Linear-quadratic two-person zero-sum differential games,necessary and sufficient conditions: Comment

We give a simpler, easier-to-check, version of the theorem of the paper referred to, i.e., a necessary and sufficient condition for the existence of a saddle point to the linear-quadratic two-person zero-sum perfect information differential game.     

关于区间上的Lorenz映射

本文引进了变换ζ．首先，用比文［４］简捷的方法证明了Ｌｏｒｅｎｚ映射的Ｓａｒｋｏｖｓｋｉ定理并更正了文［４］中的一个错误，还得到了扩张的Ｌｏｒｅｎｚ映射中周期点的存在性与其扩张常数之间的关系．其次，证明了Ｌｏｒｅｎｚ映射与其在变换ζ的像具有相等的捏制行列式．此外，给出了广义的Ｌｏｒｅｎｚ映射的周期点的存在性适合Ｓａｒｋｏｖｓｋｉ序关系的一个充分条件     

Global existence of smooth solutions to the vlasov poisson system in three dimensions

Recently Plaffelmoser has shown that solutions of the Vlasov–Poisson system in three dimensions remain smooth for all time. This paper establish the same existence theorem by a simpler method     

Convex Realizations of Planar Linear Trees

We prove a necessary condition for the existence of a convex realization of a planar linear tree. In the case of broken lines, it is shown that this condition is sufficient; a continuous algorithm constructing such a realization is found.     

Nearly perfect codes in distance-regular graphs

The idea of a nearly perfect code in a vector space over a binary field is generalized to the class of distance-regular graphs. A necessary condition for the existence of a nearly perfect code in a distance-regular graph is obtained, and it is shown that this condition implies the analogous result in the classical binary case.     

Orbits connecting singular points in the plane

This paper concerns the global structure of planar systems. It is shown that if a positively bounded system with two singular points has no closed orbits, the set of all bounded solutions is compact and simply connected. Also it is shown that for such a system the existence of connecting orbits is tightly related to the behavior of homoclinic orbits. A necessary and sufficient condition for the existence of connecting orbits is given. The number of connecting orbits is also discussed.Supported by the National Science Foundation of China Grant No. 10461003.     

Global stability in a diffusive Holling-Tanner predator-prey model

A diffusive Holling-Tanner predator-prey model with no-flux boundary condition is considered, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition.     

The Gelfand-Levitan Theory Revisited

Gelfand and Levitan in their celebrated article in 1951, and later Gasymov and Levitan in 1964 have shown that a monotone increasing function is a spectral function of a singular Sturm-Liouville problem on a half-line in the limit point case at infinity if and only if it satisfies an existence and a smoothness condition. In this article, a closer look at the original statement reveals that the existence condition in fact follows from the smoothness condition which simplifies significantly the statement of the Gelfand-Levitan theory. We also provide two sufficient and verifiable conditions for a nondecreasing function to be the spectral function of a singular Sturm Liouville operator.     

Conjugate Intervals for the Linear Fixed-Endpoint Control Problem

For certain optimal control problems with piecewise continuous controls, recently Loewen and Zheng (Ref. 1) and Zeidan (Ref. 2) defined two sets of generalized conjugate points for which, under normality assumptions, the second-order conditions in terms of the accessory problem imply their emptiness. However, simple examples show that checking the existence of such points may be more difficult than directly finding variations that make the second variation negative. In this paper, for the linear fixed-endpoint control problem, we introduce a new set whose emptiness is equivalent to the nonnegativity of the second variation along admissible variations. Moreover, we achieve by means of this set the main objective of introducing a characterization of this condition, namely, to obtain a simpler way of verifying it.     

Circuit analysis as a discrete boundary value problem

In this note, it is shown that mesh and nodal techniques for electric circuit analysis can be cast as non-autonomous discrete boundary value problems. An elegant necessary and sufficient condition for the existence of a unique solution for such problems is described. The developed condition is computationally tractable, as it only involves checking a determinant that is specially constructed from the coefficients of the underlying difference equation. Illustrative examples are presented to demonstrate the validity of the result.     

Generalizations of Hensel’s lemma and the nearest root method

Hensel’s lemma and its various modifications, such as, for instance, the Hensel–Rychlik theorem, are important tools for investigating problems of existence of roots of polynomials in valued fields. It is shown that a method proposed in the author’s previous papers can also be used to provide a simpler derivation of results in [3].     

A simple proof that every matroid is an intersection of fundamental transversal matroids

We give a new proof of a theorem of Bondy and Welsh. Our proof is simpler than previous ones in that it makes no use of Hall's theorem on the existence of a transversal of a family of sets.     

A note on matrices of zeros and ones

Gale and Ryser have given a necessary and sufficient condition for the existence of a matrix of zeros and ones with specified row and column sums. Though in general it appears quite difficult to compute the number of such matrices, it is shown in this note that it is possible to obtain a quite elementary and easily used formula in a large number of interesting cases.     

Further results on adaptive full-order and reduced-order observers for Lur’e differential inclusions

This paper is concerned with the adaptive observer design of Lur’e differential inclusions with unknown parameters. Under a relaxed assumption on nonlinear perturbation functions, a sufficient condition for the existence of an adaptive full-order observer is established. Comparing with results in the literature, the present conditions are complemented with a numerically reliable computational approach, which can be checked by means of linear matrix inequalities. Furthermore, it is shown that, under the sufficient condition, the existence of a reduced-order observer is guaranteed. Also, the reduced-order observer is designed. The effectiveness of the proposed design is illustrated via a simulation example.     

A note on economic equilibrium and financial networks

Nagurney （1999） used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney （1999）. It is a simpler model than that of Nagurney （1999）. We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney （]999）. Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results.     

The existence of a generalized fourier transform for a solution as a radiation condition for a class of problems generalizing oscillation excitation problems in regular waveguides

For a second-order inhomogeneous differential equation defined on the real axis and such that its right-hand side and solutions are functions in a Hilbert space, it is shown that the existence of a generalized Fourier transform of the solution is a correct radiation condition if the right-hand side is sufficiently smooth and compactly supported.     

Stable vector bundles with c1 = 0 on K3 surfaces

In this paper we study stable bundles with trivial first Chern class on a K3 surface. A necessary and sufficient condition for the existence of such bundles is proved and it is shown that Mukai's reflection functor gives a new compactification of the moduli space. A relation to the notion of T-duality in theoretical physics is also discussed.     

BV solutions for hyperbolic systems of balance laws with relaxation

For hyperbolic systems of balance laws with source manifesting relaxation, it is shown that the Kawashima condition, which yields global classical solutions with smooth initial values near equilibrium, is also instrumental in inducing the existence of global admissible BV solutions, accommodating shocks.     

A note on inverse mean curvature flow in cosmological spacetimes

In 12 Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in 12 that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface.We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.     

Liouvillian integration and Bernoulli foliations

Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.