In this paper, there are defined the anisotropic local Morrey-type spaces LM(p) over bar,(q) over bar(lambda) over bar and the anisotropic generalized Morrey-type spaces M-(p) over bar,(q) over bar((lambda) over bar), where (p) over bar, (q) over bar, and (lambda) over bar are vectors. The spaces LM(p) over bar,(q) over bar(lambda) over bar allow relaxation of the conditions on the parameter (lambda) over bar, namely, the components of the given vector can take any real value, i.e., -infinity < lambda(i) < infinity, i = (1, d) over bar, in contrast to previously studied spaces. The embedding properties of the defined spaces are investigated. Additionally, an anisotropic interpolation method is considered, which allows the study of the interpolation properties of these spaces.
We prove Adams theorem for the Riesz potential I gamma alpha (B-Riesz potential) in the total Morrey spaces Lp,(lambda,& micro;),gamma (total B-Morrey spaces), associated with the Laplace-Bessel differential operator AB. More precisely, we obtain necessary and sufficient conditions for the operator I gamma alpha to be bounded from the total B-Morrey space Lp,(lambda,& micro;),gamma to the total B-Morrey space Lq,(lambda,& micro;),gamma and from the total B-Morrey space L1,(lambda,& micro;),gamma to the weak total B-Morrey space W Lq,(lambda,& micro;),gamma.
In 1992 V.I. Burenkov proved some Hardy's inequalities with sharp constants in Lebesgue spaces for monotone functions for 0 < p < 1. Later R.A. Bandaliev established analogous estimates in weighted variable exponent Lebesgue spaces for monotone functions for 0 < p(x) <= q(x) < 1. In 2020 A. Senouci and A. Zanou generalized the results of R.A. Bandaliev for quasi-monotone functions. The aim of this paper is to obtain some generalizations of the previous results cited above for weighted Hardy operators by introducing a parameter alpha is an element of R. Moreover, by using the quasi-norms kfkBT Lp(x)(Omega) introduced by V.I. Burenkov and T.V. Tararykova, we obtain an improvement of constants in our previous estimates.
In this paper, we give several inequalities for the tensor product of two operators involving the p-numerical radius and the Schatten p-norms.
For a compact Hausdorff space X, we consider the space IB(X) of all idempotent probability measures on X, which are defined as set-functions on the sigma-algebra of all Borel subsets of X, and also the space IUSC(X) of all normalized max-plus linear functionals on the linear space of all upper semicontinuous functions on X, equipped with idempotent operations. In the main result it is established that a max-plus version of the Holder inequality holds on the space of upper semicontinuous functions.
We present a new method for dimension reduction that combines unsupervised dimension reduction (UDR) with sufficient dimension reduction (SDR). In unsupervised dimension reduction the goal is to find a low-dimensional linear subspace that approximates the support of a data distribution. If data is supervised, then in sufficient dimension reduction the goal is to find a low-dimensional linear subspace, called the effective subspace, such that the projection of an input vector onto that subspace maximally captures information on correlations between an input and an output. The objective that we suggest to minimize consists of two parts. The first one is responsible for the UDR part, it forces a low-dimensional probabilistic measure & micro; to approximate a distribution over inputs. The second one is responsible for the SDR part, it forces a regression function f to be consistent with supervised data. Additionally, we require the support of & micro; and the effective subspace of f to be equal. In this hybrid setting we solve two problems, UDR and SDR, so that the UDR term serves as a regularizer of the SDR term. We reformulate the problem as an optimization task of finding a k-dimensional linear subspace S and a pair of complex measures (& micro;, & micro;') supported in S. Instead of optimizing over complex measures, we suggest minimizing over ordinary functions (g1, g2) but with an additional term R that penalizes a distortion of the common support of g1, g2 from a k-dimensional linear subspace. The algorithm that we develop can be formulated for functions (g1, g2) as well as for their inverse Fourier transforms. Eventually, we report results of numerical experiments on well-known datasets.
In this paper, the main theorem is proved by establishing the reducibility to an equivalent multiperiodic linear system with a differentiation operator directed along the diagonal of the independent variables space. It is shown that helical lines on a circular cylindrical surface form periodic characteristics of the operator. The reducibility of a multiperiodic system is examined near a helix starting from the initial point of the phase circle, following the classical approach used for periodic systems. A monodromy matrix is introduced, which remains constant along the first integrals of the characteristic equations and possesses the properties of smoothness and multiperiodicity. The existence of localised positive eigenvalues consistent with the properties of this matrix is demonstrated. It is assumed that at the initial point of the phase circle, the monodromy matrix attains the maximal number of distinct eigenvalues. Their localisation on the cylindrical surface is established using the Gershgorin method.
The hypothetical possibility of building a quantum computer in the near future has forced a revision of the foundations of modern cryptography. The fact is that many difficult algorithmic problems, such as the discrete logarithm, factoring a (large) natural number into prime factors, etc., on the complexity of which many cryptographic protocols are based these days, have turned out to be relatively easy to solve using quantum algorithms. Intensive research is currently underway to find problems that are difficult even for a quantum computer and have potential applications for cryptographic protocols. Our article contains notes related to the so-called generalized Gauss algorithm, which calculates the reduced basis of a two-dimensional lattice [8], [2]. Note that researchers are increasingly putting forward difficult algorithmic problems from lattice theory as candidates for the foundation of post-quantum cryptography. The majority of algorithmic problems related to lattice reduction become NP-hard as the lattice dimension increases 3, 1. Fundamental problems such as the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), and Bounded Distance Decoding (BDD) are conjectured to remain hard even for quantum algorithms 4, 6. Although the generalized Gauss reduction algorithm applies to two-dimensional lattices, where exact analysis is feasible (dimensions 3 and 4 are studied in 171, 51), understanding such low-dimensional reductions provides important insights into the structure and complexity of lattice-based cryptographic constructions.
For the real part of the Cauchy-type integral that is known to be the logarithmic potential of the double layer, a necessary and sufficient condition for the continuous extension to the Ahlfors-regular boundary is established. Sufficient conditions involving subclasses of Ahlfors-regular curves are also considered. Illustrative examples are presented.
We present an inventory model where a manufacturer (firm) uses for "production" a "commodity" (resource), which is consumed with the unit intensity. The price of the commodity follows a stochastic process, modelled by a continuous time Markov chain with a finite number of states and known transition rates. The firm can buy this commodity at the current price or use "stored" one. The storage cost is proportional to the storage level. The goal of the firm is to minimize the long-run average cost functional. We prove the existence of a canonical triple with an optimal threshold strategy, present an algorithm for constructing optimal thresholds and the optimal value of the functional, and discuss issues of uniqueness.
The paper studies the spread of waves along the star graph. The continuation of the initial data from the graph edges for the entire numerical axis allows to represent an analogue of the d'Alembert formula for waves on the star graph. At the same time, the continuation of the initial data is closely related to the continuation of the system of its eigenfunctions of the Sturm-Liouville problem originally defined on the star graph. The continuation of the eigenfunctions defined on the star graph is based on the continuation of the initial data of the mixed problem for the wave equation. The indicated continuation of the initial data of the mixed problem was proposed by B.M. Levitan.
The objective of this paper is to establish sufficient conditions for the boundedness of the generalized Riemann-Liouville operator in local Morrey-type spaces with mixed quasi-norms on a parallelepiped and to obtain sharp estimates of the norm of this operator with respect to the lengths of the edges of this parallelepiped.
The manuscript introduces an innovative framework for establishing the existence of infinitely many nontrivial periodic solutions within a class of differential equations characterized by a piecewise alternately advanced and retarded argument. It comprehensively delineates the essential criteria required for the existence of these solutions and provides detailed procedures for their determination. Additionally, the study incorporates illustrative examples, including cases with infinitely many solutions, to demonstrate the effectiveness and applicability of the proposed approach.
In this paper we present and prove some new results concerning approximation properties of T means with respect to the Vilenkin system in Lebesgue spaces for any 1 <= p < infinity. As applications, we obtain extensions of some known approximation inequalities.
In the paper there are investigated the oscillatory properties of a 2nth order differential equation and the spectral properties of a 2nth order differential operator. These properties are established using the variational method, which relies on verifying a specific nth order differential inequality. Here, the coefficients of both the equation and the operator are the weights in this inequality. Furthermore, the characterization of the inequality occurs when the weights satisfy conditions, ensuring the existence of a certain combination of boundary values at infinity and at zero for the function involved in this inequality.
In this article, we provide various uniqueness results for inverse problems of weighted differential operator with point delta-interaction. In terms of the method of spectral mappings, we also offer step by step strategies for finding their potential and boundary conditions basing either on the Weyl function, on spectral data, or on two spectra.
We study a one-dimensional Stefan type problem which models the behavior of electromagnetic fields and heat transfer in closed electrical contacts that arises, when an instantaneous explosion of the micro-asperity occurs. This model involves vaporization, liquid and solid zones, in which the temperature satisfies a generalized heat equation with the Thomson effect. Accounting for the nonlinear thermal coefficient, the model also incorporates temperature-dependent electrical conductivity. By employing a similarity transformation, the Stefan-type problem is reduced to a system of coupled nonlinear integral equations. The existence of a solution is established using the fixed point theory in Banach spaces.
In this paper, we study the asymptotic behaviour of fundamental systems of solutions to the Sturm-Liouville equation with rapidly oscillating potentials in a two-dimensional vector-function space. We consider different cases in which the coefficients do not satisfy the regularity conditions. Additionally, we investigate the asymptotic behaviour of solutions in resonance cases.
The main objective of the work is to identify the relationship between evolution equations with potential operators and geometries of related configuration spaces of the given systems. Using the Hamilton principle, a wide class of such equations is derived. Their structural analysis is carried out, containing operator analogues of the Christoffel symbols of both the 1st and 2nd kind. It is shown that the study of the obtained evolution equations can be associated, in general, with an extended configuration space, the metric of which is determined by the kinetic energy of the given system.
Subject 17A32, 17B30, Abstract. We show that any local 1/2-derivation on solvable Leibniz algebras with model or abelian nilradicals, whose dimensions of complementary spaces are maximal, is a 1/2-derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimensional complementary spaces are 1/2-derivations. Moreover, a similar problem concerning 2-local 1/2-derivations of such algebras is investigated.
