We impose a monotonicity condition with several reversals on the moduli of the coefficients of a polynomial. We then consider three types of polynomials: (1) those satisfying the condition on all of the coefficients, (2) those satisfying the condition on the even indexed and odd indexed coefficients separately, and (3) polynomials of the form P(z) = a0+ Σnj=µ ajzj where µ ≥ 1 with the coefficients aµ; aµ+1;…; an satisfying the condition. For each type of polynomial, we give a result which puts a bound on the number of zeros in a disk (in the complex plane) centered at the origin. For each type, we give an example showing the results are best possible.

We put restrictions on the coefficients of polynomials and give bounds concerning the number of zeros in a specific region. The restrictions involve a monotonicity-type condition on the coefficients of the even powers of the variable and on the coefficients of the odd powers of the variable (treated separately). We present results by imposing the restrictions on the moduli of the coefficients, the real and imaginary parts of the coefficients, and the real parts (only) of the coefficients.

京都大学 / 0048 / 新制・課程博士 / 博士(経済学) / 甲第21523号 / 経博第591号 / 新制||経||288(附属図書館) / 京都大学大学院経済学研究科経済学専攻 / (主査)教授 梶井 厚志, 教授 原 千秋, 教授 若井 克俊 / 学位規則第4条第1項該当 / Doctor of Economics / Kyoto University / DFAM

We define the notion of monotone operations admitted by partially ordered sets, specifically monotone near-unanimity functions and Jónsson operations. We then prove a result of McKenzie's in [8] which states that if a finite, bounded poset P admits a set of monotone Jónsson operations then it admits a set of monotone Jónsson operations for which the operations with even indices do not depend on their second variable. We next define zigzags of posets and prove various useful properties about them. Using these zigzags, we proceed carefully through Zadori's proof from [12] that a finite, bounded poset P admits a monotone near-unanimity function if and only if P admits monotone Jónsson operations.

We define the notion of monotone operations admitted by partially ordered sets, specifically monotone near-unanimity functions and Jónsson operations. We then prove a result of McKenzie's in [8] which states that if a finite, bounded poset P admits a set of monotone Jónsson operations then it admits a set of monotone Jónsson operations for which the operations with even indices do not depend on their second variable. We next define zigzags of posets and prove various useful properties about them. Using these zigzags, we proceed carefully through Zadori's proof from [12] that a finite, bounded poset P admits a monotone near-unanimity function if and only if P admits monotone Jónsson operations.

We consider the constrained convex minimization problem: min_x∈C f(x) we will present gradient projection method which generates a sequence x^k according to the formula x^(k+1) = P_c(x^k − £\_k∇f(x^k)), k= 0, 1, ¡P ¡P ¡P , our ideal is rewritten the formula as a xed point algorithm: x^(k+1) = T_(£\k)x^k, k = 0, 1, ¡P ¡P ¡P is used to solve the minimization problem. In this paper, we present the gradient projection method(GPM) and different choices of the stepsize to discuss the convergence of gradient projection method which converge to a solution of the concerned problem.

In this thesis, we report recent results on existence for complementarity problems in infinite-dimensional spaces under generalized monotonicity are reported.

Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C¡÷ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach's contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann's algorithm and Halpern's algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.

In this paper we give a general framework for isotone optimization. First we discuss a generalized version of the pool-adjacent-violators algorithm (PAVA) to minimize a separable convex function with simple chain constraints. Besides of general convex functions we extend existing PAVA implementations in terms of observation weights, approaches for tie handling, and responses from repeated measurement designs. Since isotone optimization problems can be formulated as convex programming problems with linear constraints we then develop a primal active set method to solve such problem. This methodology is applied on specific loss functions relevant in statistics. Both approaches are implemented in the R package isotone. (authors' abstract)

<p>In this thesis we consider the Einstein-Vlasov system, which models a system of particles within the framework of general relativity, and where collisions between the particles are assumed to be sufficiently rare to be neglected. Here the particles are stars, galaxies or even clusters of galaxies, which interact by the gravitational field generated collectively by the particles.</p><p>The thesis consists of three papers, and the first two are devoted to cylindrically symmetric spacetimes and the third treats the spherically symmetric case.</p><p>In the first paper the time-dependent Einstein-Vlasov system with cylindrical symmetry is considered. We prove global existence in the so called polarized case under the assumption that the particles never reach a neighborhood of the axis of symmetry. In the more general case of a non-polarized metric we need the additional assumption that the derivatives of certain metric components are bounded in a vicinity of the axis of symmetry to obtain global existence.</p><p>The second paper of the thesis considers static cylindrical spacetimes. In this case we prove global existence in space and also that the solutions have finite extension in two of the three spatial dimensions. It then follows that it is possible to extend the spacetime by gluing it with a Levi-Civita spacetime, i.e. the most general vacuum solution of the static cylindrically symmetric Einstein equations.</p><p>In the third and last paper, which is a joint work with C. Uggla and M. Heinzle, the static spherically symmetric Einstein-Vlasov system is studied. We introduce a new method by rewriting the system as an autonomous dynamical system on a state space with compact closure. In this way we are able to improve earlier results and enlarge the class of distribution functions which give rise to steady states with finite mass and finite extension.</p>