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The first part of the paper presents a unique reduction of some general Riccati-type differential equation to the general Bernoulli-type equation. Furthermore four special theorems on reduction of some Riccati-type differential equations are considered in the second part of the paper. A number of examples are also presented to illustrate the considered issues. [ABSTRACT FROM AUTHOR]

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model "politics-economy" described by a nonlinear system of differential equations with a parameter and obtain conditions for the existence of a nonzero positive periodic solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider model problems of chemical kinetics that lead to singularly perturbed parabolic equations in domains with nonsmooth boundaries. The solvability of such problems depends on the properties of the inhomogeneity at corner points of the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities ∇ u q , ∂ t ∇ u q , and ∂ t 2 ∇ u q are considered. For the corresponding Cauchy problems, we obtain results on local-in-time unique solvability in the weak sense and on blow-up for a finite time. These three equations have the same critical exponent q = 3/2 since weak solutions behave differently for 1 < q ≤ 3/2 and for q > 3/2. By the method of nonlinear capacity proposed by S. I. Pokhozhaev, we obtain a priori estimates, which imply the absence of local and global weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

The paper introduces and explores the notions of ergodicity index and ergodicity exponent of a set of stochastic matrices. For the ergodicity exponent a sharp upper bound is obtained. A particular case of this bound is the well-known Paz bound. Also a connection between the ergodicity index and the Protasov–Voynov imprimitivity index is established. [ABSTRACT FROM AUTHOR]

The cyclicity index of a directed graph is defined as the least common multiple of the cyclicity indices of all its strongly connected components, and the cyclicity index of a strongly connected directed graph is equal to the greatest common divisor of the lengths of all its directed cycles. The cyclicity index of a tropical matrix is the cyclicity index of its critical subgraph, i.e., the subgraph of the adjacency graph consisting of all cycles with the largest average weight. This paper considers linear transformations of tropical matrices that preserve two values of the cyclicity index, 1 and 2. A complete characterization of such transformations is obtained. To this end, it is proved that the values 1 and 2 of the cyclicity index are preserved if and only if all its values are preserved. It is shown that there are mappings of another type that preserve only one value of the cyclicity index. [ABSTRACT FROM AUTHOR]

The paper suggests and illustrates a simple unified approach to deriving upper bounds for the dominant eigenvalues of the so-called PF matrices (or matrices with the Perron–Frobenius property) from those for the Perron root of a nonnegative matrix. [ABSTRACT FROM AUTHOR]

The paper suggests two general approaches to deriving upper bounds for ‖ A - 1 ‖ ∞ for some eventually H -matrices A. The approaches are illustrated by considering the classes of eventually SDD and eventually DSDD matrices, for which improvements of the known bounds are obtained. Also it is indicated that the approaches suggested are actually applicable to considerably larger matrix classes. [ABSTRACT FROM AUTHOR]

The paper considers the classes of GSDD1, GSDD 1 ∗ , and SD-SDD matrices, which contain the class of SDD (strictly diagonally dominant) matrices and are contained in the class of nonsingular H -matrices. New upper bounds on ‖ A - 1 ‖ ∞ for GSDD1, GSDD 1 ∗ , and SD-SDD matrices A, generalizing known upper bounds for S-SDD, SDD 1 ∗ , and GSDD1 matrices, are established and compared. [ABSTRACT FROM AUTHOR]

The paper studies quasilinear interpolation by minimal splines constructed on nonuniform grids with multiple nodes. Asymptotic representations for normalized splines are obtained. The sharpness of biorthogonal approximation and the order of accuracy of quasilinear interpolation with respect to the grid stepsize are established. Results of numerical experiments on approximating some test functions, which demonstrate the effect of choosing a generating vector function in constructing the corresponding minimal spline, are presented. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

The paper provides a characterization of linear operators preserving column majorization of the (0, 1)-vectors. In addition, such operators are characterized explicitly in the case where they are given by special matrices, namely, by (±1)-matrices of order not exceeding 10 or (0,±1)-matrices of order not exceeding 5. Several related combinatorial matrix theory results also are proved. [ABSTRACT FROM AUTHOR]

In this paper, the length of the group algebra of the direct product of a cyclic group and a cyclic p-group over a field of characteristic p is determined. A general lower bound for the length of a commutative group algebra is proved. It is shown that in the case of the direct product of a cyclic group and a cyclic p-group this bound is sharp. [ABSTRACT FROM AUTHOR]

In this paper, we define the class of demi ab-continuous operators on Banach lattices where a and b are given as n, w, o, ru, uo, un, and uaw. The relations between ab-continuous and demi ab-continuous operators are given. Moreover, the relations between the class of demi ab-continuous operators for different a, b convergence are investigated. [ABSTRACT FROM AUTHOR]

