Perhaps a better title for this chapter would be "Duality," but this has a special meaning in functional analysis: an abstraction of the notion of a space X and its dual X into a pairing ( X , Y ), where X is a vector space and Y is a space of linear functionals on X . The ...
and we are going to recall the basic concepts of convex analysis on\({\mathcal {M}}\). The central idea is to replace straight lines in the definition of convex sets in Euclidean vector spaces by geodesics.
The interpolation properties of the spaces GMp∞,w were studied by Spanne in [8]. The spaces GMpθ,r−λ were used by Lu [9] for studying the embedding theorems for vector fields of Hörmander type. As mentioned in [4], the intersection △(∁LMpθ,w{x}) of the Banach family ...
Here, tij is the hopping parameters depending on the relative coordinate vector ri−rj, where ri is the position vector of the center of the ith Wannier orbital. In the present study, for tij, we retain the nearest neighbor pair of ri and rj in each direction of a~, b~ and c~ as...
Four different machine learning algorithms, including Decision Tree (DT), Random Forest (RF), Multivariable Linear Regression (MLR), Support Vector Regressions (SVR), and Gaussian Process Regressions (GPR), were applied to predict the performance of a multi-media filter operating as a function of...
Moreover, whenever there is a vector space and an inner product there exists an associated dual space. The elements of this dual space, called dual forms, –1– must be localized to generalized unitarity cuts in order for the definition of the intersection number to make sense. Mathematically...
If{\mathcal {H}}_1and{\mathcal {H}}_2are Hilbert spaces then{\mathcal {H}}_1\mathbin {\otimes }^2 {\mathcal {H}}_2denotes their Hilbert-space tensor product. Given a complex vector spaceV, the conjugate vector space{\overline{V}}is defined to have the same underlying additive...
Let $$V_*\\otimes Vightarrow {\\mathbb {C}}$$ V V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V . The Mackey Lie algebra $${\\mathfrak {g}}=\\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ) corresponding to this...
and aims to solve LSOPs in low-dimensional spaces to reduce the difficulties that PSO has with diversity preservation [26], as evidenced by CCPSO-SH and CCPSO-SK. These two methods randomly divide the whole decision vector into K sub-components, where K is predefined by users. Then, PSO ...
Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, 2nd ed.; MOS-SIAM Series on Optimization; Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA, USA; Mathematical Optimization Society: Philadelphia, PA, USA, 2014; pp. xii+793. [Google ...