On the f- and h-triangle of the barycentric subdivision of a simplicial complex

For a simplicial complex Δ we study the behavior of its f- and h-triangle under the action of barycentric subdivision. In particular we describe the f- and h-triangle of its barycentric subdivision sd(Δ). The same has been done for f- and h-vector of sd(Δ) by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the h-triangle of Δ are nonnegative, then the entries of the h-triangle of sd(Δ) are also nonnegative. We conclude with a few properties of the h-triangle of sd(Δ).     

P-points and Q-points over a measurable cardinal

This paper presents constructions of κ-ultrafilters over a measurable cardinal κ, specifically p-points and q-points, in the continuation of works of Gitik, Kanamori and Ketonen. On the assumption of supercompactness, Kunen has shown the existence of a chain of p-points, in the Rudin–Keisler ordering, of maximal length. We shall prove a similar result for q-points. Results of Mitchell [8,9] establish connections between Rudin—Keisler chains of κ-ultrafilters and inner models of “?ν(o(ν)= ν⁺⁺)”. This shows the necessity of some strong large cardinal hypothesis. The second part of the paper is devoted to a separation property of κ- ultrafilters (cf. Kanamori and Taylor). To answer a question of Taylor concerning the existence of a non-separating p-point, we use a combination of Silver's Forcing and iterated ultrapowers; the proof itself may be of some interest.     

Some results for the primitive equations with physical boundary conditions

In this paper, we consider the (simplified) 3-dimensional primitive equations with physical boundary conditions. We show that the equations with constant forcing have a bounded absorbing ball in the H ¹-norm and that a solution to the unforced equations has its H ¹-norm decay to 0. From this, we argue that there exists an invariant measure (on H ¹) for the equations under random kick-forcing.     

Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ,u) where ρ∈C¹([0,T];H¹([0,1])) and u∈H¹([0,T];H²([0,1])) for any T>0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ,u)∈C¹([0,T];H³(R¹)) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ²utt, such as Lemmas 3.2-3.6. This leads to further regularities of (ρ,u) where ρ∈C¹([0,T];H³([0,1])), u∈H¹([0,T];H³([0,1])). It is still open whether the regularity of u could be improved to C¹([0,T];H³([0,1])) with the appearance of vacuum, since it is not obvious that the solutions in C¹([0,T];H³([0,1])) to the initial boundary value problem must blow up in finite time.     

$H^2$-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this...     

1-Triangle graphs and perfect neighborhood sets

A graph is called a 1-triangle if, for its every maximal independent set I, every edge of this graph with both endvertices not belonging to I is contained exactly in one triangle with a vertex of I. We obtain a characterization of 1-triangle graphs which implies a polynomial time recognition algorithm. Computational complexity is establishedwithin the class of 1-triangle graphs for a range of graph-theoretical parameters related to independence and domination. In particular, NP-completeness is established for the minimum perfect neighborhood set problem in the class of all graphs.     

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

Noncontinuous data boundary value problems for Schrödinger equation in Lipschitz domains

The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the L ^p boundary value problems with p > 1, and H ^p boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.     

Local controllability and non-controllability for a 1D wave equation with bilinear control

We consider a linear wave equation, on the interval (0,1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.

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For every T>2, this system is locally controllable in H³×H², in time T, with controls in L²((0,T),R).

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For T=2, this system is locally controllable up to codimension one in H³×H², in time T, with controls in L²((0,T),R): the reachable set is (locally) a non-flat submanifold of H³×H² with codimension one.

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For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L²((0,T),R), is contained in a non-flat submanifold of H³×H², with infinite codimension.

The proof of these results relies on the inverse mapping theorem and second order expansions.     

A discrete finite element galerkin method for a unidimensional single-phase Stefan problem

Based on Landau-type transformation, a Stefan problem with nonlinear free boundary condition is transformed into a system consisting of parabolic equation and the ordinary differential equations. Semidiscrete approximations are constructed. Optimal orders of convergence of semidiscrete approximation inL ₂,H ¹ andH ² normed spaces are derived.     