This paper considers the problem of constructing a classical Ford—Fulkerson network whose flow matches a stationary flow in an original resource network. We show that this problem has a solution. Some properties of stationary flows in resource networks are studied. We defined the concept of disengagement of a resource network at an arbitrary vertex to a classical Ford–Fulkerson network. This allows us to prove that the problem of constructing a classical network implementing a stationary flow in a resource network by a flow in a classical network has a solution, but it is not unique. We use the original definition of the functioning of the resource network, which is different from that used by Kuznetsov and Zhilyakova (2010), Doklady Mathematics 82(1):643-646, but equivalent to it. It has the advantage of being simpler and more precise. The necessary and sufficient conditions for the initial resource distribution to be stable at infinity are obtained. The examples illustrating both the definitions and the results are presented. [ABSTRACT FROM AUTHOR]

This paper investigates optimal control of a susceptible-infected-recovered (SIR) epidemic model using the dynamic programming approach. In fact, our study focuses on the modified model by incorporating it with a Bolza-type objective function. We use a certain refinement of Cauchy’s method of characteristics for stratified Hamilton-Jacobi equations to describe a large set of admissible trajectories and identify a domain on which the value function exists and is generated by a certain admissible control. The optimality is justified by using one of the well-known verification theorems, which provides an argument for sufficient optimality conditions. Through numerical tests, we demonstrate how adjustments to the infection and recovery rates, guided by the optimization framework provided by the guidance function, can significantly impact epidemic outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we prove vanishing theorems for the null space of the Laplacian admitting the Weizenböck decomposition and acting on the space of smooth sections of a Riemannian bundle and, in particular, the space of smooth sections of a bundle of symmetric tensors over complete and closed Riemannian manifolds. In addition, we give a general estimate for eigenvalues of the Laplacian admitting the Weizenböck decomposition and acting on the space of smooth sections of a bundle of symmetric tensors over closed Riemannian manifolds. [ABSTRACT FROM AUTHOR]

According to the Ray-Knight description, the Brownian local time in some conditional probability space is a diffusion process with respect to spatial variable. This diffusion has a local time. Thus the following definition of local time of the original Brownian local time seems to be natural: such a process is called the second-order Brownian local time. The paper studies the Laplace transform of the distribution of the Brownian local time of the second order. [ABSTRACT FROM AUTHOR]

For a convex lattice polytope P ⊂ R d of dimension d with vertices in Z d , denote by L(P) its discrete volume which is defined as the number of integer points inside P. The classical Ehrhart theorem states that given a positive integer n, the function L(nP) is a polynomial in n of degree d, whose leading coefficient is the volume of P. In particular, L(nP) approximates the volume of nP for large n. In convex geometry, one of the central notions generalizing the volume is the intrinsic volume. The main goal of the present paper is to introduce its discrete counterpart. In particular, it is proved that an analog of the Ehrhart theorem where volume is replaced by intrinsic volume holds true for it. A notion of Grassmann angle valuation generalizing both discrete volume and the solid-angle valuation introduced by Reeve and Macdonald is also introduced and studied. [ABSTRACT FROM AUTHOR]

The aim of the paper is to construct an analog of local time for the standard Wiener process multiplied by complex constant. [ABSTRACT FROM AUTHOR]

A family of measures on a given set of real algebraic numbers of arbitrary fixed degree is considered. The paper presents a method for constructing a sequence of measures that weakly converges to the Cauchy distribution. The method is based on the theory of random polynomials. [ABSTRACT FROM AUTHOR]

This paper introduces Dyck free groups that are geometric realizations of polygon cycles, the ratio of either two or three collinear points in the Desargues affine plane. A biproduct of this work is a concise presentation of Desargues affine planar cycles as a free group. In this paper, we include (1) the study of properties for the ratio of two and three points in a line in the Desargues affine plane. Also, we discuss the cases related to the “line-skew field” characteristic, when it is two and when it is different from two. (2) We have constructed the maps for ratio points-set for two and three points and have proved that these maps are bijections of the lines. (3) We observe that the set of ratio points (for two and for three points) with addition and multiplication of points forms a skew field and these skew fields are sub-skew fields of the “line-skew field” in a Desargues affine plane. (4) We also observe that every Dyck polygon cycle in the Desargues affine plane is the realization of a Dyck path cycle. (5) We prove that every Dyck polygon containing collinear ratio vertices has a free group presentation. [ABSTRACT FROM AUTHOR]

