Exact solutions and mixing in an algebraic dynamical system

Let be an n×n matrix with entries a_ij in the field . We consider two involutive operations on these matrices: the matrix inverse I: ^–1 and the entry-wise or Hadamard inverse J: a_ij a _ij ^–1 . We study the algebraic dynamical system generated by iterations of the product J. I. We construct the complete solution of this system for n 4. For n = 4, it is obtained using an ansatz in theta functions. For n 5, the same ansatz gives partial solutions. They are described by integer linear transformations of the product of two identical complex tori. As a result, we obtain a dynamical system with mixing described by explicit formulas.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 131–149, April, 2005.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

Exact L 2-norm plane separation

We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of L _p -norm distances to the plane of points lying on the wrong side of it. Despite recent progress, practical techniques for the exact solution of cases other than the L ₁ and L _∞-norm were unavailable. We propose and implement a new approach, based on non-convex quadratic programming, for the exact solution of the L ₂-norm case. We solve in reasonable computing times artificial problems of up to 20000 points (in 6 dimensions) and 13 dimensions (with 2000 points). We also observe that, for difficult real-life instances from the UCI Repository, computation times are substantially reduced by incorporating heuristic results in the exact solution process. Finally, we compare the classification performance of the planes obtained for the L ₁, L ₂ and L _∞ formulations. It appears that, despite the fact that L ₂ formulation is computationally more expensive, it does not give significantly better results than the L ₁ and L _∞ formulations.     

An everywhere convergent series representation of the distribution of Hotelling's generalized T0

A new series representation of the exact distribution of Hotelling's generalized T₀² statistic is obtained. Unlike earlier work, the series representation given here is everywhere convergent. Explicit formulae are given for both the null and the non-central distributions. Earlier results by [1], 215–225), which are convergent on the interval [0, 1), are also derived quite simply from our formulae. The paper therefore provides a solution to the long standing problem of the exact distribution of the T₀² statistic in the general case.     

Some Algebraic Features of Z2-graded KK-Theory

We study the relation of Z₂equivariant and Z₂graded KK-theory. The former is the universal stable, split exact and homotopy invariant theory on the category of Z₂graded C^*algebras and graded homomorphisms (Theorem 1). We obtain an abstract characterization for the product of the graded KK-functor (Theorem 2). We give generalizations to Z₂graded C^*algebras of the Universal Coefficent Theorem, Künneth Theorem and Künneth Theorem for tensor products. We prove some results about graded crossed products of Thom isomorphism and Pimsner-Voiculescu type (Theorem 3 and Corollary 2) and compute an example. We obtain a split surjective map KK(A,B) KK(A⁰,B⁰) commuting with products, where A⁰is a canonically defined trivially graded algebra for any Z₂-graded A.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

The Exactness Property of the Vector Exact l1 Penalty Function Method in Nondifferentiable Invex Multiobjective Programming

In this article, the vector exact l₁ penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l₁ penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l₁ penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.     

On the ordered K0 group of a simple C*-algebra

It is shown that any countable simple and unperforated partially ordered Abelian group with order unit, paired in an arbitrary way with a compact metrizable Choquet simplex, occurs as the K₀group of a simple unital ^*-algebra which is the inductive limit of a sequence of subhomogeneous ^*-algebras of a particularly simple form.     

行列式detVn-(x1,...,xn)的准确值及其应用

本文给出行列式Ｖ_n^-(x₁,...,x_n)的准确值，它是通常的Ｖａｎｄｅｒｍｏｄｅ行列式计算公式的推广，以及它在理论上的一些重要应用．     

Linearity of Zp[[t]]-Perfect Groups

Let G be a _p[[t]]-standard group of level 1. Then G is _p[[t]]-perfect if its lower central series is given by powers of the maximal ideal (p, t), i.e. if _n(G) = G((p,t)ⁿ). We prove that a _p[[t]]-perfect group is linear by imitating the proof that a _p[[t]]-standard group is linear.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

The relationship between the boundedly controlled K1 and Whitehead groups

Let Z be a boundedness control space and p: X Z be a continuous map. The boundedly controlled Whitehead group Wh ^bc (X, p) is defined to be a quotient of the boundedly controlled K ₁-group K ₁ ^bc (X, p) by a certain subgroup whose generators are explicitly given. In general, little is known about this subgroup and it is even possible that it vanishes; i.e. that the boundedly controlled K ₁ and Whitehead groups are identical. This paper examines the structure of this subgroup in the case when p is the open cone on a PL map between compact polyhedra. As a byproduct, it calculates Wh ^bc (X, p) in some of these cases.Partially supported by the NSF under grant number DMS-8803149.     

Exact sequences for the homology of the matching complex

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanishing 3-torsion in whenever , where . By results due to Bouc and to Shareshian and Wachs, is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of Mn for all n≠2. Second, we prove that is a nontrivial 3-group whenever . Third, for each k?0, we show that there is a polynomial fk(r) of degree 3k such that the dimension of , viewed as a vector space over Z₃, is at most fk(r) for all r?k+2.     

Categorical and Simplicial Constructions and Dualization in Algebraic K-Theory

Given an exact category A, we introduce a bisimplicial set W A, which is self-dual and contains both the G-construction of A and its dualization. We prove that the embeddings of G.A and G ^op .A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induced action of duality on K ₁ (A). We also survey on the self-duality property of some known constructions.     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

Exact solution to a nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation ?^μ?_μΦ + M²Φ + λ₁Φ^1?m + λ₂Φ^1?2m = 0 has the exact formal solution Φ = [u^2m ?λ₁u^m/(m ? 2)M²+λ₁²/(m?2)²M⁴?λ₂/4(m ? 1)M²]^1/mu^?1, m ≠ 0, 1, 2, where u and v^?1 are solutions of the linear Klein-Gordon equation. This equation is a simple generalization of the ordinary second order differential equation satisfied by the homogeneous function y = [au^m + b(uv)^m/2 + cv^m]^k/m, where u and v are linearly independent solutions of y″ + r(x) y′ + q(x) y = 0.     

Flip-chip-type high-Tc gradiometer for biomagnetic measurements in unshielded environment

A flip-chip-type gradiometer has been constructed with a 10 mm × 5 mm planar DCSQUID gradiometer fabricated on a SrTiO₃ bicrystal substrate and a flux transformer made from a YBCO*/YBCO/CeO₂/YSZ multilayer on ϕ50.8 mm Si wafer. The coupling coefficient between the flux transformer and the planar gradiometer is 0.18. The transformer increases effectively the resolution of the gradiometer. A magnetic field gradient resolution of 73 fT.cm⁻¹Hz^−1/2 in the white region and 596 fT.cm⁻¹Hz^−1/2 at 1 Hz has been obtained. High quality magnetocardiogram signals have been successfully measured by using this flip-chip-type gradiometer in an unshielded environment.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

On Finite Groups and the Small Square Property

Let G be a finite group written multiplicatively and k a positive integer. If X is a non-empty subset of G, write X ² = |xy | x, y X . We say that G has the small square property on k-sets if |X ²| < k ² for any k-element subset X of G. For each group G, there is a unique m = m _G such that G has the small square property on (m + 1)-sets but not on m-sets. In this paper we show that given any positive integer d, there is a finite group G with m _G = d.