Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

Contribuyendo al reto de la sostenibilidad: el sistema producto- servicio en la industria de la moda rápida

El presente proyecto de grado se inscribe bajo el proyecto de Responsabilidad Social Empresarial de la línea de investigación de Realidad Empresarial de la Facultad de Administración de la Universidad del Rosario. El problema de investigación surge de dos problemas actuales, que convergen en la insostenibilidad a largo y mediano plazo de la manera en que se hacen la mayoría de los negocios hoy en día. El primer problema, es el de suficiencia, se evidencia desde el punto de vista del consumidor, el cual pareciera estar diseñado para comprar y acumular objetos y cosas que luego de un tiempo irán a parar a la basura. El segundo problema es el modelo de negocio que extrae recursos, los transforma, los comercializa de manera repetitiva y obliga al cliente a seguir comprando, olvidando que la mayoría de cosas que produce, terminan en el basurero. La deficiencia actual del consumismo insostenible y del diseño que no piensa en las generaciones futuras son dos caras de la misma moneda. Por eso es que se estudia el problema desde la perspectiva tanto del consumidor, como del productor. Las tendencias internacionales relacionadas con la sostenibilidad están llevando a los negocios por una nueva senda, la cual les dará ventajas competitivas a los adoptadores tempranos. A raíz de los problemas planteados y con el objetivo de encontrar la sostenibilidad empresarial, se estudia una posible solución para cada actor del problema. Por el lado del consumidor, se analiza la teoría que gira alrededor del consumo responsable y de cómo los consumidores tienen el poder de causar un efecto positivo con lo que mejor saben hacer: comprando (y eligiendo que comprar). Por el lado del productor se estudia un nuevo modelo de negocios llamado el sistema producto-servicio, el cual se enfoca a satisfacer necesidades de los clientes por medio de una combinación de prestación de servicios y facilitación de productos, en vez de la comercialización de estos. Es decir que en vez de vender cosas que los consumidores quieran acumular (problema de suficiencia), se les vende soluciones que satisfagan sus necesidades, reteniendo la compañía la propiedad sobre los productos que integran al servicio. Se estudia también de que maneras este modelo particular de sistemas producto-servicio puede ser recibido por los consumidores de hoy en día, tan acostumbrados al modelo actual. Se quiere explorar el sistema de productos y servicio como una alternativa empresarial perdurable que solucione los retos ambientales del siglo XXI. Para poder probar la viabilidad de este hipotético modelo que cambia los patrones de interacción de las empresas con los usuarios, se plantea una idea de negocio que integre productos y servicios para satisfacer las necesidades del cliente. Se escogió un área viii caracterizada por ser de las más consumistas y acumuladoras en cuanto al volumen de compras anual de los clientes: el negocio de la ropa. Y no solo cualquier tipo de ropa, sino el de la moda rápida, un modelo que obliga a los consumidores a comprar ropa con una frecuencia aún mayor. Se hizo esto con el objetivo de probar que el modelo propuesto puede funcionar inclusive bajo los supuestos más “consumistas” que existen hoy en día.

Random walk radiosity with generalized absorption probabilities

The author studies random walk estimators for radiosity with generalized absorption probabilities. That is, a path will either die or survive on a patch according to an arbitrary probability. The estimators studied so far, the infinite path length estimator and finite path length one, can be considered as particular cases. Practical applications of the random walks with generalized probabilities are given. A necessary and sufficient condition for the existence of the variance is given, together with heuristics to be used in practical cases. The optimal probabilities are also found for the case when one is interested in the whole scene, and are equal to the reflectivities

Europe's single supervisory mechanism and the long journey towards banking union. Bruegel Policy Contribution 2012/16, 4 October 2012

Problems in the banking system are at the core of the current crisis. The establishment of a banking union is a necessary (though not sufficient) condition for eventual crisis resolution that respects the integrity of the euro. The European Commission’s proposal for the establishment of a Single Supervisory Mechanism and related reform of the European Banking Authority (EBA) do not and cannot create a fully-fledged banking union, but represent a broadly adequate step on the basis of the leaders’ declaration of 29 June 2012 and of the decision to use Article 127(6) of the treaty as legal basis. The proposal rightly endows the European Central Bank (ECB) with broad authority over banks within the supervisory mechanism’s geographical perimeter; however, the status of non-euro area member states willing to participate in this mechanism, and the governance and decision-making processes of the ECB in this respect, call for further elaboration. Further adjustments are also desirable in the proposed reform of the EBA, even though they must probably retain a stopgap character pending the more substantial review planned in 2014.

Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

Global exponential periodicity of a class of neural networks with recent-history distributed delays

In this paper, we propose to study a class of neural networks with recent-history distributed delays. A sufficient condition is derived for the global exponential periodicity of the proposed neural networks, which has the advantage that it assumes neither the differentiability nor monotonicity of the activation function of each neuron nor the symmetry of the feedback matrix or delayed feedback matrix. Our criterion is shown to be valid by applying it to an illustrative system. (c) 2005 Elsevier Ltd. All rights reserved.

Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series. The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.

