Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier–Stokes equations in 2<Emphasis Type="Italic">D</Emphasis>

We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|^−α |u(x)| < ∞ as |x| → ∞ for some α ∈ [0, 1/7).^† Bibliography: 10 titles.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0.     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.     

Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e ^− V in ℝ ⁿ are established. It is proved by using Prékopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures with p > 1 in ℝ ⁿ , and give out explicit estimates for their constants. This work is supported by NSFC (No. 10721091), 973-Project (No.2006CB805901) and DFMEC (NO. 20070027007).     

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]     

Accelerated hybrid conjugate gradient algorithm with modified secant condition for unconstrained optimization

An accelerated hybrid conjugate gradient algorithm represents the subject of this paper. The parameter β _k is computed as a convex combination of b_k^HS\beta_k^{HS} (Hestenes and Stiefel, J Res Nat Bur Stand 49:409–436, 1952) and b_k^DY\beta_k^{DY} (Dai and Yuan, SIAM J Optim 10:177–182, 1999), i.e. b_k^C = (1-q_k)b_k^HS + q_k b_k^DY\beta_k^C =\left({1-\theta_k}\right)\beta_k^{HS} + \theta_k \beta_k^{DY}. The parameter θ _k in the convex combinaztion is computed in such a way the direction corresponding to the conjugate gradient algorithm is the best direction we know, i.e. the Newton direction, while the pair (s _k , y _k ) satisfies the modified secant condition given by Li et al. (J Comput Appl Math 202:523–539, 2007) B _k + 1 s _k  = z _k , where z_k = y_k +(h_k / || s_k ||² )s_kz_k =y_k +\left({{\eta_k} / {\left\| {s_k} \right\|^2}} \right)s_k, h_k = 2( f_k -f_k+1 )+( g_k +g_k+1 )^Ts_k\eta_k =2\left( {f_k -f_{k+1}} \right)+\left( {g_k +g_{k+1}} \right)^Ts_k, s _k  = x _k + 1 − x _k and y _k  = g _k + 1 − g _k . It is shown that both for uniformly convex functions and for general nonlinear functions the algorithm with strong Wolfe line search is globally convergent. The algorithm uses an acceleration scheme modifying the steplength α _k for improving the reduction of the function values along the iterations. Numerical comparisons with conjugate gradient algorithms show that this hybrid computational scheme outperforms a variant of the hybrid conjugate gradient algorithm given by Andrei (Numer Algorithms 47:143–156, 2008), in which the pair (s _k , y _k ) satisfies the classical secant condition B _k + 1 s _k  = y _k , as well as some other conjugate gradient algorithms including Hestenes-Stiefel, Dai-Yuan, Polack-Ribière-Polyak, Liu-Storey, hybrid Dai-Yuan, Gilbert-Nocedal etc. A set of 75 unconstrained optimization problems with 10 different dimensions is being used (Andrei, Adv Model Optim 10:147–161, 2008).     

Multiwall carbon nanotube/epoxy composites produced by a masterbatch process

A masterbatch process based on a minicalander (three-roller mill) and a vacuum dissolver was developed in order to produce multiwall carbon nanotube/epoxy composites with loading fractions of 0.5, 1.0, and 2.0 wt.%. TEM and SEM analyses were performed to investigate the dispersion results. A contrast imaging in the SEM backscattering mode revealed a homogeneous distribution of carbon nanotubes in the whole volume of the material. Furthermore, an interesting correlation was found to exist between the network structure formed by the nanotubes in the epoxy matrix and the appearance of fracture surface of the nanocomposites. Furthermore, the nanocomposites exhibited an electrical conductivity in the regime of some 10⁻² S/m. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 567–582, September–October, 2006.     

Semidiscrete and asymptotic approximations for the nonstationary radiative–conductive heat transfer problem in a periodic system of grey heat shields

We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε ^3/4) for asymptotic approximations. Bibliography: 9 titles.     

Hitting Half-spaces by Bessel-Brownian Diffusions

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ ^n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ) ^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.     

Duality and the Poincaré–Hopf Inequalities

In this article we obtain a duality result for an n-manifold N with boundary ∂N = N + ⊔N ⁻ a disjoint union, where N ⁺ and N ⁻ are arbitrarily chosen parts in ∂N and need not be compact. This duality result is used to generalize the Poincaré–Hopf inequalities in a non-compact setting.     

Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization

We consider an operator K˚ϕ = Lϕ−: <CDU(x), Dϕ> in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, U∈W ^1,4(H, μ) and μ is the invariant measure associated with L. We show that K˚ is essentially self-adjoint in the space L ²(H, ν) where ν is the “Gibbs” measure ν(dx) = Z ^−:1 e ^−:2U(x) dx. An application to Stochastic quantization is given. Received: 13 August 1998 / Revised version: 20 September 1999 / Published online: 8 August 2000     

On Order Types of Systems of Segments in the Plane

Let r(n) denote the largest integer such that every family C\mathcal{C} of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Tóth gave a construction that shows r(n) < n ^log8/log9 (Pach and Tóth 2009). They also stated that one can apply the Erdős–Szekeres theorem for convex sets in Pach and Tóth (Discrete Comput Geom 19:437–445, 1998) to obtain r(n) > log₁₆ n. In this note, we will show that r(n) > cn ^1/4 for some absolute constant c.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Bounds for the Commutator of Parabolic Singular Integral with Rough Kernels

In this paper, the authors give the L ^p (1 < p < ∞ ) boundedness of the k-th order commutator of parabolic singular integral with the kernel function Ω ∈ L(log⁺ L) ^k + 1(S ^n − 1). The result in this paper is an extension of some known results. The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Asymptotic behavior of positive solutions to emden–fowler type equations

We study the asymptotic behavior of positive solutions to nonlinear elliptic equations of Emden–Fowler type with absorption term. For operators with variable coefficients we obtain conditions on coefficients under which the solutions have the same asymptotics as solutions to the model equation Δu = −x| ^p |u| ^σ−1 u. For positive solutions we obtain lower order terms of the asymptotic expansion at infinity. Bibliography: 10 titles.     

Stable Subconstructs: A Correction and New Results

In Lowen and Wuyts (Appl Categ Struct 8:235–245, 2000) the authors studied the simultaneously concretely reflective and concretely coreflective subconstructs of the category Ap of approach spaces. For the sake of shortness we call such subconstructs stable. Using a technique introduced in Herrlich and Lowen (1999) it was possible to explicitly describe such stable subconstructs by a condition on the objects which used certain subsets of [0, ∞ ]. Thus each stable subconstruct Ap _m described in [9] corresponds to the subset {0} ∪ [m, ∞ ] ⊂ [0, ∞ ] for m ∈ [0, ∞ ]. Although this characterization is correct, Theorem 4.7 in [9] stating that the subconstructs Ap _m were the only stable subconstructs of Ap is not. The main results, which together prove that the only stable subconstructs are those where a restriction is put on the range of the distances of the objects, are upheld, but it turns out that not only the sets {0} ∪ [m, ∞ ], but actually each closed subsemigroup of [0, ∞ ] determines a stable subconstruct (albeit again in exactly the same way as characterized in [9]). In the first part of our paper, Sections 1 and 2, we develop the general technique, which is totally different to the one from [3], and in Theorem 2.13 we prove the main result for the case of approach spaces. The technique which we develop is also applicable to other cases. Thus, in Section 3, more precisely in Theorems 3.9 and 3.11, we give the complete solution to the corresponding characterization problem for the constructs pq Met ^∞ of pseudo-quasi-metric spaces and p Met ^∞ of pseudometric spaces and in Section 4 we briefly sketch how the technique can be adapted and used to also completely solve the problem in the case of more general types of approach spaces and metric spaces. At the same time, in all cases, we are able to give necessary and sufficient conditions under which two stable subconstructs of one of these topological constructs are concretely isomorphic. It turns out that in all cases there are 2^à₀2^{\aleph_0} non-concretely isomorphic stable subconstructs.     

Projection–difference method for controlled Fourier filtering

A projection–difference method is developed for approximating controlled Fourier filtering for quasilinear parabolic functional-differential equations. The method relies on a projection–difference scheme (PDS) for the approximation of the differential problem and derives a O(τ^1/2 + h) bound on the rate of convergence of PDS in the weighted energy norm without prior assumptions of additional smoothness of the generalized solutions. The PDS leads to a natural approximation of the objective functional in the optimal Fourier filtering problem. A bound of the same order is obtained for the rate of convergence in the functional of the problems approximating the Fourier filter control problem.     

Some Exponential Inequalities for Positively Associated Random Variables and Rates of Convergence of the Strong Law of Large Numbers

We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain the rate of convergence n ^−1/2 β _n log ^3/2 n in which β _n can be particularly taken as (log log n)^1/σ with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n ^−1/2(log log n)^1/2log ² n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n ^−1/2 β _n log ^1/2 n for the above β _n under the given covariance function, which improves the relevant one n ^−1/2(log log n)^1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results, which extend and improve some known results.