Limits of Julia Sets for Sums of Power Maps and Polynomials

Suppose f_{n,c} is a complex-valued mapping of one complex variable given by f_{n,c}(z) = z^n + p(z) + c, where p is a polynomial such that p(0) = 0 and c is a complex parameter such that |c| \u3c 1. We provide necessary and sufficient conditions that the geometric limit, as n approaches infinity, of the set of points that remain bounded under iteration by f_{n,c} is the disk of radius 1 centered at the origin

Locking-free two-layer Timoshenko beam element with interlayer slip

A new locking-free strain-based finite element formulation for the numerical treatment of linear static analysis of two-layer planar composite beams with interlayer slip is proposed. In this formulation, the modified principle of virtual work is introduced as a basis for the finite element discretization. The linear kinematic equations are included into the principle by the procedure, similar to that of Lagrangian multipliers. A strain field vector remains the only unknown function to be interpolated in the finite element implementation of the principle. In contrast with some of the displacement-based and mixed finite element formulations of the composite beams with interlayer slip, the present formulation is completely locking-free. Hence, there are no shear and slip locking, poor convergence and stress oscillations in these finite elements. The generalization of the composite beam theory with the consideration of the Timoshenko beam theory for the individual component of a composite beam represents a substantial contribution in the field of analysis of non-slender composite beams with an interlayer slip. An extension of the present formulation to the non-linear material problems is straightforward. As only a few finite elements are needed to describe a composite beam with great precision, the new finite element formulations is perfectly suited for practical calculations. (c) 2007 Elsevier B.V. All rights reserved

Extremalis problémák többváltozós és súlyozott polinomokra = Extremal problems for multivariate and weighted polynomials

Jól ismert hogy a többváltozós polinomok sűrűek a d-dimenziós kompakt halmazokon folytonos függvények terében. A többváltozós polinomok egy fontos részhalmaza a homogén polinomok osztálya. Igy természetesen felmerül az a kérdés, hogy igaz-e a sűrüség a homogén polinomokra? Egy ismert sejtés szerint a konvex felületeken folytonos függvények megközelíthetőek két homogén polinom összegével. A pályázat keretében két fontos új eredmény született 1) igazoltuk a sejtést tetszőleges sima ( egyértelmü támasz sikkal rendelkező) konvex testeken egyenletes normában 2) igazoltuk a sejtést teljes általánosságban Lp normában Ezen kivül általánosított Freud súlyokra vonatkozó polinom-approximációs problémákat vizsgáltunk. Itt az általánosítás azt jelenti, hogy az eredeti Freud súlyokat megszorozzuk olyan un. általánosított polinomokkal, amelyeknek csak valós gyökeik vannak. A klasszikus polinom-egyenlotlenségek analogonjait, valamint direkt és fordított approximációs tételeket bizonyítottunk. Hibabecsléseket adtunk függvények súlyozott approximációjára Freud súlyok esetén, olyan egész függvényekkel történo approximáció esetén, amelyek véges, ill. végtelen sok pontban interpolálják a függvényt. Ezek a hibabecslések olyan súlyozott folytonossági modulusokat tartalmaznak, amelyeknél a polinom-suruség nem mindig garantált | It is well known that multivariate polynomials are dense in the space of continuous functions on compact subsets of the d-dimensional space. An important family of multivariate polynomials is the space of all homogeneous polynomials. Thus it is natural to ask if the density holds for homogeneous polynomials. It has been conjectured that any function continuous on a convex surface can be approximated by sums of two homogeneous polynomials. In the framework of the present project the above conjecture was verified in two new important cases: 1) the conjecture was verified for uniform norm on arbitrary regular convex bodies, i.e., in case when the body possesses a unique tangent plane at each point of its boundary 2) the conjecture was verified in full generality in the Lp norm We also considered polynomial approximation problems on the real line with generalized Freud weights. The generalization means multiplying these weights by so-called generalized polynomials which have real roots only. Analogues of classical polynomial inequalities, as well as direct and converse approximation theorems were proved. We gave error estimates for the weighted approximation of functions with Freud-type weights, by entire functions interpolating at finitely or infinitely many points on the real line. The error estimates involve weighted moduli of continuity corresponding to general Freud-type weights for which the density of polynomials is not always guaranteed

Algebras and non-geometric flux vacua

In this work we classify the subalgebras satisfied by non-geometric Q-fluxes in type IIB orientifolds on T^6/(Z_2 x Z_2) with three moduli (S,T,U). We find that there are five subalgebras compatible with the symmetries, each one leading to a characteristic flux-induced superpotential. Working in the 4-dimensional effective supergravity we obtain families of supersymmetric AdS_4 vacua with all moduli stabilized at small string coupling g_s. Our results are mostly analytic thanks to a judicious parametrization of the non-geometric, RR and NSNS fluxes. We are also able to leave the flux-induced C_4 and C_8 RR tadpoles as free variables, thereby enabling us to study which values are allowed for each Q-subalgebra. Another novel outcome is the appearance of multiple vacua for special sets of fluxes. However, they generically have g_s > 1 unless the net number of O3/D3 or O7/D7 sources needed to cancel the tadpoles is large. We also discuss briefly the issues of axionic shift symmetries and cancellation of Freed-Witten anomalies.Comment: 61 pages, LaTex, v2: added reference

Constructive Function Theory on Sets of the Complex Plane through Potential Theory and Geometric Function Theory

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory

Trees and the dynamics of polynomials

The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C} carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\eta)$, with limiting dynamics F : T \arrow T. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary \PT_d compactifying the moduli space of polynomials of degree $d$. We show that $(T,\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space \PT_3 is itself a tree. The metrized trees $(T,\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.Comment: 60 page