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To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems., Comment: 8 pages, 1 figure; submitted to Examples and Counterexamples

We perform a systematic study of the base change conductor for Jacobians. Through the lens of intersection theory and Deligne's Riemann-Roch theorem, we present novel computational approaches for both the tame and wild parts of the base change conductor. Our key results include a general formula of the tame part, as well as a computation of the wild part in terms of Galois quotients of semistable models of the curves. We treat in detail the case of potential good reduction when the quotient only has weak wild quotient singularities, relying on recent advances by Obus and Wewers., Comment: 45 pages. Added section 6.4, and corrected typos

Superconducting flip-chip interconnects are crucial for the three-dimensional integration of superconducting circuits in sensing and quantum technology applications. We demonstrate a simplified approach for a superconducting flip-chip device using commercially available indium microspheres and an in-house-built transfer stage for bonding two chips patterned with superconducting thin films. We use a gold-passivated niobium or niobium nitride layer as an under-bump metallization (UBM) layer between an aluminum-based superconducting wiring layer and the indium interconnect. At millikelvin temperatures, our flip-chip assembly can transport a supercurrent with tens of milliamperes, limited by the smallest geometric feature size and critical current density of the UBM layer and not by the indium interconnect. We show that the pressed indium interconnect itself can carry a supercurrent exceeding 1A due to its large size of about 500 micrometer diameter. Our flip-chip assembly does not require electroplating nor patterning of indium. The assembly process does not need a flip-chip bonder and can be realized with a transfer stage using a top chip with transparency or through-vias for alignment. These flip-chip devices can be utilized in applications that require few superconducting interconnects carrying large currents at millikelvin temperatures., Comment: 12 pages, 8 figures incl supplemental material

In this paper, we investigate the single photon response from the reflection of the Microwave Kinetic Inductance Detector (MKID) array. Reflection measurements are carried out using two configurations: one is measured simultaneously with the transmission, and the other is obtained with a single-ended MKID array terminated with an open load. Compared with the transmission, reflection measurements significantly reduce the readout noise of the single-ended MKID array. This is also reflected in the improvement of the median energy resolving power by around 20%-30% under pulsed photon illumination at $\lambda = 405$~nm, mainly due to an increase in the size of the resonance circle on the IQ plane. This method has the potential to be used to read out large MKID arrays., Comment: 8 pages, 5 figures

We report on simulations of a novel design of optical titanium nitride (TiN)- based Kinetic Inductance Detectors (KIDs) in order to improve their response to optical photons. We propose to separate the meander from the substrate to trap hot phonons generated by optical photons, preventing their rapid propagation through the substrate. These phonons would in turn contribute to the breaking of more Cooper pairs, thereby increasing the response of the detector. In our design, the meander is suspended a few hundred nanometers above the substrate. Furthermore, reflective gold (Au) or aluminum (Al)-based layers can be placed under the meander to improve photon coupling in the optical wavelengths.

In this paper, we investigate the quasi-particle (QP) relaxation of strongly disordered superconducting resonators under optical illumination at different bath temperatures with the Rothwarf and Taylor equations and the gap-broadening theory described by the Usadal equation. The analysis is validated with various single-photon responses of Titanium Nitride (TiN) microwave kinetic inductance detectors (MKIDs) under pulsed 405~nm laser illumination. The QP relaxation in TiN is dominated by QPs with energy below the energy gap smeared by the disorder, and its duration is still inversely proportional to the QP density. The QP lifetime versus temperature can be fitted. The relaxation of the resonator can be further modeled with QP diffusion. The fitted QP diffusion coefficient of TiN is significantly smaller than expected. Our result also shows a significant increase in QP generation efficiency as the bath temperature increases., Comment: Accepted by Superconductor Science and Technology

We study stability of abstract differential equations coupled by means of a general algebraic condition. Our approach is based on techniques from operator theory and systems theory, and it allows us to study coupled systems by exploiting properties of the components, which are typically much simpler to analyse. As our main results we establish resolvent estimates and decay rates for abstract boundary-coupled systems. We illustrate the power of the general results by using them to obtain rates of energy decay in coupled systems of one-dimensional wave and heat equations, and in a wave equation with an acoustic boundary condition., Comment: Accepted for publication in Journal of Functional Analysis

We consider time-harmonic elastodynamic problems in heterogeneous media.cWe focus on scattering problems in the high-frequency regime and incnearly incompressible media, where the the angular frequency $\omega$ and ratio of the Lam\'e parameters $\lambda/\mu$ may both be large. We derive stability estimates controlling the norm of the solution by the norm of the right-hand side up to a fully-explicit constant. Crucially, under natural assumptions on the domain and coefficients, this constant increases linearly with $\omega$ and is uniform in the ratio $\lambda/\mu$.

Let $\Gamma$ be a divisible subgroup of $(\mathbb{R},+)$. Our central result states that, at the level of Grothendieck groups, the classification of $\Gamma$-rational polyhedra in $\mathbb{R}^n$ up to affine transformations in $\Gamma^n\rtimes \mathrm{GL}_n(\mathbb{Z})$ is equivalent to the classification up to affine transformations in $\Gamma^n\rtimes \mathrm{GL}_n(\mathbb{Q})$. We prove this by giving an explicit description of these Grothendieck groups. This yields, in particular, a positive answer to the basic case of a question by Hrushovski and Kazhdan; all other cases are still open. As a second application, we give a simple description of the kernel of the motivic volume for non-archimedean semi-algebraic sets, which is a key ingredient of Hrushovski and Kazhdan's theory of motivic integration.

The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$ degenerates at $t=0$. In analogy with Igusa's monodromy conjecture for $p$-adic zeta functions of hypersurface singularities, we expect that the poles of the motivic zeta function of $X$ correspond to monodromy eigenvalues on the cohomology of $X$. In this paper, we prove that the motivic zeta function and the monodromy conjecture are preserved under birational equivalence, which extends the range of known cases. As a further application, we explain how the motivic zeta function acts as an obstruction to the existence of a smooth filling for $X$ at $t=0$.

SpectroPhotometric Imaging in Astronomy with Kinetic Inductance Detectors (SPIAKID) aims at designing, building, and deploying on the sky a spectrophotometric imager based on microwave kinetic inductance detectors (MKIDs) in the optical and near-infrared bands. MKIDs show a fast response and the ability to resolve photon energy compared to the conventional Charge-coupled Devices (CCDs). In this paper, we present the design and simulation of the MKID arrays for SPIAKID. The detectors consist of four arrays with each array of 20,000 lumped-element pixels, and each array will be read with 10 readout lines. %The array is designed to have resonances between 4-8GHz with a frequency spacing of 2 MHz and a coupling quality factor (Qc) of about 50000. The meander material of the resonators is trilayer TiN/Ti/TiN to have better uniformity of the critical temperature across the array. We also present the measurement result for a test array with $30\times30$ pixels which is a subset of the designed 2000-pixel array to verify the design and fabrication. The current measured best energy resolving power $R = E/\Delta E$ is 2.4 at $\lambda = 405~$nm and the current medium R is around 1.7. We have also observed the response of the TiN/Ti/TiN is much smaller than expected., Comment: 9 pages, 6 figures. Accepted by the Journal of Low Temperature Physics

We study the free Schr\"odinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the $L^1$ to $L^\infty$ time decay rate at least $t^{-1/2}$. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow between the bounded sub-graph and the unbounded ends., Comment: 34 pages, 6 figures, 29 references; major revisions with respect to the first version of october 2023

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected Elastic-Piezoelectric-Elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected Piezoelectric-Elastic design with a local damping acting on the elastic part only. Unlike the existing literature, piezoelectric layers are considered magnetizable, and therefore, a fully-dynamic PDE model, retaining interactions of electromagnetic fields (due to Maxwell's equations) with the mechanical vibrations, is considered. The design (i) is shown to have exponentially stable solutions. However, the nature of the stability of solutions of the design (ii), whether it is polynomial or exponential, is dependent entirely upon the arithmetic nature of a quotient involving all physical parameters. Furthermore, a polynomial decay rate is provided in terms of a measure of irrationality of the quotient. Note that this type of result is totally new (see Theorem 3.6 and Condition $\rm{\mathbf{(H_{Pol})}}$). The main tool used throughout the paper is the multipliers technique which requires an adaptive selection of cut-off functions together with a particular attention to the sharpness of the estimates to optimize the results.

To improve the optical coupling in Microwave Kinetic Inductance Detectors (MKIDs), we investigate the use of a reflective plate beneath the meandered absorber. We designed, fabricated and characterized high-Q factors TiN-based MKIDs on sapphire operating at optical wavelengths with a Au/Nb reflective thin bilayer below the meander. The reflector is set at a quarter-wave distance from the meander using a transparent Al$_2$O$_3$ dielectric layer to reach the peak photon absorption. We expect the plate to recover undetected photons by reflecting them back onto the absorber., Comment: Journal of Low Temperature Physics, Springer Verlag (Germany), In press

In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping and by assuming that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreover, under some geometric control condition, we prove the well-posedness and strong stability in the multi-dimensional case.

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $t^{-\frac{1}{2}}$ if the two waves have the same speed of propagation, and with rate $t^{-\frac{1}{3}}$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $t^{-1}$.

Following optical pulses ($\lambda=405~\text{nm}$) on titanium nitride (TiN) Microwave Kinetic Inductance Detectors (MKIDs) cooled down at temperatures $T \le T_c / 20$ ($T_c \simeq 4.6~\text{K}$), we observe a large phase-response highlighting two different modes simultaneously that are nevertheless related. The first corresponds to the well-known transition of cooper-pair breaking into quasi-particles which produces a known phase response. This is immediately followed by a large inverse response lasting several hundreds of microseconds to several milliseconds depending on the temperature. We propose to model this inverse pulse as the thermal perturbation of the superconductor and interaction with two level system (TLS) that reduces the dielectric constant which in turns modify the capacitance and therefore the resonance frequency. The ratio of the TLS responding to the illumination is on the order of that of the area of the inductor to the whole resonator, Comment: Accepted by Applied Physics Letters

In this paper, we obtain some stability results of (abstract) dissipative evolution equations with a nonautonomous and nonlinear damping using the exponential stability of the retrograde problem with a linear and autonomous feedback and a comparison principle. We then illustrate our abstract statements for different concrete examples, where new results are achieved. In a preliminary step, we prove some well-posedness results for some nonlinear and nonautonomous evolution equations.