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This document is comprised of a collection of updated preliminary parameters for the key parts of the muon collider. The updated preliminary parameters follow on from the October 2023 Tentative Parameters Report. Particular attention has been given to regions of the facility that are believed to hold greater technical uncertainty in their design and that have a strong impact on the cost and power consumption of the facility. The data is collected from a collaborative spreadsheet and transferred to overleaf.

We introduce a quadratic gradient type term for the Pucci extremal operators. Our analysis demonstrates that this proposed term extends the classical quadratic gradient term associated with the Laplace equation, and we investigate the impact of the Kazdan-Kramer transformation. As an application, we explore the existence, non-existence, uniqueness, Liouville-type results, and asymptotic behavior of solutions for the new class of Pucci equations under various conditions on both nonlinearity and domains.

The structure of extended affine Weyl symmetry group of higher Painlev\'e equations of $N$ periodicity depends on whether $N$ is even or odd. We find that for even $N$, the symmetry group ${\widehat A}^{(1)}_{N-1}$ contains the conventional B\"acklund transformations $s_j, j=1,{\ldots},N$, the group of automorphisms consisting of cycling permutations but also reflections on a periodic circle of $N$ points, which is a novel feature uncovered in this paper. The presence of reflection automorphisms is connected to existence of degenerated solutions and for $N=4$ we explicitly show how the reflection automorphisms around even points cause degeneracy of a class of rational solutions obtained on the orbit of translation operators of ${\widehat A}^{(1)}_{3}$. We obtain the closed expressions for solutions and their degenerated counterparts in terms of determinants of Kummer polynomials., Comment: 26 pages

We study the wave form emitted by a particle moving along an arbitrary (in general open) geodesic of the Schwarzschild geometry. The mathematical problem can be phrased in terms of quantities in ${\cal N}=2$ supersymmetric gauge theories that can be calculated by using localization and the AGT correspondence. In particular through this mapping, the post-Newtonian expansion of the wave form is expressed as a double instanton sum with rational coefficients that resums all tail contributions into Gamma functions and exponentials. The formulae we obtain are valid for generic values of the orbital quantum numbers $\ell$ and $m$. For $\ell=2,3$ we check explicitly that our results agree with the small mass ratio limit of the wave forms derived in the Multipole Post-Minkowskian and the amplitudes approaches. We show how the so-called tail and tail of tail contributions to the wave form arise in our approach. Finally, we derive a universal formula for the soft limit of the wave form that resums all logarithmic divergent terms of the form $\omega^{n-1} (\log \omega)^n$., Comment: 29 pages, references added, some typos corrected, version prepared for submission to phys.rev

We reconsider linear perturbations around general Friedmann - Lemaitre - Robertson - Walker (FLRW) cosmological backgrounds. Exploiting gauge freedom involving only time reparametrizations, we write down classical background solutions analytically, for an arbitrary number of fluid components. We then show that the time evolution of scalar and tensor adiabatic perturbations are governed by Schr\"odinger-like differential equations of generalized Heun type. After recovering known analytic results for a single-component fluid, we discuss more general situations with two and three different fluid components, with special attention to the combination of radiation, matter and vacuum energy, which is supposed to describe the $\Lambda$CDM model. The evolution of linear perturbations of a flat $\Lambda$CDM universe is described by a two-transient model, where the transitions from radiation to matter and matter to vacuum energy are governed by a Heun equation and a Hypergeometric equation, respectively. We discuss an analytic approach to the study of the general case, involving generalized Heun equations, that makes use of (quantum) Seiberg-Witten curves for ${\cal N}=2$ supersymmetric gauge theories and has proven to be very effective in the analysis of Black-Hole, fuzzball and ECO perturbations., Comment: 30 pages, 6 figures, 3 tables

In this work we study an inhomogeneous two-phase obstacle-type problem associated to the $s$-fractional $p$-Laplacian. Besides the existence and uniqueness of solutions, we study the convergence of the solutions when $s\to 1$ to the classical problem. We also study the continuous dependence with respect to the level-set function $v$, yielding an existence result for a two-phase obstacle-type problem with an implicit level-set function., Comment: 21 pages

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schr\"{o}dinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -\Delta u+\lambda u=\mu|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $\lambda\in\mathbb{R}$, $\mu>0$, $0

We consider coupled gravitational and electromagnetic perturbations of a family of five-dimensional Einstein-Maxwell solutions that describes both magnetized black strings and horizonless topological stars. We find that the odd perturbations of this background lead to a master equation with five Fuchsian singularities and compute its quasinormal mode spectrum using three independent methods: Leaver, WKB and numerical integration. Our analysis confirms that odd perturbations always decay in time, while spherically symmetric even perturbations may exhibit for certain ranges of the magnetic fluxes instabilities of Gregory-Laflamme type for black strings and of Gross-Perry-Yaffe type for topological stars. This constitutes evidence that topological stars and black strings are classically stable in a finite domain of their parameter space.

