Том 12 № 2

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  • Документ
    Додатні ряди, множини підсум яких є канторвалами
    (2019) Ярослав Виннишин, Віта Маркітан, Микола Працьовитий, Ігор Савченко
    Наводиться конструкція континуальної сім'ї додатних рядів, множини неповних сум яких є канторвалами (об'єднанням ніде не щільної множини і множини, яка є нескінченним об'єднанням відрізків). Кожен ряд даної сім'ї має властивість $$sumlimits_{n=1}^{infty}a_{n}=1,~~~overline{lim_{nrightarrowinfty}}frac{a_n}{sum_{k=1}^{infty}a_{n+k}}=+infty,$$ причому для будь-якого $varepsilon>0$ в цій сім'ї існує ряд, міра Лебега множини неповних сум якого є більшою за $1-varepsilon$.
  • Документ
    On the generalization of the Darboux theorem
    (2019) Kaveh Eftekharinasab
    Darboux theorem to more general context of Frechet manifolds we face an obstacle:  in general vector fields do not have local flows. Recently, Fr'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic  manifold $(M^{2n},omega)$ is locally symplectomorphic to $(R^{2n}, omega_0)$, where $omega_0$  is the standard symplectic form on  $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ ff_t $  generated by a time-dependent vector field $ X_t  $ on $M$ such that $ ff_1^{*} omega = omega_0$. Nevertheless,  it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we  associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the  classifying space does not depend on the point of the manifold and a suitable smoothness condition holds.  If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar.  In this paper we consider the category of so-called bounded Fr'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields  we prove the Darboux theorem.
  • Документ
    On fractal properties of Weierstrass-type functions
    (2019) Claire David
    In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {mathcal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left(2, pi,N_b^n,x right)$, where $lambda$ and $N_b$ are two real numbers such that~mbox{$0 <lambda<1$},~mbox{$ N_b,in,N$} and $ lambda,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.
  • Документ
    A (CHR)3-flat trans-Sasakian manifold
    (2019) Koji Matsumoto
    In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3  and the (CHR)3-scalar curvature τ3  in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized   ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.