Rectifiable oscillations of self-adjoint and damped linear differential equations of second-order

Asymptotic and oscillatory behaviours near x=0 of all solutions y=y(x) of self-adjoint linear differential equation (Ppq): ^′(py^′)+qy=0 on (0,T], will be studied, where p=p(x) and q=q(x) satisfy the so-called Hartman-Wintner type condition. We show that the oscillatory behaviour near x=0 of (Ppq) is characterised by the nonintegrability of on (0,T). Moreover, under this condition, we show that the rectifiable (resp. unrectifiable) oscillations near x=0 of (Ppq) are characterised by the integrability (resp. nonintegrability) of on (0,T). Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturm?s comparison type theorem for the rectifiable oscillations is stated. Furthermore, previous results are used to establish such kind of oscillations for damped linear second-order differential equation y^″+g(x)y^′+f(x)y=0, and especially, the Bessel type damped linear differential equations are considered. Finally, some open questions are posed for the further study on this subject.     

Fractal oscillations of self-adjoint and damped linear differential equations of second-order

For a prescribed real number s ∈ [1, 2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y = y(x), y ∈ C²((0, T]) of the linear differential equation (p(x)y′)′ + q(x)y = 0 on (0, T], is bounded and fractal oscillatory near x = 0 with the fractal dimension equal to s. This means that y oscillates near x = 0 and the fractal (box-counting) dimension of the graph Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(y) is finite and strictly positive. It verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x = 0 like the chirp function y_ch(x) = a(x)S(φ(x)), which often occurs in the time-frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form y″ + (μ/x)y′ + g(x)y = 0, x ∈ (0, T]. In order to prove the main results, we state a new criterion for fractal oscillations near x = 0 of real continuous functions which essentially improves related one presented in [1].     

Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Exceptional cases of Terai’s conjecture on Diophantine equations

Let a, b, c be relatively prime positive integers such that a ^p  + b ^q  = c ^r for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a ^x  + b ^y  = c ^z in positive integers has only the solution (x, y, z) = (p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases.     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

More eigenvalue estimates

Estimates of the eigenvalues of the Sturm-Liouville problem —y+q(x)y = λy₁y(0) = 0 = y(1) are obtained by making linear approximations of q(x) over subintervals of [0, 1] when q(x) has a bounded second derivative.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

A New Method for Constructing Williamson Matrices

For every prime power q 1 (mod 4) we prove the existence of (q; x, y)-partitions of GF(q) with q=x²+4y² for some x, y, which are very useful for constructing SDS, DS and Hadamard matrices. We discuss the transformations of (q; x,y)-partitions and, by using the partitions, construct generalized cyclotomic classes which have properties similar to those of classical cyclotomic classes. Thus we provide a new construction for Williamson matrices of order q².The research supported by NSF of China (No. 10071029).     

Inverse spectral problem for a generalized Sturm-Liouville equation with complex-valued coefficients

The present paper is the first to prove that one of the columns of the monodromy matrix and two of the three coefficients (piecewise analytic on the interval [0, 1]) of the equation (f(x)y′)′+(r(x)−λ ² q(x))y = 0 uniquely determine the third coefficient on this interval provided that the values of the functions f(x) and q(x) lie in the lower (or upper) open complex halfplane and on the positive part of the real axis. This unknown coefficient can be reconstructed by finding the unique zero minimum of a specially constructed functional depending on the solutions of the corresponding Cauchy problem and the given elements of the monodromy matrix.     

Equations fonctionnelles et estimations de normes de matrices

We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ²=aθ⁴+bθ²+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x _r −x _s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex _r are equidistant.

Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux     

Parity patterns associated with lifts of Hecke groups

Let q be an odd prime, m a positive integer, and let Γ _m (q) be the group generated by two elements x and y subject to the relations x ^2m =y ^qm =1 and x ²=y ^q ; that is, Γ _m (q) is the free product of two cyclic groups of orders 2m respectively qm, amalgamated along their subgroups of order m. Our main result determines the parity behaviour of the generalized subgroup numbers of Γ _m (q) which were defined in Müller (Adv. Math. 153:118–154, 2000), and which count all the homomorphisms of index n subgroups of Γ _m (q) into a given finite group H, in the case when gcd (m,| H |)=1. This computation depends upon the solution of three counting problems in the Hecke group ℋ(q)=C ₂*C _q : (i) determination of the parity of the subgroup numbers of ℋ(q); (ii) determination of the parity of the number of index n subgroups of ℋ(q) which are isomorphic to a free product of copies of C ₂ and of C _∞; (iii) determination of the parity of the number of index n subgroups in ℋ(q) which are isomorphic to a free product of copies of C _q . The first problem has already been solved in Müller (Groups: Topological, Combinatorial and Arithmetic Aspects, LMS Lecture Notes Series, vol. 311, pp. 327–374, Cambridge University Press, Cambridge, 2004). The bulk of our paper deals with the solution of Problems (ii) and (iii). Research of C. Krattenthaler partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Absolute continuity and spectral concentration for slowly decaying potentials

We consider the spectral function ϱ(μ) (μ⩾0) for the Sturm-Liouville equation y″ + (λ −qq)y=0 on [0,∞) with the boundary condition y(0)=0 and where q has slow decay O(x^−a (a > 0) as x → ∞. We develop our previous methods of locating spectral concentration for q with rapid exponential decay (this Journal 81 (1997) 333–348) to deal with the new theoretical and computational complexities which arise for slow decay.     

The sum of squared logarithms inequality in arbitrary dimensions

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝⁿ whose elementary symmetric polynomials satisfy e_k(x) ≤ e_k(y) (for 1 ≤ k < n) and e_n(x) = e_n(y) , the inequality ∑_i(log x_i)² ≤ ∑_i(log y_i)² holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂⁿ → ℝ with f(z) = ∑_i(log z_i)² has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A result concerning the stability of some difference equations and its applications

In this paper, we investigate the Hyers-Ulam stability problem for the difference equation f(x +p, y +q)- φ(x, y)f(x, y)- ψ(x, y)= 0. An erratum to this article is available at .     

An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(x, t) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

A generalized numerical range: the range of a constrained sesquilinear form

Let M_n denote the algebra of all nxn complex matrices. For a given q?C with ∣Q∣≤1, we define and denote the q-numerical range of A?M_n by

W_q (A)={x ^? Ay:x,y?C ⁿ , _{x ^? x?y ^? y=1,x ^? y=q} }

The q-numerical radius is then given by r_q (A)=sup{∣z∣:z?W _q (A)}. When q=1,W _q (A) and r _q (A) reduce to the classical numerical range of A and the classical numerical radius of A, respectively. when q≠0, another interesting quantity associated with W _q (A) is the inner q-numerical radius defined by [rtilde] _q (A)=inf{∣z∣:z?W _q (A)}

In this paper, we describe some basic properties of W _q (A), extending known results on the classical numerical range. We also study the properties of r_q considered as a norm (seminorm if q=0) on M_n .Finally, we characterize those linear operators L on M_n that leave W_q ,r_q of [rtilde]_q invariant. Extension of some of our results to the infinite dimensional case is discussed, and open problems are mentioned.     

Asymptotic forms of solutions of certain linear ordinary differential equations

We analyze the asymptotic behavior as x → ∞ of the product integral Π_x0^xe^A(s)ds, where A(s) is a perturbation of a diagonal matrix function by an integrable function on [x₀,∞). Our results give information concerning the asymptotic behavior of solutions of certain linear ordinary differential equations, e.g., the second order equation y″ = a(x)y.     

A Schrödinger singular perturbation problem

Consider the equation −ε²Δu_ε + q(x)u_ε = f(u_ε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution u_ε exists and lim_ε→0u_ε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper.