Some properties of Van Koch’s curves

We investigate the properties of an integral operator T with a Cauchy kernel. The operator acts from L ^∞(Γ, μ), where Γ is a Van Koch curve, to the space of functions ? → ?. We prove that the range of T is nontrivial and lies in the space AC(Γ) of functions continuous in ?, vanishing at ∞, and analytic outside Γ. We also show that T is injective and compact while satisfying some special functional equation. These results may be regarded as a natural continuation of our research on the problem of AC-removability of quasiconformal curves whose solution was announced in [1] for the first time and supplemented later with some other properties of Van Koch’s curves [2, 3]. In this paper the problem is discussed in a more general setting and, in particular, all important details lacking in [1] are given. Some open problems are formulated.     

Special congruence in a subspace of a finsler space

Summary G. Tsagas[5] obtained the special curves in a Riemannian hypersurface. In the present paper, twofold generalizations of these curves have been obtained and as such special and secondary special congruences with respect to a curve C in Finsler subspace are introduced. When a special congruence becomes a λ-geodesic[2] is also discussed. Entrata in Redazione il 1^o aprile 1971.     

A linear multistep numerical integration scheme for solving systems of ordinary differential equations with oscillatory solutions

In ^[1], a set of convergent and stable two-point formulae for obtaining the numerical solution of ordinary differential equations having oscillatory solutions was formulated. The derivation of these formulae was based on a non-polynomial interpolant which required the prior analytic evaluation of the higher order derivatives of the system before proceeding to the solution. In this paper, we present a linear multistep scheme of order four which circumvents this (often tedious) initial preparation. The necessary starting values for the integration scheme are generated by an adaptation of the variable order Gragg-Bulirsch-Stoer algorithm as formulated in ^[2].     

Transition distributions of young diagrams under periodically weighted plancherel measures

Kerov[16,17] proved that Wigner＇s semi-circular law in Gauss[an unitary ensembles is the transition distribution of the omega curve discovered by Vershik and Kerov[34] for the limit shape of random partitions under the Plancherel measure. This establishes a close link between random Plancherel partitions and Gauss[an unitary ensembles, In this paper we aim to consider a general problem, namely, to characterize the transition distribution of the limit shape of random Young diagrams under Poissonized Plancherel measures in a periodic potential, which naturally arises in Nekrasov＇s partition functions and is further studied by Nekrasov and Okounkov[25] and Okounkov[28,29]. We also find an associated matrix mode[ for this transition distribution. Our argument is based on a purely geometric analysis on the relation between matrix models and SeibergWitten differentials.     

Gaussian series and daubechies operators

The subspace [Mtilde] of L²(Cⁿ) which is composed of Gaussian series and contains the subspace M spanned by Gaussian functions given in the paper [6] by Du and Wong has the proporety that the product of two Daubechies operators with symbols in [Mtilde] is a Daubechies operator with symbol H in [Mtilde]. Furthermore, an explicit expression for the symbol H is given     

A development on approximation by monotone sequences of polynomials

Recently people proved that every f∈C[0,1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,… that {fx98-1}. For example, Xie and Zhou^[2] showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous function. Actually they obtained a result as {fx98-2}, which essentially improved a conclusion in Gal and Szabados^[1]. The present paper, by optimal procedure, improves this inequality to {fx98-3}, where ɛ is any positive real number.     

The Category of Directed Systems in a Category

We present two related categorical constructions. Given a category C, we construct a category C^[d], the category of directed systems in C. C embeds into C^[d], and if C has enough colimits, then C is monadic over C^[d]. Also, if E,M is a factorization structure for C, then C^[d] has a related factorization structure E_d M_d such that if E consists entirely of monic arrows, then so does E_d and the E_d-quotient poset of an object A is naturally the poset of directed downsets of the E-quotient poset of A. Similarly, if M consists entirely of monicarrows, then so does M_d and the M_d-subobject poset of an object A is naturally the poset of directed downsets of the M-subobject poset. C^[d] has completeness and cocompleteness properties at least as good as those of C, and it is abelian if C is. Dualization gives the other construction: a category C^[i], the category of inverse systems in C, into which C also embeds and which satisfies similar properties, except that directed downsets in the E-quotient and M-subobject posets are replaced by directed upsets.     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

