Ukrainian Mathematical Journal最新文献

The main aim of the paper is to determine some basic properties of the essential pseudospectrum of a bounded linear operator A defined in a Banach space X. We also prove two different versions of the essential pseudospectral mapping theorem.

We study the main properties of the Laguerre–Cayley functions and related polynomials, which can be regarded as an essential component of the mathematical apparatus of the functional-discrete (FD-) method used to solve the Cauchy problem for an abstract homogeneous evolutionary equation of fractional order.

For 0 < α ≤ 1 and λ > 0, let

({G}_{lambda ,alpha }=left{f in A: left|frac{1-alpha +alpha zf^{primeprime}left(zright)/{f}^{{^{prime}}}left(zright)}{z{f}^{{^{prime}}}left(zright)/fleft(zright)}-left(1-alpha right)right|< lambda , z in {mathbb{D}}right}. (0.1))

The general form of the Silverman class was introduced by Tuneski and Irmak [Int. J. Math. Math. Sci., 2006, Article ID 38089 (2006)]. Our differential-inequality formulation is based on several sufficient conditions for this class. Further, we consider a class Ω given by

(Omega =left{fin A:left|z{f{^{prime}}}^{left(zright)}-fleft(zright)right|<frac{1}{2},zin {mathbb{D}}right}. (0.2))

For these two classes, we establish inclusion relations involving some well-known subclasses of S^* and compute radius estimates featuring various pairings of these classes.

A generalization of the Leonardo numbers is defined and called hyper-Leonardo numbers. Infinite lowertriangular matrices whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A- and Z-sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are deduced by using the fundamental theorem on Riordan arrays.

We establish sufficient conditions for the continuous extension of a Cauchy-type integral whose density depends on the parameter to a nonsmooth integration line.

It is shown that the Bojanov–Naidenov problem ({Vert {x}^{left(kright)}Vert }_{q, delta }) → sup, k = 0, 1, . . . , r − 1, on the classes of functions ({Omega }_{p}^{r}left({A}_{0}, {A}_{r}right)) := (left{x in {L}_{infty }^{r}: {Vert {x}^{left(rright)}Vert }_{infty }le {A}_{r}, L{left(xright)}_{p}le {A}_{0}right},) where q ≥ 1 for k ≥ 1 and q ≥ p for k = 0, is equivalent to the problem of finding the sharp constant C = C(λ) in the Kolmogorov-type inequality

({Vert {x}^{left(rright)}Vert }_{q,delta }le CL{left(xright)}_{p}^{alpha }{Vert {x}^{left(rright)}Vert }_{infty }^{1-alpha }, xin {Omega }_{p,lambda }^{r}, (1))

where (alpha =frac{r-k+1/q}{r+1/p},) ({Vert xVert }_{p,delta }) := sup {({Vert xVert }_{{L}_{p}[a,b]}):a, b, ∈ R, 0 < b – a ≤ δ} δ > 0, ({Omega }_{p,lambda }^{r}) := (bigcup left{{Omega }_{p}^{r}left({A}_{0}, {A}_{r}right):{A}_{0}={A}_{r}Lleft(varphi lambda ,rright)pright},) ⋋ > 0, φ⋋,r is a contraction of the ideal Euler spline of order r, and L_(x)p : = sup {({Vert xVert }_{{L}_{p}[a,b]}:) a, b, ∈ R |x(t)| > 0, t ∈ (a,b)}. In particular, we obtain a sharp inequality of the form (1) in the classes ({Omega }_{p,lambda }^{r},) ⋋ > 0. We also prove the theorems on relationships for the Bojanov–Naidenov problems in the spaces of trigonometric polynomials and splines and establish the corresponding sharp Bernstein-type inequalities.

We study boundary-value problems for the Lyapunov operator-differential equation. By using the theory of Moore–Penrose pseudoinverse operators and its generalizations, we establish conditions for the existence of generalized solutions and propose algorithms for their construction.

In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present work, we provide an alternative proof of the existence of a positive solution in the general case. This proof is based on the application of a version of the M. A. Krasnosel’skii fixed-point theorem. The results are then extended to the equations with infinite matrices of the general type.

We prove the equivalence of two possible definitions of rotational interval exchange transformations: by the first definition, this is the first return map for the rotation of a circle onto a union of finitely many circle arcs, whereas by the second definition, this is an interval exchange with a scheme (in a sense of interval rearrangement ensemble) whose dual is also an interval exchange scheme.

Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under certain mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category (T ↓ B). As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.