Toeplitz operators with distributional symbols on Bergman spaces

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere

It is shown that, for a sufficiently large value of β, two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic (phase-space filling) on the phase-space surface of constant energy and enstrophy. A corresponding result holds for flow on the surface of a rotating sphere, for a sufficiently rapid rotation rate Ω. This implies that the higher-order, non-quadratic invariants are exerting a significant influence on the statistical evolution of the flow. The proof relies on the existence of a finite-amplitude Liapunov stability theorem for zonally symmetric basic states with a non-vanishing absolute-vorticity gradient. When the domain size is much larger than the size of a typical eddy, then a sufficient condition for non-ergodicity is that the wave steepness ε < 1, where ε = 2[surd radical]2Z/βU in the planar case and $\epsilon = 2^{\frac{1}{4}} a^{\frac{5}{2}}Z^{\frac{7}{4}}/\Omega U^{\frac{5}{2}}$ in the spherical case, and where Z is the enstrophy, U the r.m.s. velocity, and a the radius of the sphere. This result may help to explain why numerical simulations of unforced beta-plane turbulence (in which ε decreases in time) seem to evolve into a non-ergodic regime at large scales.

Analysis of the seasonal cycle within the first international urban land-surface model comparison

A number of urban land-surface models have been developed in recent years to satisfy the growing requirements for urban weather and climate interactions and prediction. These models vary considerably in their complexity and the processes that they represent. Although the models have been evaluated, the observational datasets have typically been of short duration and so are not suitable to assess the performance over the seasonal cycle. The First International Urban Land-Surface Model comparison used an observational dataset that spanned a period greater than a year, which enables an analysis over the seasonal cycle, whilst the variety of models that took part in the comparison allows the analysis to include a full range of model complexity. The results show that, in general, urban models do capture the seasonal cycle for each of the surface fluxes, but have larger errors in the summer months than in the winter. The net all-wave radiation has the smallest errors at all times of the year but with a negative bias. The latent heat flux and the net storage heat flux are also underestimated, whereas the sensible heat flux generally has a positive bias throughout the seasonal cycle. A representation of vegetation is a necessary, but not sufficient, condition for modelling the latent heat flux and associated sensible heat flux at all times of the year. Models that include a temporal variation in anthropogenic heat flux show some increased skill in the sensible heat flux at night during the winter, although their daytime values are consistently overestimated at all times of the year. Models that use the net all-wave radiation to determine the net storage heat flux have the best agreement with observed values of this flux during the daytime in summer, but perform worse during the winter months. The latter could result from a bias of summer periods in the observational datasets used to derive the relations with net all-wave radiation. Apart from these models, all of the other model categories considered in the analysis result in a mean net storage heat flux that is close to zero throughout the seasonal cycle, which is not seen in the observations. Models with a simple treatment of the physical processes generally perform at least as well as models with greater complexity.

On the dominance of roots of characteristic equations for neutral functional differential equations

We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail. (C) 2009 Elsevier Inc. All rights reserved.

Benders, metric and cutset inequalities for multicommodity capacitated network design

Solving multicommodity capacitated network design problems is a hard task that requires the use of several strategies like relaxing some constraints and strengthening the model with valid inequalities. In this paper, we compare three sets of inequalities that have been widely used in this context: Benders, metric and cutset inequalities. We show that Benders inequalities associated to extreme rays are metric inequalities. We also show how to strengthen Benders inequalities associated to non-extreme rays to obtain metric inequalities. We show that cutset inequalities are Benders inequalities, but not necessarily metric inequalities. We give a necessary and sufficient condition for a cutset inequality to be a metric inequality. Computational experiments show the effectiveness of strengthening Benders and cutset inequalities to obtain metric inequalities.

Towards a maximal extension of Pelczynski`s decomposition method in Banach spaces

l Suppose that X, Y. A and B are Banach spaces such that X is isomorphic to Y E) A and Y is isomorphic to X circle plus B. Are X and Y necessarily isomorphic? In this generality. the answer is no, as proved by W.T. Cowers in 1996. In the present paper, we provide a very simple necessary and sufficient condition on the 10-tuples (k, l, m, n. p, q, r, s, u, v) in N with p+q+u >= 3, r+s+v >= 3, uv >= 1, (p,q)$(0,0), (r,s)not equal(0,0) and u=1 or v=1 or (p. q) = (1, 0) or (r, s) = (0, 1), which guarantees that X is isomorphic to Y whenever these Banach spaces satisfy X(u) similar to X(p)circle plus Y(q), Y(u) similar to X(r)circle plus Y(s), and A(k) circle plus B(l) similar to A(m) circle plus B(n). Namely, delta = +/- 1 or lozenge not equal 0, gcd(lozenge, delta (p + q - u)) divides p + q - u and gcd(lozenge, delta(r + s - v)) divides r + s - v, where 3 = k - I - in + n is the characteristic number of the 4-tuple (k, l, m, n) and lozenge = (p - u)(s - v) - rq is the discriminant of the 6-tuple (p, q, r, s, U, v). We conjecture that this result is in some sense a maximal extension of the classical Pelczynski`s decomposition method in Banach spaces: the case (1, 0. 1, 0, 2. 0, 0, 2. 1. 1). (C) 2009 Elsevier Inc. All rights reserved.

Cantor-Bernstein Sextuples for Banach Spaces

Let X and Y be Banach spaces isomorphic to complemented subspaces of each other with supplements A and B. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain a necessary and sufficient condition on the sextuples (p, q, r, s, u, v) in N with p + q >= 1, r + s >= 1 and u, v is an element of N*, to provide that X is isomorphic to Y, whenever these spaces satisfy the following decomposition scheme A(u) similar to X(P) circle plus Y(q) B(v) similar to X(r) circle plus Y(s). Namely, Phi = (p - u)(s - v) - (q + u)(r + v) is different from zero and Phi divides p + q and r + s. These sextuples are called Cantor-Bernstein sextuples for Banach spaces. The simplest case (1, 0, 0, 1, 1, 1) indicates the well-known Pelczynski`s decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder-Bernstein problem become evident.