Petrovskii elliptic systems in the extended Sobolev scale

Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated in the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. Theorems of solvability of the elliptic systems in the extended Sobolev scale are proved. An a priori estimate for solutions is obtained, and their local regularity is studied.     

The best quadrature based on given Hermite information for the Sobolev class KW r[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r ? 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r ? 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

The bifurcation of limit cycles in Zn-equivariant vector fields

In this paper, we study the bifurcation of limit cycles from fine focus in Z_n-equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z₈-equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z₈-equivariant systems, our results are good and interesting.     

The scalar Nevanlinna-Pick interpolation problem with boundary conditions

We show that if the Nevanlinna-Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna-Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H_∞ norm solutions. Finally, some numerical examples are given.     

Cardinal interpolation,submodules and the 4-direction mesh

This paper introduces the idea of cardinal interpolation on submodules of Z^d by translates of box splines if the condition of global linear independence fails to hold. In particular, the special case of the 4-direction box splines is discussed, where the pertinent submodule is given by the pairs (k, l) of integersk, l withk+l even. For this case, one obtains results that parallel the known results for the 3-direction box splines.     

Stability for Parabolic Quasiminimizers

This paper studies parabolic quasiminimizers which are solutions to parabolic variational inequalities. We show that, under a suitable regularity condition on the boundary, parabolic Q-quasiminimizers related to the parabolic p-Laplace equations with given boundary values are stable with respect to parameters Q and p. The argument is based on variational techniques, higher integrability results and regularity estimates in time. This shows that stability does not only hold for parabolic partial differential equations but it also holds for variational inequalities.     

Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX _t = a _θ (X _t )dt + dZ _t with a Lévy process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation $\text{u′(t) ? A(t) u(t) = ?(t), t ∈ [0, T]}$, with the initial datum u(0) = x, in the space C([0, T], E), where E is a Banach space and {A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D(A(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u′ and A(·)u(·) with values in the interpolation spaces D_A(0)(θ, ∞) and D_A(0)(θ) between D(A(0)) and E, is studied. A characterization of such spaces in a concrete case is also given.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.     

Maximum principle and propagation for intrinsicly regular solutions of differential inequalities structured on vector fields

We prove suitable versions of the weak maximum principle and of the maximum propagation for solutions u of a differential inequality Hu?0. Here H=_∑i,jai,j(z)ZiZj+Z₀ is a differential operator structured on the vector fields Zj's, whereas u belongs to an appropriate intrinsic class of regularity modelled on the Zj's.     

On p-adic differentiability

A p-adic-valued function on the p-adic integers has a continuous derivative, Mahler showed, whenever its interpolation coefficients decay at a certain rate. It is shown here that Mahler's decay condition is equivalent to the strict differentiability of the function. There is a discussion of the Banach-space structure of the space of strictly differentiable functions. It is shown, moreover, that there is no rate of decay common to all functions with continuous derivative. Specifically, given any decay condition, there exists a function with derivative identically zero, whose interpolation coefficients decay more slowly.     

Best interpolation in seminorm with convex constraints

Implicit and explicit characterizations of the solutions to the following constrained best interpolation problem $$\min \left\{ {\left\| {Tx - z} \right\|:x \in C \cap A^{ - 1} d} \right\}$$ are presented. Here,T is a densely-defined, closed, linear mapping from a Hilbert spaceX to a Hilbert spaceY, A: X→Z is a continuous, linear mapping withZ a locally, convex linear topological space,C is a closed, convex set in the domain domT ofT, andd∈AC. For the case in whichC is a closed, convex cone, it is shown that the constrained best interpolation problem can generally be solved by finding the saddle points of a saddle function on the whole space, and, if the explicit characterization is applicable, then solving this problem is equivalent to solving an unconstrained minimization problem for a convex function.     

An asymptotically optimal cubic spline

In this paper we consider the interpolation problem for a sufficiently smooth function on the segment [0, 1]. The values of the function under consideration are defined at given mesh nodes. We construct a cubic spline asymptotically optimal with respect to the growing number of nodes. Then we estimate interpolation errors for the constructed spline in the uniform and L ₂ metrics.     

Entropy function spaces and interpolation

We associate to every function space, and to every entropy function E, a scale of spaces Λp,q(E) similar to the classical Lorentz spaces Lp,q. Necessary and sufficient conditions for they to be normed spaces are proved, their role in real interpolation theory is analyzed, and a number of applications to functional and interpolation properties of several variants of Lorentz spaces and entropy spaces are given.     

Darstellung durch definite ternäre quadratische Formen

We study the representation behaviour of a $\text{Z}$-lattice L on a positive definite ternary quadratic space V over $\text{Q}$. As a new tool for this we use the Bruhat-Tits building of the spingroup of the completion of V at a suitable prime p. In Section 2 we show how this can be described in an elementary way as a graph whose vertices are the $\text{Z}$_p-maximal lattices on V_p, and in Section 4 we let this graph induce a graph, whose vertices are lattices on V, which differ from L only at the prime p. In Section 3 we investigate which lattices from the graph defined in Section 2 have a given vector in common. The results are used in Sections 5 and 6 to obtain information on the representation behaviour of some special lattices. In Section 5 we get a list of lattices, which represent all numbers they represent locally everywhere; this list contains that given by Watson in [16]. In Section 6 we sharpen a result of Jones and Pall from [6].     

Semialgebraic complexity of functions

In this paper we study the rate of the best approximation of a given function by semialgebraic functions of a prescribed “combinatorial complexity”. We call this rate a “Semialgebraic Complexity” of the approximated function. By the classical Approximation Theory, the rate of a polynomial approximation is determined by the regularity of the approximated function (the number of its continuous derivatives, the domain of analyticity, etc.). In contrast, semialgebraic complexity (being always bounded from above in terms of regularity) may be small for functions not regular in the usual sense. We give various natural examples of functions of low semialgebraic complexity, including maxima of smooth families, compositions, series of a special form, etc. We show that certain important characteristics of the functions, in particular, the geometry of their critical values (Morse–Sard Theorem) are determined by their semialgebraic complexity, and not by their regularity.     

The universality of ? 1 as a dual space

Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a ${{\mathcal L}_\infty}$ space Z whose dual is isomorphic to ? ₁. If, moreover, U is a space with separable dual, so that U and X are totally incomparable, then we construct such a Z, so that Z and U are totally incomparable. If X is separable and reflexive, we show that Z can be made to be somewhat reflexive.     

Descendant-homogeneous digraphs

The descendant setdesc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets, and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant-homogeneous digraphs with finite out-valency, specially those which are also highly arc-transitive. We show that these digraphs must be imprimitive. In particular, we study those which can be mapped homomorphically onto Z and show that their descendant sets have only one end.There are examples of descendant-homogeneous digraphs whose descendant sets are rooted trees. We show that these are highly arc-transitive and do not admit a homomorphism onto Z. The first example (Evans (1997) [6]) known to the authors of a descendant-homogeneous digraph (which led us to formulate the definition) is of this type. We construct infinitely many other descendant-homogeneous digraphs, and also uncountably many digraphs whose descendant sets are rooted trees but which are descendant-homogeneous only in a weaker sense, and give a number of other examples.