Global existence and decay of energy to systems of wave equations with damping and supercritical sources

This paper is concerned with a system of nonlinear wave equations with supercritical interior and boundary sources and subject to interior and boundary damping terms. It is well-known that the presence of a nonlinear boundary source causes significant difficulties since the linear Neumann problem for the single wave equation is not, in general, well-posed in the finite-energy space H ¹(Ω) × L ²(?Ω) with boundary data from L ²(?Ω) (due to the failure of the uniform Lopatinskii condition). Additional challenges stem from the fact that the sources considered in this article are non-dissipative and are not locally Lipschitz from H ¹(Ω) into L ²(Ω) or L ²(?Ω). With some restrictions on the parameters in the system and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution and establish (depending on the behavior of the dissipation in the system) exponential and algebraic uniform decay rates of energy. Moreover, we prove a blow-up result for weak solutions with nonnegative initial energy.     

On an exact control problem for a semilinear wave equation with an unknown source

The exact controllability of a semilinear wave equation, with Dirichlet boundary control on a part of the boundary and an unknown source, is shown. The nonlinear term has at most a linear growth, the initial and target spaces are L²(Ω)×H⁻¹(Ω).     

Espaces Hp au dessus de l'espace hermitien hyperbolique de Cn (n > 1), II

Growth of harmonic functions and boundary values in a Sobolev space; comparison between the H^p space associated to Calderón Lusin's area integral and the H^p space associated to the maximal admissible function; study of Littlewood-Paley functions for the harmonic functions for the invariant laplacian of the hermitian hyperbolic space; relation between a Littlewood-Paley function and the bounded mean oscillation functions; study of H¹ and H¹ B.M.O. duality.     

Solution of Band Matrix Equations by Projection-Recurrence

The solution space of the matrix equation Hx=u is decomposed by a projection which leads to a recurrence for H^-1. For tridiagonal H it is shown that the elements of H^-1 can always be completely factorized. An explicit expression is obtained for H^-1 when H is tridiagonal with constant interior elements and arbitrary boundary elements. This result reduces to a particularly simple form when H is Toeplitz. For general N×N tridiagonal H, the maximum number of arithmetic operations to obtain any element of H^-1 is asymptotic to 6N, and to obtain H^-1 it is N²+14N. To obtain the vector x the number of operations ranges from 8N to 11N.     

Einige weitere Resultate in Analogie zu einem Gleichverteilungssatz von Koksma

Continuing former investigations by the authors (see the references) the present paper contains metric results on the distribution modulo 1 of the powers of special kinds of real matricesA, namely of (2×2)- and (3×3)-triangle matrices, symmetric (2×2)-matrices and so-called “cosymmetric” (2×2)-matrices (i. e. matrices, symmetric with respect to the secondary diagonal). For almost all such matricesA (in the sense of the Lebesgue measure in ?³ resp. ?⁶) possessing no eigenvalue of modulus smaller than 1 the inequality $$D(N) \leqslant C(A, \varepsilon ) N^{ - 1/2} (\log N)^{d + 3/2 + \varepsilon } (\varepsilon > 0)$$ is proved as an estimate for the discrepancy of the sequence (A ^s(n) ) where (s(n)) _n ^=1/∞ is an arbitrary fixed strictly increasing sequence of positive integers andd is the dimension of the appropriate space ? ^d (d=3 or 6).     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.     

Thin sequences in the corona of H ∞

In this paper we consider several conditions for sequences of points in M(H ^∞) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $M(H^\infty )\backslash \mathbb{D}$ of H ^∞ is a thin sequence for H ^∞, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ^∞ necessarily satisfy the same condition.     

Uniqueness for the harmonic map flow from surfaces to general targets

LetM be a two-dimensional compact Riemannian manifold with smooth (possibly empty) boundary,N an arbitrary compact manifold. Ifu andv are weak solutions of the harmonic map flow inH ¹(Mx[0,T]; N) whose energy is non-increasing in time and having the same initial datau ₀∈H¹(M, N) (and same boundary values if ?M≠Ø) thenu=v. Combined with a result of M. Struwe, this shows any suchu is smooth in the complement of a finite subset ofM×(0,T)c.     

Meromorphic solutions of a Riccati differential equation with a doubly periodic coefficient

We treat a Riccati differential equation w^′+w²+p(z)=0, where p(z) is a nonconstant doubly periodic meromorphic function. Under certain assumptions, every solution is meromorphic in the whole complex plane. We show that the growth order of it is equal to 2, and examine the frequency of α-points and poles. Furthermore, the number of doubly periodic solutions is discussed.     

La monotonia di rappresentazioni del gruppo libero

Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC ^*-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC ^*-algebra Γ×_λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv ₀∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv ₀ exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect.