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In this paper, finite element methods for the problems of wave propagation in a fluid-saturated porous medium are discussed. The medium is composed of a porous elastic solid (soil, rock, etc.) saturated by a compressible viscous fluid (oil, water, etc.), and the fluid may flow relatively to the solid. Biot's lowfrequency dynamic equations are chosen to describe the problems mentioned above, with stress-given boundary conditions, ABGs (Absorbing Boundary Conditions) on artificial boundaries and conditions on interfaces between the fluid-saturated porous medium and elastic solids. In the paper, a new kind of discrete ABCs is presented, and a discrete-time Galerkin method are utilized for obtaining approximate solutions. The numerical results show that they both are effective. Two dilatational waves (fast wave P1 and slow wave P2) and one rotational wave (S wave) are clearly visible in the figures of computational results, which coincide with theoretical analysis very well. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for an elliptic equation in a plane domain. The problem is ill-posed. Using the method of quasi-reversibility, an approximation to the exact solution is given. Using Carleman's inequatily, we derive a sharp error estimate. [ABSTRACT FROM AUTHOR]

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\mathcal {J}(u)=\int_{\partial\Omega}f(x)u\,\mathrm {d}\mathcal {H}^{N-1}$ over some admissible class of loads f where u is the (unique) solution to the problem −Δ p u+| u| p−2 u=0 in Ω with | ∇ u| p−2 u ν= f on ∂Ω. [ABSTRACT FROM AUTHOR]

We found a minor error in the proof of paper “Universal Alignment Probability Revisited” by S.Y. Lin and Y.C. Ho (J. Optim. Theory Appl. 113(2):399–407, ). In this note, we give a counterexample and explain the reason. We also show that the conclusion of that paper is still correct despite this minor error. A new proof of the conclusion is given. [ABSTRACT FROM AUTHOR]

Abstract  The seminal paper by Kasner (Am. J. Math. 43, 217–221, 1921) is reprinted here, together with an editorial comment on its lasting scientific relevance, and a biography of the author. [ABSTRACT FROM AUTHOR]

This paper studies a class of transport equations arising from stochastic models in congestion control. This class contains two cases of loss models as particular cases: the rate-independent case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where it depends on it. This class of equations covers both the case of persistent and of non-persistent flows. For the first time, we give a direct proof of the fact that there is a unique density solving the associated differential equation. This density and its mean value are provided as closed form expressions. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds ( M, g0) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when ( M, g0) is not conformally equivalent to the unit sphere S n and that the set of all such metrics is compact. When ( M, g0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlocal ∞-Laplacian type diffusion equation obtained as the limit as p → ∞ to the nonlocal analogous to the p-Laplacian evolution, We prove exist ence and uniqueness of a limit solution that verifies an equation governed by the subdifferential of a convex energy functional associated to the indicator function of the set $${K = \{ u \in L^2(\mathbb{R}^N) \, : \, | u(x) - u(y)| \le 1, \mbox{ when } x-y \in {\rm supp} (J)\}}$$ . We also find some explicit examples of solutions to the limit equation. If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞(0, T; L2 ( Ω)) to the limit solution of the local evolutions of the p-Laplacian, v t = Δ p v. This last limit problem has been proposed as a model to describe the formation of a sandpile. Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large. Finally, we give an interpretation of the limit problem in terms of Monge–Kantorovich mass transport theory. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy. [ABSTRACT FROM AUTHOR]

Using the criterion of this paper, we solve the substitution problem and obtain an algorithm for determining the solvability of logical equations in the modal logic S4. α N . Another corollary of the criterion is the solvability of the corresponding quasiequational theory of the free modal algebra whose signature is enriched with constants for the free generators. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of solutions for two–dimensional Cahn–Hilliard equation with non–constant mobility. Basing on the L p type estimates and Schauder type estimates, we prove the global existence of classical solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [Σ( V× W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. [ABSTRACT FROM AUTHOR]

This paper continues the recent study of an Allen-Cahn model PDE [1] by eliminating a strong spatial reversibility condition and by weakening certain nondegeneracy conditions on families of basic heteroclinic solutions, enabling us to obtain multibump solutions in a much more general setting. As in [1], novel minimization arguments play a key role in finding solutions. [ABSTRACT FROM AUTHOR]

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples. [ABSTRACT FROM AUTHOR]

We consider a parabolic equation with a drift term Δu+b∇u-ut=0. Under the condition divb=0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected. [ABSTRACT FROM AUTHOR]

Nontransitive additive conjoint measurement for a binary relation » on a set X1 ⊗ X2 ⊗ ··· ⊗ Xn of n-tuples (x1, …, xn), (y1, …, yn), … is concerned with the representation (x1, …, xn) » (y1, …, yn) ⇔ [This symbol cannot be presented in UNICODE format] φi(xi, yi) »0, where φi is a skew symmetric real valued function on Xi ⊗ Xi. The representation is said to be unique if, whenever (φ1, …, φn) and (φ′1, …, φ′n) satisfy it, there is a positive constant λ such that φ′i = λφi for all i. This paper investigates unique representation for nontransitive additive conjoint measurement when every Xi is finite. It begins with background on measurement theory, followed by conditions for uniqueness that are based on equations ∑φi(xi, yi) = 0 that correspond to the symmetric complement of ». We then examine aspects of sets of unique solutions for n = 2, for arbitrary n with ¦Xi¦ = 2 for all i, and for n = 3. Various combinatorial and number-theoretic arguments are used to derive results that count and characterize sets of unique solutions. [ABSTRACT FROM AUTHOR]

We prove the boundary Harnack principle for ratios of solutions u/ v of non-divergence second order elliptic equations Lu = a D u + b D u = 0 in a bounded domain Ω ⊂ $ {\mathbb R} $. We assume that b ∈ L(Ω) and Ω is a twisted Hölder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Hölder regularity of u/ v for uniform domains. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

The article discusses an inverse problem of viscoelasticity. It considers two settings of the problem of specifying a function which differ in form and information given by the solution of a direct problem. It notes that in one setting, it is relegated with many observation questions. It points out the determination of a kernel function under assumption similar to the viscolelasticity system equations. Furthermore, the assumption considerably simplifies analysis on it and does not justified from the point of view of mathematical applications.

The article investigates some delay integral inequalities on time scales which provide explicit bounds on unknown functions. Researchers have used Gronwall's inequality and comparison theorem. Results of the study unify and extend some delay integral inequalities and their corresponding discrete analogues. It was found that the numerical solution agrees with the analytical solution for some discrete inequalities. The inequalities given can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales.

Using the criterion of this paper, we solve the substitution problem and obtain an algorithm for determining the solvability of logical equations in the modal logic S4. α N . Another corollary of the criterion is the solvability of the corresponding quasiequational theory of the free modal algebra whose signature is enriched with constants for the free generators. [ABSTRACT FROM AUTHOR]

We define the topological degree for a class of operators and use it to find generalized zeros of operators. Using these results we get the existence of solutions for singular nonlinear elliptic equations. [ABSTRACT FROM AUTHOR]

Many characterizations of semipreconnected (=β-connected) spaces are obtained by Aho and Nieminen [2]. In this paper, we obtain further characterizations of semipreconnected spaces. We introduce β-separated and βs-connected sets and by using these sets we obtain several properties of β-connected subspaces. [ABSTRACT FROM AUTHOR]

The article presents an abstract of the article on measurement models for content analysis. This article conceptualizes in terms of structural equation models several methodological problems in computer-aided content analysis. The principal frame of reference is the analysis of covariance structures as developed by statistician Karl Joreskog. The article suggests how the estimation of specific structural-measurement models can be used to study the reliabilities, validates and other properties of various content-analysis models.

The article presents the formulas for the Lyapunov dimension of generalized Lorenz system attractors. It develops new classes of Lyapunov functions used in obtaining Lyapunov dimension formulas for the attractors. It points out that Lorenz system's attractor originates from three equilibria. It presents and explains the formula for the generalized Lorenz system's dimension. The equations, formulas, theorems, explanations and solutions used in analyzing the attractors of the system are also provided.

For an order-preserving map f : L → Q between two complete lattices L and Q, there exists a largest residuated map ρ under f, which is called the residuated approximation of f. Andreka, Greechie, and Strecker introduced the notion of the shadow σ of f Iterations of the shadow are called the umbral mappings. The umbral mappings form a decreasing net that converges to the residuated approximation ρ of f. The umbral number u of f is the smallest ordinal number α such that the equation $${\sigma^{(\alpha)}_{f} = \rho_{f}}$$ holds. In order to speed up the computation of the umbral number u of f and find some relation between the structure of L and u, we present the concept of the order skeleton of a lattice $${L, \tilde{L} = L/\sim}$$, determined by a certain congruence relation ~ on L where each equivalence class [ x] is the maximal autonomous chain containing x. If [ x] is finite for each $${x \in L}$$, then $${L_{o} := \{ \Lambda [x]\,|\, x \in L \}}$$ is a join-subcomplete sub-semilattice of L isomorphic to the order skeleton $${\tilde{L}}$$ of L; for every order-preserving mapping f : L → Q from such a lattice L to a complete lattice Q, we define f : L → Q by $${f_{o} := f|_{{L}_{o}}}$$ and prove that $${u_{f} = u_{{f}_{o}}}$$. For a lattice L with no infinite chains, the order skeleton $${\tilde{L}}$$ of L is distributive if and only if the shadow σ of f is residuated for every complete lattice Q and every mapping f : L → Q. Related topics are discussed. [ABSTRACT FROM AUTHOR]

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1, K≫1, s>1, we construct smooth initial data u0 with $\|u_{0}\|_{{H}^{s}}<\delta$, so that the corresponding time evolution u satisfies $\|u(T)\|_{{H}^{s}}>K$ at some time T. This growth occurs despite the Hamiltonian’s bound on $\|u(t)\|_{\dot{H}^{1}}$ and despite the conservation of the quantity $\|u(t)\|_{L^{2}}$. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982. [ABSTRACT FROM AUTHOR]

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of ℝ n. Our analysis is based on the framework of weak- L p spaces. [ABSTRACT FROM AUTHOR]

We study the evolution of a weakly convex surface $${\Sigma_0}$$ in $${\mathbb {R}^3}$$ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class C k, γ, for some $${k\in \mathbb{N}}$$ and 0 < γ ≤ 1, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for t > 0 does not depend on the regularity of the initial data. [ABSTRACT FROM AUTHOR]

The notions of (ϵ, ϵ∨ q)-fuzzy BCI-positive implicative (resp., BCI-implicative, BCI-commutative) ideals in BCI-algebras are introduced, and related properties are investigated. Some characterization theorems of these generalized fuzzy ideals are derived. The relationship among these generalized fuzzy ideals of BCI-algebras is considered. We also consider the concept of implication-based fuzzy implicative ideals in BCI-algebras. In particular, we discuss the implication operators in Lukasiewicz system of continuous-valued logic. [ABSTRACT FROM AUTHOR]

We use a version of the Fair-Luke algorithm to find the Padé approximate solutions of the Painlevé I and II equations. We find the distributions of poles for the well-known Ablowitz-Segur and Hastings-McLeod solutions of the Painlevé II equation. We show that the Boutroux tritronquée solution of the Painleé I equation has poles only in the critical sector of the complex plane. The algorithm allows checking other analytic properties of the Painlevé transcendents, such as the asymptotic behavior at infinity in the complex plane. [ABSTRACT FROM AUTHOR]

Zero-sum ergodic semi-Markov games with weakly continuous transition probabilities and lower semicontinuous, possibly unbounded, payoff functions are studied. Two payoff criteria are considered: the ratio average and the time average. The main result concerns the existence of a lower semicontinuous solution to the optimality equation and its proof is based on a fixed-point argument. Moreover, it is shown that the ratio average as well as the time average payoff stochastic games have the same value. In addition, one player possesses an ε-optimal stationary strategy ( ε>0), whereas the other has an optimal stationary strategy. [ABSTRACT FROM AUTHOR]

We find the inverse and direct recursion operator for the intrinsic generalized sine-Gordon equation in any number n > 2 of independent variables. Among the flows generated by the direct operator we identify a higher-dimensional analogue of the pmKdV equation. [ABSTRACT FROM AUTHOR]

