The zero-divisor graph of a reduced ring

In this paper the zero-divisor graph Γ(R) of a commutative reduced ring R is studied. We associate the ring properties of R, the graph properties of Γ(R) and the topological properties of . Cycles in Γ(R) are investigated and an algebraic and a topological characterization is given for the graph Γ(R) to be triangulated or hypertriangulated. We show that the clique number of Γ(R), the cellularity of and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and ; Γ(R) is complemented if and only if is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ(R) is between the density and the weight of . We show that Γ(R) is not triangulated and the set of centers of Γ(R) is a dominating set if and only if the set of isolated points of is dense in .     

On the Laplacian spectral radii of trees

Let Δ(T) and μ(T) denote the maximum degree and the Laplacian spectral radius of a tree T, respectively. Let T_n be the set of trees on n vertices, and . In this paper, we determine the two trees which take the first two largest values of μ(T) of the trees T in when . And among the trees in , the tree which alone minimizes the Laplacian spectral radius is characterized. We also prove that for two trees T₁ and T₂ in , if Δ(T₁)>Δ(T₂) and , then μ(T₁)>μ(T₂). As an application of these results, we give a general approach about extending the known ordering of trees in T_n by their Laplacian spectral radii.     

Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process given by the SDE , t∈[0,T), with initial condition , where T∈(0,∞], α∈R, (Bt)_t∈[0,T) is a standard Wiener process, b:[0,T)→R?{0} and σ:[0,T)→(0,∞) are continuously differentiable functions. Assuming , t∈[0,T), with some K∈R, we derive an explicit formula for the joint Laplace transform of and for all t∈[0,T) and for all α∈R. Our motivation is that the maximum likelihood estimator (MLE) of α can be expressed in terms of these random variables. As an application, we show that in case of α=K, K≠0,

     

Doubly nonlinear periodic problems with unbounded operators

The solvability of the evolution system v′(t)+B(t)u(t)∋f(t), v(t)∈A(t)u(t), 0<t<T, with the periodic condition v(0)=v(T) is investigated in the case where are bounded, possibly degenerate, subdifferentials and are unbounded subdifferentials.     

Toeplitz algebras on the disk

Let B be a Douglas algebra and let B be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that B is generated by H^∞(D) and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra TB generated by Toeplitz operators (on the Bergman space) with symbols in B has a canonical decomposition for some R in the commutator ideal CTB; and S is in CTB iff the Berezin transform vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set M₁ of trivial Gleason parts.     

Extremal energies of trees with a given domination number

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let T(n,γ) be the set of trees of order n and with domination number γ. In this paper, we characterize the tree from T(n,γ) with the minimal energy, and determine the tree from T(n,γ) where n=kγ with maximal energy for .     

The chain rule as a functional equation

We consider operators T from C¹(R) to C(R) satisfying the “chain rule”

     

Characterizing the derivative and the entropy function by the Leibniz rule

Consider an operator T:C¹(R)→C(R) satisfying the Leibniz rule functional equation

     

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.     

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T(a)∈L(?p), 1<p<∞, where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections Tn(a)=PnT(a)Pn depends heavily on the Fredholm properties of the operators T(a) and . In particular, if the operators T(a) and are Fredholm on ?p, then the approximation numbers of Tn(a) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than .     

Calderón-Zygmund operators on product Hardy spaces

Let T be a product Calderón-Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of Hp(Rn×Rm) and Journé's geometric covering lemma, R. Fefferman proved the remarkable Hp(Rn×Rm)−Lp(Rn×Rm) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood-Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journé's covering lemma to show that T is bounded on product Hp(Rn×Rm) for if and only if , where ε is the regularity exponent of the kernel of T.     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

