Ukrainian Mathematical Journal最新文献

Let n ∈ ℕ, n ≥ 2. An element (x₁,…,x_n) ∈ E_n is called a norming point of T ∈ (mathcal{L}left({}^{n}Eright)) if ||x₁|| = … = ||x_n|| = 1 and |T(x₁,…,x_n)| = ||T||, where ℒ(ⁿE) denotes the space of all continuous n-linear forms on E. For T ∈ ℒ (ⁿE), we define

The set Norm(T) is called the norming set of T. For m ∈ ℕ, m ≥ 2, we characterize Norm(T) for any T ∈ (mathcal{L}left({}^{m}{l}_{1}^{n}right)), where ({l}_{1}^{n}={mathbb{R}}^{n}) with the l₁-norm. As applications, we classify Norm(T) for every T ∈ (mathcal{L}left({}^{m}{l}_{1}^{n}right)) with n = 2, 3 and m = 2.

We introduce the spaces ℓ_∞(𝒞_α), f(𝒞_α), and f₀(𝒞_α) of Cesàro bounded, Cesàro almost convergent, and Cesàro almost null sequences of order α, respectively. Moreover, we establish some inclusion relations for these spaces and determine the α -, β- and γ-duals of the spaces ℓ_∞ (𝒞_α) and f(𝒞_α). Finally, we characterize the classes of matrix transformations from the space f(𝒞_α) to any sequence space Y and from any sequence space Y to the space f(𝒞_α).

We consider a parabolic-hyperbolic equation with multiplicative control and nonlocal boundary conditions. By using the Riesz biorthogonal basis, the problem is reduced to a sequence of one-dimensional problems with alternative representations of their solutions. Conditions guaranteeing the existence and uniqueness of the solution to the analyzed problem are established.

Let [·] be the floor function. We show that if 1 < c < (frac{3849}{3334}), then there exist infinitely many prime numbers of the form [n^c], where n is square free.

We propose an approach to the generalized k-Bessel function defined by

({text{U}}_{p,q,r}^{text{k}}left(zright)=sum_{n=0}^{infty }frac{{left(-rright)}^{n}}{{Gamma }_{k}left(nk+p+frac{q+1}{2}text{k}right)n!}{left(frac{z}{2}right)}^{2n+frac{p}{text{k}}},)

where k > 0 and p, q, r ∈ ({mathbb{C}}). We discuss the uniform convergence of ({text{U}}_{p,q,r}^{text{k}}) (z). Moreover, we prove that the analyzed function is entire and determine its growth order and type. We also find its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane. The integral representation for ({text{U}}_{p,q,r}^{text{k}}) (z) is found by using the representation for k-beta functions. We also prove that the specified function is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions. In addition, some interesting properties, such as recurrence and differential relations, are demonstrated. Some of these properties can be used to establish Turán-type inequalities for this function. Ultimately, we study the monotonicity and log-convexity of the normalized form of the modified k-Bessel function ({text{T}}_{p,q,1}^{text{k}}) defined by ({text{T}}_{p,q,1}^{text{k}}) (z) = (i{-}^frac{p}{k}{text{U}}_{p,q,1}^{text{k}}) (iz), as well as the quotient of the modified k-Bessel function, exponential, and k-hypergeometric functions. In this case, the leading concept of the proofs comes from the monotonicity of the ratio of two power series.

We propose a method for the construction of exact solutions to the nonlinear heat equation with a source based on the classical method of separation of variables, its generalization, and the method of reduction. We consider substitutions reducing the nonlinear heat equation to ordinary differential equations and to a system of two ordinary differential equations. The classes of exact solutions of the analyzed equation are constructed by the method of generalized separation of variables.

We discuss some algebraic identities related to multiplicative (generalized) derivations and multiplicative (generalized)-(α, β)-derivations on appropriate subsets in prime rings.

We obtain a parametric description of elements of complete linear groups of the second and third orders over an arbitrary field. It is based on their canonical (single-valued) representation as a product of elements from the commutators of certain Jordan matrices and representatives of the left cosets of these groups.

In the space L₂[(0, 1); x], by using a system of functions ({left{{widehat{J}}_{v}left({mu }_{k,v}xright)right}}_{kin {mathbb{N}}}, vge 0,) orthonormal with weight x and formed by a Bessel function of the first kind of index v and its positive roots, we construct generalized finite differences of the mth order ({Delta }_{gamma left(hright)}^{m}left(fright),) m ∈ ℕ, h ∈ (0, 1), and the generalized characteristics of smoothness ({Phi }_{gamma left(hright)}^{left(gamma right)}left(f,tright)=left(1/tright)underset{0}{overset{t}{int }}Vert {Delta }_{gamma left(tau right)}^{m}left(fright)Vert dtau .) For the classes ({mathcal{W}}_{2}^{r,v}{Phi }_{m}^{left(gamma right)},left(uppsi right)) defined by using the differential operator ({D}_{v}^{r},) the function ({Phi }_{m}^{left(gamma right)}left(fright),) and the majorant ψ, we establish lower and upper estimates for the values of a series of n-widths. We established the condition for ψ, which enables us to compute the exact values of n-widths. To illustrate our exact results, we present several specific examples. We also consider the problems of absolute and uniform convergence of Fourier–Bessel series on the interval (0, 1).

We obtain several new integral inequalities in terms of fractional integral operators for the functions whose first derivatives satisfy either the conditions of the Lagrange theorem or the Lipschitz condition. In some special cases, the obtained results provide better upper estimates than the results known in the literature for the Bullen-type inequality and the Hadamard-type right-hand side inequality. Finally, some error estimates for the trapezoidal formula are discussed.