Simultaneous approximation to values of exponential function at algebraic points

This paper applies the methods of Chudnovsky^[2], Osgood^[4] and Baker^[1] to give a simultaneous approximation theorem to the values of exponential function at algebraic points, which slightly sharpens and generalises the results of Chudnovsky^[2],[3]. This work is supported by the National Natural Science Foundation of China.     

Real-analytic solvability for differential complexes associated to locally integrable structures

Inspired by the work of Suzuki [12] on the concept of real-analytic solvability for first-order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one. We prove that such notion of solvability is related to the smooth solvability condition introduced by F. Treves [13] in 1983. In our arguments the natural extension to closed forms of the well-known Baouendi–Treves approximation formula, the so-called “Approximate Poincaré Lemma” (cf. [1], [14]), plays a key role.     

D-optimality andD n -optimal designs for mixtures regression models with logarithmic terms

In this paper the authors put forward a new mixtures regression models with Logarithmic terms^[14] to generalize Draper's models^{[1, 2 or 6]} by using Kiefer-Wolfowitz's equivalence theorem^[3], Fedorov's and Wynn's method^[5]. And we also suggest a method for computer-aided design of combinatorial search^[13].In this study, we have proved and constructed the approximateD-optimal (measure) andD _n -optimal (exact) designs by the use of the first and second order mixtures regression models with logarithmic terms in three and four components.Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

On the approximation by the mixed exponential type integral operators

In 1984, L. C. Hsu, X. J. Yang and the auther^[1] defined a class of summation-integral operators. Later, the author^[2]–[5], Mazhar-Totik^[6] and Sahai-Prased^[7] studied some specific exponential type integral operators, separately. The purpose of this paper is to introduce and study some more general mixed exponential type integral operators.     

Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a ₀+a ₁ z ^?1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0<α<2).     

Linear forms and quadratic uniformity for functions on ℤ<Subscript><Emphasis Type="Italic">N</Emphasis></Subscript>

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW10a] and [GW10b], [GW10c], we gave a complete answer to this question for groups of the form G = F _p ⁿ , provided p is not too small. In ℤ _N , substantial extra difficulties arise, of which the most important is that an “inverse theorem” even for the uniformity norm || ·||_U³{\left\| \cdot \right\|_{{U^3}}} requires a more sophisticated “local” formulation. When N is prime, ℤ _N is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW10a]. Moreover, we obtain a doubly exponential bound.     

On the mean curvatures sharp estimates of hypersurfaces

Sharp estimates for the mean curvatures of hypersurfaces in Riemannian manifolds are known from the works of Jorge-Xavier ^[3], Markvorsen ^[6] and Vlachos ^[11]. We first give a simplified proof of these estimates. This proof shows that a similar original result holds for hypersurfaces in Einstein manifolds which are warped product of by Ricci-flat manifolds.     

On the existence of positive solutions for a class of singular boundary value problems

In this paper, by the use of a fixed point theorem, many new necessary and sufficient conditions for the existence of positive solutions in C[0,1]∩C¹[0,1]∩C²(0,1) or C[0,1]∩C²(0,1) are presented for singular superlinear and sublinear second-order boundary value problems. Singularities at t=0, t=1 will be discussed.     

A nonlinear diffusion model for image restoration

In this paper, we study the ill posed Perona-Malik equation of image processing^[14] and the regularized P-M model i.e. C-model proposed by Catte et al.^[4]. The authors present the convex compound of these two models in the form of the system of partial differential equations. The weak solution for the equations is proved in detail. The additive operator splitting (AOS) algorithm for the proposed model is also given. Finally, we show some numeric experimental results on images.