Perspective Graves triads and triads of perspective inscribed triangles

Any Pappus configuration can be regarded (in six ways) as a triad of triangles, each inscribed in the previous one; such a triad of triangles is called aGraves triad. Graves triads in which each pair of triangles is in perspective may be calledperspective Graves triads; such triads occur in connection with various results in both projective and metrical geometry. In this paper we give a synthetic proof of the existence of perspective Graves triads; the proof is of interest because it holds in allharmonic planes and requires neither the axiom of Pappus nor the full axiom of Desargues. The paper also includes synthetic proofs of other related results.(Results involving perspective Graves triads can be found in [l, p.149] and [2]. This paper is the result of recent correspondence with Guinand concerning [2] and [3].)     

Graves triads,Möbius pairs,and related matrices

A Graves triad is a cyclic triad of triangles, each circumscribing the next, forming a Pappus configuration. If two of the triangles are in perspective, then so are the other two pairs. In a non-cyclic triad of such circumscribing triangles, if two pairs are in perspective, then so is the third pair. These results correspond to certain properties of 3×3 zero-diagonal matrices. There is also an analogous result for Möbius pairs of mutually circumscribing tetrahedra and 4×4 zero-diagonal matrices.     

The conflict triad dynamical system

A dynamical model of the natural conflict triad is investigated. The conflict interacting substances of the triad are: some biological population, a living resource, and a negative factor (e.g., infection diseases). We suppose that each substance is multi-component. The main coexistence phases for substances are established: the equilibrium point (stable state), the local cyclic orbits (attractors), the global periodic oscillating trajectories, and the evolution close to chaotic. The bifurcation points and obvious thresholds between phases are exhibited in the computer simulations.     

Coalitions and payoffs in three‐person sequential games: Initial tests of two formal models

In this paper we develop two formal models predicting coalitions and payoffs among rank striving players in a sequential three‐person game. We test the models’ predictions with data from a laboratory study of eleven male triads. Each triad plays a sequence of games; in each game a two‐person coalition forms and divides the coalition's point value between the two coalition partners. Participants know that the sequence of games will end without warning at a randomly chosen time; at the sequence's end each player's monetary payoff is a linear function of the rank of his accumulated point score, relative to those of the other members of his triad. The complexity of this situation prevents players and analysts from representing it as a single game; thus they are unable to use n‐person game theory to identify optimal strategies. Consequently, we assume that players, unable to develop strategies that are demonstrably optimal in the long run, adopt certain bargaining heuristics and surrogate short run objectives.

The two models follow the same basic outline; they differ, however, in the planning horizon they assume players to use. Proceeding from a priori assumptions concerning each player's decision calculus and the bargaining process, the two models state the probability that each coalition forms and predict the point divisions in the winning coalition. The laboratory data provide consistently strong support for the predictions of both models.     

Stochastic leadtimes in a one-warehouse,N-retailer inventory system with the warehouse not carrying stock

This study extends upon a multi-echelon inventory model developed by Graves, introducing in the one-warehouse, N-retailer case—as Graves suggested—stochastic leadtimes between the warehouse and the retail sites in place of the original deterministic leadtimes. Effects of stochastic leadtimes on required base stock levels at the retail sites in the case where the warehouse carries no stock (e.g., serves as a cross-dock point) were investigated analytically. Two alternative treatments of stochastic leadtime distributions were considered. Using as a baseline Graves’ computational study under deterministic leadtimes, results of the current study suggest that it may be better to use the deterministic model with an accurately estimated mean leadtime than a stochastic model with a poorly estimated mean leadtime.     

A mathematical study of the linear theory for orthotropic elastic simple shells

The theory of simple shells is a surface‐related Cosserat model for thin elastic shells. In this direct approach, each material point is connected with a triad of rigidly rotating directors. This paper presents a study of the governing equations for orthotropic elastic simple shells in the framework of the linearized theory. We establish the uniqueness of classical solutions, without any restrictive assumption on the strain energy function. The continuous dependence of solutions on the body loads and initial data is proved. Also, the existence of weak solutions to the equations of simple shells is proved by means of an inequality of Korn's type established for such directed surfaces. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on Robinson–Ursescu and Lyusternik–Graves theorem

The aim of this note is twofold. First, we prove an analogue of the well-known Robinson–Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings. The approximated non-linear mapping is restricted to a closed convex subset of a Banach space.     

Metrically Regular Vector Field and Iterative Processes for Generalized Equations in Hadamard Manifolds

This paper is focused on the problem of finding a singularity of the sum of two vector fields defined on a Hadamard manifold, or more precisely, the study of a generalized equation in a Riemannian setting. We extend the concept of metric regularity to the Riemannian setting and investigate its relationship with the generalized equation in this new context. In particular, a version of Graves’s theorem is presented and we also define some concepts related to metric regularity, including the Aubin property and the strong metric regularity of set-valued vector fields. A conceptual method for finding a singularity of the sum of two vector fields is also considered. This method has as particular instances: the proximal point method, Newton’s method, and Zincenko’s method on Hadamard manifolds. Under the assumption of metric regularity at the singularity, we establish that the methods are well defined in a suitable neighborhood of the singularity. Moreover, we also show that each sequence generated by these methods converges to this singularity at a superlinear rate.     

