Stability criteria of the Bendixon type

Summary In this paper we consider a system of two first order differential equations {x′=P(x,y). y′=Q(x,y)}. We usually assume that ∂P/∂x+∂Q/∂y vanish identically in a certain region. A number of conditions are then given to insure boundedness of solutions or asymptotic stability of the zero solution. Entrata in Redazione il 5 gennaio 1972.     

On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian

The displacement map related to small polynomial perturbations of the planar Hamiltonian systemdH=0 is studied in the elliptic caseH=1/2y ²+1/2x ²−1/3x ³. An estimate of the number of isolated zeros for each of the successive Melnikov functionsM _k(h),k=1, 2,…is given in terms of the orderk and the maximal degreen of the perturbation. This sets up an upper bound to the number of limit cycles emerging from the periodic orbits of the Hamiltonian system under polynomial perturbations. Research partially supported by grant MM810/98 from the NSF of Bulgaria and MURST, Italy.     

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits.More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x~2+ y~2+ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincar′e disc.     

On the polynomial limit cycles of polynomial differential equations

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution.     

Nonexistence of Isochronous Centers in Planar Polynomial Hamiltonian Systems of Degree Four

In this paper we study centers of planar polynomial Hamiltonian systems and we are interested in the isochronous ones. We prove that every center of a polynomial Hamiltonian system of degree four (that is, with its homogeneous part of degree four not identically zero) is nonisochronous. The proof uses the geometric properties of the period annulus and it requires the study of the Hamiltonian systems associated to a Hamiltonian function of the form H(x, y)=A(x)+B(x) y+C(x) y²+D(x) y³.     

The number of components of complements to level surfaces of partially harmonic polynomials

In this paperk-harmonic polynomials in ℝ ⁿ , i.e. polynomials satisfying the Laplace equation with respect tok variables: ∂²/∂x ₁ ² +...+∂²/∂x _k ² F=0 are considered; here 1≤k≤n andn≥2. For a polynomialF (of degreem) of this type, it is proved that the number of components of the complements of its level sets does not exceed 2m ⁿ⁻¹+O(m ⁿ⁻²). Under the assumptions that the singular set of the level surface is compact or that the leading homogeneous part of thek-harmonic polynomialF is nondegenerate, sharper estimates are also established. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 831–835, December, 1997 Translated by S. S. Anisov     

La riflessione analitica delle funzioni biarmoniche attorno a un cerchio ed alcuni problemi di elasticità piana

Sunto Due funzionif(x, y) e ϕ(x, y), biarmoniche (e cioè soddisfacenti all'equazione ΔΔ=0), rispettivamente definite nei semipianix>0 edx<0, le cui derivate seconde si annullano all'infinito, e tali che nei punti dell'assey risultif=ϕ e∂f/∂x=∂ϕ/∂x, si dicono l'una ? riflessa ? dell'altra attorno all'assey. Da ognuna delle due funzionif e ϕ l'altra si ottiene con sole operazioni di sostituzione e derivazione (indicando, precisamente, con{f}, {∂f/∂x} e{Δf} le funzioni che si ottengono daf, ∂f/∂x eΔf ponendo, in queste, in luogo dix il suo contrario −x, si ha ϕ={f}+2x{∂f/∂x}+x ² {Δf} e, reciprocamente,f={ϕ}+2x{∂ϕ/∂x}+x ²{Δϕ}). In modo analogo si definisce una operazione di riflessione analitica attorno a un cerchio. La retta potendosi riguardare come cerchio degenere (di raggio infinito) l'operazione di rifiessione analitica attorno alla retta viene ottenuta, nel testo, come caso limite di quella di riflessione attorno al cerchio. L'operazione di riflessione analitica trova applicazione in alcuni problemi di elasticità piana (perturbazione prodotta da un foro circolare nella sollecitazione di un sistema piano; determinazione degli sforzi in un semipiano elastico sollecitato da una forza applicata in un punto interno).     

A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Given a domain Ω of class C ^k,1, k ∈ ℕ, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of in the sense that (∂/∂x _n )α(x′, 0) = − N(x′) and that still is of class C ^k,1. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C ^k,1. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.     

Global integration of differential equations through Lobatto quadrature

For the numerical solution of the initial value problemy=f(x,y), –1x1;y(–1)=y ₀ a global integration method is derived and studied. The method goes as follows.At first the system of nonlinear equations is solved. The matrix (A _i,k ⁽ⁿ⁾ ) of quadrature coefficients is nearly lower left triangular and the pointsx _k,n ,k=1,2,...,n are the zeros ofP _n –P _n–2, whereP _n is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x _i,n ,y _i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.     

One sufficient condition for Hamiltonian graphs involving distances

In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

The infinitesimal 16th Hilbert problem in the quadratic case

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let X _H =H _y -H _x be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X _H in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles. Oblatum 23-III-2000 & 19-VI-2000?Published online: 11 October 2000     

Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum

A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a complete bipartite graph. In this paper, we study the existence of cycles of various lengths in a Hamiltonian graph G given the existence of a pair of vertices that have a high degree sum but are not adjacent on any Hamiltonian cycle in G.     

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

A note on the congruences of the nakano superlattice and some properties of the associated quotients

In this paper we explore theNakano superlattice (H, ⊔, ⊓), where ⊔, ⊓ are the Nakanohyperoperations x⊔y={z:x∨z=y∨z=x∨y},x⊓y={z:x∧z=y∧z=x∧y}. In particular, we study the properties of congruences on the Nakano superlattice and the associated quotients. New hyperoperations are introduced on the quotient and their properties studied.     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

A counterexample in copositive approximation

The present paper gives a converse result by showing that there exists a functionf ∈C _[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE _n ⁽⁰⁾ (f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E _n(f) is the ordinary best polynomial approximation off of degreen.     

Strongly Closed Subgraphs in a Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method

A third derivative method (TDM) with continuous coefficients is derived and used to obtain a main and additional methods, which are simultaneously applied to provide all approximations on the entire interval for initial and boundary value problems of the form y′′ = f(x, y, y′). The convergence analysis of the method is discussed. An algorithm involving the TDMs is developed and equipped with an automatic error estimate based on the double mesh principle. Numerical experiments are performed to show efficiency and accuracy advantages.