A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

On the cone of a finite dimensional compact convex set at a point

Four equivalent conditions for a convex cone in a Euclidean space to be an F_σ-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by Brønsted [2] and Larman [3].     

Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen

In [8] the author extended the concept of neighbouring functions (cp. [9]) to the case of several variables. Using these results it is shown that under some weak conditions a multiplicative functionf in two variables has a mean-value different from zero if and only if the two multiplicative functionsf ₁(n)=f(n, 1) andf ₂(n)=f(1,n) have mean-values different from zero. Applications to theorems ofDelange [3],Elliott [6] andDaboussi [1] are given.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

The spectrum of Hill's equation

Letq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=?d ²/dx ²+q(x) in the class of functions of period 2 is a discrete series - ∞<λ₀<λ₁≦λ₂<λ₃≦λ₄<...<λ_2i?1≦λ_2i ↑∞. Let the numer of simple eigenvalues be 2n+1<=∞. Borg [1] proved thatn=0 if and only ifq is constant. Hochstadt [21] proved thatn=1 if and only ifq=c+2p with a constantc and a Weierstrassian elliptic functionp. Lax [29] notes thatn=m if¹ q=4k ² K ² m(m+1)sn ²(2Kx,k). The present paper studies the casen<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardneret al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax [28–30] in various directions. The content may be summed up in the statement thatq is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperelliptic irrationality \(\ell (\lambda ) = \sqrt { - (\lambda - \lambda _0 )(\lambda - \lambda _1 )...(\lambda - \lambda _{2n} )} .\) The casen=∞ requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [34], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [22].     

Symmetry in Generalized Quadrangles

In this paper, we describe some aspects of a Lenz(-Barlotti)-type classification of finite generalized quadrangles, which is being prepared by the author. Some new points of view are given. We also prove that each span-symmetric generalized quadrangle of order s > 1 with s even is isomorphic to $ \mathcal{Q} $ (4, s), without using the canonical connection (obtained by S. E. Payne in [15] between groups of order s ³ ? s with a 4-gonal basis and span-symmetric generalized quadrangle of order s. (The latter result was obtained for general s independently by W. M. Kantor in [10], and the author in [30] Finally, we obtain a classification program for all finite translation generalized quadrangles, which is suggested by the main results of [27], [30], [32], [35], [38] and [37].     

On the Minimum Weight of Codes over F5 Constructed from Certain Conference Matrices

In this paper, codes over F₅ with parameters [36, 18, 12], [48, 24, 15], [60, 30, 18], [64, 32, 18] and [76, 38, 21] which improve the previously known bounds on the minimum weight for linear codes over F₅ are constructed from conference matrices. Through shortening and truncating, the above codes give numerous new codes over F₅ which improve the previously known bounds on minimum weights.     

Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A

In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12] and [13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20], [19], [17] and [18], etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16]). These operators have product functions of the forms _m+1F_m and _m+1Ψ_m and integral representations by means of the Meijer G- and Fox H-functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok-Srivastava operator ([8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.103, No 1 (1999), pp. 1-13) defined as a Hadamard product with an arbitrary generalized hypergeometric function _pF_q. Similar conditions are suggested also for its extension involving the Wright _pΨ_q-function and called the Srivastava-Wright operator (Srivastava, [36]). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders p ? q + 1 of the generalized hypergeometric functions and to their parameters.     

FBI transforms in Gevrey classes

In this work we develop the FBI Transform tools in Gevrey classes. Our goal is to extend to a Gevrey-s obstacle withs < 3 the localization of poles result obtained by Sjöstrand [10] in the analytic class. In that work, the author proved that the pole-free zone is controlled by a constantC _0,a (which was only implicit in Bardos-Lebeau-Rauch [1]), improving the constantC _0,∞ of the results of Hargé-Lebeau [13] and Sjöstrand-Zworski [13] valid in C^∞ The works [3], [13] and [10] feature an adapted complex scaling for convex obstacles, but in [10] there is the addition of a small complex “G³ deformation”. The study of such Gevrey deformations for operators with symbols in Gevrey classes is the central point of this work.     

Classification of Modules with Two Distinguished Pure Submodules and Bounded Quotients

In this paper, we consider modules over principal ideal domains R. The objects are free R- modules F with two distinguished pure submodules F ₀ and F₁ with F ₀ ∩ F₁ = 0 and bounded quotient F/(F ₀ ⊕ F ₁) and morphisms are the usual R-homomorphisms which preserve the distinguished submodules. This category is denoted by cRep₂.R and its objects, we say the cR₂-modules are denoted by F = (F, F₀, F ₁). The rank of a cR₂-module F is the rank of the free R-module F. We will show that cR₂ -@#@ modules are direct sums of indecomposable cR₂-modules of rank 1 or 2. The infinite series of indecomposable cR₂-modules is well-known and given explicitly after our Main Theorem 1.4. The result was first shown for cR₂modules of finite rank in Arnold and Dugas [4], then for countable rank, using heavy machinery due to Hill and Megibben [25] in Files and Göbel [20]. Our proof for arbitrary rank is based on [20] and illustrates the importance of Hill’s notion of an axiom-3 family of modules. The Main Theorem is applied to a classification of Butler groups with two critical types. ^{1 2}     

