Refinements of the β-core and the strong equilibrium and the Aumann proposition

The Aumann proposition establishes that an outcome in the -core of the one-shot gameG is the same as the strong equilibrium utility allocations of the associated supergame. We refine the -core by requiring that the threat strategies used to deter deviations be credible and examine the consequences for the Aumann proposition in the case of the refinement. We show that a refinement of the strong-equilibrium notion is always in the refined -core. The converse is not necessarily true unless the utility allocations are linear.I am grateful to T. Ichiishi for several discussions that helped in the development of this paper. This is part of the author's Ph.D. dissertation submitted at the university of Iowa in December 1985.     

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.     

On the Quaternion Ball and the Quaternion Projective Space

Since the quaternion ball was used to study the AdS/CFT problems tor spinor fields, it is worthwhile to study further the geometry （in sense of Klein） and analysis on it and on its extended space （in the sense of Behnke-Thullen）, the quaternion projective space.     

On the Semivalues and the Least Square Values Average Per Capita Formulas and Relationships

In this paper, it is shown that both the Semivalues and the Least Square Values of cooperative transferable utilities games can be expressed in terms of n^2 averages of values of the characteristic function of the game, by means of what we call the Average per capita formulas. Moreover, like the case of the Shapley value earlier considered, the terms of the formulas can be computed in parallel, and an algorithm is derived. From these results, it follows that each of the two values mentioned above are Shapley values of games easily obtained from the given game, and this fact gives another computational opportunity, as soon as the computation of the Shapley value is efficiently done.     

Bridging the Gap Between the Dense and the Discrete: The Number Line and the “Rubber Line” Bridging Analogy

In two experiments we explored the instructional value of a cross‐domain mapping between “number” and “line” in secondary school students' understanding of density. The first experiment investigated the hypothesis that density would be more accessible to students in a geometrical context (infinitely many points on a straight line segment) compared to a numerical context (infinitely many numbers in an interval). The participants were 229 seventh to eleventh graders. The results supported this hypothesis but also showed that students' conceptions of the line segment were far from that of a dense array of points. We then designed a text-based intervention that attempted to build the notion of density in a geometrical context, making explicit reference to the number-to-points correspondence and using the “rubber line” bridging analogy (the line as an imaginary unbreakable rubber band) to convey the no-successor principle. The participants were 149 eighth and tenth graders. The text intervention improved student performance in tasks regarding the infinity of numbers in an interval; the “rubber line” bridging analogy further improved performance successfully conveying the idea that these numbers can never be found one immediately next to the other.     

A Remark on the Weak Topology and the Degree

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Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

δ - and δ ′-shocks and the transportation and concentration processes

Using the definitions of δ - and δ ′-shocks for some systems of conservation laws, the corresponding Rankine–Hugoniot conditions are derived. We also derive the balance laws describing area, volume, mass and momentum transportation between the area outside the wave front and the wave front. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourier series and the δ process

We discuss the effect of a particular sequence acceleration method, the δ²${\delta }^2$ process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points.     

Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott–Suppes Theorem

This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to has an interval representation. The second is that any finite poset with no restriction isomorphic to or to has a unit interval representation. Both proofs are straightforward and inductive.     

On global analytic and Gevrey hypoellipticity on the torus and the Métivier inequality

We obtain a global version in the N-dimensional torus of the Métivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context.     

Leonardo da Vinci and the Ellipse

Leonardo da Vinci was the ultimate Renaissance manartist, architect, scientist, inventor, mathematician, philosopher, and sculptor. His notebooks are full of sketches and innovative ideas spanning a variety of subjects, sometimes too advanced for his time. He invented various types of special compasses capable of producing parabolas, ellipses, and proportional figures. He is also credited with the invention of perspectograph, used by artists, such as Albrecht Dürer, to help draw objects in perspective. He strove to learn and master subjects whose understanding he knew was essential or influenced his     

Leonardo da Vinci and the Ellipse

Leonardo da Vinci was the ultimate Renaissance man- artist．architect,scientist,inventor,mathematician, philosopher,sculptor．His note-books are full of sketches and inno- vative ideas spanning a variety of subjects, sometimes too ad- vanced for his time．He invented various types of special com- pass capable of produ- cing parabolas,ellip- ses and proportional figures．He is also     

Surgery and the Spinorial τ-Invariant

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then τ(N) ≥ min{τ(M), Λ _n }, where Λ _n is a positive constant depending only on n = dim M. Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ _n . Also, below Λ _n the values of τ cannot accumulate from above when varied over all manifolds of dimension n.     

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

L-fuzzy ⋎ and ⋏ hyperoperations and the associated l-fuzzy hyperalgebras

In this paper we study two fuzzy hyperoperations, denoted by ⋎ (which can be seen as a generalization of ∨) and ⋏ (which can be seen as a generalization of ∧). ⋎ is obtained from a family of crisp ∨; _p hyperoperations and ⋏ is obtained from a family of crisp ∧ _p hyperoperations. The hyperstructure (X, ⋎, ∧) resembles ahyperlattice and the hyperstructure (X, ∨, ⋏) resembles adual hyperlattice