Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

The Banach-Lie algebra of multiplication operators on a JBW-triple

The Banach-Lie algebra L(A) of multiplication operators on the JB^∗-triple A is introduced and it is shown that the hermitian part Lh(A) of L(A) is a unital GM-space the base of the dual cone in the dual GL-space ^∗(Lh(A)) of which is affine isomorphic and weak^∗-homeomorphic to the state space of L(A). In the case in which A is a JBW^∗-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space Lh(A) satisfy

0?D(u,u)+D(v,v)?idA,     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations

This paper investigates the effects of a degenerate diffusion term in reaction-diffusion models u_t=[D(u)u_x]_x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c∗ and we show the appearance of a sharp-type profile when c=c∗. These results solve recent conjectures formulated by Sánchez-Garduño and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339).     

Smooth stable invariant manifolds and arbitrary growth rates

For nonautonomous linear equations v^′=A(t)v with a generalized exponential dichotomy, we show that there is a smooth stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v) provided that f is sufficiently small. The generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates for some function ρ(t). We consider the general case of nonuniform exponential dichotomies, and the result is obtained in Banach spaces. Moreover, we show that for an equivariant system, the dynamics on the stable manifold in a certain class of graphs is also equivariant. We emphasize that this result cannot be obtained by averaging over the symmetry.     

Balanced Leonard pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A^∗ : V → V that satisfy the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity

In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wavefronts with the speed c>c_∗, where c_∗>0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x→−∞, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c>c_∗ with sufficiently small delay time r≈0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c>c_∗ and for an arbitrary time-delay r>0.     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.     

Elementary operators on self-adjoint operators

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that and are surjective maps such that M(AM^∗(B)A)=M(A)BM(A) and M^∗(BM(A)B)=M^∗(B)AM^∗(B) for every pair A∈As, B∈Bs. Then there exist an invertible bounded linear or conjugate-linear operator and a constant c∈{−1,1} such that M(A)=cTAT^∗, A∈As, and M^∗(B)=cT^∗BT, B∈Bs.