Abstract.  We prove that for any variety $$\mathcal{V}$$, the existence of an edge-term (defined in [1]) and Jónsson terms is equivalent to the existence of a near-unanimity term. We also characterize the idempotent Maltsev conditions which are defined by a system of linear absorption equations and which imply congruence distributivity. [ABSTRACT FROM AUTHOR]

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

The article examines the regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure. An interpolation inequality was used to refine when the solution is smooth. Deriving a priori estimates for smooth solutions is measured through multiplying the regularity condition of weak solutions to the Navier-Stokes equations by the interpolation inequality. Since the Leray weak solution satisfies the energy inequality, Serrin's uniqueness criterion can be applied.

The article presents a study which examines the stability of a new pexider-type functional equation. This functional equation is f1(x+y+z)+f2(x-y)+f3(z-x)f4(x-y-z)-f5(x+y)-f6(x+z) = 0, a mixed of a quadratic and an additive functional equations. It established the generalized Hyers-Ulam-Rassias stability for new pexider-type functional equation on the punctured domain V\ {0} and obtain its general solution from the stability results. It also let V and X be normed space and Banach space, respectively. Results of the applications and stability of the equation are also given.

Abstract   We define canonical representations R λ , , for the Lobachevsky space ℒ=G/K of dimension n−1 where G=SO0(n−1,1), K=SO(n−1), as the restriction to G of maximal degenerate series representations of the overgroup . We determine explicitly the interaction of Lie operators of with operators intertwining canonical representations and representations of G associated with a cone. [ABSTRACT FROM AUTHOR]

We construct a new loop algebra $\widetilde{A_{3}}$ , which is used to set up an isospectral problem. Then a new integrable couplings of the generalized AKNS hierarchy is derived, which possesses bi-Hamiltonian structure and contains an arbitrary spatial function. As its reduction, we gain the integrable couplings of the Schrödinger equation. Furthermore, many conserved quantities of the integrable couplings are obtained. [ABSTRACT FROM AUTHOR]

We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: $${-div(|x|^\alpha |\nabla u|^{p-2}\nabla u)=|x|^\beta u^{p(\alpha,\beta)-1}, u(x) > 0,\,x\in\Omega\subset\mathbb{R}^N} (N\geq 3), 1 p}$$ . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in rather general, possibly degenerate or singular settings. [ABSTRACT FROM AUTHOR]

We introduce new proof systems G[ β] and G ext[ β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization ( Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction [ABSTRACT FROM AUTHOR]

Constitutive equations that describe the experimentally observed failure waves are proposed to model inelastic strains of brittle materials. The complete system of equations is hyperbolic, each equation of this system has divergent form. The model is based on the assumption that continual failure is the process of transition from an intact state to a “fully damaged” state described by the kinetics of the order parameter. The structure of stationary traveling compressive waves is analyzed using a simplified model. It is shown that in a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave. [ABSTRACT FROM AUTHOR]

The article examines the study on the Cauchy problem for a fourth-order equation. The investigation of the Cauchy problem for general hyperbolic equations and systems was initiated. Furthermore, mixed problems for a fourth-order equation with a biwave operator in a cylindrical domain were considered.

We consider the problem of testing whether a given system of equations over a fixed finite semigroup S has a solution. For the case where S is a monoid, we prove that the problem is computable in polynomial time when S is commutative and is the union of its subgroups but is NP-complete otherwise. When S is a monoid or a regular semigroup, we obtain similar dichotomies for the restricted version of the problem where no variable occurs on the right-hand side of each equation. We stress connections between these problems and constraint satisfaction problems. In particular, for any finite domain D and any finite set of relations Γ over D, we construct a finite semigroup SΓ such that CSP(Γ) is polynomial-time equivalent to the satifiability problem for systems of equations over SΓ. [ABSTRACT FROM AUTHOR]

The article presents a discussion that focuses on the Kantorovich and the Durrmeyer-type modifications of the generalized favard operators. The discussion is also said to be aimed at proving an inverse approximation theorem for functions f. An analogous inequality for the rth weighted modulus of smoothness of the function f is also presented. The theorems, equations, and calculations associated with such mathematical concept are also included in the discussion. Mathematical proofs linked to such equations are also provided.