Lithuanian Mathematical Journal最新文献

Let β be a Pisot or Salem number with β > 1, and let α₁ and α₂ be elements in ({mathbb{Q}})(β) ∩ [0, 1). In this note, we prove that α₁ and α₂ have either the same tail greedy expansions in a base β or independent random greedy expansions in a base β.

Let (mathcal{K})(r) and arctanh r for r ∈ (0, 1) be the complete elliptic integral of the first kind and the inverse tangent hyperbolic function, respectively. In this paper, we prove that the double inequality

({Phi }_{p}left({r}{prime}right)frac{text{arctanh}r}{r}<frac{2}{pi }mathcal{K}left(rright)<{Phi }_{q}left({r}{prime}right)frac{text{arctanh}r}{r})

holds for r ∈ (0, 1) if and only if q ⩽ 56 543/20 976 and 23(90π − 233)/(10(69π − 178)) ⩽ p ⩽ 3, where r′ (sqrt{1-{r}^{2}}) and

({Phi }_{q}left(xright)=60frac{left(17q-41right){x}^{2}+6qx+69-23q}{left(620q-1521right){x}^{2}+2left(580q-1079right)x+5359-1780q})

for q ⩽ 3 and x ∈ (0, 1). This improves some known results and yields several new bounds for the Gauss arithmetic–geometric mean.

In this paper, we study the frequency of different types of arithmetic word problems (AWP) in Lithuanian textbooks. The results show the lack of variety among types of AWP. We propose the framework for analysis of the frequency of types of AWP in a textbook and apply it to a particular set of primary school textbooks. We use a statistical method to compare the sample from the textbook rather than from the entire textbook. Also, we compare the proportions of types of AWP in Lithuanian textbooks with those in Singaporean and Spanish textbooks. The approach adopted in the paper can be used to analyze other textbooks from different countries.

The generalizations of some results of H. Bor, L. Leindler, and H. Sevli pertaining to absolute summability are examined.

Subsets of rational numbers are specified as preimages of values of arithmetical functions. The uniformity of distribution of elements of these sets is proved and interpreted in the context of Diophantine approximation to real numbers.

Let d be a positive integer, and let A be a set in ({mathbb{Z}}^{d}) that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on ({mathbb{Z}}^{d}), whereas the transition probabilities from the points inside A are different. We investigate the impact of the perturbation on a scaling limit of X. It turns out that if d ⩾ 2, then in a typical situation the scaling limit of X coincides with that of the underlying standard random walk. This is unlike the case d = 1, in which the scaling limit of X is usually a skew Brownian motion, a skew stable Lévy process, or some other “skew” process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in ({mathbb{Z}}^{d}) for d ⩾ 3. More interestingly, in the situation where the standard random walk in ({mathbb{Z}}^{2}) is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of X to the set A is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum, we have the situation in which the standard random walk admits a Donsker’s scaling limit, whereas its locally perturbed version does not because of huge jumps from the set A, which occur early enough.

The paper reviews the scientific and pedagogical activities of Julius Kruopis, the pioneer of Lithuanian applied statistics. His contribution to statistics and cooperation with Lithuanian companies in the application of statistical methods in various fields of human activity is described in the most detail, especially in the quality control area and industrial process optimization. His works in probability theory are also mentioned, emphasizing important contributions to approximations of distributions. Peculiarities of his pedagogical activity, textbooks and monographs, and supervision of students’ theses and dissertations are discussed.

Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λ_Sym^m _f (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Sym^m f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λ_Sym^m _f (n) and establish its asymptotic formula.

Inspired by unfading popularity of the Turán–Kubilius inequality for additive number theoretic functions within the last decades, we examine the variance of additive functions defined on random permutations uniformly taken from the symmetric group. Extending the optimal estimate achieved in 2018 by Klimavičius and Manstavičius for the case of completely additive functions, we obtain asymptotically sharp upper and lower bounds when the functions are strongly additive. The upper estimates are analogous to that established in number theory by Kubilius in 1985.

We consider the quasilinear Schrödinger equation involving a general nonlinearity at critical growth. By using Jeanjean’s monotonicity trick and the Pohozaev identity we get the existence results that generalize an earlier work [H. Liu and L. Zhao, Existence results for quasilinear Schrödinger equations with a general nonlinearity, Commun. Pure Appl. Anal., 19(6):3429–3444, 2020] about the subcritical case to the critical case.