Numerical investigation of interactions of multiple spherical shock waves between themselves and with the underlying surface

Within the investigation of various aspects of asteroid and comet danger and, in particular, the explosion of several fragments of meteoroids in the atmosphere above the Earth surface, the toy problem about four point explosions in the case of their special arrangement above the underlying surface is numerically solved. Complex interactions of primary and secondary shock waves between themselves, with the hard surface, and with tangential discontinuities are examined. The structure of flow inside gas regions disturbed by the explosions—the occurrence of eddy structures in them and the influence of reflected shocks waves on them—are investigated. The tendency of the external wave fronts of each explosion to form a unified front and the tendency of their internal hot domains to merge into a joined configuration (where the second process proceeds a little later than the first one) is revealed. This unified front and joined configuration are qualitatively identical to the external internal structure for the solitary explosion. The specially arranged explosions are chosen because the effects of multiple diffraction, interference, and, the main thing, cumulation of spherical waves are manifested more clearly in this caseTwo variants with different altitude of the explosions above the surface are calculated.     

非锥序的有效点的存在性与外在稳定性

本文讨论了非锥序的有效点的存在性与外在稳定性，并且各给出了一个充要条件，在此基础上，对序－半紧集的有效点的存在性与外在稳定性给出了一些充分条件，最后，在决策空间中给出了相应的结果。     

On resonant interactions of gravity‐capillary waves without energy exchange

We consider resonant triad interactions of gravity‐capillary waves and investigate in detail special resonant triads that exchange no energy during their interactions so that the wave amplitudes remain constant in time. After writing the resonance conditions in terms of two parameters (or two angles of wave propagation), we first identify a region in the two‐dimensional parameter space, where resonant triads can be always found, and then describe the variations of resonant wavenumbers and wave frequencies over the resonance region. Using the amplitude equations recovered from a Hamiltonian formulation for water waves, it is shown that any resonant triad inside the resonance region can interact without energy exchange if the initial wave amplitudes and relative phase satisfy the two conditions for fixed point solutions of the amplitude equations. Furthermore, it is shown that the symmetric resonant triad exchanging no energy forms a transversely modulated traveling wave field, which can be considered a two‐dimensional generalization of Wilton ripples.     

Constrained open mapping theorem with applications

We prove a generalization of Graves? Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a set-valued one having particular properties. The nonlinear mapping is restricted to a closed convex subset of a Banach space. We apply the results to derive necessary and sufficient conditions ensuring the existence of a differentiable selection for the inverse mapping. A slight generalization of a sufficient condition by J. Klamka on so-called constrained exact local controllability of nonlinear and semi-linear dynamic systems is also proved.     

On the theory of orthotropic porous elastic rods

We employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. In this purpose, we consider the direct approach to the theory of rods, in which the thin body is modeled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. For orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches (directed curves and three-dimensional rods). (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Triad count estimation in digraphs

One may use information about a random sample of network members to estimate quantities related to the triad census of a network. Various kinds of information about the graph may be observable from the sample; four observation schemes involving the local networks of the sampled vertices are considered here. Unbiased triad count estimators are defined, and their variances (and unbiased estimators of these variances) are derived. A main result is that under one of the observation schemes, the estimator can be written as a sum of vertex attributes; standard estimation formulas for various sampling designs, such as stratified sampling, are therefore effortlessly applied. The estimator properties are compared in a simulation study.     

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997     

Pattern memory analysis of second-order neural networks with time-delay

In this paper, the pattern memory of the second-order neural networks with time-delay are investigated based on the stability theory. Several sufficient conditions are obtained such that the equilibrium point is locally exponential stable when the equilibrium point located in the saturation region. These conditions can be directly derived from the synaptic connection weights and the external input of the second-order neural networks, and be verified easily. In addition, four examples are given to show the effectiveness of the results.     

Sociometric triads as multi‐variate systems†

Binary choices (choose v. not choose) among three persons may be viewed as variables and the possible choices in a triad as a 64 celled contingency table, amenable to analysis in terms of log linear effects. Fifteen non‐redundant effects occur, of which four appear consistently in a data bank of 384 sociograms. Consequently, it is claimed one may reproduce the triad frequencies in most sociograms by the effects implied by four propositions: (A) P is more likely to choose O when O chooses P, net of all other variables; (B) P is more likely to choose O when X chooses O. This association is particularly strong when imitation produces transitivity; (C) when X chooses O (O chooses X), P tends to choose both or neither; and (D) P tends to choose an X who favors P's choices. P tends to not choose an X who favors P's non‐choices.     

A solvable mixed charge ensemble on the line: global results

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2:1 in the presence of an external field. With the total charge fixed and the system held at temperature corresponding to β = 1, it is proved that the particles form a Pfaffian point process. When the external field is quadratic (the harmonic oscillator potential), we produce the explicit family of skew-orthogonal polynomials necessary to simplify the related matrix kernels. In this setting a variety of limit theorems are proved on the distribution of the number as well as the spatial density of each species of particle as the total charge increases to infinity. Connections to Ginibre’s real ensemble of random matrix theory are highlighted throughout.     

On the motion of two-dimensional vortices with mass

Summary The Helmholtz-Kirchhoff ODEs governing the planar motion ofN point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two “massive” vortices totwo degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently by Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the Helmholtz-Kirchhoff equations.