the x-axis, and ending at ( n,0). The area is the region bounded by the path and the x-axis. We first establish a bijection between the area of paths in M k n and some lattice paths of length n+1. Then, by using a rejection technique, we obtain a linear algorithm with an average time complexity ( k mod 2 +1)( n+1). [ABSTRACT FROM AUTHOR]

In the literature, the conditions on history variables or forward simulations that are related to liveness are expressed in terms of behaviours, and are stronger than convenient and necessary. In this paper, we propose alternative conditions on the simulation relation, which are expressed in terms of the next state relation, and are closely tied to the weak or strong fairness conditions of the specifications. The proof of soundness of this proposal is based on a new theorem that asserts the existence of a strongly fair scheduler for infinitely many alternatives. The theory is extended to simulations in which the concrete specification (occasionally) does fewer steps than the abstract specification it implements. [ABSTRACT FROM AUTHOR]

The problem of projective reconstruction by minimization of the 2D reprojection error in multiple images is considered. Although bundle adjustment techniques can be used to minimize the 2D reprojection error, these methods being based on nonlinear optimization algorithms require a good starting point. Quasi-linear algorithms with better global convergence properties can be used to generate an initial solution before submitting it to bundle adjustment for refinement. In this paper, we propose a factorization-based method to integrate the initial search as well as the bundle adjustment into a single algorithm consisting of a sequence of weighted least-squares problems, in which a control parameter is initially set to a relaxed state to allow the search of a good initial solution, and subsequently tightened up to force the final solution to approach a minimum point of the 2D reprojection error. The proposed algorithm is guaranteed to converge. Our method readily handles images with missing points. [ABSTRACT FROM AUTHOR]

Local image features or interest points provide compact and abstract representations of patterns in an image. In this paper, we extend the notion of spatial interest points into the spatio-temporal domain and show how the resulting features often reflect interesting events that can be used for a compact representation of video data as well as for interpretation of spatio-temporal events. To detect spatio-temporal events, we build on the idea of the Harris and Förstner interest point operators and detect local structures in space-time where the image values have significant local variations in both space and time. We estimate the spatio-temporal extents of the detected events by maximizing a normalized spatio-temporal Laplacian operator over spatial and temporal scales. To represent the detected events, we then compute local, spatio-temporal, scale-invariant N-jets and classify each event with respect to its jet descriptor. For the problem of human motion analysis, we illustrate how a video representation in terms of local space-time features allows for detection of walking people in scenes with occlusions and dynamic cluttered backgrounds. [ABSTRACT FROM AUTHOR]

In Part I of this paper we developed the theory and algorithms for performing Shape-From-Silhouette (SFS) across time. In this second part, we show how our temporal SFS algorithms can be used in the applications of human modeling and markerless motion tracking. First we build a system to acquire human kinematic models consisting of precise shape (constructed using the temporal SFS algorithm for rigid objects), joint locations, and body part segmentation (estimated using the temporal SFS algorithm for articulated objects). Once the kinematic models have been built, we show how they can be used to track the motion of the person in new video sequences. This marker-less tracking algorithm is based on the Visual Hull alignment algorithm used in both temporal SFS algorithms and utilizes both geometric (silhouette) and photometric (color) information. [ABSTRACT FROM AUTHOR]

This paper discusses analytic algorithms and software for the enumeration of all integer flows inside a network. Concrete applications abound in graph theory, representation theory, and statistics. Our methods are based on the study of rational functions with poles on arrangements of hyperplanes; they surpass traditional exhaustive enumeration and can even yield formulas when the input data contains some parameters. We also discuss the calculation of chambers in detail because it is a necessary subroutine. [ABSTRACT FROM AUTHOR]

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed. [ABSTRACT FROM AUTHOR]

The paper deals with the three-dimensional dynamic problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, considering the contact interaction of the crack faces. An asymmetric solution is obtained using an iteration algorithm developed earlier. Numerical results are presented [ABSTRACT FROM AUTHOR]

An approach to analyzing and predicting systems described by time-varying series is developed on the basis of identification of intervals in which time-varying processes do not change their parameters and construction of models for cointegration relations in these intervals and for tracking the changes both in the processes themselves and in the long-range cointegration relations formed by them. CUSUM-type detection algorithms are constructed for detecting the changes in time-varying processes and identifying stationary intervals. If the process parameters are known exactly, these algorithms are optimal by the quickest detection criterion. For the case studied in the paper, the process parameters are not known after the changes in properties, only certain assumptions concerning the type of parameter changes can be made. Therefore, property changes are detected with the help of modified CUSUM-algorithms, which are not optimal, but guarantee a reasonable detection quality. [ABSTRACT FROM AUTHOR]

Effective design of parallel matrix multiplication algorithms relies on the consideration of many interdependent issues based on the underlying parallel machine or network upon which such algorithms will be implemented, as well as, the type of methodology utilized by an algorithm. In this paper, we determine the parallel complexity of multiplying two (not necessarily square) matrices on parallel distributed-memory machines and/or networks. In other words, we provided an achievable parallel run-time that can not be beaten by any algorithm (known or unknown) for solving this problem. In addition, any algorithm that claims to be optimal must attain this run-time. In order to obtain results that are general and useful throughout a span of machines, we base our results on the well-known LogP model. Furthermore, three important criteria must be considered in order to determine the running time of a parallel algorithm; namely, (i) local computational tasks, (ii) the initial data layout, and (iii) the communication schedule. We provide optimality results by first proving general lower bounds on parallel run-time. These lower bounds lead to significant insights on (i)–(iii) above. In particular, we present what types of data layouts and communication schedules are needed in order to obtain optimal run-times. We prove that no one data layout can achieve optimal running times for all cases. Instead, optimal layouts depend on the dimensions of each matrix, and on the number of processors. Lastly, optimal algorithms are provided. [ABSTRACT FROM AUTHOR]

This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly notable for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties. [ABSTRACT FROM AUTHOR]

This paper presents a new proof method about the paper (Yuan et al. in Appl. Math. Model. 47:811–825, 2017). In the proof, the global convergence of the Polak–Ribière–Polak algorithm is established without these two assumptions d k T g k < 0 and g k + 1 d k ≤ − σ 1 g k T d k which are needed in the above paper. This means that this paper has the same results under weaker conditions. More dimension functions for practical problems are tested to show the performance of the modified algorithm and the normal algorithm. An application of the fact engineering model is done to show the effectiveness of the given conjugate gradient algorithm. [ABSTRACT FROM AUTHOR]

The problem in this paper is motivated by physical problems concerned with the case when a class of continuous and equimeasurable densities of a string is given then how to find minimal frequencies among these given densities, that is, what kind of densities minimize the frequencies. By taking Dirichlet eigenvalues into account, given a certain weight function ω, we will show the minimizer of the mth eigenvalue is the m-degree continuous symmetrical decreasing rearrangement of ω. The main result of this paper can be viewed as complementary to Schwarz’s work (Schwarz in J. Math. Mech. 10:401-422, 1961). [ABSTRACT FROM AUTHOR]