Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem

We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion

We study the problem of minimizing the functional \(I(\phi ) = \int\limits_\Omega {W(x,D\phi )dx}\) on a new class of mappings. We relax summability conditions for admissible deformations to φ ∈ W _n ¹ (Ω) and growth conditions on the integrand W(x, F). To compensate for that, we require the condition \(\frac{{\left| {D\phi (x)} \right|^n }} {{J(x,\phi )}} \leqslant M(x) \in L_s (\Omega )\), s > n ? 1, on the characteristic of distortion. On assuming that the integrand W(x, F) is polyconvex and coercive, we obtain an existence theorem for the problem of minimizing the functional I(φ) on a new family of admissible deformations A.     

Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra

An element a of a complex Banach algebra with unit \(1I\) and with standard conditions on the norm (‖ab‖ ? ‖a‖ · ‖b‖ and ‖\(1I\)‖ = 1) is said to be Hermitian if ‖e ^ita ‖ = 1 for any real number t. An element is said to be decomposable if it admits a representation of the form a + ib in which a and b are Hermitian. The decomposable elements form a Banach Lie algebra (with respect to the commutator). The Hermitian components are determined uniquely, and hence this Lie algebra has the natural involution a + ib = x → x* = a ? ib. One can readily see that ‖x*‖ ? 2‖x‖. Among other things, we prove that ‖ x*‖ ? γ‖x‖, where γ < 2. In fact, the situation is treated in more detail: the original problem is included in a continuous family parametrized by the numerical radius of the element. Finding the exact value of the constant γ is reduced to a variational problem in the theory of entire functions of exponential type. Approximately, γ is equal to 1.92 ± 0.04.     

Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem

We investigate the convergence rates for Tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the Neumann problem for the elliptic equation \({{-{\rm div}(q \nabla u) + au = f \;{\rm in}\; \Omega, q{\partial u}/{\partial n} = g}}\) on the boundary \({{\partial\Omega, \Omega \subset \mathbb{R}^d, d \geq 1}}\) , when u is imprecisely given by \({{{z^\delta} \in {H^1}(\Omega), \|u-z^\delta\|_{H^1(\Omega)}\le\delta, \delta > 0}}\). We regularize this problem by minimizing the strictly convex functional of (q, a)

$\begin{array}{lll}\int\limits_{\Omega}\left(q| \nabla (U(q,a)-z^{\delta})|^2 + a(U(q,a)-z^{\delta})^2\right)dx\\\quad+\rho\left(\|q-q^*\|^2_{L^2(\Omega)} + \|a-a^*\|^2_{L^2(\Omega)}\right)\end{array}$

over the admissible set K, where ρ > 0 is the regularization parameter and (q*, a*) is an a priori estimate of the true pair (q, a) which is identified, and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem. We obtain the convergence rate \({{{\mathcal {O}}(\sqrt{\delta})}}\), as δ → 0 and ρ ~ δ, for the regularized solutions under the simple and weak source condition

${\rm there\;is\;a\;function}\;w^* \in V^*\;{\rm such\;that}\;{U^\prime (q^ \dagger, a^\dagger)}^*w^* = (q^\dagger - q^*, a^\dagger - a^*)$

with \({{(q^\dagger, a^\dagger)}}\) being the (q*, a*)-minimum norm solution of the coefficient identification problem, U′(·, ·) the Fréchet derivative of U(·, ·), V the Sobolev space on which the boundary value problem is considered. Our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill-posed problems. Furthermore, some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces.

     

Regularity in the Local CR Embedding Problem

We consider a formally integrable, strictly pseudoconvex CR manifold M of hypersurface type, of dimension 2n?1≥7. Local CR, i.e., holomorphic, embeddings of M are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces C ^a (M), a∈R. If the structure of M is of class C ^m , m∈Z, 4≤m≤∞, we construct a local CR embedding near each point of M. This embedding is of class C ^a , for every a, 0≤a<m+(1/2). Our method is based on Henkin’s local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash–Moser argument due to the second author.     

