Boundary displacement control at one end of a string in the presence of a nonlocal boundary condition of one of four types,and its optimization

In the present paper, we exhaustively solve the problem of boundary control by the displacement u(0, t) = µ(t) at the end x = 0 of the string in the presence of a model nonlocal boundary condition of one of four types relating the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point \(\mathop x\limits^ \circ\).     

Boundary control by an elastic force at one endpoint of the string in the presence of a model nonlocal boundary condition of one of four types,and its optimization

In the present paper, in terms of a generalized solution of the wave equation, we perform an exhaustive study of the problem on the boundary control by an elastic force u _x (0, t) = µ(t) at one endpoint x = 0 of a string in the presence of a model nonlocal boundary condition of one of four types relating (with the sign “+” or “?”) the values of the displacement u(x, t) or its derivative u _x (x, t) at the boundary point x = l of the string to their values at some interior point \(\mathop x\limits^ \circ \) of the string (0 < \(\mathop x\limits^ \circ \) < l). We prove necessary and sufficient conditions for the existence of such boundary controls. Under these conditions, we optimize the controls by minimizing the boundary energy integral and then write out the optimal boundary controls in closed analytic form.     

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.     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

Flat coordinates for Saito Frobenius manifolds and string theory

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type A _n . We also discuss a possible generalization of our proposed approach to SU(N) _k /(SU(N) _k+1 × U(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.     

Matrix Mechanics of the Relativistic Point Particle and String in Clifford Space

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two N × N Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw–Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the \({SL(2.\mathbb{C})}\) charge and space-time momentum of the quantized string satisfy the Poincaré algebra.     

Exploring the vacuum structure near identity-based solutions

We explore the vacuum structure in the bosonic open string field theory expanded near an identity-based solution parameterized by a (≥ ?1/2). Analyzing the expanded theory using the level-truncation approximation up to the level 20, we find that the theory has the tachyon vacuum solution for a ≥ ?1/2. We also find that at a = ?1/2, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the hypothesis that the identity-based solution is a trivial pure gauge configuration for a > ?1/2, but it can be regarded as the tachyon vacuum solution at a = ?1/2.     

Kolmogorov complexity and cryptography

We consider (in the framework of algorithmic information theory) questions of the following type: construct a message that contains different amounts of information for recipients that have (or do not have) certain a priori information. Assume, for example, that a recipient knows some string a and we want to send him some information that allows him to reconstruct some string b (using a). On the other hand, this information alone should not allow the eavesdropper (who does not know a) to reconstruct b. This is indeed possible (if the strings a and b are not too simple). Then we consider more complicated versions of this question. What if the eavesdropper knows some string c? How long should our message be? We provide some conditions that guarantee the existence of a polynomial-size message; we show then that without these conditions this is not always possible.     

On the Automorphism Group of a Smooth Schubert Variety

Let G be a simple algebraic group of adjoint type over the field \(\mathbb {C}\) of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and let X(w) be the Schubert variety in G/B corresponding to w. Let α ₀ denote the highest root of G with respect to T and B. Let P be the stabiliser of X(w) in G. In this paper, we prove that if G is simply laced and X(w) is smooth, then the connected component of the automorphism group of X(w) containing the identity automorphism equals P if and only if w ^?1(α ₀) is a negative root (see Theorem 4.2). We prove a partial result in the non simply laced case (see Theorem 6.6).     

On the complexity of the gradient of a rational function

The Baur-Strassen method implies L(?f) ? 4L(f), where L(f) is the complexity of computing a rational function f by arithmetic circuits, and ?f is the gradient of f. We show that L(? f) ? 3L(f) + n, where n is the number of variables in f. In addition, the depth of a circuit for the gradient is estimated.     

Optimal boundary control by a displacement in <Emphasis Type="Italic">W</Emphasis>′<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of a string with free end

We consider an optimal boundary control of a string with free end by a displacement of the other end in W′ _p (Q, T). For p ≠ 2, we prove that the optimal control depends on the initial and terminal conditions nonlinearly.     

On the number of edges in a uniform hypergraph with a range of permitted intersections

The paper studies the quantity p(n, k, t ₁, t ₂) equal to the maximum number of edges in a k-uniform hypergraph with the property that the size of the intersection of any two edges lies in the interval [t ₁, t ₂]. Previously known upper and lower bounds are given. New bounds for p(n, k, t ₁, t ₂) are obtained, and the relationship between these bounds and known estimates is studied. For some parameter values, the exact values of p(n, k, t ₁, t ₂) are explicitly calculated.     

On the Stability of a k-Cubic Functional Equation in Intuitionistic Fuzzy n-Normed Spaces

In this paper, we investigate some stability results concerning the k-cubic functional equation f(kx + y) + f(kx?y) = kf(x + y) + kf(x?y) + 2k(k²?1)f(x) in the intuitionistic fuzzy n-normed spaces.     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

Calculating the Isotropic Subspace of a Symmetric Quasi-Definite Matrix

Solutions to the sesquilinear matrix equation X*DX + AX + X*B + C = 0, where all matrices are of size n × n, are put in correspondence with n-dimensional neutral (or isotropic) subspaces of the associated matrix M of order 2n. A way of constructing such subspaces is proposed for when M is a symmetric quasi-definite matrix of the (n, n) type.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

On the norm property of the Hilbert symbol for polynomial formal modules in a multidimensional local field

In a two-dimensional local field K containing the pth root of unity, a polynomial formal group F _c (X, Y) = X + Y + cXY acting on the maximal ideal M of the ring of integers б _K and a constructive Hilbert pairing {·, ·} _c : K ₂(K) × F _c (M) → <ξ> _c , where <ξ> _c is the module of roots of [p] _c (pth degree isogeny of F _c ) with respect to formal summation are considered. For the extension of two-dimensional local fields L/K, a norm map of Milnor groups Norm: K ₂(L) → K ₂(K) is considered. Its images are called norms in K ₂(L). The main finding of this study is that the norm property of pairing {·, ·}c: {x,β} _c : = 0 ? x is a norm in K ₂(K([p] _c ^-1 (β))), where [p] _c ^-1 (β) are the roots of the equation [p] _c = β, is checked constructively.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

Probabilities of high extremes for a Gaussian stationary process in a random environment

Let ξ(t) be a zero-mean stationary Gaussian process with the covariance function r(t) of Pickands type, i.e., r(t) = 1 ? |t| ^α + o(|t| ^α ), t → 0, 0 < α ≤ 2, and η(t), ζ(t) be periodic random processes. The exact asymptotic behavior of the probabilities P(max _t∈[0,T] η(t)ξ(t) > u), P(max _t∈[0,T] (ξ(t) + η(t)) > u) and P(max _t∈[0,T] (η(t)ξ(t) + ζ(t)) > u) is obtained for u → ∞ for any T > 0 and independent ξ(t), η(t), ζ(t).