In conventional binaural rendering a pair of head-related impulse responses (HRIR), measured from source direction to left and right ears, is convolved with a source signal to create the impression of a virtual 3D sound source when played on headphones. It is well known that using HRIRs measured in a real room, which includes a natural reverberant decay, increases the externalization and realis...

We extend the notion of singular vectors to context Diophantine approximation real numbers with elements a totally number field K. For $$m\ge 1$$ , we establish version Dani’s correspondence in fields and prove that under class ‘friendly measures’ $$K_S^m$$ set has measure zero. Here S is Archimedean valuations K $$K_S$$ product completions $$\sigma (K)$$ \in S$$ . On other hand, show existence...

In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), et al. (J Optim Theory 169(1):1–16, which defines extremal varieties \(\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) \), we define a more general concept real Grassmannian \({G_p}\left( {{\mathbb \) ^p}{\mathbb \). This is based on minimization sums squares quadratic Plücker r...