Dubins’ problem on surfaces. I. nonnegative curvature

Let M be a complete, connected, two-dimensional Riemannian manifold. Consider the following question: Given any (p₁,v₁) and (p₂, v₂) in T M, is it possible to connect p₁ to P₂ by a curve y in M with arbitrary small geodesic curvature such that, for i = 1, 2, y is equal to v_i at p_i? In this article, we bring a positive answer to the question if M verifies one of the following three conditions: (a) M is compact, (b) M is asymptotically flat, and (c) M has bounded nonnegative curvature outside a compact subset.     

The Arcsine Law

Let N _n denote the number of positive sums in the first n trials in a random walk (S _i) and let L _n denote the first time we obtain the maximum in S ₀,..., S _n. Then the classical equivalence principle states that N _n and L _n have the same distribution and the classical arcsine law gives necessary and sufficient condition for (1/n) L _n or (1/n) N _n to converge in law to the arcsine distribution. The objective of this note is to provide a simple and elementary proof of the arcsine law for a general class of integer valued random variables (T _n) and to provide a simple an elementary proof of the equivalence principle for a general class of integer valued random vectors (N _n, L _n).     

On Edge Semi-Isomorphisms and Semi-Dualities of Graphs

Let G₁ and G₂ be undirected graphs, and ?₁(G ₁) and ?₂(G ₂) be families of edge sets of G₁ and G₂, respectively. An (?₁,?₂)-semi-isomorphism ofG ₁ ontoG ₂ is an edge bijection between G₁ and G₂ that induces an injection from ?₁(G ₁) to ?₂(G ₂). This concept generalizes a well known concept of a circuit isomorphism of graphs due to H. Whitney. If has a “dual nature” with respect to ?₂(G ₂) then the concept of (?₁,?₂)-semi-isomorphism of graphs turns into a concept of a (?₁,?₂)-semi-duality of graphs. This gives a natural generalization of the circuit duality of graphs due to H. Whitney. In this paper we investigate (?₁,?₂)-semi-isomorphisms and (?₁,?₂)-semi-dualities of graphs for various families ?₁(G ₁) and ?₂(G ₂). In particular, we consider families of circuits and cocircuits of graphs from this point of view, and obtain some strengthenings of Whitney’s 2-isomorphism theorem and Whitney’s planarity criterion for 3-connected graphs.     

非扩张映射和广义变分不等式的粘滞逼近法

应用已提出的非扩张映射的粘滞逼近方法,给定初值x_0∈C,考虑一般迭代过程{x_n},g(x_(n+1))=α_nf(x_n)+(1-α_n)SP_C(g(x_n)-λ_nAx_n),n≥0,其中{α_n}■(0,1),S:C→C是非扩张映射,C是实Hilbert空间H的非空闭凸子集.在{α_n}满足合适的条件下可证明,{x_n}强收敛到非扩张映射的不动点集和广义变分不等式解的公共元,且满足某变分不等式.     

Finite quotients of the automorphism group of a free group

The automorphism group AutF_n of a free group F_n of rank n acts on the product of n copies of a group G by substituting n elements of G into the words defining an automorphism of the free group. This gives rise to an antihomomorphism from AutF_nto a permutation group. We determine this antihomomorphic image of AutF_n when G is the semidirect product Z_p x Z_q     

A Theta Lift for Spin7

We establish an example of a functorial lift from generic cuspidal representations of a similitude group of the type A ₁×C ₂ to generic representations of Spin₇. Our construction uses the theta correspondence associated to the dual pair of the type (A ₁×C ₂ B ₃) inside E ₇. We also consider another theta correspondence associated to the dual pair of type (A ₁×C ₂ A ₁×A ₁) in D ₆ and show that these two pairs fit into a tower and the standard properties of a tower of theta correspondences hold.     

Minima of initial segments of infinite sequences of reals

Suppose that 〈x_k〉_k∈? is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA₀ suffices to prove the existence of a sequence of reals 〈u_k〉_k∈? such that for each k, u_k is the minimum of {x₀, x₁, …, x_k}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈x_k〉_k∈?, the stronger subsystem WKL₀ is required. Following the presentation of these reverse mathematics results, we will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of generalized inverses to interval linear programs in hilbert spaces

Let H₁ and H₂ be real Hilbert spaces. Suppose H₂ is partially ordered, a, b ? H₂ c ? H₁ and A :H₁ → H₂ is a continuous linear map. We consider the following interval linear program: Maximize subject to a ≤ Ax ≤ b.

Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of A. Several examples are given to illustrate the main results.     

Primitivity in the free groups of the variety UmUn

For each m,n >= 0, let G_m,n denote the free group of rank r in the variety U_mU_n. The main results in this paper are: (i) a necessary and sufficient condition for a system of r elements {v₁,…, v_r} to form a basis of G_m,ni (ii) necessary and sufficient conditions for a system of l elements {v₁,…, v_l}, l <= r, to be included in a basis of G_m,0. In particular, (i), (ii) yield corresponding results for the free metabelian group of rank r.     