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $g$-Laplacian $\mathcal{L}_g^s$, with $0

We present direct N-body simulations of black-hole-only clusters with up to $2 \cdot 10^4$ compact objects, zero natal spin and no primordial binaries as predicted by various primordial black hole (PBH) Dark Matter models. The clusters' evolution is computed using ${\tt NBODY6\!+\!+GPU}$, including the effects of the tidal field of the galaxy, the kicks of black hole mergers and orbit-averaged energy loss by gravitational radiation of binaries. We investigate clusters with four initial mass distributions, three of which attempt to model a generic PBH scenario using a lognormal mass distribution and a fourth one that can be directly linked to a monochromatic PBH scenario when accretion is considered. More specifically, we dive into the clusters' internal dynamics, describing their expansion and evaporation, along with the resultant binary black hole mergers. We also compare several simulations with and without black hole merger kicks and find modelling implications for the probability of hierarchical mergers., Comment: 14 pages, 9 figures, 2 tables

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-\Delta_p^s$ operator, when the fractional parameter $s\to\sigma$ for $0<\sigma\leq1$, converge to the solution of the corresponding obstacle problem for $-\Delta_p^\sigma$, being $\sigma=1$ the classical obstacle problem for the local $p$-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when $s\nearrow 1$ under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when $s\nearrow 1$, under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local $p$-Laplacian, essentially when the limit coincidence set is the closure of its interior.

We study the gravitational tidal response of D1D5, Top Star and (1,0,n) strata horizonless geometries. We find that the tidal interactions in fuzzball geometries, unlike in the case of black holes, exhibits a sequence of resonant peaks associated to the existence of metastable bound states. The spectrum of resonant frequencies is computed by semi-analytical and numerical methods., Comment: 23 pages, 8 figures

A follow-up of a subsolar black hole candidate identified in the second part of the third observing run of the LIGO-Virgo-KAGRA collaboration is carried out. With a search signal-to-noise ratio of $8.90$ and a false-alarm rate of 1 per 5 years, close to the usual thresholds for claiming a gravitational-wave event, we cannot exclude a noise origin. A complete Bayesian parameter estimation of this candidate, denoted SSM200308, reveals that if the signal originates from a compact binary coalescence, the component masses are $m_1= 0.62^{+0.46}_{-0.20} M_{\odot}$ and $m_2 = 0.27^{+0.12}_{-0.10} M_{\odot}$ (90% credible intervals) with at least one component being firmly subsolar, below the minimum mass of a neutron star. This discards the hypothesis that the signal comes from a standard binary neutron star. The signal coherence test between the two LIGO detectors brings support to a compact object coalescence origin., Comment: 7 pages, 4 figures, comments welcome

We derive a gauge inspired combinatorial formula based on localization for the Post-Newtonian expansion of the gravitational wave form luminosity of binary systems made of objects with very different masses orbiting at large distances and small velocities. The results are tested against previous formulae in the literature for Schwarschild and Kerr black holes at the 5th and 3rd Post Newtonian order respectively beyond the quadrupole approximation. Tidal effects show up in the wave form at the 5th PN order, providing a quantitative measure of the blackness/compactness properties of the heavy object., Comment: 29 pages, few notations changed hoping to make the paper more readable. few typos corrected. version to appear in JHEP

In the search for improved permanent magnets, fueled by the geostrategic and environmental issues associated with rare-earth-based magnets, magnetically hard (high anisotropy)-soft (high magnetization) composite magnets hold promise as alternative magnets that could replace modern permanent magnets, such as rare-earth-based and ceramic magnets, in certain applications. However, so far, the magnetic properties reported for hard-soft composites have been underwhelming. Here, an attempt to further understand the correlation between magnetic and microstructural properties in strontium ferrite-based composites, hard SrFe$_{12}$O$_{19}$ (SFO) ceramics with different contents of Fe particles as soft phase, both in powder and in dense injection molded magnets, is presented. In addition, the influence of soft phase particle dimension, in the nano- and micron-sized regimes, on these properties is studied. While Fe and SFO are not exchange-coupled in our magnets, a remanence that is higher than expected is measured. In fact, in composite injection molded anisotropic (magnetically oriented) magnets, remanence is improved by 2.4% with respect to a pure ferrite identical magnet. The analysis of the experimental results in combination with micromagnetic simulations allows us to establish that the type of interaction between hard and soft phases is of a dipolar nature, and is responsible for the alignment of a fraction of the soft spins with the magnetization of the hard. The mechanism unraveled in this work has implications for the development of novel hard-soft permanent magnets.

We consider weak solutions for the obstacle-type viscoelastic ($\nu>0$) and very weak solutions for the obstacle inviscid ($\nu=0$) Dirichlet problems for the heterogeneous and anisotropic wave equation in a fractional framework based on the Riesz fractional gradient $D^s$ ($0

We study the problem of valuing and hedging a vulnerable derivative claim with bilateral cash flows between two counterparties in the presence of asymmetric funding costs, defaults and wrong way risk (WWR). We characterize the pre-default claim value as the solution to a non-linear Cauchy problem. We show an explicit stochastic representation of the solution exists under a funding policy which linearises the Cauchy PDE. We apply this framework to the valuation of a vulnerable equity forward and show it can be represented as a portfolio of European options. Despite the complexity of the model, we prove the forward's value admits an analytical formula involving only elementary functions and Gaussian integrals. Based on this explicit formula, numerical analysis demonstrates WWR has a significant impact even under benign assumptions: with a parameter configuration less punitive than that representative of Archegos AM default, we find WWR can shift values for vulnerable forwards by 100bps of notional, while peak exposures increase by 25% of notional. This framework is the first to apply to contracts with bilateral cash flows in the presence of credit, funding and WWR, resulting in a non-linear valuation formula which admits a closed-form solution under a suitable funding policy., Comment: 44 pages, 4 figures

Reduced Order Quadrature (ROQ) methods can greatly reduce the computational cost of Gravitational Wave (GW) likelihood evaluations, and therefore greatly speed up parameter estimation analyses, which is a vital part to maximize the science output of advanced GW detectors. In this paper, we do an in-depth study of ROQ techniques applied to GW data analysis and present novel algorithms to enhance different aspects of the ROQ bases construction. We improve upon previous ROQ construction algorithms allowing for more efficient bases in regions of parameter space that were previously challenging. In particular, we use singular value decomposition (SVD) methods to characterize the waveform space and choose a reduced order basis close to optimal and also propose improved methods for empirical interpolation node selection, greatly reducing the error added by the empirical interpolation model. To demonstrate the effectiveness of our algorithms, we construct multiple ROQ bases ranging in duration from 4s to 256s for compact binary coalescence (CBC) waveforms including precession and higher order modes. We validate these bases by performing likelihood error tests and P-P tests and explore the speed up they induce both theoretically and empirically with positive results. Furthermore, we conduct end-to-end parameter estimation analyses on several confirmed GW events, showing the validity of our approach in real GW data., Comment: 21 pages, 9 figures

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator.

We establish a supercritical Trudinger-Moser type inequality for the $k$-Hessian operator on the space of the $k$-admissible radially symmetric functions $\Phi^{k}_{0,\mathrm{rad}}(B)$, where $B$ is the unit ball in $\mathbb{R}^{N}$. We also prove the existence of extremal functions for this new supercritical inequality.

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the $\Omega$ background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators., Comment: 37 pages, references added

Conformal theories of the Argyres-Douglas type are notoriously hard to study given that they are isolated and strongly coupled thus lacking a lagrangian description. In flat space, an exact description is provided by the Seiberg-Witten theory. Turning on a $\Omega$-background makes the geometry ``quantum" and tractable only in the weak curvature limit. In this paper we use the AGT correspondence to derive $\Omega$-exact formulae for the partition function, in the nearby of monopole points where the dynamics is described by irregular conformal blocks of the CFT. The results are checked against those obtained by the recursion relations coming from a conformal anomaly in the region where the two approaches overlap. The Nekrasov-Shatashvili limit is also discussed. Finally, we comment on the existence of black holes in De Sitter space whose low energy dynamics is described by an Argyres-Douglas theory., Comment: 26 pages, 1 reference added, many typos corrected as suggested from the referee

