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Sets, Relations and Functions
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Index Set: Given a non-empty set I, we say that I serves as an index set for the family of sets F={Aα} $ \mathcal F =\{A_{\alpha }\} $ , if to each α∈I $ \alpha \in I $ , there is a set Aα $ A_{\alpha } $ in the family of sets F $ \mathcal F $ . The index set I can be any set, finite or infinite. Typically the index set is I={1,2,…,n} $ I=\{\,1,2,\ldots ,n\,\} $ .
Simple random walks
Published in Henry C. Tuckwell, Elementary Applications of Probability Theory, 2018
Mathematically, a random process is defined as a collection of random variables. The various members of the family are distinguished by different values of a parameter, a, say. The entire set of values of a, which we shall denote by A, is called an index set or parameter set. A random process is then a collection such as {Xα,α∈A}
Feasible rounding approaches for equality constrained mixed-integer optimization problems
Published in Optimization, 2023
Christoph Neumann, Oliver Stein
Finding all solutions to the system is then equivalent to solving the two systems with , and , . Here, for a vector v and an index set I, denotes the subvector with the entries , , of v. The notation with is shorthand for the index set .
Local optimality for stationary points of group zero-norm regularized problems and equivalent surrogates
Published in Optimization, 2023
Shaohua Pan, Ling Liang, Yulan Liu
Notation. For every , write and , and let be the mapping consisting of those with , where for . For each , write . For any , denote by its group support and write . For every , represents the -norm of vectors. For a matrix and an index set , is a matrix consisting of those for . For a closed set , denotes the indicator function of C, i.e. if , otherwise ; means the projection mapping onto C which may be multi-valued; and denotes the distance of x from C on the norm . For a given , denotes the closed ball of radius δ centred at x on the norm . For a proper function , write and denote by the conjugate of h, i.e. . For a differentiable mapping , denotes the transpose of , the Jacobian of F at x. The notation e denotes the vector of all ones with dimensions known from the context.