Main Results Clause Samples

Main Results. ‌ ^√ ^ In this paper we focus on two optimization problems, namely the sparse Minimum Bisection Problem (sMBP) and the ▇▇▇▇▇▇ Quadratic Assignment Problem (LQAP). Formal definitions are given below. However, we should add that many of our results hold for a larger class of optimization problems as long as log m = o(N ) (see ▇▇▇▇▇▇▇▇▇▇▇, 1995). In the rest of the paper we will utilize the temperature rescaling β = β log m/N with β = O(1) which together with log m = o(N ) explains β 0 limit. This rescaling was justified in (▇▇▇▇▇▇▇ et al., 2014). ^ For these two problems we shall provide tight asymptotics for the free en- ergy (5), and compute asymptotically the log-posterior agreement as well as β∗ that maximizes the posterior kernel. 2.1. Minimum bisection and quadratic assignment optimization problems‌ This section introduces combinatorial optimization problems that will be used to describe our findings. These problems fall into the log m = o(N ) class specified in Sec. 1.2 and cover a wide range of practical applications in signal processing and neural information processing. Minimum bisection problem (MBP). Consider a complete undirected weighted graph G = (V, E, X) of n vertices, where n is an even number. The input data instance X is represented by (random) weights (Wi)i∈E of the graph edges. Σ ∈S

Main Results. In this paper we prove the following theorem. Theorem 1.1. For k < n, there is no deterministic wait-free protocol in the shared atomic registers model which solves the k-set agreement problem in a system of n processors. (i) two schedules are “indistinguishable” if for any protocol they exhibit the same output behavior, (ii) a set S of schedules is “knowable” if there is a protocol which “recognizes” it, in the sense that for some specified output symbol, the protocol pro- duces that symbol during an execution if and only if the execution proceeds according to some schedule from S. ∈ − These two concepts lead naturally to the definition of a topology on the set of schedules, and Theorem 1.1 is proved by analyzing this topology. Our approach reveals and exploits a close analogy between the impossibility of wait-free k-set agreement and a lemma of Knaster, ▇▇▇▇▇▇▇▇▇▇, and ▇▇▇▇▇▇▇▇▇▇▇▇ (KKM lemma) [1], which is equivalent to the fixed point theorem for the closed unit ▇▇▇▇ ▇▇ in m-dimensional Euclidean space: if f is a continuous map from Bm to itself, then there exists a point x Bm such that f (x) = x. Very roughly, f corresponds to a distributed protocol Π, and the fixed point x corresponds to the schedule for which Π fails to solve the k-set agreement. The increase in difficulty of the k-set agreement proof in going from the case k = 1 to the case k > 1 corresponds to the increase in difficulty in going from the fixed point theorem for the interval [ 1, 1], which is very simple, to the theorem for balls in higher dimension, which, while elementary, is considerably harder. An additional obstacle in our work is that, while the topological structure of Bm is well understood, we must develop the topological structure for the set of schedules from scratch. While the explicit use of topology can be avoided, we have retained the topological structure of the proof, because this is what drove the proof and it provides important insight into what is going on. Our topological structure has an intuitive interpretation in terms of the information about an execution which is “public knowledge.” We believe that it will be worthwhile to explore the connection with the formal theory of distributed knowledge [14]. The inspiration for the topological approach came from ▇▇▇▇▇▇▇▇▇’▇ work [10], in which the combinatorial properties of triangulations in Rk were used to obtain certain reductions among various decision problems. There is a considerable literature con- cerning topologies u...

Main Results. Figure 1 shows the fringe visibility loss resulting from the mirror vibrations in var- ious operational conditions and for three different observing wavelengths: visible (0.6 μm), near infrared (2.2 μm) and ther- mal infrared (10 μm). The VLTI error budgets call for a 1% visibility loss due to vibrations inside the telescope for any of these observing wavelengths. This cor- responds respectively to an OPL varia- tion of 14, 50 and 215 nanometers r.m.s.

Main Results. ‌ In this chapter, we study the Best-of-k protocol in the particular case k = 3 under the two-party setting, where initially each vertex is blue independently with probability 1/2 − δ > 0 , otherwise red. By applying two models to analyse the process of a vertex updating its opinions, we prove convergence to majority in O(log log n) time- steps with high probability under certain conditions. Our main result is the following. 1. Consider a graph G of n vertices with minimum degree d = nα , where α = Ω((log log n)−1) , and suppose the initial opinion of each vertex is blue inde- pendently with probability 1/2 − δ , otherwise red, where δ ⩾ (log d)−C for some C > 0 . Then, w.

Main Results. We start with some rather general admissible conditions that simplify the secret key agreement scheme significantly without loss of optimality. Theorem 1 cS(r) remains unchanged even if we set omniscience is no smaller than the communication complexity. We say that cS can be achieved via omniscience of zV .‌ ℓ(ui) = 0 ∀i ∈ V, and K = k ∈ ⟪zV ⟫, (14a)‌ (14b)