Topological stars, or top stars for brevity, are smooth horizonless static solutions of Einstein-Maxwell theory in 5-d that reduce to spherically symmetric solutions of Einstein-Maxwell-Dilaton theory in 4-d. We study linear scalar perturbations of top stars and argue for their stability and deformability. We tackle the problem with different techniques including WKB approximation, numerical analysis, Breit-Wigner resonance method and quantum Seiberg-Witten curves. We identify three classes of quasi-normal modes corresponding to prompt-ring down modes, long-lived meta-stable modes and what we dub `blind' modes. All mode frequencies we find have negative imaginary parts, thus suggesting linear stability of top stars. Moreover we determine the tidal Love and dissipation numbers encoding the response to tidal deformations and, similarly to black holes, we find zero value in the static limit but, contrary to black holes, we find non-trivial dynamical Love numbers and vanishing dissipative effects at linear order. For the sake of illustration in a simpler context, we also consider a toy model with a piece-wise constant potential and a centrifugal barrier that captures most of the above features in a qualitative fashion.

The popularity of open-source software (OSS) projects has grown significantly over the last few years with more organizations relying on them. As these projects become larger, the need for higher quality also increases. DevOps practices have been shown to improve quality and performance. The DORA benchmarking reports provide useful information to compare DevOps practices performance between organizations, but they focus on continuous deployment and delivery to production, while OSS projects focus on the continuous release of code and its impact on third parties. The DORA reports mention the increasing presence of OSS projects as they are widely used in the industry, but they have never been used to measure OSS projects performance levels. This study reveals that the DORA benchmark cannot be applied to OSS projects and proposes benchmarking metrics for OSS projects, being the first one that adapts the DORA metrics and applies them in OSS projects. The metrics proposed in this study for benchmarking OSS projects include Release Frequency and Lead Time For Released Changes to measure throughput, and Time To Repair Code and Bug Issues Rate to assess stability. In contrast to the DORA reports, where data is collected through manual surveys, in our proposal, data is collected automatically by a tool we developed that retrieves information from public GitHub repositories. This reduces the risk of survey-based data collection. Our study also shows the benchmark feasibility by applying it to four popular OSS projects: Angular, Kubernetes, Tensorflow, and VS Code. In addition, we proposed challenges that address the topics and future works to expand the knowledge and findings of this study. Overall, the findings of the study can help to improve future research on OSS projects and provide a better understanding and challenges of the role of DevOps practices in OSS projects., Comment: 18 pages, 10 figures

We perform an exhaustive follow-up analysis of a subsolar-mass (SSM) gravitational wave (GW) candidate reported by Phukon et al. from the second observing run of Advanced LIGO. This candidate has a reported signal-to-noise ratio (SNR) of $8.6$ and false alarm rate of $0.41$ yr which are too low to claim a clear gravitational-wave origin. When improving on the search by using more accurate waveforms, extending the frequency range from 45 Hz down to 20 Hz, and removing a prominent blip glitch, we find that the posterior distribution of the network SNR lies mostly below the search value, with the $90\%$ confidence interval being $7.94^{+0.70}_{-1.05}$. Assuming that the origin of the signal is a compact binary coalescence (CBC), the secondary component is $m_2 = \text{$0.76^{+0.50}_{-0.14}$} M_\odot$, with $m_2 < 1 M_\odot$ at $84\%$ confidence level, suggesting an unexpectedly light neutron star or a black hole of primordial or exotic origin. The primary mass would be $m_1 = \text{$4.71^{+1.57}_{-2.18}$} M_\odot$, likely in the hypothesized lower mass gap and the luminosity distance is measured to be $D_{\rm L}=124^{+82}_{-48}$Mpc. We then probe the CBC origin hypothesis by performing the signal coherence tests, obtaining a log Bayes factor of $4.96 \pm 0.13$ for the coherent vs. incoherent hypothesis. We demonstrate the capability of performing a parameter estimation follow-up on real data for an SSM candidate with moderate SNR. The improved sensitivity of O4 and subsequent LIGO-Virgo-KAGRA observing runs could make it possible to observe similar signals, if present, with a higher SNR and a more precise measurement of the parameters of the binary., Comment: 10 pages, 6 figures

In this paper, we propose a gradient type term for the $k$-Hessian equation that extends for $k>1$ the classical quadratic gradient term associated with the Laplace equation. We prove that such as gradient term is invariant by the Kazdan-Kramer change of variables. As applications, we ensure the existence of solutions for a new class of $k$-Hessian equation in the sublinear and superlinear cases for Sobolev type growth. The threshold for existence is obtained in some particular cases. In addition, for the Trudinger-Moser type growth regime, we also prove the existence of solutions under either subcritical or critical conditions.