A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part I

In this paper, we prove a result of Ambrosetti–Prodi type for the problem x′=f(t,x)+λx, where f(t,x) is T-periodic in t, f(t,0)≡0 and f(t,x) has “cubic nonlinearities”. Received: February 4, 2000?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

A result of Ambrosetti–Prodi type for first-order ODEs with cubic non-linearities. Part II

In this paper we study the scalar equation x′=f(t,x), where f(t,x) has cubic non-linearities in x and we prove that this equation has at most three bounded separate solutions. We say that λ∈ℝ is a critical value of the equation x′=f(t,x)+λx if this equation has a degenerate bounded solution and we exhibit two classes of functions f such that the above equation has a unique critical value. Received: February 4, 2000; in final form: March 19, 2002?Published online: April 14, 2003 RID="*" ID="*"This paper was partially supported by CDCHT, Universidad de los Andes.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

On the existence of entire functions of boundedl-index andl-regular growth

We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM _f(r)lnL(r),r→∞, whereM _f (r)=max {|f(z)|: |z|=r} andL(r)=∫ ₀ ^r l(t)dt. Ifl(r)=r ^p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM _f (r)≈r ^p . Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996.     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation

Commutator equations in free groups

Letf ₁, …,f _n be free generators of a free groupF. We consider the equation [z ₁, …,z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, …,z _n]_ω and [f ₁, …,f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solutionz ₁=r₁, …,z _z=r _n we have <r ₁, …,r _n>=〈f ₁, …f _n〉.     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

On estimation of a density and its derivatives

Summary Letf _n ^(p) be a recursive kernel estimate off ^(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of and show that the rate of almost sure convergence of to zero isO(n ^−α), α<(r−p)/(2r+1), iff ^(r),r>p≧0, is a continuousL ₂(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of to zero under different conditions onf. This work was supported in part by the Research Foundation of SUNY.     

The line analog of Ramsey numbers

For positive integersr ands, f′(r, s) is defined as the smallest positive integerp such that every connected (ordinary) graph of orderp contains eitherr mutually adjacent lines ors mutually disjoint lines. It is found thatf’(r,s) =(r−1) (s−1)+2 unlessr=2 and s ≠ 1, in which casef′(2,s)=3. Definitions not given here can be found in [7, 8].     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Differentiable functions defined in closed sets. A problem of Whitney

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ℝ ⁿ is the restriction of a function of class 𝒞 ^p . A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a “tame topology”. We prove that, if X is a compact subanalytic set, then there exists q=q _X (p) such that the criterion of order q implies that f is 𝒞 ^p . The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces. Oblatum 21-XI-2001 & 3-VII-2002?Published online: 8 November 2002 RID="*" ID="*"Research partially supported by the following grants: E.B. – NSERC OGP0009070, P.M. – NSERC OGP0008949 and the Killam Foundation, W.P. – KBN 5 PO3A 005 21.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Positivity and conditional positivity of Loewner matrices

We give elementary proofs of the fact that the Loewner matrices [\fracf(p_i) - f (p_j)p_i-p_j]{[\frac{f(p_i) - f (p_j)}{p_i-p_j}]} corresponding to the function f(t) = t ^r on (0, ∞) are positive semidefinite, conditionally negative definite, and conditionally positive definite, for r in [0, 1], [1, 2], and [2, 3], respectively. We show that in contrast to the interval (0, ∞) the Loewner matrices corresponding to an operator convex function on (−1, 1) need not be conditionally negative definite.     

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

Letf be a non-decreasing C¹-function such that andF(t)/f ² a(t)→ 0 ast → ∞, whereF(t)=∫ ₀ ^t f(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu| ^a =p(x)f(u) in a smooth bounded domain Ω ⊂R^N, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.     

Weyl’s theorem for algebraically wF(p, r, q) operators with p, r?>?0 and q?≥?1

If T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1 acting in an infinite-dimensional separable Hilbert space, then we prove that Weyl’s theorem holds for f(T) for any f ∈ Hol(σ(T)), where Hol(σ(T)) is the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is a wF(p, r, q) operator with p, r > 0 and q ≥ 1, then the a-Weyl’s theorem holds for f(T). In addition, if T (or T*) is an algebraically wF(p, r, q) operator with p, r > 0 and q ≥ 1, then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f ∈ Hol(σ(T)), respectively. Finally, we examine the stability of Weyl’s theorem and the a-Weyl’s theorem under commutative perturbations by finite-rank operators.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

The Clique Numbers of Regular Graphs

 Let ω(G) be the clique number of a graph G. We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values ω(G) completely cover a line segment [a,b] of positive integers. For an arbitrary graphic degree sequence d, we define min(ω,d) and max(ω,d) as follows:

Large holes in sparse random graphs

We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G _{n, p} ,p=c/n almost surely has a hole of size at least β(c)n asn→∞.