On the k-polygonal numbers and the mean value of Dedekind sums

For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are a_n(k) = (2n+n(n?1)(k?2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(a_n(k)ā_m(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p ? 1, and give an interesting computational formula for it.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

On solutions of a boundary value problem for the biharmonic equation

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.     

Optimal combinations bounds of root-square and arithmetic means for Toader mean

We find the greatest value α ₁ and α ₂, and the least values β ₁ and β ₂, such that the double inequalities α ₁ S(a,b)?+?(1???α ₁) A(a,b)?T(a,b)?β ₁ S(a,b)?+?(1???β ₁) A(a,b) and \(S^{\alpha_{2}}(a,b)A^{1-\alpha_{2}}(a,b)< T(a,b)< S^{\beta_{2}}(a,b)A^{1-\beta_{2}}(a,b)\) hold for all a,b?>?0 with a?≠?b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b)?=?[(a ²?+?b ²)/2]^1/2, A(a,b)?=?(a?+?b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.     

A unified approach to singular problems arising in the membrane theory

We consider the singular boundary value problem \(({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,\) where f(t, x) is a given continuous function defined on the set (0, 1]×(0,∞) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n ∈ ?, n ? >2, and a ₀, a ₁, A are real constants such that a ₀ ∈ (0,1), whereas a ₁,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.     

On solutions of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains

We study the unique solvability of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x| ^a . Depending on the value of the parameter a, we prove uniqueness theorem or present exact formulas for the dimension of the solution space of the mixed Dirichlet-Steklov problem in the exterior of a compact set.     

Tong-type identity and the mean square of the error term for an extended Selberg class

In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

On Lehmer’s problem and Dedekind sums

Let p be an odd prime and c a fixed integer with (c, p) = 1. For each integer a with 1 ≤ a ≤ p ? 1, it is clear that there exists one and only one b with 0 ? b ? p ? 1 such that ab ≡ c (mod p). Let N(c, p) denote the number of all solutions of the congruence equation ab ≡ c (mod p) for 1 ? a, b ? p?1 in which a and \(\overline b \) are of opposite parity, where \(\overline b \) is defined by the congruence equation b\(\overline b \) ≡ 1 (mod p). The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L-functions to study the hybrid mean value problem involving N(c, p)?½φ(p) and the Dedekind sums S(c, p), and to establish a sharp asymptotic formula for it.     

Lie Triple Derivations on von Neumann Algebras

Let A be a von Neumann algebra with no central abelian projections. It is proved that if an additive map δ :A → A satisfies δ([[a, b], c]) = [[δ(a), b], c] + [[a, δ(b)], c] +[[a, b], δ(c)] for any a, b, c∈ A with ab = 0(resp. ab = P, where P is a fixed nontrivial projection in A), then there exist an additive derivation d from A into itself and an additive map f :A → ZA vanishing at every second commutator [[a, b], c] with ab = 0(resp.ab = P) such that δ(a) = d(a) + f(a) for any a∈ A.     

A sufficient condition for nilpotency of the commutator subgroup

Let G be a finite group with the property that if a and b are commutators of coprime orders, then |ab| = |a||b|. We show that G′ is nilpotent.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

The Existence of a Solution to the Multi-Index Problem

The multi-index problem can be described as minimizing the cost of moving a set of p different commodities (k = 1, 2,..., p) from n origins (i = 1, 2,..., n) to m destinations (j = 1, 2,..., m). The equations then give rise to the conditions on the amount of the various types of combination that is available and required. Alternatively, the same set of restrictions arise when a single commodity has to be moved by different methods, e.g. road, rail, sea, canal, air, etc. Similarly the use of intermediate depots may require the use of a multi-index formulation. A third special type of problem where the method can be used is the capacitated transportation problem (each variable has an upper bound).