A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.     

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ?³ with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C _? r ^?1+ε |t|^?ε/2 for ? T ₀ ≤ t < 0 and 0 < C _? < ∞ allowed to be large. We prove that v is regular at time zero.     

When is the value of public information positive in a game?

The value of public information is studied by considering the equilibrium selections that maximize the weighted sum of players' payoffs. We show that the value of information can be deduced from the deterministic games where the uncertain parameters have given values. If the maximal weighted sum of equilibrium payoffs in deterministic games is convex then the value of information in any Bayesian game derived from the deterministic games is positive with respect to the selection. We also show the converse result that positive value of information implies convexity. Hence, the convexity of maximal weighted sum of payoffs in deterministic games fully characterizes the value of information with respect to considered selections. We also discuss the implications of our results when positive value of information means that for any equilibrium in a game with less information there is a Pareto dominant equilibrium in any game with more information.     

Sharp Bounds on the Aα-index of Graphs in Terms of the Independence Number

Acta Mathematicae Applicatae Sinica, English Series - Given a graph G, the adjacency matrix and degree diagonal matrix of G are denoted by A(G) and D(G), respectively. In 2017, Nikiforov[24]...     

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.     

Predicting the diffusion of information to mass audiences†

This paper considers the statistical inference of structural properties in organizations. Specifically, the sampling probability distributions of the number of links and the cumulative distribution of the number of components are determined. Probability estimates of both the number of links and the number of components are then found.     

Multiple positive solutions of four point boundary value problems for finite difference equations†

In this paper, we consider the following second-order four-point boundary-value problems Δ² u(k ? 1)+f(k,u(k), Δu(k)) = 0,k ∈ {1,2,…,T}, u(0) = au(l ₁), u(T+1) = bu(l ₂). We give conditions on f to ensure the existence of at least three positive solutions of the given problem by applying a new fixed-point theorem of functional type in a cone. The emphasis is put on the nonlinear term involved with the first-order difference.     

The effect of order and co‐ordination of the problem quantities on the difficulty of missing value proportion problems

One variable of the solution process of missing value proportion problems is the order co‐ordination of the missing value and the units of measure. A scheme is devised to analyse the syntactical structure of such problems in a standard self‐paced college arithmetic text. Student performance is assessed to determine the impact of this co‐ordination on problem difficulty.     

Estimates for the Upper Bounds of the First Eigenvalue on Submanifolds

ＥｓｔｉｍａｔｅｓｆｏｒｔｈｅＵｐｐｅｒＢｏｕｎｄｓｏｆｔｈｅＦｉｒｓｔＥｉｇｅｎｖａｌｕｅｏｎＳｕｂｍａｎｉｆｏｌｄｓ祁锋，于霖泉，雒秋明ＥｓｔｉｍａｔｅｓｆｏｒｔｈｅＵｐｐｅｒＢｏｕｎｄｓｏｆｔｈｅＦｉｒｓｔＥｉｇｅｎｖａｌｕｅｏｎＳｕｂｍａｎｉ...     

Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube*

Gomorys and Chvátals cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The Chvátal rank of the polyhedron is the number of rounds needed to obtain all valid inequalities. It is well known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is 2-dimensional, and if its integer hull is a 0/1-polytope.We show that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, we prove that the rank of every polytope contained in the n-dimensional 0/1-cube is at most n ² (1+log n). Moreover, we also demonstrate that the rank of any polytope in the 0/1-cube whose integer hull is defined by inequalities with constant coefficients is O(n).Finally, we provide a family of polytopes contained in the 0/1-cube whose Chvátal rank is at least (1 + ) n, for some > 0.* An extended abstract of this paper appeared in the Proceedings of the 7th International Conference on Integer Programming and Combinatorial Optimization [20].     

Extremal problems on the Hua domain of the first type

In this paper we study some extremal problems between the Hua domain of the first type and the unit ball.     

Issues arising on the use of hand‐held calculators in schools†

This paper considers the use of hand‐held calculators (HHCs) in schools from the pedagogical and sociological viewpoints. Using arguments based on the observed pattern of mathematical education, we discuss the effect of the use of HHCs, from the learning viewpoint and from the viewpoint of providing students with a ready competence in arithmetical manipulation. We shall also discuss the effects of the adoption of HHCs in undeveloped countries and for deprived minorities.

The topics are developed from philosophical/psychopedagogical reasoning and from developmental policy. Strategies designed to make mathematical education more immediately useful are discussed. An example is provided, and the relationship between Polya's problem‐solving approach and the use of HHCs is examined.     

Bounds on the vertex–edge domination number of a tree

A vertex–edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex–edge domination number of a graph G  , denoted by γ_ve(T)${\gamma }_{\mathrm{ve}}(T)$, is the minimum cardinality of a vertex–edge dominating set of G. We prove that for every tree T   of order n?3$n?3$ with l leaves and s   support vertices, we have (n−l−s+3)/4?γ_ve(T)?n/3$(n-l-s+3)/4?{\gamma }_{\mathrm{ve}}(T)?n/3$, and we characterize the trees attaining each of the bounds.     

Ultrafilters on ω and atoms in the lattice of uniformities II

Interaction between ultrafilters and uniformities on a countable set is investigated. Various ultrafilters are constructed such that atoms in the lattice of uniformities refining the corresponding ultrafilter uniformities have special properties.     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

On boundary value problems for the ϕ-Laplacian

For the φ-Laplacian, we consider a boundary value problem with functional boundary conditions. The Dirichlet problem is a special case of this problem.     

Bounds on the Non-real Spectrum of a Singular Indefinite Sturm-Liouville Operator on ℝ

A simple explicit bound on the absolute values of the non-real eigenvalues of a singular indefinite Sturm-Liouville operator on the real line with the weight function sgn(·) and an integrable, continuous potential q is obtained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)