The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.     

On the computation of the hull number of a graph

Let G be a graph. If u,v∈V(G), a u-vshortest path of G is a path linking u and v with minimum number of edges. The closed interval I[u,v] consists of all vertices lying in some u-v shortest path of G. For S⊆V(G), the set I[S] is the union of all sets I[u,v] for u,v∈S. We say that S is a convex set if I[S]=S. The convex hull of S, denoted I_h[S], is the smallest convex set containing S. A set S is a hull set of G if I_h[S]=V(G). The cardinality of a minimum hull set of G is the hull number of G, denoted by hn(G). In this work we prove that deciding whether hn(G)≤k is NP-complete.We also present polynomial-time algorithms for computing hn(G) when G is a unit interval graph, a cograph or a split graph.     

On ideals whose adic and symbolic topologies are linearly equivalent

Ideals whose adic and symbolic topologies are linearly equivalent are characterized in terms of analytic spread and u-essential prime divisors. Using this characterization, under certain conditions on a Noetherian ring R and an ideal I of R it is shown that the I-adic and the I-symbolic topologies are linearly equivalent iff gr(I,R)_red is a domain, and locally unmixed rings are characterized as those rings in which the adic and the symbolic topologies of every ideal of the principal class are linearly equivalent.     

Convergence of convex integrals in Lp spaces

We consider functionals of the form: I_f(u) = ∝_Tf[t, u(t)]μ(dt), which are defined on spaces L^p(T, R^k), and we study for these functionals the properties of a convergence for which the conjugacy $\text{I}{}_{\mathrm{f}}\text{→ I}{}_{\mathrm{f1}}$ is a continuous operator.     

Continuous dependence on the data in abstract control problems

We consider the following problems: minimize $$I_n (u) = |u - \hat u|^p + |L_n u - \hat y|^p , n \geqslant 0,$$ whereL _n are equibounded linear operators. If we callu _n,u ₀ the minimum points ofI _n, we characterize the strong convergence ofu _n tou ₀ in terms of the pointwise convergence ofL _n and their adjoint operatorsL _n* toL ₀ andL ₀*, respectively. Then, we apply this result to the case of a problem governed by a linear differential equation. Finally, we conclude by studying perturbations of an abstract constrained minimum problem.     

Commutator estimates in von Neumann algebras

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c ₀ such that, given any ? > 0, |[a, u _ε ]| ? (1 ? ε)|aε ? c ₀| for some unitary operator u _ε ∈ M. In particular, every derivation δ: M → I (where I is an ideal in M) is inner, and δ = δ _a for a ∈ I.     

Moment methods in Padé approximation: The unitary case

Given a unitary operator T in a Hilbert space H = (H, 〈·, ·〉) convergence results for two sequences of ($\text{(n ? 1)}\text{n)}$ two-point Padé approximants to the function f(z) = 〈(I ? zT)^?1u₀, u₀〉, (u₀ ∈ H, ∥ u₀∥ = 1, z regular for T) are given. An elementary proof is also given of the well-known operator version of the trigonometric moment problem, not using the solution of the classical trigonometric moment problem.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications

Assume that I is the functional defined on the Hilbert space concerning the problem: in and u on , where f is sublinear at and superlinear at 0, that is, , and is the first eigenvalue of in . Under very general conditions, I has at least two local minimizers u ₁ and u ₂ and one mountain pass point u ₃ , and . Assuming that u ₁ , u ₂ and u ₃ are the only three nontrivial critical points of I, we prove that the level set I _b is contractible for all . Using this conclusion, we extend one of Hofer's result concerning existence of four nontrivial solutions of the above problem to the case where I is not and the trivial critical point 0 may be degenerate. Since I is not , the local topological degree and the critical groups of u ₃ can not be clearly computed. The lack of topological information about 0 and u ₃ makes it impossible to use topological degree theory or Morse theory in obtaining the fourth nontrivial solution. To overcome these difficulties, we explore a new technique in this paper.     

Direct and inverse asymptotic scattering problems for Dirac-Krein systems

The asymptotic scattering matrix s _ε(λ) for a Dirac-Krein system with signature matrix J = diag{ I _p ,-I _p }, integrable potential, and the boundary condition u ₁(0, λ) = u ₂(0, λ)ε(λ) with a coefficient ε(λ) that belongs to the Schur class of holomorphic contractive p × p matrix-valued functions in the open upper half-plane is defined. The inverse asymptotic scattering problem for a given s _ε is analyzed by Krein’s method. Earlier studies by Krein and others focused on the case in which ε = I _p (or a constant unitary matrix).     

Tame parts of free summands in coproducts of Priestley spaces

It is well known that a sum (coproduct) of a family of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully understood.In this article we study a certain dense subset satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the Xi's. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a ?ech-Stone compactification of the disjoint union I_?Xi. In this case, all the Xu's with u free turn out to be isomorphic under the Continuum Hypothesis.     

Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Let S = K[x₁; x₂;...; x_n] be the polynomial ring in n variables over a field K; and let I be a squarefree monomial ideal minimally generated by the monomials u₁; u₂;...; u_m: Let w be the smallest number t with the property that for all integers 1 6 i₁ < i₂ <... < i _t 6 m such that \(lcm({u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_t}}}) = lcm({u_1},{u_2},...,{u_m})\) We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I: As a corollary, the projective dimension of I is bounded by the number w.     

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa-tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (▽h ( u-Ihu )1, ▽hvh) h may be estimated as order O ( h2 ) when u ∈ H3 (Ω), where Ihu denotes the bilinear interpolation of u , vh is a polynomial belongs to quasi-Wilson finite element space and ▽h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O ( h2 ) /O ( h3 ) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H3 (Ω) /H4 (Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O ( h3 ), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.     

Eventual partition of conserved quantities in wave motion

Let u be a classical solution to the wave equation in an odd number n of space dimensions, with compact spatial support at each fixed time. Duffin (J. Math. Anal. Appl.32 (1970), 386–391) uses the Paley-Wiener theorem of Fourier analysis to show that, after a finite time, the (conserved) energy of u is partitioned into equal kinetic and potential parts. The wave equation actually has $\text{(n + 2)(n + 3)}\text{2}$ independent conserved quantities, one for each of the standard generators of the conformal group of (n + 1)-dimensional Minkowski space. Of concern in this paper is the “zeroth inversional quantity” I₀, which is commonly used to improve decay estimates which are obtained using conservation of energy. We use Duffin's method to partition I₀ into seven terms, each of which, after a finite time, is explicitly given as a constant-coefficient quadratic function of the time. Zachmanoglou has shown that under the above assumptions if n ? 3, the spatial L² norm of u is eventually constant. A consequence of the analysis here is a bound on this constant in terms of the energy and the radius of the support of the Cauchy data of u at a fixed time.     

On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u₁,u₂,u₃) is regular on (0,T] if there exist two solution components, for example, u₂ and u₃, satisfying the condition

     

Steiner intervals, geodesic intervals, and betweenness

The concept of the k-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping S:V×?×V?2^V such that S(u₁,…,u_k) consists of all vertices in G that lie on some Steiner tree with respect to a multiset W={u₁,…,u_k} of vertices from G. In this paper we obtain, for each k, a characterization of the class of graphs in which every k-Steiner interval S has the so-called union property, which says that S(u₁,…,u_k) coincides with the union of geodesic intervals I(u_i,u_j) between all pairs from W. It turns out that, as soon as k>3, this class coincides with the class of graphs in which the k-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, S satisfies (m), if x₁,…,x_k∈S(u₁,…,u_k) implies S(x₁,…,x_k)⊆S(u₁,…,u_k), and S satisfies (b2) if x∈S(u₁,u₂,…,u_k) implies S(x,u₂,…,u_k)⊆S(u₁,…,u_k). In the case k=3, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval S satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class of graphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.     

Nash-solvable two-person symmetric cycle game forms

A two-person positional game form g (with perfect information and without moves of chance) is modeled by a finite directed graph (digraph) whose vertices and arcs are interpreted as positions and moves, respectively. All simple directed cycles of this digraph together with its terminal positions form the set A of the outcomes. Each non-terminal position j is controlled by one of two players i∈I={1,2}. A strategy x_i of a player i∈I involves selecting a move (j,j^′) in each position j controlled by i. We restrict both players to their pure positional strategies; in other words, a move (j,j^′) in a position j is deterministic (not random) and it can depend only on j (not on preceding positions or moves or on their numbers). For every pair of strategies (x₁,x₂), the selected moves uniquely define a play, that is, a directed path form a given initial position j₀ to an outcome (a directed cycle or terminal vertex). This outcome a∈A is the result of the game corresponding to the chosen strategies, a=a(x₁,x₂). Furthermore, each player i∈I={1,2} has a real-valued utility function u_i over A. Standardly, a game form g is called Nash-solvable if for every u=(u₁,u₂) the obtained game (g,u) has a Nash equilibrium (in pure positional strategies).A digraph (and the corresponding game form) is called symmetric if (j,j^′) is its arc whenever (j^′,j) is. In this paper we obtain necessary and sufficient conditions for Nash-solvability of symmetric cycle two-person game forms and show that these conditions can be verified in linear time in the size of the digraph.     

Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

Let (W, H, μ) be the classical Wiener space. Assume that U = I _W  + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if ${u\in {\rm ID}_{p,1}(H)}$ is adapted and if ${\exp(\frac{1}{2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)}$ , where p ^?1 + q ^?1 = 1, then I _W  + u is almost surely invertible. With the help of this result it is shown that if ${\nabla u\in L^\infty(\mu,H\otimes H)}$ , then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I _W  +  u . As a consequence, if, there exists an integer k ≥  1 such that ${\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)}$ , then I _W  +  u is again almost surely invertible under the almost sure continuity hypothesis of ${t\to\nabla^i \dot{u}_t}$ for i ≤  k ? 1.