The rising impact of software development in globally distributed teams strengthens the need for strategies that establish a clear separation of concerns in software models. Large, weakly modularized models are hard to comprehend and to analyse. A further maintainance obstacle is introduced by conflicting changes of the model. In our recent work, we propose a structured process for distributed ...

We give a new, combinatorial proof for the necklace splitting problem for two thieves using only Tucker's lemma (a combinatorial version of the Borsuk-Ulam theorem). We show how this method can be applied to obtain a related recent result of Simonyi and even generalize it. 1 Necklace Splitting This paper was inspired by the combinatorial proof of Matou2ek [7] of the Lovász-Kneser theorem [6]. H...

A standard problem within universities is that of teaching space allocation which can be thought of as the assignment of rooms and times to various teaching activities. The focus is usually on courses that are expected to fit into one room. However, it can also happen that the course will need to be broken up, or ‘split’, into multiple sections. A lecture might be too large to fit into any one ...

This paper shows that with B = {1, 2, . . . , n}, the smallest k such that (B ×B)− {(j, j) | j ∈ B} = k ⋃ i=1 (Ci ×Di) is s(n), where s(n) is the smallest integer k such that n 6 ( k b k 2 c ) . This provides a simple set-based formulation and a new proof of a result for boolean ranks [2] and biclique covering of bipartite graphs [1, 5], making these intricate results more accessible.

Given a directed network G = (V, A) with positive capacity for each a ∈ A, and a specified set of source-sink pairs of vertices, the objective is to remove a set of arcs with minimum capacity so that the resulting network stops all communication from sources to their respective sinks. We study the facial structure of the polytope associated with the solutions of this problem and identify a gen...

In the previous lecture, we covered a series of online/offline edge-weighted Steiner tree/forest problems. This lecture extends the discussion to the node-weighted scope. In particular, we will study the nodeweighted Steiner tree/forest problem and introduce an offline O(logn)−approximation polynomial-time algorithm[KR95]. It is well known there is no polynormial-time algorithm that achieves o(...

So, suppose that i ◦ f : Z → X is continuous. Since for every α, the projection πα : X → Xα is continuous, πα ◦ (i ◦ f) : Z → Xα is continuous. The map πα ◦ (i ◦ f) = iα ◦ (ρα ◦ f). By the universal property of the subspace topology on Yα, the map iα ◦ (ρα ◦ f) being continuous implies ρα ◦ f : Z → Yα is continuous. So, we have a collection of continuous maps: ρα ◦ f : Z → Yα so by the universa...

(b) Usually (in practice), the k used for signing is not generated completely at random, but by using a pseudorandom generator. Again, in practice, people use a simple “pseudorandom” generator such as the linear congruential generator. A linear congruential generator LCG, on input a seed s0 outputs a sequence (s1, s2, . . . , sn) such that si = asi−1 + b(mod q). (where a and b are assumed to be...