Main Results. The main results of this paper are summarized in the following. For the secret key generation scenario among m terminals that have access to a “deterministic broadcast chan- nel,” we completely characterize the key generation capacity. This result can be considered as the generalization of the result of [8], [24] for “packet erasure broadcast channels” (see The- orem 1). For a “state-dependent Gaussian broadcast channel,” we provide upper and lower bounds for the key generation capacity and show that these bounds will match in the high- dynamic range, high-SNR regime. Furthermore, the achievable secrecy rate by our proposed scheme for the Gaussian model is described by a non-convex power optimization problem. Although this problem is non-convex, by exploiting its special structure, we find the optimal power allocation that leads to the best secrecy rate achievable by the proposed scheme. Theorem 1. The SKG capacity among m terminals that have access to a state-dependent deterministic broadcast channel, defined in Section II-C, is given by ΣC = [rank F − rank F ] θ (1 − θ ) log q,i i−1 i i det Σ −where ▇ ¾ ▇ .▇ ▇ ▇▇▇▇▇▇▇ 1 is proved in Section V (see Lemma 2 and Lemma 3). Notice that the result of [8] is a special case of Theorem 1 when s = 1. where “=· ” defined in Section II, is used to denote for the exponential equality with respect to some scaling parameter

Main Results. Here we prove the cohomological rigidity for small covers and quasitoric mani- folds over 3-polytopes in the Pogorelov class P. We start with a crucial lemma. Lemma 5.1. In the notation of Theorem 2.3, consider the set D (M ) = {±[vi] ∈ H2(M ), i = 1, . . . , m} of cohomology classes. If P ∈ P , then for any cohomology ring isomorphism ϕ : H∗(M ) −∼=→ H∗(M′) of quasitoric manifolds over P and P′ , ϕ(D (M )) = D (M′). Moreover, each of the sets D (M ) and D (M′) consists of 2m distinct elements. Proof. The idea is to reduce the statement to Lemma 4.11. The ring isomorphism ϕ is uniquely determined by the isomorphism H2(M ) −∼=→ H2(M′) of free abelian groups. Let ϕ([vi]) = Σm Aij[v′ ] for some Aij ∈ Z, 1 ≤ i, j ≤ m.

Main Results. The main result of the paper is the formalization of agreements of continuous-time games. In Section 3, I define an agreement as a collection of continuation outcomes: it specifies an initial outcome that should be effective after the initial history; for any other finite history possible under the play of the agreement, it specifies a continuation outcome that should be effective after that history. In any continuation outcome, a player’s strategy is an admissible path of her future actions, from the beginning of the continuation outcome until the end of the game. If a player uses a strategy in a continuation outcome, the path of her opponents’ reactions is determined by the following convolution formula: at any time, the opponents’ actions played in response to the strategy equal the actions prescribed to them at that time in the effective continuation outcome. That is, the convolution formula is a consequence of deviations being unilateral. For each strategy, the convolution formula defines a unique path of play that will be induced by that strategy. I impose two restrictions on continuation outcomes comprising an agreement, admissibility and coherency. Admissibility requires that for each strategy, the total induced path of play must be possible under the rules of the game. Coherency is the requirement that in spells of time when players do not deviate from effective outcomes, the agreement must be recommending the same continuation path of play. These two restrictions are natural, and are sufficient for the play under an agreement to be well-defined. Coherency ensures that the following promise keeping property is satisfied in agreements: if a player decides to follow a continuation outcome, her opponents will react by also following that continuation outcome. The value of a strategy is the payoff the player receives from the path induced by that strategy. A player’s strategy in a continuation outcome is called a profitable deviation if its value exceeds the payoff promised to her in that continuation outcome. Self-enforcing agreements are agreements in which no player has a profitable unilateral deviation after any possible history of play. By promise keeping, the value of the strategy that prescribes to follow a continuation outcome coincides with the value promised in that continuation outcome. Hence, in self-enforcing agreements, it is optimal for players to always keep following the agreement’s recommendations. Another main result of the paper is...

Main Results. The IEA TEM#96 gives a contribution to better understand the issues, the challenges and the opportunities related to the large amount of wind power capacity that is reaching its end-of- life. A community of experts from very different kind of organizations actively contributed to the discussion on the three sub-topics: decommissioning, repowering and recycling. In the General Framework, Decommissioning and Repowering session, it was underlined that most of the wind plants at the end-of-life will probably benefit of a life-extension, but in a medium to long term scenario the amount of repowering interventions is going to grow significantly. The repowering interventions offer many benefits such as: • Higher energy production in the same area • Better support from new turbines to the power grid • Environmental and visual impact: less turbines = less impacts • Areas already used for wind energy: better social acceptance • Reduction of national energy price • Increase of (temporary) jobs in the sector However, there are still many issues/challenges such as: • No specific regulation for repowering • Administrative matter: complex, unclear and long permitting • Grid connection: lack of available grid capacity for repowered plants • Transport to site of the new bigger wind turbines (especially in high complex terrain) • (New) environmental and landscape contraints • Difficult to allocate new incentives for RES and need of new mechanisms • Difficult to develop wind plant at grid parity • Dismantling and recycling of the components (turbine component - cables - foundations) The most of the capacity facing today the end-of-life is based onshore, however in few years the number of offshore wind plants reaching the end-of-life is going to grow very fast according to the boom of offshore installations in the last decade. Some specific challenges, such as dismantling the offshore foundations, will have to be considered in this case. Decommissioning and repowering of a great amount of wind power capacity means dealing with a high number of “waste” turbine components. The challenge of recycling these components, in particular the blades, has been discussed also in the vision of a circular economy. In particular, different recycling methods for composite materials have been presented underlining also the issue of the traceability in each step of the process. The breakout sessions identified research gaps and needs for future collaboration for each subtopic. Concerning recyc...