Convergent series solution of nonlinear equations

The author's decomposition method [1] provides a new, efficient computational procedure for solving large classes of nonlinear (and/or stochastic) equations. These include differential equations containing polynomial, exponential, and trigonometric terms, negative or irrational powers, and product nonlinearities [2]. Also included are partial differential equations [3], delay-differential equations [4], algebraic equations [5], and matrix equations [6] which describe physical systems. Essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance. The procedure depends on calculation of the author's A_n, a finite set of polynomials [1,13] in terms of which the nonlinearities can be expressed. This paper shows important properties of the A_n which ensure an accurate and computable convergent solution by the author's decomposition method [1]. Since the nonlinearities and/or stochasticity which can be handled are quite general, the results are potentially extremely useful for applications and make a number of common approximations such as linearization, unnecessary.     

A multipoint method of third order

LetF be a mapping of the Banach spaceX into itself. A convergence theorem for the iterative solution ofF(x)=0 is proved for the multipoint algorithmx _n+1=x _n ?ø(x _n ), where $$\phi (x) = F\prime_x^{ - 1} \left[ {F(x) + F\lgroup {x - F\prime_x^{ - 1} F(x)} \rgroup} \right]$$ andF′x is the Frechet derivative ofF. The theorem guarantees that, under appropriate conditions onF, the multipoint sequence {x _n } generated by ø converges cubically to a zero ofF. The algorithm is applied to the nonlinear Chandrasekhar integral equation $$\frac{1}{2}\omega _0 x(t)\int_0^1 {\frac{{tx(s)}}{{s + t}}ds - x(t) + 1 = 0}$$ where ω₀>0. A discretization of the equations of iteration is discussed, and some numerical results are given.     

On separation axioms in hyperspaces

It is shown that the well-known characterizations of separation axiomsT ₂ andT ₃, respectively of hit-and-miss hyperspace topologies withT ₁ base space (cf. [9], [10]) are valid with no preliminary conditions on the base space.     

On the Mellin transform of products of Bessel and generalized hypergeometric functions

We deduce in an elementary way representations for the Mellin transform of a product of Bessel functions ₀F₁[−a²x²] and generalized hypergeometric functions _pF_p+1[−b²x²] for a,b>0. As a corollary we obtain a transformation formula for _p+1F_p[1] which was discovered by Wimp in 1987 by using Bailey's method for the specialization ₃F₂[1].     

Flows and generalized coloring theorems in graphs

An (oriented) graph H is said to be F_k(k ≥ 2) iff there exists an integer flow in H with all edge-values in $\text{[1 ? k, ?1] ? [1, k ? 1]}$. It is known that a plane 2-edge-connected graph is face-colorable with k colors (k ≥ 2) iff it is F_k; W. T. Tutte has proposed [1] to seek for extensions to general graphs of coloring results known for planar graphs through the use of the F_k property. In this direction, we prove among other results that every 2-edge-connected graph is F₈.     

Imbedded operators with finite Blaschke product symbol

LetF=(f _ij )be ann×n matrix ofH ^∞entries, and define $$S_F = (T_{2 \otimes ln}^* \oplus T_{2 \otimes ln}^* )\left| {_{Graph T_F^* } } \right..$$ This type of operator plays an important role in Cowen-Douglas theory. We call it the imbedded operator with symbolF. In the paper [L], we have already shown thatS _F is a compact pertubation ofT _z? ^* I _n by BDF Theorem. In this paper, we begin to investigate the compact part ofS _F . We give a practical method to calculate this compact part whenn=1 andF is any finite Blaschke product.     

Asymptotic distribution of a Cramér-von mises type statistic for testing symmetry when the center is estimated

Summary In this paper we investigate the effect of estimating the center of symmetry on a Cramér-von Mises type statistic for testing the symmetry of a distribution function. The test statistic is defined by whereF _n is the empirical distribution function andS[F _n] is an estimator of the center ofF which is consistent with the ordern ^1/2 and has von Mises derivative. The asymptotic distribution ofnT ₀[F_n] under the null hypothesis is obtained. The distribution depends on the distributionF and on the estimatorS[F _n]. The Institute of Statistical Mathematics     

A problem of P. Seymour on nonbinary matroids

The following statement fork=1, 2, 3 has been proved by Tutte [4], Bixby [1] and Seymour [3] respectively: IfM is ak-connected non-binary matroid andX a set ofk-1 elements ofM, thenX is contained in someU ₄ ² minor ofM. Seymour [3] asks whether this statement remains true fork=4; the purpose of this note is to show that it does not and to suggest some possible alternatives. Supported in part by the National Science Foundation