Learning embeddings of words and knowledge base elements is a promising approach for open domain question answering. Based on the remark that relations and entities are distinct object types lying in the same embedding space, we analyze the benefit of adding a regularizer favoring the embeddings of entities to be orthogonal to those of relations. The main motivation comes from the observation t...

Carbon nanotubes provide a remarkably versatile system in which to explore the effects of Coulomb interactions in one dimension. The most dramatic effects of strong electron-electron repulsion are orthogonality catastrophes. These orthogonality catastrophes come in different varieties, and can be observed both in low-bias transport and tunneling measurements on nanotubes. This article contains ...

The 'canonical model' of semicircular canal orientation in mammals assumes that 1) the three ipsilateral canals of an inner ear exist in orthogonal planes (i.e., orthogonality), 2) corresponding left and right canal pairs have equivalent angles (i.e., angle symmetry), and 3) contralateral synergistic canals occupy parallel planes (i.e., coplanarity). However, descriptions of vestibular anatomy ...

Combinatorial representation in terms of Schröder paths and other weighted plane paths are given of Laurent biorthogonal polynomials (LBPs) and a linear functional with which LBPs have orthogonality and biorthogonality. Particularly, it is clarified that quantities to which LBPs are mapped by the corresponding linear functional can be evaluated by enumerating certain kinds of Schröder paths, wh...

A four-parameter family of multivariable big q-Jacobi polynomials and a threeparameter family of multivariable little q-Jacobi polynomials are introduced. For both families, full orthogonality is proved with the help of a second-order q-difference operator which is diagonalized by the multivariable polynomials. A link is made between the orthogonality measures and R. Askey’s q-extensions of Sel...

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.

We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

We show that the approximate quasi-orthogonality of two operator algebras is equivalent to being approximately private relative their conditional expectation quantum channels. Our analysis based on a characterization measure orthogonality in terms Choi matrices and Kraus operators for completely positive maps. present examples drawn from different areas information.