By Fourier‐series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two‐dimensional formulations of the three‐dimensional boundary‐value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform‐approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid‐plane displacement. The resulting second‐order theory, considered as a first‐order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear‐rigid as a second‐order theory. Therefore, it is modified by an a‐priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first‐order theory, but is in addition equivalent to Kienzler's theory as a second‐order theory for the special case of isotropy, which implies further equivalences to established shear‐deformable theories, especially the Reissner‐Mindlin theory and Zhilin's plate theory. The presented new second‐order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edgeweighted tree T , and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problems for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests.Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for trees of arbitrary (fixed) depth. The competitive ratio is O(D 4 2 D ), where D is the depth of T . The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We include several additional results in the paper. We show that a standard lower-bound technique for MLAP, based on so-called Single-Phase instances, cannot give super-constant lower bounds (as a function of the tree depth). This result is established by giving an online algorithm with optimal competitive ratio 4 for such instances on arbitrary trees. We prove that, in the offline case, these instances can be solved to optimality in polynomial time. We also study the MLAP variant when the tree is a path, for which we give a lower bound of 4 on the competitive ratio, improving the lower bound known for general MLAP. We complement this with a matching upper bound for the deadline setting. In addition, for arbitrary trees, we give a simple 2-approximation algorithm for offline MLAP with deadlines. accessing a resource. For example, costs may be associated with waiting until the resource is accessible, or, if the resource is not in the desired state, a costly setup or retooling may be required.1-level aggregation. A simple example of an aggregation problem is the TCP Acknowledgment Problem (TCP-AP), where control messages ("agents") waiting for transmission across a network link can be aggregated and transmitted in a single packet ("resource"). Such aggregation can reduce network traffic, but it also results in undesirable delays. A reasonable compromise is to balance the two costs, namely the number of transmitted packets and the total delay, by minimi...

Abstract. Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of 11/8 = 1.375 for three bins. Additionally, we present a lower bound of 45/33 = 1.36 for Online Bin Stretching on three bins and a lower bound of 19/14 for four and five bins that were discovered using a computer search.