Multiplicatively range-preserving maps between Banach function algebras

Let A and B be two Banach function algebras on locally compact Hausdorff spaces X and Y, respectively. Let T be a multiplicatively range-preserving map from A onto B in the sense that (TfTg)(Y)=(fg)(X) for all f,g∈A. We define equivalence relations on appropriate subsets and of X and Y, respectively, and show that T induces a homeomorphism between the quotient spaces of and by these equivalence relations. In particular, if all points in the Choquet boundaries of A and B are strong boundary points, then and are equal to the Choquet boundaries of A and B, respectively, and moreover, there exist a continuous function h on the Choquet boundary of B taking its values in {−1,1} and a homeomorphism φ from the Choquet boundary of B onto the Choquet boundary of A such that Tf(y)=h(y)f(φ(y)) for all f∈A and y in the Choquet boundary of B. For certain Banach function algebras A and B on compact Hausdorff spaces X and Y, respectively, we can weaken the surjectivity assumption and give a representation for maps belonging 2-locally to the family of all multiplicatively range-preserving maps from A onto B.     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

Multi-dimensional explosion due to a concentrated nonlinear source

Let D be a bounded n-dimensional domain, ∂D be its boundary, be its closure, T be a positive real number, B be an n-dimensional ball {x∈D:|x−b|<R} centered at b∈D with a radius R, be its closure, ∂B be its boundary, ν denote the unit inward normal at x∈∂B, and χ_B(x) be the characteristic function. This article studies the following multi-dimensional parabolic first initial-boundary value problem with a concentrated nonlinear source occupying :

     

Operator Hölder-Zygmund functions

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to Hölder classes. Namely, we prove that if f belongs to the Hölder class Λα(R) with 0<α<1, then for arbitrary self-adjoint operators A and B. We prove a similar result for functions f in the Zygmund class Λ₁(R): for arbitrary self-adjoint operators A and K we have . We also obtain analogs of this result for all Hölder-Zygmund classes Λα(R), α>0. Then we find a sharp estimate for ‖f(A)−f(B)‖ for functions f of class for an arbitrary modulus of continuity ω. In particular, we study moduli of continuity, for which for self-adjoint A and B, and for an arbitrary function f in Λω. We obtain similar estimates for commutators f(A)Q−Qf(A) and quasicommutators f(A)Q−Qf(B). Finally, we estimate the norms of finite differences for f in the class Λω,m that is defined in terms of finite differences and a modulus continuity ω of order m. We also obtain similar results for unitary operators and for contractions.     

On the dual rigidity matrix

A novel characterization of bar-and-joint framework rigidity was introduced in [A.Y. Alfakih. Graph rigidity via Euclidean distance matrices. Linear Algebra Appl., 310 (2000) 149-165; A.Y. Alfakih. On rigidity and realizability of weighted graphs. Linear Algebra Appl., 325 (2001) 57-70]. This characterization uses the notion of normal cones of convex sets to define a matrix whose rank determines whether or not a given generic framework is rigid. Furthermore, this characterization was derived under the assumption that the framework of interest G(p) has an equivalent framework G(q) in R^n-1, where n is the number of vertices of G(p). In this paper we show that the matrix corresponding to a framework G(p) contains the same information as the well-known rigidity matrix R. Whereas the entries of R are a function of the positions of the vertices of G(p), the entries of are a function of the Gale matrix corresponding to G(p). Furthermore, while the number of rows of R is equal to the number of edges of G(p), the number of columns of is equal to the number of missing edges of G(p). We also show that the assumption of the existence of an equivalent framework G(q) in R^n-1 can be dropped and we give the precise relation between the left-nullspaces, and consequently the nullspaces, of R and .     

The reduced effect of a single scattering with a low-mass particle via a point interaction

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in λ to the map , where G∈B(L²(Rn)) is a heavy-particle observable, ρ∈B₁(Rn) is the density matrix corresponding to the state of the light particle, is the mass ratio of the light particle to the heavy particle, Sλ∈B(L²(Rn)⊗L²(Rn)) is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on G.     

On commutators of vector BMO functions and multilinear singular integrals with non-smooth kernels

Let T be a bounded multilinear operator on some product of Lq(Rn) spaces. Assume that T has a non-smooth associated kernel which satisfies certain weak regularity conditions but not regular enough to fall under the scope of the standard multilinear Calderón-Zygmund theory. The main aim of this paper is to establish a sufficient condition on the kernel of T so that the commutator of a vector BMO function and T is bounded on certain product Lp(Rn) spaces. We obtain boundedness of the commutator of and T by first proving certain pointwise estimates on the Fefferman-Stein sharp maximal operator. An important example of multilinear operators which satisfy our kernel conditions is the maximal mth order Calderón commutator.