This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an initial value problem. Our main tools include polynomial recursive formalism, hyperelliptic curve with finite number genus, Baker-Akhiezer functions, meromorphic function, Dubrovin-type equations auxiliary ...

We use the hypergraph container theory of Balogh–Morris–Samotij and Saxton–Thomason to obtain general rectilinear approximations volume estimates for sequences bodies closed under certain families projections. give a number applications our results, including multicolour generalisation theorem Hatami, Janson Szegedy on entropy graph limits. Finally, we raise questions geometric analytic approac...

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) the form $$\mathrm{d} X= u(\omega,t,X)\, \mathrm{d} t + \frac12 \sigma(\omega,t,X)\sigma'(\omega,t,X)\,\mathrm{d} \sigma(\omega,t,X) \, \mathrm{d}W(t), $$ where drift coefficient $u$ is random and irregular. The regular noise $\sigma$ may vanish. main contribution pathwise uniquen...

This paper describes an approach to coherent aperture synthesis and fine-resolution image reconstruction employing phase retrieval from intensity measurements. A Gerchberg-Saxton algorithm is used on a pair of intensity measurements to obtain the fields for each telescope at each position of the telescopes without a reference beam. These are combined into a larger synthetic aperture, and a cros...