The behavior of \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-\nulldelimiterspace} n}} for particular values of θ

Let ζ be a primitive q′-root of unity. We prove that the series $ \sum\nolimits_{n = 1}^\infty {{{\zeta ^{ \llcorner n\theta \lrcorner } } \mathord{\left/ {\vphantom {{\zeta ^{ \llcorner n\theta \lrcorner } } n}} \right. \kern-0em} n}} $ for θ ∈ Q converges if and only if θ = p/q with (p,q) = 1 and q′ ? p, and that there exists an uncountable set S of Liouville’s numbers such that the series does not converge when θ ∈ S.     

The elliptic functions and integrals of the “{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}” problem

We describe the functions needed in the determination of the rate of convergence of best $L^\infty $ rational approximation to $\exp ( - x)$ on [0,∞) when the degree n of the approximation tends to ∞ (“1/9” problem).     

On the exponential diophantine equation $$\left( am^{2}+1\right) ^{x}+\left( bm^{2}-1\right) ^{y}=(cm)^{z}$$ with $$ c\mid m $$c∣m

Let \(a,\ b,\ c,\ m\) be positive integers such that \(a+b=c^2, 2\mid a, 2\not \mid c\) and \(m>1\). In this paper we prove that if \(c\mid m \) and \(m>36c^3 \log c\), then the equation \((am^2+1)^x+(bm^2-1)^y=(cm)^z\) has only the positive integer solution \((x,\ y,\ z)\)=\((1,\ 1,\ 2)\).     

Menger’s theorem in {{\Pi^1_1\tt{-CA}_0}}

We prove Menger’s theorem for countable graphs in ${{\Pi^1_1\tt{-CA}_0}}$ . Our proof in fact proves a stronger statement, which we call extended Menger’s theorem, that is equivalent to ${{\Pi^1_1\tt{-CA}_0}}$ over ${{\tt{RCA}_0}}$ .     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

Some problems of the approximation of functions by “hyperbolic” Fourier-Hermite sums in the space L_2 \left( {{\mathbb{R}}^2 ,e^{ - x^2 - y^2 } } \right)

We obtain, among others, an exact estimate of the deviation of the “hyperbolic” Fourier-Hermite sums of the functions in the class L ₂ ^r (D) of the space indicated in the title.     

Poincaré polynomial of the space $$\overline {{M_{0,n}}} \left( \mathbb{C} \right)$$ and the number of points of the space $$\overline {{M_{0,n}}} \left( {{F_q}} \right)$$M0,n¯(Fq)

A combinatorial proof that the number of points of the space \(\overline {{M_{0,n}}} \left( {{F_q}} \right)\) satisfies the recurrent formula for the Poincaré polynomials of the space \(\overline {{M_{0,n}}} \left( \mathbb{C} \right)\) is obtained.     

Criterion for the Denseness of Algebraic Polynomials in the Spaces $$L_p \left( {{\mathbb{R}},d {\mu }} \right)$$ , 1 ≤ p < ∞

The criterion for the denseness of polynomials in the space established by Hamburger in 1921 is extended to the spaces , 1 p < .     

Infinite-Dimensional Representations of the Lie Algebra \mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right) Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL\left( {n,\mathbb{C}} \right)

Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z ^±, where Z ⁺ and Z ^? are the triangular unipotent subgroups. Representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.     

Using Fourier–Legendre expansions to derive series for \frac{1}{\pi} and \frac{1}{\pi^{2}}

In this paper, we derive some series for $\frac{1}{\pi}$ and $\frac{1}{\pi^{2}}$ from the Fourier–Legendre expansions of odd powers of $\sqrt{1-x^{2}}$ .     

�˻�ͼ\mathbb{Z}^2\times\{0,1,\ldots,l-1\}�ij����Եij���֤��

It is well known that the two dimensional integer lattice \mathbb{Z}^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\} is recurrent. The proof approach only utilizes the elementary methods in probability theory (without the words of electric networks).     

A note on Ramanujan’s identities involving the hypergeometric function {_{2}}F_{1}(\frac{1}{6},\frac{5}{6};1;z)

We study a class of elliptic functions associated with the hypergeometric function ${_{2}}F_{1}(\frac{1}{6},\frac{5}{6};1;z)$ . From the perspective of the properties of conformal mappings and differential equations, we provide new insight into a set of identities of Ramanujan associated with the above hypergeometric function.     

关于数论函数方程φ(x_1…x_(n-1)x_n)=m(φ(x_1)+…+φ(x_(n-1))+φ(x_n))

运用Euler函数的性质证明了:当n>1时,方程φ(x_1…x_(n-1)x_n)=m(φ(x_1)+…+φ(x_(n-1))+φ(x_n))仅有有限多组正整数解(x_1,…,x_(n-1),x_n),得到了这些解都满足max{x_1,…,x_(n-1),x_n}≤2m⁴(n-1)4(n-1)²n2n².     

A Structure Theorem for \displaystyle\mathbb{P}^{{1}} — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.     

涉及无限族非扩张映象$\{T_n\}_{n=1}^\infty$的迭代算法$x_{n 1}=\alpha_{n 1}f(x_n) (1-\alpha_{n 1})T_{n 1}x_n$

Under the framework of uniformly smooth Banach spaces, Chang proved in 2006 that the sequence {xn} generated by the iteration xn＋1 =αn＋1f（xn） ＋ （1 - αn＋1）Tn＋1xn converges strongly to a common fixed point of a finite family of nonexpansive maps {Tn}, where f ： C → C is a contraction. However, in this paper, the author considers the iteration in more general case that {Tn} is an infinite family of nonexpansive maps, and proves that Chang＇s result holds still in the setting of reflexive Banach spaces with the weakly sequentially continuous duality mapping.     

Approximating Characteristics of the Nikol’skii–Besov Classes S1,θrBℝd$$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$

Ukrainian Mathematical Journal - We establish the exact-order estimates for the approximation of the classes $$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$ by entire functions of...