Axiomatizing Provable <Emphasis Type="Italic">n</Emphasis>-Provability

The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an ε₀-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Convergence analysis of the rectangular Morley element scheme for second order problem in arbitrary dimensions

We present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of O(h) order in energy norm and of O(h²) order in L² norm on general d-rectangular triangulations. Moreover, when the triangulation is uniform, the convergence rate can be of O(h²) order in energy norm, and the convergence rate in L² norm is still of O(h²) order, which cannot be improved. Numerical examples are presented to demonstrate our theoretical results.     

Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y ^'. In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ^?(t) + ? ² y(t) + µ y ^'(t)=0 with µ<2?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ^?(t) + ? ² y(t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity.     

The Folded (2<Emphasis Type="Italic">D</Emphasis> + 1)-cube and Its Uniform Posets

Let Γ denote the folded (2D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X. We first define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ?(x, y) + ?(y, z) = ?(x, z). Let R (resp. L) denote the raising matrix (resp. lowering matrix) of Γ. Next we show that there exists a certain linear dependency among RL², LRL,L²R and L for each given Q-polynomial structure of Γ. Finally, we determine whether the above linear dependency structure gives this poset a uniform structure or strongly uniform structure.     

Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.     

An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v_t = v_xv_y - ?_x^-1?_y[v_y + v_x²], where the formal integral ?_x^?1 becomes the asymmetric integral \( - \int_x^\infty {dx'} \). We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f(X, Y) over a parabola in the plane (X, Y) can be expressed in terms of the integrals of f(X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

Waiting Time in a Two Station Series Queueing System: The Effect of Dependent Interarrival Times

An analysis taking into account the dependencies in the departure process from the first station of the M/E _k /1→./M/1 system is conducted. Arrivals to the second station are approximated as a general independent distribution, and the waiting times in this station are compared to those found through computer simulation.     

Uniform continuity in {\omega_\mu}-metric spaces and uc {\omega_\mu}-metric extendability

Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.     

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f₀, 0} of the utility function (integrand) f₀ is relaxed to the requirement that the integrals of f₀ over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.     

Tomography models of random convex polygons

Combinatorial integral geometry possesses some results that can be interpreted as belonging to the field of Geometric Tomography. The main purpose of the present paper is to present a case of parallel X-ray approach to tomography of random convex polygons. However, the Introduction reviews briefly some earlier results by the author that refer to reconstruction of (non-random) convex domains by means of a point X-ray. The main tool in treating the parallel X-rays is disintegrated Pleijel identity, or rather, its averaged version, whose derivation is represented in complete detail. The paper singles out a class of random polygons called tomography models, that offer essential advantages for the analysis. The definition of a tomography model is given in terms of stochastic independence. Fortunately, random translation-invariant Poisson processes of lines in IR ² suggest a class of examples. We recall that each such line process is determined by its rose of directions ρ(?). For rather general ρ(?), the number weighted typical polygon in the polygonal partition of the plane generated by the corresponding Poisson line process happens to be a tomography model. For general tomography models, a differential equation is derived for the Laplace transform for parallel X-rays, that rises several interesting computational problems.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

Largest <Emphasis Type="Italic">H</Emphasis>-eigenvalue of uniform <Emphasis Type="Italic">s</Emphasis>-hypertrees

The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1 ≤ s ≤ k - 1; and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δ ^s/k ).     

Functor <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">τ</Emphasis></Subscript>, Lipschitzian and uniformly continuous mappings

The paper is devoted to questions on lifting of the functor M _τ : Tych → Tych to the categories of metric and uniform spaces. Similar problems were solved for the functor U _τ of the unit ball of τ-additive measures. The main difference between the functor M _τ and the functor U _τ is that the space M _τ (X) is compact only for X = Ø. A more delicate distinction is expressed by Theorem 2 which implies that the functor M _τ does not always preserve the uniform continuity of mappings of metric spaces (even in the case of compacta). Nevertheless, the problem of lifting the functor M _τ to the category Unif turns to be solvable.     

On generalized CS-modules

An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules.     

Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local L-splines with uniform knots (such splines were constructed in the authors’ earlier papers) corresponding to a linear differential operator L of order r with constant coefficients and real pairwise different roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which an L-spline locally inherits the property of the generalized k-monotonicity (k ≤ r ? 1) of the input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the L-spline. The parameters of an L-spline that is exact on the kernel of the operator L are written explicitly.