Acta et Commentationes Universitatis Tartuensis de Mathematica

On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series

2021-08-09T01:56:45+03:00

Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τ^λ_n;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and via entries of a matrix whose indices form a strictly increasing sequence of positive integers λ := {λ(n)}^∞_n=1.

Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas

2021-08-09T01:57:16+03:00

We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.

On Riemann problem in weighted Smirnov classes with general weight

2021-08-09T01:57:13+03:00

Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered,
and conditions on degeneration order are found.

Geometry of multilinear forms on l_1

2021-08-09T01:57:10+03:00

We characterize extreme, exposed and smooth points in the Banach space L(ⁿE) of continuous n-linear forms on E, and in its subspace Ls(ⁿE) of symmetric n-linear forms on E when E = l1 and E = l1^mfor n,m ∈ N with n,m ≥ 2 .

Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements

2021-08-09T01:57:03+03:00

We propose asymptotic approximations for the probabilities of misclassification in linear discriminant analysis when the group means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of several multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain asymptotic approximation of misclassification errors for the two group case. Finally, we perform Monte Carlo simulations to evaluate the reliability of the proposed results.

Natural vibrations of nanostrips with cracks

2021-08-09T01:57:01+03:00

Employing the main equations of the theory of plates accounting for the rotational inertia the transverse vibrations of nanobeams and nanostrips are investigated. The nano strips under consideration have piecewise constant dimensions of cross sections. The nanosheets are weakened by cracks at re-entrant corners of steps. While the material behavior corresponds to the Eringen’s nonlocal theory of elasticity it is assumed that the cracks produce additional local compliance, which can be evaluated with the aid of the stress intensity factor at the crack
tip. A numerical algorithm for determination of natural frequencies of nanosheets is developed.

About the number of τ-numbers relative to polynomials with integer coefficients

2021-08-09T01:56:58+03:00

We show that for all polynomials Q(x) with integer coefficients, that satisfy the extra condition |Q(0) · Q(1) | ≠ 1, there are infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). We also find some examples of polynomials Q(x) for which 1 is the only τ-number relative to the polynomial Q(x) and some examples of polynomials Q(x) with |Q(0) · Q(1)|= 1, which have infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). In addition, we prove one result about the generators of a τ-number.

Module bundles and module amenability

2021-08-09T01:56:56+03:00

Let X be a compact Hausdorff space, and let {Ax : x ∈ X} and {Bx : x ∈ X} be collections of Banach algebras such that each Ax is a Bx-bimodule. Using the theory of bundles of Banach spaces as a tool, we investigate the module amenability of certain algebras of Ax-valued functions on X over algebras of Bx-valued functions on X.

On generalized fractional integral inequalities of Ostrowski type

2021-08-09T01:56:54+03:00

We obtain new generalizations of Ostrowski inequality by using generalized Riemann{Liouville fractional integrals. Some special cases are also discussed.

Thue's equation as a tool to solve two different problems

2021-08-09T01:56:52+03:00

A Thue equation is a Diophantine equation of the form f(x; y) = r, where f is an irreducible binary form of degree at least 3, and r is a given nonzero rational number. A set S of at least three positive integers is called a D1³-set if the product of any of its three distinct elements is a perfect cube minus one. We prove that any D1³-set is finite and, for any positive integer a, the two-tuple {a, 2a} is extendible to a D1³-set 3-tuple, but not to a 4-tuple. Using the well-known Thue equation 2x³ - y³= 1, we show that the only cubic-triangular number is 1.

On the Fibonacci quaternion sequence with quadruple-produce components

2021-08-09T01:56:48+03:00

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.