L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .     

More Efficient Parallel Totally Monotone Matrix Searching

We give a parallel algorithm for computing all row minima in a totally monotonen × nmatrix which is simpler and more work efficient than previous polylog-time algorithms. It runs inO(lg n lg lg n) time doingwork on aCRCW PRAM, inO(lg n(lg lg n)²) time doingwork on aCREW PRAM, and intime doingwork on anEREW PRAM. Since finding the row minima of a totally monotone matrix has been shown to be fundamental in the efficient solution of a host of geometric and combinatorial problems, our algorithm leads directly to improved parallel solutions of many algorithms in terms of their work efficiency.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext¹_A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H². One condition involves an extension problem for the Hankel operator with symbolB,Γ_B, but viewed as a module map from H²into (W; B). The group Ext¹_A()(, (W; B))=(0) precisely whenΓ_Bextends to a module map from L²into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓ_B and the operator corona problem.     

Ray shooting and intersection searching amidst fat convex polyhedra in 3-space

We present a data structure for ray-shooting queries in a set of convex fat polyhedra of total complexity n in . The data structure uses O(n^2+ε) storage and preprocessing time, and queries can be answered in O(log²n) time. A trade-off between storage and query time is also possible: for any m with n<m<n², we can construct a structure that uses O(m^1+ε) storage and preprocessing time such that queries take time.We also describe a data structure for simplex intersection queries in a set of n convex fat constant-complexity polyhedra in . For any m with n<m<n³, we can construct a structure that uses O(m^1+ε) storage and preprocessing time such that all polyhedra intersecting a query simplex can be reported in O((n/m^1/3)logn+k) time, where k is the number of answers.     

Unconditional constants and polynomial inequalities

If P is a polynomial on R^m of degree at most n, given by , and P_n(R^m) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on P_n(R^m), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

On the complexity of submodular function minimisation on diamonds

Let (L;?,?) be a finite lattice and let n be a positive integer. A function f:Lⁿ→R is said to be submodular if for all . In this article we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding such that as efficiently as possible. We establish

Nonuniqueness for the Yang-Mills heat flow

We prove nonuniqueness for the Yang-Mills heat flow on bundles over manifolds of dimension m?5. For 5?m?9 and any there is an initial connection on the trivial bundle which, when evolved by the Yang-Mills heat flow, develops a point singularity in finite time, such that there are at least n different smooth continuations after the singular time. Moreover, the solution to the Yang-Mills heat flow may continue on a different bundle after the singular time, and for m∈{6,8} not even the topology of the bundle is determined uniquely.     

The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as . We investigate the case , ν∈N^?, which encompasses the gravitational fluctuations, ν=4, and the electromagnetic waves, ν=2. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as , and we get global Lp estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L²−L^∞ estimates in suitable weighted Sobolev spaces.     

Global positivity estimates and Harnack inequalities for the fast diffusion equation

We investigate local and global properties of positive solutions to the fast diffusion equation u_t=Δu^m in the range (d−2)₊/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; we use them to derive sharp global positivity estimates and a global Harnack principle. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak and elliptic Harnack inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.     

Note on the cost of the approximate controllability for the heat equation with potential

We prove the approximate controllability for the heat equation with potential with a cost of order e^c/ε when the target is in with a precision in L2(Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2(Ω) of the heat equation with potential is established.     

Uniqueness of Piecewise Smooth Weak Solutions of Multidimensional Degenerate Parabolic Equations

We study the degenerate parabolic equationu_t + ∇ · f = ∇ · (Q∇u) + g, where (x, t) ∈ ^N × ⁺, the fluxf, the viscosity coefficientQ, and the source termgdepend on (x, t, u) andQis nonnegative definite. Due to the possible degeneracy, weak solutions are considered. In general, these solutions are not uniquely determined by the initial data and, therefore, additional conditions must be imposed in order to guarantee uniqueness. We consider here the subclass of piecewise smooth weak solutions, i.e., continuous solutions which areC²-smooth everywhere apart from a closed nowhere dense collection of smooth manifolds. We show that the solution operator isL¹-stable in this subclass and, consequently, that piecewise smooth weak solutions are uniquely determined by the initial data.     

Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

Given a control region Ω on a compact Riemannian manifold M, we consider the heat equation with a source term g localized in Ω. It is known that any initial data in L²(M) can be steered to 0 in an arbitrarily small time T by applying a suitable control g in L2([0,T]×Ω), and, as T tends to 0, the norm of g grows like exp(C/T) times the norm of the data. We investigate how C depends on the geometry of Ω. We prove C?d²/4 where d is the largest distance of a point in M from Ω. When M is a segment of length L controlled at one end, we prove for some . Moreover, this bound implies where is the length of the longest generalized geodesic in M which does not intersect Ω. The control transmutation method used in proving this last result is of a broader interest.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,

On the eccentric connectivity index of a graph

If G is a connected graph with vertex set V, then the eccentric connectivity index of G, ξ^C(G), is defined as where is the degree of a vertex v and is its eccentricity. We obtain an exact lower bound on ξ^C(G) in terms of order, and show that this bound is sharp. An asymptotically sharp upper bound is also derived. In addition, for trees of given order, when the diameter is also prescribed, precise upper and lower bounds are provided.