Let G be a group with identity e. Let R be a G-graded commutative ring. In this paper, we introduce and study the concept of graded uniformly n-ideal (in short, gr-u-n-ideal). Many results concerning gr-u-n-ideals are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the existence and uniqueness of solutions for a class of initial value problems in differential equations with maxima, involving the ψ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\psi$$\end{document}-Caputo fractional order on time scales. Our main theorem is established through the application of the contraction principle of Banach, employing a combination of Picard and weak Picard operator techniques to obtain data dependence results on the parameters. Furthermore, an illustrative example is provided to demonstrate the practical utility of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we have established the existence of conformal Ricci soliton as self similar solution of conformal Ricci flow and characterized the potential function f of almost conformal gradient Ricci soliton on a Sasakian manifold in terms of the non-dynamical scalar field p. We have found the necessary condition for the potential function f to be constant. Furthermore, a relation between λ and the potential function f has been established. Finally, we prove a sufficient condition for an almost conformal Ricci soliton to be an almost conformal gradient Ricci soliton and characterize it in terms of shrinking, steady or expanding soliton. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving ψ -Caputo fractional derivative. We establish the existence and uniqueness for a new class of impulsive fractional boundary value problems with nonlocal and boundary hybrid conditions. Our main theorem is demonstrated using fixed point theorems, including the Banach fixed point theorem and the Leray-Schauder alternative fixed point theorem. Some examples are also constructed to demonstrate the application of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider one-parameter regularly varying discrete distribution generated by Levy probability (RDLP). Some useful plots of the models are illustrated. Mathematically, to propose the RDLP model as a discrete distribution in bioinformatics, some common statistical facts such as unimodality, skewness to the right, upward/downward convexity, regular variation at infinity, and asymptotically constant slowly varying component are established for the model. Based on the Monte Carlo method and limited-memory Broyden–Fletcher–Goldfarb–Shanno for bound-constrained optimization, simulation studies are done to estimate bias, mean square error, and maximum likelihood estimation for the unknown parameter of the model. Some asymptotic properties and asymptotic expansions are given for useful functions. Moreover, we shall attempt to examine the RDLP model with a real count data set to show its application, and also compare it with two rival models. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and uniqueness of solution for a new class of ψ -Hilfer-type fractional differential equation with two fractional derivatives of different order. By making use of topological degree theory for condensing maps we established the existence result, for the reason that the previously mentioned theory softens the criteria of strong compactness by weakening it. To deal with the uniqueness result we use Banach's contraction principle. As application, we give an example to demonstrate our theoretical result. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

One of the most important subjects in statistics is the theory of estimation. In this paper, we consider the generalized Bayes shrinkage estimator of the mean vector for multivariate normal distribution with the unknown covariance matrix under balanced quadratic loss. Also, we find the minimax and admissible estimator of the mean vector based on the generalized Bayes estimator. We obtain Stein's unbiased risk estimator (SURE) threshold with a generalized Bayes shrinkage estimator and we find the wavelet shrinkage generalized Bayes SURE. Finally, we present a simulation study to test the validity of the generalized Bayes SURE and the concrete compressive strength dataset is given to test the efficiency of this estimator in denoising. [ABSTRACT FROM AUTHOR]

This is a preface for the special issue "Selected papers of the XIII Annual International Conference of the Georgian Mathematical Union. [ABSTRACT FROM AUTHOR]

An operator T acting on a separable complex Banach space B is said to be hypercyclic if there exists f ∈ B such that the orbit { T n f : n ∈ N } is dense in B . Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are hypercyclic, in the commutant of the Hardy backward shift. In this paper, we study some dynamical properties of operators X that λ -commute with the Hardy backward shift B, that is, B X = λ X B . [ABSTRACT FROM AUTHOR]

In this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev homeomorphism if f belongs to W loc 1 , 1 ∩ Hom (Ω ; Ω ′) , and f is a bi-Sobolev map if and only if f and f - 1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268 2017). [ABSTRACT FROM AUTHOR]

The paper describes the closure of infinitely differentiable compactly supported functions in certain second-order weighted Sobolev space depending on the degree of singularity of weight functions at zero and at infinity. [ABSTRACT FROM AUTHOR]

We study an L2-decay estimate of solutions to the dissipative nonlinear Schrödinger equation in a one-dimensional space. In the known results, the so-called strong dissipative condition is applied to the decay estimate for the initial data without size restriction. In this paper, we succeed in obtaining the L2-decay rate under a weaker dissipative condition. [ABSTRACT FROM AUTHOR]

Estimating of the efficiency of iterative methods is determined by results of computational experiments on solving typical test and applied problems. In this paper, we present information on the numerical implementation of typical gradient methods in comparison with modifications proposed by the authors for the approximate solution of certain applied problems from chemical technology and electrical engineering. [ABSTRACT FROM AUTHOR]

In this paper, we examine the optimal control problem for a hyperbolic system with differential constraints on the boundary, taking into account the delay. Controls are selected from the class of smooth functions that satisfy pointwise constraints. Problems of this type arise, in particular, in modeling the processes of population dynamics. The approach proposed in this paper is based on the use of "internal variations" of the control, which preserves the smoothness of the control function and ensures the fulfillment of pointwise constraints. We obtain an estimate of the state increment, prove a necessary optimality condition, and develop a scheme of an iterative method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new approach to optimization of nonlinear control systems based on the representation of optimality conditions and improvement of the control in the form of special fixed-point problems for control operators. We propose algorithms for approximate solution of optimal control problems based on iterative methods for finding fixed points. The effectiveness of the optimization methods proposed is illustrated by model and test problems. [ABSTRACT FROM AUTHOR]

The vector or pest control is essential to reduce the risk of vector-borne diseases or crop losses. Among the available biological control tools, the sterile insect technique (SIT) is one of the most promising. However, SIT-control campaigns must be carefully planned in advance in order to render desirable outcomes. In this paper, we propose SIT-control intervention programs that can avoid the real-time monitoring of the wild population and require to mass-rear a minimal overall number of sterile insects, in order to induce a local elimination of the wild population in the shortest time. Continuous-time release programs are obtained by applying an optimal control approach, and then laying the groundwork of more practical SIT-control programs consisting of periodic impulsive releases. [ABSTRACT FROM AUTHOR]

In this paper, we consider problems for partial differential equations that contain a small parameter in the principal part. We construct exact solutions of the original singularly perturbed problems and, passing to the limit, we state a correspondence between these solutions and solutions of the limit problems obtained from the original problems if the small parameter is set equal to zero. Examples given in this paper show that among singularly perturbed problems for partial differential equations, there are problems that possess characteristic properties of regularly perturbed problems and allow constructing asymptotic solutions by methods of the theory of regular perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we discuss properties of the singular value decomposition (SVD-decomposition) within the framework of the analysis of the numerical rank in ill-posed problems for determining frequency properties of discrete sequences consisting of trigonometric binomials. The properties of the numerical rank of the SVD-decomposition are indicated. We propose an algorithm for determining the frequencies of trigonometric binomials involved in the original function that forms a discrete sequence; this algorithm is based on estimates of the numerical rank. We obtain a stable criterion for estimating the numerical rank based on the arithmetic-mean estimators of the pseudo-null-space of the trajectory matrix. Also, we present the results of numerical experiments that demonstrate the consistency of the arithmetic-mean estimator of the pseudo-zero-space for the analysis of the spectrum of noisy sequences. [ABSTRACT FROM AUTHOR]

In this paper, we examine the synthesis problem for a distributed thermal control system in the case where a Fredholm integral operator is involved in the boundary-value problem. We use the method proposed by A. I. Egorov and develop it based on the Bellman scheme. Using the notions of a generalized solution of a boundary-value problem and the Fréchet differential for the Bellman functional, we obtain a partial integro-differential equation. For synthesizing a distributed optimal control, we propose a numerical algorithm. [ABSTRACT FROM AUTHOR]

This paper is a review of the results on the real-time optimal control problem for linear systems obtained by the Minsk school by mathematical methods of optimal control. We consider optimal control problems for dynamical objects, their deterministic mathematical models and perfect measurements of states, objects with disturbances and imperfect measurements of the observed input and output signals, the problem of optimal decentralized control of groups of interconnected dynamical objects, and application of real-time optimal control problems and the control principle to solving stabilization problems. [ABSTRACT FROM AUTHOR]

In this paper, we examine differential equations with nonclassical initial conditions and noninvertible operators in their principal parts. We find necessary and sufficient conditions for the existence of unbounded solutions with a pole of order p at points where the operator in the principal part of the differential equation is noninvertible. Based on the alternative Lyapunov–Schmidt method and Laurent expansions, we propose a two-stage method for constructing expansion coefficients of the solution in a neighborhood of a pole. We develop the techniques of skeleton chains of linear operators in Banach spaces and discuss its applications to the statement of initial conditions for differential equations. The results obtained develop the theory of degenerate differential equations. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

In this paper, the authors studied sufficient conditions to understand the oscillatory behavior of solutions of nonlinear neutral delay (and advanced) differential equation (E) with deviating arguments. Employing Banach fixed point theorem, the authors established the existence of bounded positive solutions of (E). The authors solved various examples using MATLAB software to understand the applications of the main theorems. Moreover, the authors analyzed the effect of delay terms on the behavior of solutions of (E). This paper has improved the results obtained in Basu [2] and Chatzarakis [11]. [ABSTRACT FROM AUTHOR]

In this paper, the Kampé de Fériet functions of arbitrary orders and their Euler-type integral representations are studied. The general form of the integral representations for a Kampé de Fériet function are proved. Conditions, under which these representations are expressed in terms of products of two gen-eralized hypergeometric functions, are found. Examples are identified in which the integral representation of the Kampé de Fériet function contains an elementary function or a known second-order hypergeometric function of two variables. [ABSTRACT FROM AUTHOR]