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Conditions for Unique Localizability in Cooperative Localization of Wireless ad hoc and Sensor Networks
Published in Chao Gao, Guorong Zhao, Hassen Fourati, Cooperative Localization and Navigation, 2019
Let us consider the graph G = (V,E). Let us consider a family of subsets of E and call it I. If I satisfies the following three matroid axioms, then it forms the independent sets of a matroid M(G): Axiom 1: Empty set is an element of I.Axiom 2: If E' is in I and E″ is a subset of E', then E″ is in I.Axiom 3: If E1 and E2 are in I, and the cardinality of E1 is less than the cardinality of E2, then there is an edge e in the difference set between E2 and E1 such that the union of E1 and {e} is in I. We refer the reader to Oxley (1992) for more information on matroid theory.
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Let G be afinite group of order v (see page 106). Asubset D of size k is a(v,k,λ)- $ a(v, k, \lambda ) - $ difference set in G if every non-identity element of G can be written λ $ \lambda $ times as a “difference” d1d2-1 $ d_{1} d_{2}^{ - 1} $ with d1 and d2 in D. If G is the cyclic group Zv $ {\mathbb{Z}}_{v} $ , then the difference set is a cyclic difference set. The order of a difference set is n=k-λ. $ n = k - \lambda . $
Corrosion of reinforcement (B)
Published in Brian Cherry, Green Warren, Corrosion and Protection of Reinforced Concrete, 2021
Because of the potential difference set up across the interface, energy is required to cause a positive ion to move against the potential gradient in order to go from lattice into solution. The system comes to equilibrium and there ceases to be a nett flow of ions into solution when the potential that is set up is such that the gain in electrical potential energy as the ion moves from lattice to solution is equal to the loss in chemical potential energy as it moves from lattice to solution. This equivalence may be written: ΔG=−zFEWhere E is the potential developed across the interface and ΔG the loss in Chemical Free Energy when 1 gm.mole of the ion enters the solution. zF is the charge on 1 gm.mole of the ion. On this basis it is possible to derive an expression for the potential difference across the interface. The value of the potential difference in volts across an interface is given by an expression known as the Nernst equation. By convention the potential of the metal is measured with respect to the solution. (American usage is sometimes different and this can give rise to confusion, but throughout these sections this International convention will be used.) A positive (>0) potential would therefore indicate that as the metal is positive with respect to the solution and that there is a gain in electrical potential energy as a positive ion leaves the lattice and goes into the solution. The change in Gibbs Free Energy (chemical potential energy) in going from the solution to the metal must therefore because of Equation 5a, be negative, that is, the spontaneous direction of the reaction is: MZ++ze−→M
A new scalarization approach and applications in set optimization
Published in Optimization, 2023
Lam Quoc Anh, Pham Thanh Duoc, Tran Thi Thuy Duong
It is well known that the scalarization method is an effective approach in studying vector optimization problems [1–3]. Recently, this approach has been applied to discuss qualitative properties of efficient solutions of set optimization problems. From the existing works in the literature, there have been two main scalarization methods, namely linear scalarization and nonlinear scalarization methods. Let us present a brief overview on applicability of these methods for set optimization problems. In [4], the authors applied the linear scalarization function and convexity conditions to consider the arcwise connectedness of minimal solutions and weak minimal solutions of set optimization problems under the lower set less order relation. Then, under suitable adjustments, in [5] this method was used again to discuss the connectedness of a weak minimal solution of set optimization problems with respect to the Minkowski difference set less order relation. For the second one, let us start with the Hiriart-Urruty oriented distance function introduced to build optimality conditions of non-smooth optimization problems in [6]. In [7], employing a generalized Hiriart-Urruty oriented distance function, the author introduced a new concept of slope for a set-valued map, and then used it to establish criteria for error bounds and the existence of weak optimal solutions of a set optimization problem under the lower set less oder relation. Recently, the authors in [8,9] have proposed six set scalarization functions, extensions of the oriented distance of Hiriart-Urruty. Based on properties of these functions, the authors have provided necessary and sufficient optimality conditions of weak minimality sets of the six set optimization problems introduced in [10]. Another important nonlinear scalarization function is the Gerstewitz's function proposed in [11]. Then, Hernández and Rodríguez-Marín [12] and Araya [13] introduced generalizations of this function and used them to study optimality conditions as well as characterizations of both minimal and weak minimal solutions of set optimization problems with respect to lower and upper set less order relations. Very recently, in [14] the Gerstewitz function has been extended again to study the Ekeland's variational principle, the Caristi's fixed point theorem and the Takahashi's minimization theorem with weighted set order relations for set-valued maps.