Geometry of the nets in equiaffine spaces

The derivation formulae of the vector fields of an arbitrary net belonging to the n-dimensional equiaffine spaceEqA _n are introduced and the conditions which satisty their coefficients are found. The following special nets: Chebyshev of the first and second kind, strongly parallel of the first kind, geodesic, generalized metrical Chebyshev and symmetric nets are studied. Their characteristics by the coefficients of the derivation equations are obtained. Chebyshev and geodesic curvatures of the lines of the net belonging toEqA _n and Chebyshev and geodesic vectors of the nets are introduced. Equiaffine spaces containing above mention special nets are defined.The present investigation is partially supported by the Nacional Science Fund of the Ministry of Science and Education, Republic of Bulgaria under grant MM 64.     

On the Chebyshev nets in a hypersurface of a Weyl space

In [1], Zlatanov introduced the Chebyshev vector fields of the first and second kind and the geodesic vector fields for an n-dimensional net in the Weyl spaceW _n . After having defined, in [2], the Chebyshev and geodesic curvatures of the lines of an arbitrary net,the b-nets and the c-nets, Tsareva and Zlatanov studied, among other things, some properties of the Chebyshev nets. In this paper, we consider an n-dimensional net in the hypersurfaceW _n of the Weyl SpaceW _n+1 and study some properties of the Chebyshev vector fields of the first and second kind and the geodesic vector fields of this net. Finally, two theorems concerning the b-nets and c-nets inW _n are obtained.     

On the nets of equi-conjugate pairs of curves in a Weyl hypersurface

In this paper, equi-conjugate pairs of curves in a hypersurfaceW _n(g_ij,T_k) are defined and the Chebyshev nets of the first kind, the second kind and the geodesic nets which are formed by the tangent vector fields of these pairs of curves are investigated.     

Nets in Weyl spaces and their metrically equipped subspaces

Let the surfaceW _m⊂W_n be metrically equipped by a normal planeR _n−m. A correspondence between the nets belonging toW _m andW _n is established. Relations between the coefficients of the derivation equations for the fields of directions of the corresponding nets are found. The apparatus for studying the connection between the properties of the corresponding nets is constructed. The influence of the choice that the net belonging toW _m be Chebyshev of the first kind or geodesic or c-net on the corresponding net belonging toW _n is investigated and the results are formulated in theorems 2.6, 2.9, 2.10, 2.11, 2.12 being a generalization of the results received in [2].

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Sunto Supponiamo che la superficieW _m⊂W_n sia dotata di un piano normaleR _n−m. Si stabilisce una corrispondenza tra le reti diW _m e quelle diW _n e si propone un metodo per studiare la relazione tra le proprietà delle reti corrispettive.

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The present investigation is partially supported by the National Science Fund of the Ministry of Science and Education, Republic of Bulgaria under grant MM 528.     

Nets and their codes

A finitek-net of ordern is an incidence structure ofnk lines andn ² points, with any two lines either meeting once or being parallel, havingk parallel classes ofn lines each, and havingn points on each line. Finite nets are important to the study of finite planes and Latin squares.In this paper finite nets will be studied using the following linear codes: the row space of the incidence vectors of lines, the intersection of this code with its orthogonal, the code generated by differences of parallel lines, and the orthogonal to these codes. Using these codes we are able to recast the Moorhouse conjecture in terms of subcodes of the codes he uses, determine coding-theoretic reasons for a net's being maximal, and generalize a theorem of Assmus and Key which uses codes to classify finite planes of prime order.     

Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.     

Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}     

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind. Postdoctoral researcher FWO-Flanders.     

Konjugierte Netze mit p-ebenen u-Kurven,welche von den v-Kurven projektiv aufeinander bezogen werden

In this paper we consider conjugate nets in projective space P ⁿwith the following property: The u-curves of the net belong to p-dimensional subspaces of P ⁿand are projectively related to each other by the v-curves. We show that these nets form two classes. The first class consists of conjugate nets for which the u-curves from conic shadow boundaries. The u-curves of the nets of the second class, which we call C _p-nets, are rational normal curves of order p. Each u-curve possesses the characteristic (p–1)-space of its ambient p-space as a (p–1)-dimensional osculating space. This generalizes a result found by Degen [3] in the case p=2, n=3. By means of the Laplace transformation we get a construction of C _p-nets without integration.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesl _n ,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,l _n = _n , the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research     

具有特殊坐标曲线网的B\'ezier曲面

在计算机辅助几何设计中, B\''ezier曲面是一类重要的参数曲面.在微分几何中,坐标曲线网也是重要的研究内容.本文中,我们对具有特殊坐标曲线网(如正交曲线网、曲率曲线网、共轭曲线网等)的B\''ezier曲面进行研究.此外,我们还构造了满足能量约束的特殊B\''ezier曲面,给出了基于控制结构的充分条件并给出具体实例.     

Schauder decomposition and its application to integral equations of the second kind

The rates of convergence of Schauder decompositions in L_p and W_p⁽ⁿ⁾ are established. An application of Schauder decompositions to numerical solutions for integral of the second kind is made.     

Cayley’s decomposition and pólya’sW-property of ordinary linear differential equations

A proof is given for the equivalence of Pólya’sW- property of a linear differential equationL _n (D) y=0 to the possibility of decomposingL _n (D) ≡ Π _n ¹ [D+λ_i(x)] in a given interval. In this case a set ofn independent solutions form a Chebyshev system in the interval. An application determines intervals of non-oscillation for solutions of linear equations of the second order. Research supported by the National Science Foundation Grant No. GP-3897 with the University of Maryland.     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.     

Area,coarea, and approximation in <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript>

Let Ω⊂ℝ ⁿ be an arbitrary open set. We characterize the space W ^1,1 _loc(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space W ^1,1(ℝ ⁿ ).     

Near-best L1 approximations on circular and elliptical contours

Polynomial approximations are obtained to analytic functions on circular and elliptical contours by forming partial sums of order n of their expansions in Taylor series and Chebyshev series of the second kind, respectively. It is proved that the resulting approximations converge in the L₁ norm as n → ∞, and that they are near-best L₁ approximations within relative distances of the order of log n. Practical implications of the results are discussed, and they are shown to provide a theoretical basis for polynomial approximation methods for the evaluation of indefinite integrals on contours.