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This is the first of our papers on quasi-split affine quantum symmetric pairs$\big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big )$, focusing on the real rank one case, i.e.,$\mathfrak g = \mathfrak {sl}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type$A_2^{(2)}$on the affine$\imath$quantum group$\widetilde {{\mathbf U}}^\imath$. Real and imaginary root vectors for$\widetilde {{\mathbf U}}^\imath$are constructed, and a Drinfeld type presentation of$\widetilde {{\mathbf U}}^\imath$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine$\imath$quantum groups in the sequels.

 

The ı \imath Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the projective line. This ı \imath Hall algebra is shown to realize the universal q q -Onsager algebra (i.e., ı \imath quantum group of split affine A 1 A_1 type) in its Drinfeld type presentation. The ı \imath Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two ı \imath Hall algebras, explaining the isomorphism of the q q -Onsager algebra under the two presentations. 

Abstract In this paper, a new technique is presented for parametrically studying the steady-state dynamics of piecewise-linear nonsmooth oscillators. This new method can be used as an efficient computational tool for analyzing the nonlinear behavior of dynamic systems with piecewise-linear nonlinearity. The new technique modifies and generalizes the bilinear amplitude approximation method, which was created for analyzing proportionally damped structural systems, to more general systems governed by state-space models; thus, the applicability of the method is expanded to many engineering disciplines. The new method utilizes the analytical solutions of the linear subsystems of the nonsmooth oscillators and uses a numerical optimization tool to construct the nonlinear periodic response of the oscillators. The method is validated both numerically and experimentally in this work. The proposed computational framework is demonstrated on a mechanical oscillator with contacting elements and an analog circuit with nonlinear resistance to show its broad applicability. 

Thin-film solid-state interfacial dealloying (thin-film SSID) is an emerging technique to design nanoarchitecture thin films. The resulting controllable 3D bicontinuous nanostructure is promising for a range of applications including catalysis, sensing, and energy storage. Using a multiscale microscopy approach, we combine X-ray and electron nano-tomography to demonstrate that besides dense bicontinuous nanocomposites, thin-film SSID can create a very fine (5–15 nm) nanoporous structure. Not only is such a fine feature among one of the finest fabrications by metal-agent dealloying, but a multilayer thin-film design enables creating nanoporous films on a wider range of substrates for functional applications. Through multimodal synchrotron diffraction and spectroscopy analysis with which the materials’ chemical and structural evolution in this novel approach is characterized in details, we further deduce that the contribution of change in entropy should be considered to explain the phase evolution in metal-agent dealloying, in addition to the commonly used enthalpy term in prior studies. The discussion is an important step leading towards better explaining the underlying design principles for controllable 3D nanoarchitecture, as well as exploring a wider range of elemental and substrate selections for new applications.