Applications of a distance formula in hyperbolic plane geometry

This note may be used in model-based courses on the classical geometries. Given two points P₁(x₁, y₁) and P₂(x₂, y₂) in the Poincaré upper half-plane model of hyperbolic plane geometry with x₁≠x₂, a Cartesian equation of thebowed geodesic passing through P₁ and P₂ and an integral expression for the hyperbolic distance between P₁ and P₂ are developed. These formulas depend only on the coordinates of P₁ and P₂, and it is easy to implement them with the numerical integrators of modern technology. The distance formula is used to find the coordinates of points of division along a bowed geodesic and, thus, leads to activities discovering that results such as Ceva's Theorem are valid in hyperbolic geometry. The distance formula is also used in verifying that the model satisfies the axioms of absolute geometry.     

Intuitive Rules and Comparison Tasks

Our work in science and mathematics education has led us to observe that students react similarly to a wide variety of conceptually unrelated situations. Our work suggests that many responses that literature describes as alternative conceptions are interpreted as if they evolved from a small number of intuitive rules. Two such rules are manifested in comparison tasks. The first, more A-more B, is reflected in students' responses to tasks in which 2 objects that differ in a certain, salient quantity A are described (A₁ > A₂). Students are then asked to compare the 2 objects with respect to another quantity B (B₁ = B₂ or B₁ < B₂). In these cases, a substantial number of students responded inadequately according to the more A (the salient quantity)-more B (the quantity in question) rule. The second, same amount of A-same amount of B, is activated in situations in which A₁ = A₂, but B₁ ≠ B₂. In such situations, students often incorrectly claim that B₁ = B₂ because A₁ = A₂. In this article, we demonstrate the explanatory and predictive power of these intuitive rules.     

On Dilatations and Substantial Boundary Points of Homeomorphisms of Jordan Curves

We study the relation between the dilatations K_h and K_h* of a homeomorphism h of Jordan curves. We show that if K_h= K_h*, then either h is induced by an affine map or there is a substantial boundary point for h. In particular, we prove that if h is symmetric (in the sense of Gardiner and Sullivan), then K_h* > K_h. This is quite contrary to a previously conjectured relation between K_h and K_h*.     

A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula

By multidimensional matrix inversion, combined with an A _r extension of Jackson’s ₈ φ ₇ summation formula by Milne, a new multivariable ₈ φ ₇ summation is derived. By a polynomial argument this ₈ φ ₇ summation is transformed to another multivariable ₈ φ ₇ summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating ₆ φ ₅ summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised ₆ ψ ₆ summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).     

Structure and crystallization of bulk amorphous Pd41Ni10Cu28P21 alloy

Bulk amorphous Pd₄₁,Ni₁₀Cu₂₈P₂₁, alloy has been prepared by water quenching method. The system shows excellent glass forming ability (GFA) with a high value of reduced glass transition temperatureT _rg 0.714. Structural analyses indicate that the Pd₄₁,Ni₁₀Cu₂₈,P₂₁, alloy has a dense packing structure closer to “frozen liquid” than that of amorphous Pd₄₀Ni₄₀,P₂₀, alloy. Experiments on crystallization reveal that several crystalline phases simultaneously precipitate in the early part of crystallization. Below 710 K, a metastable phase forms, and subsequently disappears at elevated temperatures. In addition, the influence of partial substitute of Cu for Ni on GFA has been discussed with regard to thermodynamics and kinetics     

Formes de torsion analytique et familles de submersions

Let π₁ : W → V (resp. π₂ : V → S) be a holomorphic submersion of complex varieties with compact fibre. Let ξ be a holomorphic vector bundle on W. In this Note, we announce a result which relates a combination of the analytic torsion forms associated to π₁, π₂ and π₂ o π₁ in terms of Bott-Chern classes. This result extends to a relative situation a result by Berthomieu-Bismut on the behaviour of submersion of Quillen metrics.     

Geometry of the motion of robot manipulators

A p-parametric robot is a mapping g of R^p into the homogeneous space P=C₆×C₆/Diag(C₆×C₆) given by the formula g(u₁,...,u_p=exp u₁X₁..... exp u_pX_p, where C₆, is the Lie group of all congruences of E₃ and X₁,..., X_p are fixed vectors from the Lie algebra of C₆. We characterize the set g(R^p) locally by a system of PDE and give some geometrical properties of g as a p-dimensional motion for p<6. We also characterize the Frenet frame of g and show how to construct it for the robot manipulator given by its axes X₁,...,X_p.     

A New Functor from E 6-mod to E 7-mod

We find a new representation of the simple Lie algebra of type E ₇ on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor from E ₆-Mod to E ₇-Mod. A condition for the functor to map a finite-dimensional irreducible E ₆-module to an infinite-dimensional irreducible E ₇-module is obtained. Our general framework also gives a direct polynomial extension from irreducible E ₆-modules to irreducible E ₇-modules, which can be used to derive Gel'fand–Zetlin bases for E ₇ from those for E ₆ that can be obtained from those for D ₅ according to our earlier work.     

Going in Circles: Variations on the Money-Coutts Theorem

Given a polygon A ₁,...,A _n, consider the chain of circles: S ₁ inscribed in the angle A ₁, S ₂ inscribed in the angle A ₂ and tangent to S ₁, S ₃ inscribed in the angle A ₃ and tangent to S ₂, etc. We describe a class of n-gons for which this process is 2n-periodic. We extend the result to the case when the sides of a polygon are arcs of circles. The case of triangles is known as the Money-Coutts theorem.